unique about the scientific method of problem solving

What is the Scientific Method: How does it work and why is it important?

The scientific method is a systematic process involving steps like defining questions, forming hypotheses, conducting experiments, and analyzing data. It minimizes biases and enables replicable research, leading to groundbreaking discoveries like Einstein's theory of relativity, penicillin, and the structure of DNA. This ongoing approach promotes reason, evidence, and the pursuit of truth in science.

Updated on November 18, 2023

Beginning in elementary school, we are exposed to the scientific method and taught how to put it into practice. As a tool for learning, it prepares children to think logically and use reasoning when seeking answers to questions.

Rather than jumping to conclusions, the scientific method gives us a recipe for exploring the world through observation and trial and error. We use it regularly, sometimes knowingly in academics or research, and sometimes subconsciously in our daily lives.

In this article we will refresh our memories on the particulars of the scientific method, discussing where it comes from, which elements comprise it, and how it is put into practice. Then, we will consider the importance of the scientific method, who uses it and under what circumstances.

What is the scientific method?

The scientific method is a dynamic process that involves objectively investigating questions through observation and experimentation . Applicable to all scientific disciplines, this systematic approach to answering questions is more accurately described as a flexible set of principles than as a fixed series of steps.

The following representations of the scientific method illustrate how it can be both condensed into broad categories and also expanded to reveal more and more details of the process. These graphics capture the adaptability that makes this concept universally valuable as it is relevant and accessible not only across age groups and educational levels but also within various contexts.

Steps in the scientific method

While the scientific method is versatile in form and function, it encompasses a collection of principles that create a logical progression to the process of problem solving:

• Define a question : Constructing a clear and precise problem statement that identifies the main question or goal of the investigation is the first step. The wording must lend itself to experimentation by posing a question that is both testable and measurable.
• Gather information and resources : Researching the topic in question to find out what is already known and what types of related questions others are asking is the next step in this process. This background information is vital to gaining a full understanding of the subject and in determining the best design for experiments. 
• Form a hypothesis : Composing a concise statement that identifies specific variables and potential results, which can then be tested, is a crucial step that must be completed before any experimentation. An imperfection in the composition of a hypothesis can result in weaknesses to the entire design of an experiment.
• Perform the experiments : Testing the hypothesis by performing replicable experiments and collecting resultant data is another fundamental step of the scientific method. By controlling some elements of an experiment while purposely manipulating others, cause and effect relationships are established.
• Analyze the data : Interpreting the experimental process and results by recognizing trends in the data is a necessary step for comprehending its meaning and supporting the conclusions. Drawing inferences through this systematic process lends substantive evidence for either supporting or rejecting the hypothesis.
• Report the results : Sharing the outcomes of an experiment, through an essay, presentation, graphic, or journal article, is often regarded as a final step in this process. Detailing the project's design, methods, and results not only promotes transparency and replicability but also adds to the body of knowledge for future research.
• Retest the hypothesis : Repeating experiments to see if a hypothesis holds up in all cases is a step that is manifested through varying scenarios. Sometimes a researcher immediately checks their own work or replicates it at a future time, or another researcher will repeat the experiments to further test the hypothesis.

Where did the scientific method come from?

Oftentimes, ancient peoples attempted to answer questions about the unknown by:

• Making simple observations
• Discussing the possibilities with others deemed worthy of a debate
• Drawing conclusions based on dominant opinions and preexisting beliefs

For example, take Greek and Roman mythology. Myths were used to explain everything from the seasons and stars to the sun and death itself.

However, as societies began to grow through advancements in agriculture and language, ancient civilizations like Egypt and Babylonia shifted to a more rational analysis for understanding the natural world. They increasingly employed empirical methods of observation and experimentation that would one day evolve into the scientific method . 

In the 4th century, Aristotle, considered the Father of Science by many, suggested these elements , which closely resemble the contemporary scientific method, as part of his approach for conducting science:

• Study what others have written about the subject.
• Look for the general consensus about the subject.
• Perform a systematic study of everything even partially related to the topic.

By continuing to emphasize systematic observation and controlled experiments, scholars such as Al-Kindi and Ibn al-Haytham helped expand this concept throughout the Islamic Golden Age . 

In his 1620 treatise, Novum Organum , Sir Francis Bacon codified the scientific method, arguing not only that hypotheses must be tested through experiments but also that the results must be replicated to establish a truth. Coming at the height of the Scientific Revolution, this text made the scientific method accessible to European thinkers like Galileo and Isaac Newton who then put the method into practice.

As science modernized in the 19th century, the scientific method became more formalized, leading to significant breakthroughs in fields such as evolution and germ theory. Today, it continues to evolve, underpinning scientific progress in diverse areas like quantum mechanics, genetics, and artificial intelligence.

Why is the scientific method important?

The history of the scientific method illustrates how the concept developed out of a need to find objective answers to scientific questions by overcoming biases based on fear, religion, power, and cultural norms. This still holds true today.

By implementing this standardized approach to conducting experiments, the impacts of researchers’ personal opinions and preconceived notions are minimized. The organized manner of the scientific method prevents these and other mistakes while promoting the replicability and transparency necessary for solid scientific research.

The importance of the scientific method is best observed through its successes, for example: 

• “ Albert Einstein stands out among modern physicists as the scientist who not only formulated a theory of revolutionary significance but also had the genius to reflect in a conscious and technical way on the scientific method he was using.” Devising a hypothesis based on the prevailing understanding of Newtonian physics eventually led Einstein to devise the theory of general relativity .
• Howard Florey “Perhaps the most useful lesson which has come out of the work on penicillin has been the demonstration that success in this field depends on the development and coordinated use of technical methods.” After discovering a mold that prevented the growth of Staphylococcus bacteria, Dr. Alexander Flemimg designed experiments to identify and reproduce it in the lab, thus leading to the development of penicillin .
• James D. Watson “Every time you understand something, religion becomes less likely. Only with the discovery of the double helix and the ensuing genetic revolution have we had grounds for thinking that the powers held traditionally to be the exclusive property of the gods might one day be ours. . . .” By using wire models to conceive a structure for DNA, Watson and Crick crafted a hypothesis for testing combinations of amino acids, X-ray diffraction images, and the current research in atomic physics, resulting in the discovery of DNA’s double helix structure .

Final thoughts

As the cases exemplify, the scientific method is never truly completed, but rather started and restarted. It gave these researchers a structured process that was easily replicated, modified, and built upon. 

While the scientific method may “end” in one context, it never literally ends. When a hypothesis, design, methods, and experiments are revisited, the scientific method simply picks up where it left off. Each time a researcher builds upon previous knowledge, the scientific method is restored with the pieces of past efforts.

By guiding researchers towards objective results based on transparency and reproducibility, the scientific method acts as a defense against bias, superstition, and preconceived notions. As we embrace the scientific method's enduring principles, we ensure that our quest for knowledge remains firmly rooted in reason, evidence, and the pursuit of truth.

The AJE Team

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Scientific Method

Science is an enormously successful human enterprise. The study of scientific method is the attempt to discern the activities by which that success is achieved. Among the activities often identified as characteristic of science are systematic observation and experimentation, inductive and deductive reasoning, and the formation and testing of hypotheses and theories. How these are carried out in detail can vary greatly, but characteristics like these have been looked to as a way of demarcating scientific activity from non-science, where only enterprises which employ some canonical form of scientific method or methods should be considered science (see also the entry on science and pseudo-science ). Others have questioned whether there is anything like a fixed toolkit of methods which is common across science and only science. Some reject privileging one view of method as part of rejecting broader views about the nature of science, such as naturalism (Dupré 2004); some reject any restriction in principle (pluralism).

Scientific method should be distinguished from the aims and products of science, such as knowledge, predictions, or control. Methods are the means by which those goals are achieved. Scientific method should also be distinguished from meta-methodology, which includes the values and justifications behind a particular characterization of scientific method (i.e., a methodology) — values such as objectivity, reproducibility, simplicity, or past successes. Methodological rules are proposed to govern method and it is a meta-methodological question whether methods obeying those rules satisfy given values. Finally, method is distinct, to some degree, from the detailed and contextual practices through which methods are implemented. The latter might range over: specific laboratory techniques; mathematical formalisms or other specialized languages used in descriptions and reasoning; technological or other material means; ways of communicating and sharing results, whether with other scientists or with the public at large; or the conventions, habits, enforced customs, and institutional controls over how and what science is carried out.

While it is important to recognize these distinctions, their boundaries are fuzzy. Hence, accounts of method cannot be entirely divorced from their methodological and meta-methodological motivations or justifications, Moreover, each aspect plays a crucial role in identifying methods. Disputes about method have therefore played out at the detail, rule, and meta-rule levels. Changes in beliefs about the certainty or fallibility of scientific knowledge, for instance (which is a meta-methodological consideration of what we can hope for methods to deliver), have meant different emphases on deductive and inductive reasoning, or on the relative importance attached to reasoning over observation (i.e., differences over particular methods.) Beliefs about the role of science in society will affect the place one gives to values in scientific method.

The issue which has shaped debates over scientific method the most in the last half century is the question of how pluralist do we need to be about method? Unificationists continue to hold out for one method essential to science; nihilism is a form of radical pluralism, which considers the effectiveness of any methodological prescription to be so context sensitive as to render it not explanatory on its own. Some middle degree of pluralism regarding the methods embodied in scientific practice seems appropriate. But the details of scientific practice vary with time and place, from institution to institution, across scientists and their subjects of investigation. How significant are the variations for understanding science and its success? How much can method be abstracted from practice? This entry describes some of the attempts to characterize scientific method or methods, as well as arguments for a more context-sensitive approach to methods embedded in actual scientific practices.

1. Overview and organizing themes

2. historical review: aristotle to mill, 3.1 logical constructionism and operationalism, 3.2. h-d as a logic of confirmation, 3.3. popper and falsificationism, 3.4 meta-methodology and the end of method, 4. statistical methods for hypothesis testing, 5.1 creative and exploratory practices.

• 5.2 Computer methods and the ‘new ways’ of doing science

6.1 “The scientific method” in science education and as seen by scientists

6.2 privileged methods and ‘gold standards’, 6.3 scientific method in the court room, 6.4 deviating practices, 7. conclusion, other internet resources, related entries.

This entry could have been given the title Scientific Methods and gone on to fill volumes, or it could have been extremely short, consisting of a brief summary rejection of the idea that there is any such thing as a unique Scientific Method at all. Both unhappy prospects are due to the fact that scientific activity varies so much across disciplines, times, places, and scientists that any account which manages to unify it all will either consist of overwhelming descriptive detail, or trivial generalizations.

The choice of scope for the present entry is more optimistic, taking a cue from the recent movement in philosophy of science toward a greater attention to practice: to what scientists actually do. This “turn to practice” can be seen as the latest form of studies of methods in science, insofar as it represents an attempt at understanding scientific activity, but through accounts that are neither meant to be universal and unified, nor singular and narrowly descriptive. To some extent, different scientists at different times and places can be said to be using the same method even though, in practice, the details are different.

Whether the context in which methods are carried out is relevant, or to what extent, will depend largely on what one takes the aims of science to be and what one’s own aims are. For most of the history of scientific methodology the assumption has been that the most important output of science is knowledge and so the aim of methodology should be to discover those methods by which scientific knowledge is generated.

Science was seen to embody the most successful form of reasoning (but which form?) to the most certain knowledge claims (but how certain?) on the basis of systematically collected evidence (but what counts as evidence, and should the evidence of the senses take precedence, or rational insight?) Section 2 surveys some of the history, pointing to two major themes. One theme is seeking the right balance between observation and reasoning (and the attendant forms of reasoning which employ them); the other is how certain scientific knowledge is or can be.

Section 3 turns to 20 th century debates on scientific method. In the second half of the 20 th century the epistemic privilege of science faced several challenges and many philosophers of science abandoned the reconstruction of the logic of scientific method. Views changed significantly regarding which functions of science ought to be captured and why. For some, the success of science was better identified with social or cultural features. Historical and sociological turns in the philosophy of science were made, with a demand that greater attention be paid to the non-epistemic aspects of science, such as sociological, institutional, material, and political factors. Even outside of those movements there was an increased specialization in the philosophy of science, with more and more focus on specific fields within science. The combined upshot was very few philosophers arguing any longer for a grand unified methodology of science. Sections 3 and 4 surveys the main positions on scientific method in 20 th century philosophy of science, focusing on where they differ in their preference for confirmation or falsification or for waiving the idea of a special scientific method altogether.

In recent decades, attention has primarily been paid to scientific activities traditionally falling under the rubric of method, such as experimental design and general laboratory practice, the use of statistics, the construction and use of models and diagrams, interdisciplinary collaboration, and science communication. Sections 4–6 attempt to construct a map of the current domains of the study of methods in science.

As these sections illustrate, the question of method is still central to the discourse about science. Scientific method remains a topic for education, for science policy, and for scientists. It arises in the public domain where the demarcation or status of science is at issue. Some philosophers have recently returned, therefore, to the question of what it is that makes science a unique cultural product. This entry will close with some of these recent attempts at discerning and encapsulating the activities by which scientific knowledge is achieved.

Attempting a history of scientific method compounds the vast scope of the topic. This section briefly surveys the background to modern methodological debates. What can be called the classical view goes back to antiquity, and represents a point of departure for later divergences. [ 1 ]

We begin with a point made by Laudan (1968) in his historical survey of scientific method:

Perhaps the most serious inhibition to the emergence of the history of theories of scientific method as a respectable area of study has been the tendency to conflate it with the general history of epistemology, thereby assuming that the narrative categories and classificatory pigeon-holes applied to the latter are also basic to the former. (1968: 5)

To see knowledge about the natural world as falling under knowledge more generally is an understandable conflation. Histories of theories of method would naturally employ the same narrative categories and classificatory pigeon holes. An important theme of the history of epistemology, for example, is the unification of knowledge, a theme reflected in the question of the unification of method in science. Those who have identified differences in kinds of knowledge have often likewise identified different methods for achieving that kind of knowledge (see the entry on the unity of science ).

Different views on what is known, how it is known, and what can be known are connected. Plato distinguished the realms of things into the visible and the intelligible ( The Republic , 510a, in Cooper 1997). Only the latter, the Forms, could be objects of knowledge. The intelligible truths could be known with the certainty of geometry and deductive reasoning. What could be observed of the material world, however, was by definition imperfect and deceptive, not ideal. The Platonic way of knowledge therefore emphasized reasoning as a method, downplaying the importance of observation. Aristotle disagreed, locating the Forms in the natural world as the fundamental principles to be discovered through the inquiry into nature ( Metaphysics Z , in Barnes 1984).

Aristotle is recognized as giving the earliest systematic treatise on the nature of scientific inquiry in the western tradition, one which embraced observation and reasoning about the natural world. In the Prior and Posterior Analytics , Aristotle reflects first on the aims and then the methods of inquiry into nature. A number of features can be found which are still considered by most to be essential to science. For Aristotle, empiricism, careful observation (but passive observation, not controlled experiment), is the starting point. The aim is not merely recording of facts, though. For Aristotle, science ( epistêmê ) is a body of properly arranged knowledge or learning—the empirical facts, but also their ordering and display are of crucial importance. The aims of discovery, ordering, and display of facts partly determine the methods required of successful scientific inquiry. Also determinant is the nature of the knowledge being sought, and the explanatory causes proper to that kind of knowledge (see the discussion of the four causes in the entry on Aristotle on causality ).

In addition to careful observation, then, scientific method requires a logic as a system of reasoning for properly arranging, but also inferring beyond, what is known by observation. Methods of reasoning may include induction, prediction, or analogy, among others. Aristotle’s system (along with his catalogue of fallacious reasoning) was collected under the title the Organon . This title would be echoed in later works on scientific reasoning, such as Novum Organon by Francis Bacon, and Novum Organon Restorum by William Whewell (see below). In Aristotle’s Organon reasoning is divided primarily into two forms, a rough division which persists into modern times. The division, known most commonly today as deductive versus inductive method, appears in other eras and methodologies as analysis/​synthesis, non-ampliative/​ampliative, or even confirmation/​verification. The basic idea is there are two “directions” to proceed in our methods of inquiry: one away from what is observed, to the more fundamental, general, and encompassing principles; the other, from the fundamental and general to instances or implications of principles.

The basic aim and method of inquiry identified here can be seen as a theme running throughout the next two millennia of reflection on the correct way to seek after knowledge: carefully observe nature and then seek rules or principles which explain or predict its operation. The Aristotelian corpus provided the framework for a commentary tradition on scientific method independent of science itself (cosmos versus physics.) During the medieval period, figures such as Albertus Magnus (1206–1280), Thomas Aquinas (1225–1274), Robert Grosseteste (1175–1253), Roger Bacon (1214/1220–1292), William of Ockham (1287–1347), Andreas Vesalius (1514–1546), Giacomo Zabarella (1533–1589) all worked to clarify the kind of knowledge obtainable by observation and induction, the source of justification of induction, and best rules for its application. [ 2 ] Many of their contributions we now think of as essential to science (see also Laudan 1968). As Aristotle and Plato had employed a framework of reasoning either “to the forms” or “away from the forms”, medieval thinkers employed directions away from the phenomena or back to the phenomena. In analysis, a phenomena was examined to discover its basic explanatory principles; in synthesis, explanations of a phenomena were constructed from first principles.

During the Scientific Revolution these various strands of argument, experiment, and reason were forged into a dominant epistemic authority. The 16 th –18 th centuries were a period of not only dramatic advance in knowledge about the operation of the natural world—advances in mechanical, medical, biological, political, economic explanations—but also of self-awareness of the revolutionary changes taking place, and intense reflection on the source and legitimation of the method by which the advances were made. The struggle to establish the new authority included methodological moves. The Book of Nature, according to the metaphor of Galileo Galilei (1564–1642) or Francis Bacon (1561–1626), was written in the language of mathematics, of geometry and number. This motivated an emphasis on mathematical description and mechanical explanation as important aspects of scientific method. Through figures such as Henry More and Ralph Cudworth, a neo-Platonic emphasis on the importance of metaphysical reflection on nature behind appearances, particularly regarding the spiritual as a complement to the purely mechanical, remained an important methodological thread of the Scientific Revolution (see the entries on Cambridge platonists ; Boyle ; Henry More ; Galileo ).

In Novum Organum (1620), Bacon was critical of the Aristotelian method for leaping from particulars to universals too quickly. The syllogistic form of reasoning readily mixed those two types of propositions. Bacon aimed at the invention of new arts, principles, and directions. His method would be grounded in methodical collection of observations, coupled with correction of our senses (and particularly, directions for the avoidance of the Idols, as he called them, kinds of systematic errors to which naïve observers are prone.) The community of scientists could then climb, by a careful, gradual and unbroken ascent, to reliable general claims.

Bacon’s method has been criticized as impractical and too inflexible for the practicing scientist. Whewell would later criticize Bacon in his System of Logic for paying too little attention to the practices of scientists. It is hard to find convincing examples of Bacon’s method being put in to practice in the history of science, but there are a few who have been held up as real examples of 16 th century scientific, inductive method, even if not in the rigid Baconian mold: figures such as Robert Boyle (1627–1691) and William Harvey (1578–1657) (see the entry on Bacon ).

It is to Isaac Newton (1642–1727), however, that historians of science and methodologists have paid greatest attention. Given the enormous success of his Principia Mathematica and Opticks , this is understandable. The study of Newton’s method has had two main thrusts: the implicit method of the experiments and reasoning presented in the Opticks, and the explicit methodological rules given as the Rules for Philosophising (the Regulae) in Book III of the Principia . [ 3 ] Newton’s law of gravitation, the linchpin of his new cosmology, broke with explanatory conventions of natural philosophy, first for apparently proposing action at a distance, but more generally for not providing “true”, physical causes. The argument for his System of the World ( Principia , Book III) was based on phenomena, not reasoned first principles. This was viewed (mainly on the continent) as insufficient for proper natural philosophy. The Regulae counter this objection, re-defining the aims of natural philosophy by re-defining the method natural philosophers should follow. (See the entry on Newton’s philosophy .)

To his list of methodological prescriptions should be added Newton’s famous phrase “ hypotheses non fingo ” (commonly translated as “I frame no hypotheses”.) The scientist was not to invent systems but infer explanations from observations, as Bacon had advocated. This would come to be known as inductivism. In the century after Newton, significant clarifications of the Newtonian method were made. Colin Maclaurin (1698–1746), for instance, reconstructed the essential structure of the method as having complementary analysis and synthesis phases, one proceeding away from the phenomena in generalization, the other from the general propositions to derive explanations of new phenomena. Denis Diderot (1713–1784) and editors of the Encyclopédie did much to consolidate and popularize Newtonianism, as did Francesco Algarotti (1721–1764). The emphasis was often the same, as much on the character of the scientist as on their process, a character which is still commonly assumed. The scientist is humble in the face of nature, not beholden to dogma, obeys only his eyes, and follows the truth wherever it leads. It was certainly Voltaire (1694–1778) and du Chatelet (1706–1749) who were most influential in propagating the latter vision of the scientist and their craft, with Newton as hero. Scientific method became a revolutionary force of the Enlightenment. (See also the entries on Newton , Leibniz , Descartes , Boyle , Hume , enlightenment , as well as Shank 2008 for a historical overview.)

Not all 18 th century reflections on scientific method were so celebratory. Famous also are George Berkeley’s (1685–1753) attack on the mathematics of the new science, as well as the over-emphasis of Newtonians on observation; and David Hume’s (1711–1776) undermining of the warrant offered for scientific claims by inductive justification (see the entries on: George Berkeley ; David Hume ; Hume’s Newtonianism and Anti-Newtonianism ). Hume’s problem of induction motivated Immanuel Kant (1724–1804) to seek new foundations for empirical method, though as an epistemic reconstruction, not as any set of practical guidelines for scientists. Both Hume and Kant influenced the methodological reflections of the next century, such as the debate between Mill and Whewell over the certainty of inductive inferences in science.

The debate between John Stuart Mill (1806–1873) and William Whewell (1794–1866) has become the canonical methodological debate of the 19 th century. Although often characterized as a debate between inductivism and hypothetico-deductivism, the role of the two methods on each side is actually more complex. On the hypothetico-deductive account, scientists work to come up with hypotheses from which true observational consequences can be deduced—hence, hypothetico-deductive. Because Whewell emphasizes both hypotheses and deduction in his account of method, he can be seen as a convenient foil to the inductivism of Mill. However, equally if not more important to Whewell’s portrayal of scientific method is what he calls the “fundamental antithesis”. Knowledge is a product of the objective (what we see in the world around us) and subjective (the contributions of our mind to how we perceive and understand what we experience, which he called the Fundamental Ideas). Both elements are essential according to Whewell, and he was therefore critical of Kant for too much focus on the subjective, and John Locke (1632–1704) and Mill for too much focus on the senses. Whewell’s fundamental ideas can be discipline relative. An idea can be fundamental even if it is necessary for knowledge only within a given scientific discipline (e.g., chemical affinity for chemistry). This distinguishes fundamental ideas from the forms and categories of intuition of Kant. (See the entry on Whewell .)

Clarifying fundamental ideas would therefore be an essential part of scientific method and scientific progress. Whewell called this process “Discoverer’s Induction”. It was induction, following Bacon or Newton, but Whewell sought to revive Bacon’s account by emphasising the role of ideas in the clear and careful formulation of inductive hypotheses. Whewell’s induction is not merely the collecting of objective facts. The subjective plays a role through what Whewell calls the Colligation of Facts, a creative act of the scientist, the invention of a theory. A theory is then confirmed by testing, where more facts are brought under the theory, called the Consilience of Inductions. Whewell felt that this was the method by which the true laws of nature could be discovered: clarification of fundamental concepts, clever invention of explanations, and careful testing. Mill, in his critique of Whewell, and others who have cast Whewell as a fore-runner of the hypothetico-deductivist view, seem to have under-estimated the importance of this discovery phase in Whewell’s understanding of method (Snyder 1997a,b, 1999). Down-playing the discovery phase would come to characterize methodology of the early 20 th century (see section 3 ).

Mill, in his System of Logic , put forward a narrower view of induction as the essence of scientific method. For Mill, induction is the search first for regularities among events. Among those regularities, some will continue to hold for further observations, eventually gaining the status of laws. One can also look for regularities among the laws discovered in a domain, i.e., for a law of laws. Which “law law” will hold is time and discipline dependent and open to revision. One example is the Law of Universal Causation, and Mill put forward specific methods for identifying causes—now commonly known as Mill’s methods. These five methods look for circumstances which are common among the phenomena of interest, those which are absent when the phenomena are, or those for which both vary together. Mill’s methods are still seen as capturing basic intuitions about experimental methods for finding the relevant explanatory factors ( System of Logic (1843), see Mill entry). The methods advocated by Whewell and Mill, in the end, look similar. Both involve inductive generalization to covering laws. They differ dramatically, however, with respect to the necessity of the knowledge arrived at; that is, at the meta-methodological level (see the entries on Whewell and Mill entries).

3. Logic of method and critical responses

The quantum and relativistic revolutions in physics in the early 20 th century had a profound effect on methodology. Conceptual foundations of both theories were taken to show the defeasibility of even the most seemingly secure intuitions about space, time and bodies. Certainty of knowledge about the natural world was therefore recognized as unattainable. Instead a renewed empiricism was sought which rendered science fallible but still rationally justifiable.

Analyses of the reasoning of scientists emerged, according to which the aspects of scientific method which were of primary importance were the means of testing and confirming of theories. A distinction in methodology was made between the contexts of discovery and justification. The distinction could be used as a wedge between the particularities of where and how theories or hypotheses are arrived at, on the one hand, and the underlying reasoning scientists use (whether or not they are aware of it) when assessing theories and judging their adequacy on the basis of the available evidence. By and large, for most of the 20 th century, philosophy of science focused on the second context, although philosophers differed on whether to focus on confirmation or refutation as well as on the many details of how confirmation or refutation could or could not be brought about. By the mid-20 th century these attempts at defining the method of justification and the context distinction itself came under pressure. During the same period, philosophy of science developed rapidly, and from section 4 this entry will therefore shift from a primarily historical treatment of the scientific method towards a primarily thematic one.

Advances in logic and probability held out promise of the possibility of elaborate reconstructions of scientific theories and empirical method, the best example being Rudolf Carnap’s The Logical Structure of the World (1928). Carnap attempted to show that a scientific theory could be reconstructed as a formal axiomatic system—that is, a logic. That system could refer to the world because some of its basic sentences could be interpreted as observations or operations which one could perform to test them. The rest of the theoretical system, including sentences using theoretical or unobservable terms (like electron or force) would then either be meaningful because they could be reduced to observations, or they had purely logical meanings (called analytic, like mathematical identities). This has been referred to as the verifiability criterion of meaning. According to the criterion, any statement not either analytic or verifiable was strictly meaningless. Although the view was endorsed by Carnap in 1928, he would later come to see it as too restrictive (Carnap 1956). Another familiar version of this idea is operationalism of Percy William Bridgman. In The Logic of Modern Physics (1927) Bridgman asserted that every physical concept could be defined in terms of the operations one would perform to verify the application of that concept. Making good on the operationalisation of a concept even as simple as length, however, can easily become enormously complex (for measuring very small lengths, for instance) or impractical (measuring large distances like light years.)

Carl Hempel’s (1950, 1951) criticisms of the verifiability criterion of meaning had enormous influence. He pointed out that universal generalizations, such as most scientific laws, were not strictly meaningful on the criterion. Verifiability and operationalism both seemed too restrictive to capture standard scientific aims and practice. The tenuous connection between these reconstructions and actual scientific practice was criticized in another way. In both approaches, scientific methods are instead recast in methodological roles. Measurements, for example, were looked to as ways of giving meanings to terms. The aim of the philosopher of science was not to understand the methods per se , but to use them to reconstruct theories, their meanings, and their relation to the world. When scientists perform these operations, however, they will not report that they are doing them to give meaning to terms in a formal axiomatic system. This disconnect between methodology and the details of actual scientific practice would seem to violate the empiricism the Logical Positivists and Bridgman were committed to. The view that methodology should correspond to practice (to some extent) has been called historicism, or intuitionism. We turn to these criticisms and responses in section 3.4 . [ 4 ]

Positivism also had to contend with the recognition that a purely inductivist approach, along the lines of Bacon-Newton-Mill, was untenable. There was no pure observation, for starters. All observation was theory laden. Theory is required to make any observation, therefore not all theory can be derived from observation alone. (See the entry on theory and observation in science .) Even granting an observational basis, Hume had already pointed out that one could not deductively justify inductive conclusions without begging the question by presuming the success of the inductive method. Likewise, positivist attempts at analyzing how a generalization can be confirmed by observations of its instances were subject to a number of criticisms. Goodman (1965) and Hempel (1965) both point to paradoxes inherent in standard accounts of confirmation. Recent attempts at explaining how observations can serve to confirm a scientific theory are discussed in section 4 below.

The standard starting point for a non-inductive analysis of the logic of confirmation is known as the Hypothetico-Deductive (H-D) method. In its simplest form, a sentence of a theory which expresses some hypothesis is confirmed by its true consequences. As noted in section 2 , this method had been advanced by Whewell in the 19 th century, as well as Nicod (1924) and others in the 20 th century. Often, Hempel’s (1966) description of the H-D method, illustrated by the case of Semmelweiss’ inferential procedures in establishing the cause of childbed fever, has been presented as a key account of H-D as well as a foil for criticism of the H-D account of confirmation (see, for example, Lipton’s (2004) discussion of inference to the best explanation; also the entry on confirmation ). Hempel described Semmelsweiss’ procedure as examining various hypotheses explaining the cause of childbed fever. Some hypotheses conflicted with observable facts and could be rejected as false immediately. Others needed to be tested experimentally by deducing which observable events should follow if the hypothesis were true (what Hempel called the test implications of the hypothesis), then conducting an experiment and observing whether or not the test implications occurred. If the experiment showed the test implication to be false, the hypothesis could be rejected. If the experiment showed the test implications to be true, however, this did not prove the hypothesis true. The confirmation of a test implication does not verify a hypothesis, though Hempel did allow that “it provides at least some support, some corroboration or confirmation for it” (Hempel 1966: 8). The degree of this support then depends on the quantity, variety and precision of the supporting evidence.

Another approach that took off from the difficulties with inductive inference was Karl Popper’s critical rationalism or falsificationism (Popper 1959, 1963). Falsification is deductive and similar to H-D in that it involves scientists deducing observational consequences from the hypothesis under test. For Popper, however, the important point was not the degree of confirmation that successful prediction offered to a hypothesis. The crucial thing was the logical asymmetry between confirmation, based on inductive inference, and falsification, which can be based on a deductive inference. (This simple opposition was later questioned, by Lakatos, among others. See the entry on historicist theories of scientific rationality. )

Popper stressed that, regardless of the amount of confirming evidence, we can never be certain that a hypothesis is true without committing the fallacy of affirming the consequent. Instead, Popper introduced the notion of corroboration as a measure for how well a theory or hypothesis has survived previous testing—but without implying that this is also a measure for the probability that it is true.

Popper was also motivated by his doubts about the scientific status of theories like the Marxist theory of history or psycho-analysis, and so wanted to demarcate between science and pseudo-science. Popper saw this as an importantly different distinction than demarcating science from metaphysics. The latter demarcation was the primary concern of many logical empiricists. Popper used the idea of falsification to draw a line instead between pseudo and proper science. Science was science because its method involved subjecting theories to rigorous tests which offered a high probability of failing and thus refuting the theory.

A commitment to the risk of failure was important. Avoiding falsification could be done all too easily. If a consequence of a theory is inconsistent with observations, an exception can be added by introducing auxiliary hypotheses designed explicitly to save the theory, so-called ad hoc modifications. This Popper saw done in pseudo-science where ad hoc theories appeared capable of explaining anything in their field of application. In contrast, science is risky. If observations showed the predictions from a theory to be wrong, the theory would be refuted. Hence, scientific hypotheses must be falsifiable. Not only must there exist some possible observation statement which could falsify the hypothesis or theory, were it observed, (Popper called these the hypothesis’ potential falsifiers) it is crucial to the Popperian scientific method that such falsifications be sincerely attempted on a regular basis.

The more potential falsifiers of a hypothesis, the more falsifiable it would be, and the more the hypothesis claimed. Conversely, hypotheses without falsifiers claimed very little or nothing at all. Originally, Popper thought that this meant the introduction of ad hoc hypotheses only to save a theory should not be countenanced as good scientific method. These would undermine the falsifiabililty of a theory. However, Popper later came to recognize that the introduction of modifications (immunizations, he called them) was often an important part of scientific development. Responding to surprising or apparently falsifying observations often generated important new scientific insights. Popper’s own example was the observed motion of Uranus which originally did not agree with Newtonian predictions. The ad hoc hypothesis of an outer planet explained the disagreement and led to further falsifiable predictions. Popper sought to reconcile the view by blurring the distinction between falsifiable and not falsifiable, and speaking instead of degrees of testability (Popper 1985: 41f.).

From the 1960s on, sustained meta-methodological criticism emerged that drove philosophical focus away from scientific method. A brief look at those criticisms follows, with recommendations for further reading at the end of the entry.

Thomas Kuhn’s The Structure of Scientific Revolutions (1962) begins with a well-known shot across the bow for philosophers of science:

History, if viewed as a repository for more than anecdote or chronology, could produce a decisive transformation in the image of science by which we are now possessed. (1962: 1)

The image Kuhn thought needed transforming was the a-historical, rational reconstruction sought by many of the Logical Positivists, though Carnap and other positivists were actually quite sympathetic to Kuhn’s views. (See the entry on the Vienna Circle .) Kuhn shares with other of his contemporaries, such as Feyerabend and Lakatos, a commitment to a more empirical approach to philosophy of science. Namely, the history of science provides important data, and necessary checks, for philosophy of science, including any theory of scientific method.

The history of science reveals, according to Kuhn, that scientific development occurs in alternating phases. During normal science, the members of the scientific community adhere to the paradigm in place. Their commitment to the paradigm means a commitment to the puzzles to be solved and the acceptable ways of solving them. Confidence in the paradigm remains so long as steady progress is made in solving the shared puzzles. Method in this normal phase operates within a disciplinary matrix (Kuhn’s later concept of a paradigm) which includes standards for problem solving, and defines the range of problems to which the method should be applied. An important part of a disciplinary matrix is the set of values which provide the norms and aims for scientific method. The main values that Kuhn identifies are prediction, problem solving, simplicity, consistency, and plausibility.

An important by-product of normal science is the accumulation of puzzles which cannot be solved with resources of the current paradigm. Once accumulation of these anomalies has reached some critical mass, it can trigger a communal shift to a new paradigm and a new phase of normal science. Importantly, the values that provide the norms and aims for scientific method may have transformed in the meantime. Method may therefore be relative to discipline, time or place

Feyerabend also identified the aims of science as progress, but argued that any methodological prescription would only stifle that progress (Feyerabend 1988). His arguments are grounded in re-examining accepted “myths” about the history of science. Heroes of science, like Galileo, are shown to be just as reliant on rhetoric and persuasion as they are on reason and demonstration. Others, like Aristotle, are shown to be far more reasonable and far-reaching in their outlooks then they are given credit for. As a consequence, the only rule that could provide what he took to be sufficient freedom was the vacuous “anything goes”. More generally, even the methodological restriction that science is the best way to pursue knowledge, and to increase knowledge, is too restrictive. Feyerabend suggested instead that science might, in fact, be a threat to a free society, because it and its myth had become so dominant (Feyerabend 1978).

An even more fundamental kind of criticism was offered by several sociologists of science from the 1970s onwards who rejected the methodology of providing philosophical accounts for the rational development of science and sociological accounts of the irrational mistakes. Instead, they adhered to a symmetry thesis on which any causal explanation of how scientific knowledge is established needs to be symmetrical in explaining truth and falsity, rationality and irrationality, success and mistakes, by the same causal factors (see, e.g., Barnes and Bloor 1982, Bloor 1991). Movements in the Sociology of Science, like the Strong Programme, or in the social dimensions and causes of knowledge more generally led to extended and close examination of detailed case studies in contemporary science and its history. (See the entries on the social dimensions of scientific knowledge and social epistemology .) Well-known examinations by Latour and Woolgar (1979/1986), Knorr-Cetina (1981), Pickering (1984), Shapin and Schaffer (1985) seem to bear out that it was social ideologies (on a macro-scale) or individual interactions and circumstances (on a micro-scale) which were the primary causal factors in determining which beliefs gained the status of scientific knowledge. As they saw it therefore, explanatory appeals to scientific method were not empirically grounded.

A late, and largely unexpected, criticism of scientific method came from within science itself. Beginning in the early 2000s, a number of scientists attempting to replicate the results of published experiments could not do so. There may be close conceptual connection between reproducibility and method. For example, if reproducibility means that the same scientific methods ought to produce the same result, and all scientific results ought to be reproducible, then whatever it takes to reproduce a scientific result ought to be called scientific method. Space limits us to the observation that, insofar as reproducibility is a desired outcome of proper scientific method, it is not strictly a part of scientific method. (See the entry on reproducibility of scientific results .)

By the close of the 20 th century the search for the scientific method was flagging. Nola and Sankey (2000b) could introduce their volume on method by remarking that “For some, the whole idea of a theory of scientific method is yester-year’s debate …”.

Despite the many difficulties that philosophers encountered in trying to providing a clear methodology of conformation (or refutation), still important progress has been made on understanding how observation can provide evidence for a given theory. Work in statistics has been crucial for understanding how theories can be tested empirically, and in recent decades a huge literature has developed that attempts to recast confirmation in Bayesian terms. Here these developments can be covered only briefly, and we refer to the entry on confirmation for further details and references.

Statistics has come to play an increasingly important role in the methodology of the experimental sciences from the 19 th century onwards. At that time, statistics and probability theory took on a methodological role as an analysis of inductive inference, and attempts to ground the rationality of induction in the axioms of probability theory have continued throughout the 20 th century and in to the present. Developments in the theory of statistics itself, meanwhile, have had a direct and immense influence on the experimental method, including methods for measuring the uncertainty of observations such as the Method of Least Squares developed by Legendre and Gauss in the early 19 th century, criteria for the rejection of outliers proposed by Peirce by the mid-19 th century, and the significance tests developed by Gosset (a.k.a. “Student”), Fisher, Neyman & Pearson and others in the 1920s and 1930s (see, e.g., Swijtink 1987 for a brief historical overview; and also the entry on C.S. Peirce ).

These developments within statistics then in turn led to a reflective discussion among both statisticians and philosophers of science on how to perceive the process of hypothesis testing: whether it was a rigorous statistical inference that could provide a numerical expression of the degree of confidence in the tested hypothesis, or if it should be seen as a decision between different courses of actions that also involved a value component. This led to a major controversy among Fisher on the one side and Neyman and Pearson on the other (see especially Fisher 1955, Neyman 1956 and Pearson 1955, and for analyses of the controversy, e.g., Howie 2002, Marks 2000, Lenhard 2006). On Fisher’s view, hypothesis testing was a methodology for when to accept or reject a statistical hypothesis, namely that a hypothesis should be rejected by evidence if this evidence would be unlikely relative to other possible outcomes, given the hypothesis were true. In contrast, on Neyman and Pearson’s view, the consequence of error also had to play a role when deciding between hypotheses. Introducing the distinction between the error of rejecting a true hypothesis (type I error) and accepting a false hypothesis (type II error), they argued that it depends on the consequences of the error to decide whether it is more important to avoid rejecting a true hypothesis or accepting a false one. Hence, Fisher aimed for a theory of inductive inference that enabled a numerical expression of confidence in a hypothesis. To him, the important point was the search for truth, not utility. In contrast, the Neyman-Pearson approach provided a strategy of inductive behaviour for deciding between different courses of action. Here, the important point was not whether a hypothesis was true, but whether one should act as if it was.

Similar discussions are found in the philosophical literature. On the one side, Churchman (1948) and Rudner (1953) argued that because scientific hypotheses can never be completely verified, a complete analysis of the methods of scientific inference includes ethical judgments in which the scientists must decide whether the evidence is sufficiently strong or that the probability is sufficiently high to warrant the acceptance of the hypothesis, which again will depend on the importance of making a mistake in accepting or rejecting the hypothesis. Others, such as Jeffrey (1956) and Levi (1960) disagreed and instead defended a value-neutral view of science on which scientists should bracket their attitudes, preferences, temperament, and values when assessing the correctness of their inferences. For more details on this value-free ideal in the philosophy of science and its historical development, see Douglas (2009) and Howard (2003). For a broad set of case studies examining the role of values in science, see e.g. Elliott & Richards 2017.

In recent decades, philosophical discussions of the evaluation of probabilistic hypotheses by statistical inference have largely focused on Bayesianism that understands probability as a measure of a person’s degree of belief in an event, given the available information, and frequentism that instead understands probability as a long-run frequency of a repeatable event. Hence, for Bayesians probabilities refer to a state of knowledge, whereas for frequentists probabilities refer to frequencies of events (see, e.g., Sober 2008, chapter 1 for a detailed introduction to Bayesianism and frequentism as well as to likelihoodism). Bayesianism aims at providing a quantifiable, algorithmic representation of belief revision, where belief revision is a function of prior beliefs (i.e., background knowledge) and incoming evidence. Bayesianism employs a rule based on Bayes’ theorem, a theorem of the probability calculus which relates conditional probabilities. The probability that a particular hypothesis is true is interpreted as a degree of belief, or credence, of the scientist. There will also be a probability and a degree of belief that a hypothesis will be true conditional on a piece of evidence (an observation, say) being true. Bayesianism proscribes that it is rational for the scientist to update their belief in the hypothesis to that conditional probability should it turn out that the evidence is, in fact, observed (see, e.g., Sprenger & Hartmann 2019 for a comprehensive treatment of Bayesian philosophy of science). Originating in the work of Neyman and Person, frequentism aims at providing the tools for reducing long-run error rates, such as the error-statistical approach developed by Mayo (1996) that focuses on how experimenters can avoid both type I and type II errors by building up a repertoire of procedures that detect errors if and only if they are present. Both Bayesianism and frequentism have developed over time, they are interpreted in different ways by its various proponents, and their relations to previous criticism to attempts at defining scientific method are seen differently by proponents and critics. The literature, surveys, reviews and criticism in this area are vast and the reader is referred to the entries on Bayesian epistemology and confirmation .

5. Method in Practice

Attention to scientific practice, as we have seen, is not itself new. However, the turn to practice in the philosophy of science of late can be seen as a correction to the pessimism with respect to method in philosophy of science in later parts of the 20 th century, and as an attempted reconciliation between sociological and rationalist explanations of scientific knowledge. Much of this work sees method as detailed and context specific problem-solving procedures, and methodological analyses to be at the same time descriptive, critical and advisory (see Nickles 1987 for an exposition of this view). The following section contains a survey of some of the practice focuses. In this section we turn fully to topics rather than chronology.

A problem with the distinction between the contexts of discovery and justification that figured so prominently in philosophy of science in the first half of the 20 th century (see section 2 ) is that no such distinction can be clearly seen in scientific activity (see Arabatzis 2006). Thus, in recent decades, it has been recognized that study of conceptual innovation and change should not be confined to psychology and sociology of science, but are also important aspects of scientific practice which philosophy of science should address (see also the entry on scientific discovery ). Looking for the practices that drive conceptual innovation has led philosophers to examine both the reasoning practices of scientists and the wide realm of experimental practices that are not directed narrowly at testing hypotheses, that is, exploratory experimentation.

Examining the reasoning practices of historical and contemporary scientists, Nersessian (2008) has argued that new scientific concepts are constructed as solutions to specific problems by systematic reasoning, and that of analogy, visual representation and thought-experimentation are among the important reasoning practices employed. These ubiquitous forms of reasoning are reliable—but also fallible—methods of conceptual development and change. On her account, model-based reasoning consists of cycles of construction, simulation, evaluation and adaption of models that serve as interim interpretations of the target problem to be solved. Often, this process will lead to modifications or extensions, and a new cycle of simulation and evaluation. However, Nersessian also emphasizes that

creative model-based reasoning cannot be applied as a simple recipe, is not always productive of solutions, and even its most exemplary usages can lead to incorrect solutions. (Nersessian 2008: 11)

Thus, while on the one hand she agrees with many previous philosophers that there is no logic of discovery, discoveries can derive from reasoned processes, such that a large and integral part of scientific practice is

the creation of concepts through which to comprehend, structure, and communicate about physical phenomena …. (Nersessian 1987: 11)

Similarly, work on heuristics for discovery and theory construction by scholars such as Darden (1991) and Bechtel & Richardson (1993) present science as problem solving and investigate scientific problem solving as a special case of problem-solving in general. Drawing largely on cases from the biological sciences, much of their focus has been on reasoning strategies for the generation, evaluation, and revision of mechanistic explanations of complex systems.

Addressing another aspect of the context distinction, namely the traditional view that the primary role of experiments is to test theoretical hypotheses according to the H-D model, other philosophers of science have argued for additional roles that experiments can play. The notion of exploratory experimentation was introduced to describe experiments driven by the desire to obtain empirical regularities and to develop concepts and classifications in which these regularities can be described (Steinle 1997, 2002; Burian 1997; Waters 2007)). However the difference between theory driven experimentation and exploratory experimentation should not be seen as a sharp distinction. Theory driven experiments are not always directed at testing hypothesis, but may also be directed at various kinds of fact-gathering, such as determining numerical parameters. Vice versa , exploratory experiments are usually informed by theory in various ways and are therefore not theory-free. Instead, in exploratory experiments phenomena are investigated without first limiting the possible outcomes of the experiment on the basis of extant theory about the phenomena.

The development of high throughput instrumentation in molecular biology and neighbouring fields has given rise to a special type of exploratory experimentation that collects and analyses very large amounts of data, and these new ‘omics’ disciplines are often said to represent a break with the ideal of hypothesis-driven science (Burian 2007; Elliott 2007; Waters 2007; O’Malley 2007) and instead described as data-driven research (Leonelli 2012; Strasser 2012) or as a special kind of “convenience experimentation” in which many experiments are done simply because they are extraordinarily convenient to perform (Krohs 2012).

5.2 Computer methods and ‘new ways’ of doing science

The field of omics just described is possible because of the ability of computers to process, in a reasonable amount of time, the huge quantities of data required. Computers allow for more elaborate experimentation (higher speed, better filtering, more variables, sophisticated coordination and control), but also, through modelling and simulations, might constitute a form of experimentation themselves. Here, too, we can pose a version of the general question of method versus practice: does the practice of using computers fundamentally change scientific method, or merely provide a more efficient means of implementing standard methods?

Because computers can be used to automate measurements, quantifications, calculations, and statistical analyses where, for practical reasons, these operations cannot be otherwise carried out, many of the steps involved in reaching a conclusion on the basis of an experiment are now made inside a “black box”, without the direct involvement or awareness of a human. This has epistemological implications, regarding what we can know, and how we can know it. To have confidence in the results, computer methods are therefore subjected to tests of verification and validation.

The distinction between verification and validation is easiest to characterize in the case of computer simulations. In a typical computer simulation scenario computers are used to numerically integrate differential equations for which no analytic solution is available. The equations are part of the model the scientist uses to represent a phenomenon or system under investigation. Verifying a computer simulation means checking that the equations of the model are being correctly approximated. Validating a simulation means checking that the equations of the model are adequate for the inferences one wants to make on the basis of that model.

A number of issues related to computer simulations have been raised. The identification of validity and verification as the testing methods has been criticized. Oreskes et al. (1994) raise concerns that “validiation”, because it suggests deductive inference, might lead to over-confidence in the results of simulations. The distinction itself is probably too clean, since actual practice in the testing of simulations mixes and moves back and forth between the two (Weissart 1997; Parker 2008a; Winsberg 2010). Computer simulations do seem to have a non-inductive character, given that the principles by which they operate are built in by the programmers, and any results of the simulation follow from those in-built principles in such a way that those results could, in principle, be deduced from the program code and its inputs. The status of simulations as experiments has therefore been examined (Kaufmann and Smarr 1993; Humphreys 1995; Hughes 1999; Norton and Suppe 2001). This literature considers the epistemology of these experiments: what we can learn by simulation, and also the kinds of justifications which can be given in applying that knowledge to the “real” world. (Mayo 1996; Parker 2008b). As pointed out, part of the advantage of computer simulation derives from the fact that huge numbers of calculations can be carried out without requiring direct observation by the experimenter/​simulator. At the same time, many of these calculations are approximations to the calculations which would be performed first-hand in an ideal situation. Both factors introduce uncertainties into the inferences drawn from what is observed in the simulation.

For many of the reasons described above, computer simulations do not seem to belong clearly to either the experimental or theoretical domain. Rather, they seem to crucially involve aspects of both. This has led some authors, such as Fox Keller (2003: 200) to argue that we ought to consider computer simulation a “qualitatively different way of doing science”. The literature in general tends to follow Kaufmann and Smarr (1993) in referring to computer simulation as a “third way” for scientific methodology (theoretical reasoning and experimental practice are the first two ways.). It should also be noted that the debates around these issues have tended to focus on the form of computer simulation typical in the physical sciences, where models are based on dynamical equations. Other forms of simulation might not have the same problems, or have problems of their own (see the entry on computer simulations in science ).

In recent years, the rapid development of machine learning techniques has prompted some scholars to suggest that the scientific method has become “obsolete” (Anderson 2008, Carrol and Goodstein 2009). This has resulted in an intense debate on the relative merit of data-driven and hypothesis-driven research (for samples, see e.g. Mazzocchi 2015 or Succi and Coveney 2018). For a detailed treatment of this topic, we refer to the entry scientific research and big data .

6. Discourse on scientific method

Despite philosophical disagreements, the idea of the scientific method still figures prominently in contemporary discourse on many different topics, both within science and in society at large. Often, reference to scientific method is used in ways that convey either the legend of a single, universal method characteristic of all science, or grants to a particular method or set of methods privilege as a special ‘gold standard’, often with reference to particular philosophers to vindicate the claims. Discourse on scientific method also typically arises when there is a need to distinguish between science and other activities, or for justifying the special status conveyed to science. In these areas, the philosophical attempts at identifying a set of methods characteristic for scientific endeavors are closely related to the philosophy of science’s classical problem of demarcation (see the entry on science and pseudo-science ) and to the philosophical analysis of the social dimension of scientific knowledge and the role of science in democratic society.

One of the settings in which the legend of a single, universal scientific method has been particularly strong is science education (see, e.g., Bauer 1992; McComas 1996; Wivagg & Allchin 2002). [ 5 ] Often, ‘the scientific method’ is presented in textbooks and educational web pages as a fixed four or five step procedure starting from observations and description of a phenomenon and progressing over formulation of a hypothesis which explains the phenomenon, designing and conducting experiments to test the hypothesis, analyzing the results, and ending with drawing a conclusion. Such references to a universal scientific method can be found in educational material at all levels of science education (Blachowicz 2009), and numerous studies have shown that the idea of a general and universal scientific method often form part of both students’ and teachers’ conception of science (see, e.g., Aikenhead 1987; Osborne et al. 2003). In response, it has been argued that science education need to focus more on teaching about the nature of science, although views have differed on whether this is best done through student-led investigations, contemporary cases, or historical cases (Allchin, Andersen & Nielsen 2014)

Although occasionally phrased with reference to the H-D method, important historical roots of the legend in science education of a single, universal scientific method are the American philosopher and psychologist Dewey’s account of inquiry in How We Think (1910) and the British mathematician Karl Pearson’s account of science in Grammar of Science (1892). On Dewey’s account, inquiry is divided into the five steps of

(i) a felt difficulty, (ii) its location and definition, (iii) suggestion of a possible solution, (iv) development by reasoning of the bearing of the suggestions, (v) further observation and experiment leading to its acceptance or rejection. (Dewey 1910: 72)

Similarly, on Pearson’s account, scientific investigations start with measurement of data and observation of their correction and sequence from which scientific laws can be discovered with the aid of creative imagination. These laws have to be subject to criticism, and their final acceptance will have equal validity for “all normally constituted minds”. Both Dewey’s and Pearson’s accounts should be seen as generalized abstractions of inquiry and not restricted to the realm of science—although both Dewey and Pearson referred to their respective accounts as ‘the scientific method’.

Occasionally, scientists make sweeping statements about a simple and distinct scientific method, as exemplified by Feynman’s simplified version of a conjectures and refutations method presented, for example, in the last of his 1964 Cornell Messenger lectures. [ 6 ] However, just as often scientists have come to the same conclusion as recent philosophy of science that there is not any unique, easily described scientific method. For example, the physicist and Nobel Laureate Weinberg described in the paper “The Methods of Science … And Those By Which We Live” (1995) how

The fact that the standards of scientific success shift with time does not only make the philosophy of science difficult; it also raises problems for the public understanding of science. We do not have a fixed scientific method to rally around and defend. (1995: 8)

Interview studies with scientists on their conception of method shows that scientists often find it hard to figure out whether available evidence confirms their hypothesis, and that there are no direct translations between general ideas about method and specific strategies to guide how research is conducted (Schickore & Hangel 2019, Hangel & Schickore 2017)

Reference to the scientific method has also often been used to argue for the scientific nature or special status of a particular activity. Philosophical positions that argue for a simple and unique scientific method as a criterion of demarcation, such as Popperian falsification, have often attracted practitioners who felt that they had a need to defend their domain of practice. For example, references to conjectures and refutation as the scientific method are abundant in much of the literature on complementary and alternative medicine (CAM)—alongside the competing position that CAM, as an alternative to conventional biomedicine, needs to develop its own methodology different from that of science.

Also within mainstream science, reference to the scientific method is used in arguments regarding the internal hierarchy of disciplines and domains. A frequently seen argument is that research based on the H-D method is superior to research based on induction from observations because in deductive inferences the conclusion follows necessarily from the premises. (See, e.g., Parascandola 1998 for an analysis of how this argument has been made to downgrade epidemiology compared to the laboratory sciences.) Similarly, based on an examination of the practices of major funding institutions such as the National Institutes of Health (NIH), the National Science Foundation (NSF) and the Biomedical Sciences Research Practices (BBSRC) in the UK, O’Malley et al. (2009) have argued that funding agencies seem to have a tendency to adhere to the view that the primary activity of science is to test hypotheses, while descriptive and exploratory research is seen as merely preparatory activities that are valuable only insofar as they fuel hypothesis-driven research.

In some areas of science, scholarly publications are structured in a way that may convey the impression of a neat and linear process of inquiry from stating a question, devising the methods by which to answer it, collecting the data, to drawing a conclusion from the analysis of data. For example, the codified format of publications in most biomedical journals known as the IMRAD format (Introduction, Method, Results, Analysis, Discussion) is explicitly described by the journal editors as “not an arbitrary publication format but rather a direct reflection of the process of scientific discovery” (see the so-called “Vancouver Recommendations”, ICMJE 2013: 11). However, scientific publications do not in general reflect the process by which the reported scientific results were produced. For example, under the provocative title “Is the scientific paper a fraud?”, Medawar argued that scientific papers generally misrepresent how the results have been produced (Medawar 1963/1996). Similar views have been advanced by philosophers, historians and sociologists of science (Gilbert 1976; Holmes 1987; Knorr-Cetina 1981; Schickore 2008; Suppe 1998) who have argued that scientists’ experimental practices are messy and often do not follow any recognizable pattern. Publications of research results, they argue, are retrospective reconstructions of these activities that often do not preserve the temporal order or the logic of these activities, but are instead often constructed in order to screen off potential criticism (see Schickore 2008 for a review of this work).

Philosophical positions on the scientific method have also made it into the court room, especially in the US where judges have drawn on philosophy of science in deciding when to confer special status to scientific expert testimony. A key case is Daubert vs Merrell Dow Pharmaceuticals (92–102, 509 U.S. 579, 1993). In this case, the Supreme Court argued in its 1993 ruling that trial judges must ensure that expert testimony is reliable, and that in doing this the court must look at the expert’s methodology to determine whether the proffered evidence is actually scientific knowledge. Further, referring to works of Popper and Hempel the court stated that

ordinarily, a key question to be answered in determining whether a theory or technique is scientific knowledge … is whether it can be (and has been) tested. (Justice Blackmun, Daubert v. Merrell Dow Pharmaceuticals; see Other Internet Resources for a link to the opinion)

But as argued by Haack (2005a,b, 2010) and by Foster & Hubner (1999), by equating the question of whether a piece of testimony is reliable with the question whether it is scientific as indicated by a special methodology, the court was producing an inconsistent mixture of Popper’s and Hempel’s philosophies, and this has later led to considerable confusion in subsequent case rulings that drew on the Daubert case (see Haack 2010 for a detailed exposition).

The difficulties around identifying the methods of science are also reflected in the difficulties of identifying scientific misconduct in the form of improper application of the method or methods of science. One of the first and most influential attempts at defining misconduct in science was the US definition from 1989 that defined misconduct as

fabrication, falsification, plagiarism, or other practices that seriously deviate from those that are commonly accepted within the scientific community . (Code of Federal Regulations, part 50, subpart A., August 8, 1989, italics added)

However, the “other practices that seriously deviate” clause was heavily criticized because it could be used to suppress creative or novel science. For example, the National Academy of Science stated in their report Responsible Science (1992) that it

wishes to discourage the possibility that a misconduct complaint could be lodged against scientists based solely on their use of novel or unorthodox research methods. (NAS: 27)

This clause was therefore later removed from the definition. For an entry into the key philosophical literature on conduct in science, see Shamoo & Resnick (2009).

The question of the source of the success of science has been at the core of philosophy since the beginning of modern science. If viewed as a matter of epistemology more generally, scientific method is a part of the entire history of philosophy. Over that time, science and whatever methods its practitioners may employ have changed dramatically. Today, many philosophers have taken up the banners of pluralism or of practice to focus on what are, in effect, fine-grained and contextually limited examinations of scientific method. Others hope to shift perspectives in order to provide a renewed general account of what characterizes the activity we call science.

One such perspective has been offered recently by Hoyningen-Huene (2008, 2013), who argues from the history of philosophy of science that after three lengthy phases of characterizing science by its method, we are now in a phase where the belief in the existence of a positive scientific method has eroded and what has been left to characterize science is only its fallibility. First was a phase from Plato and Aristotle up until the 17 th century where the specificity of scientific knowledge was seen in its absolute certainty established by proof from evident axioms; next was a phase up to the mid-19 th century in which the means to establish the certainty of scientific knowledge had been generalized to include inductive procedures as well. In the third phase, which lasted until the last decades of the 20 th century, it was recognized that empirical knowledge was fallible, but it was still granted a special status due to its distinctive mode of production. But now in the fourth phase, according to Hoyningen-Huene, historical and philosophical studies have shown how “scientific methods with the characteristics as posited in the second and third phase do not exist” (2008: 168) and there is no longer any consensus among philosophers and historians of science about the nature of science. For Hoyningen-Huene, this is too negative a stance, and he therefore urges the question about the nature of science anew. His own answer to this question is that “scientific knowledge differs from other kinds of knowledge, especially everyday knowledge, primarily by being more systematic” (Hoyningen-Huene 2013: 14). Systematicity can have several different dimensions: among them are more systematic descriptions, explanations, predictions, defense of knowledge claims, epistemic connectedness, ideal of completeness, knowledge generation, representation of knowledge and critical discourse. Hence, what characterizes science is the greater care in excluding possible alternative explanations, the more detailed elaboration with respect to data on which predictions are based, the greater care in detecting and eliminating sources of error, the more articulate connections to other pieces of knowledge, etc. On this position, what characterizes science is not that the methods employed are unique to science, but that the methods are more carefully employed.

Another, similar approach has been offered by Haack (2003). She sets off, similar to Hoyningen-Huene, from a dissatisfaction with the recent clash between what she calls Old Deferentialism and New Cynicism. The Old Deferentialist position is that science progressed inductively by accumulating true theories confirmed by empirical evidence or deductively by testing conjectures against basic statements; while the New Cynics position is that science has no epistemic authority and no uniquely rational method and is merely just politics. Haack insists that contrary to the views of the New Cynics, there are objective epistemic standards, and there is something epistemologically special about science, even though the Old Deferentialists pictured this in a wrong way. Instead, she offers a new Critical Commonsensist account on which standards of good, strong, supportive evidence and well-conducted, honest, thorough and imaginative inquiry are not exclusive to the sciences, but the standards by which we judge all inquirers. In this sense, science does not differ in kind from other kinds of inquiry, but it may differ in the degree to which it requires broad and detailed background knowledge and a familiarity with a technical vocabulary that only specialists may possess.

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The 6 Scientific Method Steps and How to Use Them

General Education

When you’re faced with a scientific problem, solving it can seem like an impossible prospect. There are so many possible explanations for everything we see and experience—how can you possibly make sense of them all? Science has a simple answer: the scientific method.

The scientific method is a method of asking and answering questions about the world. These guiding principles give scientists a model to work through when trying to understand the world, but where did that model come from, and how does it work?

In this article, we’ll define the scientific method, discuss its long history, and cover each of the scientific method steps in detail.

What Is the Scientific Method?

At its most basic, the scientific method is a procedure for conducting scientific experiments. It’s a set model that scientists in a variety of fields can follow, going from initial observation to conclusion in a loose but concrete format.

The number of steps varies, but the process begins with an observation, progresses through an experiment, and concludes with analysis and sharing data. One of the most important pieces to the scientific method is skepticism —the goal is to find truth, not to confirm a particular thought. That requires reevaluation and repeated experimentation, as well as examining your thinking through rigorous study.

There are in fact multiple scientific methods, as the basic structure can be easily modified.  The one we typically learn about in school is the basic method, based in logic and problem solving, typically used in “hard” science fields like biology, chemistry, and physics. It may vary in other fields, such as psychology, but the basic premise of making observations, testing, and continuing to improve a theory from the results remain the same.

The History of the Scientific Method

The scientific method as we know it today is based on thousands of years of scientific study. Its development goes all the way back to ancient Mesopotamia, Greece, and India.

The Ancient World

In ancient Greece, Aristotle devised an inductive-deductive process , which weighs broad generalizations from data against conclusions reached by narrowing down possibilities from a general statement. However, he favored deductive reasoning, as it identifies causes, which he saw as more important.

Aristotle wrote a great deal about logic and many of his ideas about reasoning echo those found in the modern scientific method, such as ignoring circular evidence and limiting the number of middle terms between the beginning of an experiment and the end. Though his model isn’t the one that we use today, the reliance on logic and thorough testing are still key parts of science today.

The Middle Ages

The next big step toward the development of the modern scientific method came in the Middle Ages, particularly in the Islamic world. Ibn al-Haytham, a physicist from what we now know as Iraq, developed a method of testing, observing, and deducing for his research on vision. al-Haytham was critical of Aristotle’s lack of inductive reasoning, which played an important role in his own research.

Other scientists, including Abū Rayhān al-Bīrūnī, Ibn Sina, and Robert Grosseteste also developed models of scientific reasoning to test their own theories. Though they frequently disagreed with one another and Aristotle, those disagreements and refinements of their methods led to the scientific method we have today.

Following those major developments, particularly Grosseteste’s work, Roger Bacon developed his own cycle of observation (seeing that something occurs), hypothesis (making a guess about why that thing occurs), experimentation (testing that the thing occurs), and verification (an outside person ensuring that the result of the experiment is consistent).

After joining the Franciscan Order, Bacon was granted a special commission to write about science; typically, Friars were not allowed to write books or pamphlets. With this commission, Bacon outlined important tenets of the scientific method, including causes of error, methods of knowledge, and the differences between speculative and experimental science. He also used his own principles to investigate the causes of a rainbow, demonstrating the method’s effectiveness.

Scientific Revolution

Throughout the Renaissance, more great thinkers became involved in devising a thorough, rigorous method of scientific study. Francis Bacon brought inductive reasoning further into the method, whereas Descartes argued that the laws of the universe meant that deductive reasoning was sufficient. Galileo’s research was also inductive reasoning-heavy, as he believed that researchers could not account for every possible variable; therefore, repetition was necessary to eliminate faulty hypotheses and experiments.

All of this led to the birth of the Scientific Revolution , which took place during the sixteenth and seventeenth centuries. In 1660, a group of philosophers and physicians joined together to work on scientific advancement. After approval from England’s crown , the group became known as the Royal Society, which helped create a thriving scientific community and an early academic journal to help introduce rigorous study and peer review.

Previous generations of scientists had touched on the importance of induction and deduction, but Sir Isaac Newton proposed that both were equally important. This contribution helped establish the importance of multiple kinds of reasoning, leading to more rigorous study.

As science began to splinter into separate areas of study, it became necessary to define different methods for different fields. Karl Popper was a leader in this area—he established that science could be subject to error, sometimes intentionally. This was particularly tricky for “soft” sciences like psychology and social sciences, which require different methods. Popper’s theories furthered the divide between sciences like psychology and “hard” sciences like chemistry or physics.

Paul Feyerabend argued that Popper’s methods were too restrictive for certain fields, and followed a less restrictive method hinged on “anything goes,” as great scientists had made discoveries without the Scientific Method. Feyerabend suggested that throughout history scientists had adapted their methods as necessary, and that sometimes it would be necessary to break the rules. This approach suited social and behavioral scientists particularly well, leading to a more diverse range of models for scientists in multiple fields to use.

The Scientific Method Steps

Though different fields may have variations on the model, the basic scientific method is as follows:

#1: Make Observations 

Notice something, such as the air temperature during the winter, what happens when ice cream melts, or how your plants behave when you forget to water them.

#2: Ask a Question

Turn your observation into a question. Why is the temperature lower during the winter? Why does my ice cream melt? Why does my toast always fall butter-side down?

This step can also include doing some research. You may be able to find answers to these questions already, but you can still test them!

#3: Make a Hypothesis

A hypothesis is an educated guess of the answer to your question. Why does your toast always fall butter-side down? Maybe it’s because the butter makes that side of the bread heavier.

A good hypothesis leads to a prediction that you can test, phrased as an if/then statement. In this case, we can pick something like, “If toast is buttered, then it will hit the ground butter-first.”

#4: Experiment

Your experiment is designed to test whether your predication about what will happen is true. A good experiment will test one variable at a time —for example, we’re trying to test whether butter weighs down one side of toast, making it more likely to hit the ground first.

The unbuttered toast is our control variable. If we determine the chance that a slice of unbuttered toast, marked with a dot, will hit the ground on a particular side, we can compare those results to our buttered toast to see if there’s a correlation between the presence of butter and which way the toast falls.

If we decided not to toast the bread, that would be introducing a new question—whether or not toasting the bread has any impact on how it falls. Since that’s not part of our test, we’ll stick with determining whether the presence of butter has any impact on which side hits the ground first.

#5: Analyze Data

After our experiment, we discover that both buttered toast and unbuttered toast have a 50/50 chance of hitting the ground on the buttered or marked side when dropped from a consistent height, straight down. It looks like our hypothesis was incorrect—it’s not the butter that makes the toast hit the ground in a particular way, so it must be something else.

Since we didn’t get the desired result, it’s back to the drawing board. Our hypothesis wasn’t correct, so we’ll need to start fresh. Now that you think about it, your toast seems to hit the ground butter-first when it slides off your plate, not when you drop it from a consistent height. That can be the basis for your new experiment.

#6: Communicate Your Results

Good science needs verification. Your experiment should be replicable by other people, so you can put together a report about how you ran your experiment to see if other peoples’ findings are consistent with yours.

This may be useful for class or a science fair. Professional scientists may publish their findings in scientific journals, where other scientists can read and attempt their own versions of the same experiments. Being part of a scientific community helps your experiments be stronger because other people can see if there are flaws in your approach—such as if you tested with different kinds of bread, or sometimes used peanut butter instead of butter—that can lead you closer to a good answer.

A Scientific Method Example: Falling Toast

We’ve run through a quick recap of the scientific method steps, but let’s look a little deeper by trying again to figure out why toast so often falls butter side down.

#1: Make Observations

At the end of our last experiment, where we learned that butter doesn’t actually make toast more likely to hit the ground on that side, we remembered that the times when our toast hits the ground butter side first are usually when it’s falling off a plate.

The easiest question we can ask is, “Why is that?”

We can actually search this online and find a pretty detailed answer as to why this is true. But we’re budding scientists—we want to see it in action and verify it for ourselves! After all, good science should be replicable, and we have all the tools we need to test out what’s really going on.

Why do we think that buttered toast hits the ground butter-first? We know it’s not because it’s heavier, so we can strike that out. Maybe it’s because of the shape of our plate?

That’s something we can test. We’ll phrase our hypothesis as, “If my toast slides off my plate, then it will fall butter-side down.”

Just seeing that toast falls off a plate butter-side down isn’t enough for us. We want to know why, so we’re going to take things a step further—we’ll set up a slow-motion camera to capture what happens as the toast slides off the plate.

We’ll run the test ten times, each time tilting the same plate until the toast slides off. We’ll make note of each time the butter side lands first and see what’s happening on the video so we can see what’s going on.

When we review the footage, we’ll likely notice that the bread starts to flip when it slides off the edge, changing how it falls in a way that didn’t happen when we dropped it ourselves.

That answers our question, but it’s not the complete picture —how do other plates affect how often toast hits the ground butter-first? What if the toast is already butter-side down when it falls? These are things we can test in further experiments with new hypotheses!

Now that we have results, we can share them with others who can verify our results. As mentioned above, being part of the scientific community can lead to better results. If your results were wildly different from the established thinking about buttered toast, that might be cause for reevaluation. If they’re the same, they might lead others to make new discoveries about buttered toast. At the very least, you have a cool experiment you can share with your friends!

Key Scientific Method Tips

Though science can be complex, the benefit of the scientific method is that it gives you an easy-to-follow means of thinking about why and how things happen. To use it effectively, keep these things in mind!

Don’t Worry About Proving Your Hypothesis

One of the important things to remember about the scientific method is that it’s not necessarily meant to prove your hypothesis right. It’s great if you do manage to guess the reason for something right the first time, but the ultimate goal of an experiment is to find the true reason for your observation to occur, not to prove your hypothesis right.

Good science sometimes means that you’re wrong. That’s not a bad thing—a well-designed experiment with an unanticipated result can be just as revealing, if not more, than an experiment that confirms your hypothesis.

Be Prepared to Try Again

If the data from your experiment doesn’t match your hypothesis, that’s not a bad thing. You’ve eliminated one possible explanation, which brings you one step closer to discovering the truth.

The scientific method isn’t something you’re meant to do exactly once to prove a point. It’s meant to be repeated and adapted to bring you closer to a solution. Even if you can demonstrate truth in your hypothesis, a good scientist will run an experiment again to be sure that the results are replicable. You can even tweak a successful hypothesis to test another factor, such as if we redid our buttered toast experiment to find out whether different kinds of plates affect whether or not the toast falls butter-first. The more we test our hypothesis, the stronger it becomes!

What’s Next?

Want to learn more about the scientific method? These important high school science classes will no doubt cover it in a variety of different contexts.

Test your ability to follow the scientific method using these at-home science experiments for kids !

Need some proof that science is fun? Try making slime

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scientific method

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• University of Nevada, Reno - College of Agriculture, Biotechnology and Natural Resources Extension - The Scientific Method
• World History Encyclopedia - Scientific Method
• LiveScience - What Is Science?
• Verywell Mind - Scientific Method Steps in Psychology Research
• WebMD - What is the Scientific Method?
• Chemistry LibreTexts - The Scientific Method
• National Center for Biotechnology Information - PubMed Central - Redefining the scientific method: as the use of sophisticated scientific methods that extend our mind
• Khan Academy - The scientific method
• Simply Psychology - What are the steps in the Scientific Method?
• Stanford Encyclopedia of Philosophy - Scientific Method

scientific method , mathematical and experimental technique employed in the sciences . More specifically, it is the technique used in the construction and testing of a scientific hypothesis .

The process of observing, asking questions, and seeking answers through tests and experiments is not unique to any one field of science. In fact, the scientific method is applied broadly in science, across many different fields. Many empirical sciences, especially the social sciences , use mathematical tools borrowed from probability theory and statistics , together with outgrowths of these, such as decision theory , game theory , utility theory, and operations research . Philosophers of science have addressed general methodological problems, such as the nature of scientific explanation and the justification of induction .

The scientific method is critical to the development of scientific theories , which explain empirical (experiential) laws in a scientifically rational manner. In a typical application of the scientific method, a researcher develops a hypothesis , tests it through various means, and then modifies the hypothesis on the basis of the outcome of the tests and experiments. The modified hypothesis is then retested, further modified, and tested again, until it becomes consistent with observed phenomena and testing outcomes. In this way, hypotheses serve as tools by which scientists gather data. From that data and the many different scientific investigations undertaken to explore hypotheses, scientists are able to develop broad general explanations, or scientific theories.

See also Mill’s methods ; hypothetico-deductive method .

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What Is the Scientific Method?

The scientific method is a systematic way of conducting experiments or studies so that you can explore the things you observe in the world and answer questions about them. The scientific method, also known as the hypothetico-deductive method, is a series of steps that can help you accurately describe the things you observe or improve your understanding of them.

Ultimately, your goal when you use the scientific method is to:

• Find a cause-and-effect relationship by asking a question about something you observed
• Collect as much evidence as you can about what you observed, as this can help you explore the connection between your evidence and what you observed
• Determine if all your evidence can be combined to answer your question in a way that makes sense

Francis Bacon and René Descartes are usually credited with formalizing the process in the 16th and 17th centuries. The two philosophers argued that research shouldn’t be guided by preset metaphysical ideas of how reality works. They supported the use of inductive reasoning to come up with hypotheses and understand new things about reality.

Scientific Method Steps

The scientific method is a step-by-step problem-solving process. These steps include:

Observe the world around you. This will help you come up with a topic you are interested in and want to learn more about. In many cases, you already have a topic in mind because you have a related question for which you couldn't find an immediate answer.

Either way, you'll start the process by finding out what people before you already know about the topic, as well as any questions that people are still asking about. You may need to look up and read books and articles from academic journals or talk to other people so that you understand as much as you possibly can about your topic. This will help you with your next step.

Ask questions. Asking questions about what you observed and learned from reading and talking to others can help you figure out what the "problem" is. Scientists try to ask questions that are both interesting and specific and can be answered with the help of a fairly easy experiment or series of experiments. Your question should have one part (called a variable) that you can change in your experiment and another variable that you can measure. Your goal is to design an experiment that is a "fair test," which is when all the conditions in the experiment are kept the same except for the one you change (called the experimental or independent variable).

Form a hypothesis and make predictions based on it.  A hypothesis is an educated guess about the relationship between two or more variables in your question. A good hypothesis lets you predict what will happen when you test it in an experiment. Another important feature of a good hypothesis is that, if the hypothesis is wrong, you should be able to show that it's wrong. This is called falsifiability. If your experiment shows that your prediction is true, then your hypothesis is supported by your data.

Test your prediction by doing an experiment or making more observations.  The way you test your prediction depends on what you are studying. The best support comes from an experiment, but in some cases, it's too hard or impossible to change the variables in an experiment. Sometimes, you may need to do descriptive research where you gather more observations instead of doing an experiment. You will carefully gather notes and measurements during your experiments or studies, and you can share them with other people interested in the same question as you. Ideally, you will also repeat your experiment a couple more times because it's possible to get a result by chance, but it's less possible to get the same result more than once by chance.

Draw a conclusion. You will analyze what you already know about your topic from your literature research and the data gathered during your experiment. This will help you decide if the conclusion you draw from your data supports or contradicts your hypothesis. If your results contradict your hypothesis, you can use this observation to form a new hypothesis and make a new prediction. This is why scientific research is ongoing and scientific knowledge is changing all the time. It's very common for scientists to get results that don't support their hypotheses. In fact, you sometimes learn more about the world when your experiments don't support your hypotheses because it leads you to ask more questions. And this time around, you already know that one possible explanation is likely wrong.

Use your results to guide your next steps (iterate). For instance, if your hypothesis is supported, you may do more experiments to confirm it. Or you could come up with a hypothesis about why it works this way and design an experiment to test that. If your hypothesis is not supported, you can come up with another hypothesis and do experiments to test it. You'll rarely get the right hypothesis in one go. Most of the time, you'll have to go back to the hypothesis stage and try again. Every attempt offers you important information that helps you improve your next round of questions, hypotheses, and predictions.

Share your results. Scientific research isn't something you can do on your own; you must work with other people to do it.   You may be able to do an experiment or a series of experiments on your own, but you can't come up with all the ideas or do all the experiments by yourself .

Scientists and researchers usually share information by publishing it in a scientific journal or by presenting it to their colleagues during meetings and scientific conferences. These journals are read and the conferences are attended by other researchers who are interested in the same questions. If there's anything wrong with your hypothesis, prediction, experiment design, or conclusion, other researchers will likely find it and point it out to you.

It can be scary, but it's a critical part of doing scientific research. You must let your research be examined by other researchers who are as interested and knowledgeable about your question as you. This process helps other researchers by pointing out hypotheses that have been proved wrong and why they are wrong. It helps you by identifying flaws in your thinking or experiment design. And if you don't share what you've learned and let other people ask questions about it, it's not helpful to your or anyone else's understanding of what happens in the world.

Scientific Method Example

Here's an everyday example of how you can apply the scientific method to understand more about your world so you can solve your problems in a helpful way.

Let's say you put slices of bread in your toaster and press the button, but nothing happens. Your toaster isn't working, but you can't afford to buy a new one right now. You might be able to rescue it from the trash can if you can figure out what's wrong with it. So, let's figure out what's wrong with your toaster.

Observation. Your toaster isn't working to toast your bread.

Ask a question. In this case, you're asking, "Why isn't my toaster working?" You could even do a bit of preliminary research by looking in the owner's manual for your toaster. The manufacturer has likely tested your toaster model under many conditions, and they may have some ideas for where to start with your hypothesis.

Form a hypothesis and make predictions based on it. Your hypothesis should be a potential explanation or answer to the question that you can test to see if it's correct. One possible explanation that we could test is that the power outlet is broken. Our prediction is that if the outlet is broken, then plugging it into a different outlet should make the toaster work again.

Test your prediction by doing an experiment or making more observations. You plug the toaster into a different outlet and try to toast your bread.

If that works, then your hypothesis is supported by your experimental data. Results that support your hypothesis don't prove it right; they simply suggest that it's a likely explanation. This uncertainty arises because, in the real world, we can't rule out the possibility of mistakes, wrong assumptions, or weird coincidences affecting the results. If the toaster doesn’t work even after plugging it into a different outlet, then your hypothesis is not supported and it's likely the wrong explanation.

Use your results to guide your next steps (iteration). If your toaster worked, you may decide to do further tests to confirm it or revise it. For example, you could plug something else that you know is working into the first outlet to see if that stops working too. That would be further confirmation that your hypothesis is correct.

If your toaster failed to toast when plugged into the second outlet, you need a new hypothesis. For example, your next hypothesis might be that the toaster has a shorted wire. You could test this hypothesis directly if you have the right equipment and training, or you could take it to a repair shop where they could test that hypothesis for you.

Share your results. For this everyday example, you probably wouldn't want to write a paper, but you could share your problem-solving efforts with your housemates or anyone you hire to repair your outlet or help you test if the toaster has a short circuit.

What the Scientific Method Is Used For

The scientific method is useful whenever you need to reason logically about your questions and gather evidence to support your problem-solving efforts. So, you can use it in everyday life to answer many of your questions; however, when most people think of the scientific method, they likely think of using it to:

Describe how nature works . It can be hard to accurately describe how nature works because it's almost impossible to account for every variable that's involved in a natural process. Researchers may not even know about many of the variables that are involved. In some cases, all you can do is make assumptions. But you can use the scientific method to logically disprove wrong assumptions by identifying flaws in the reasoning.

Do scientific research in a laboratory to develop things such as new medicines.

Develop critical thinking skills.  Using the scientific method may help you develop critical thinking in your daily life because you learn to systematically ask questions and gather evidence to find answers. Without logical reasoning, you might be more likely to have a distorted perspective or bias. Bias is the inclination we all have to favor one perspective (usually our own) over another.

The scientific method doesn't perfectly solve the problem of bias, but it does make it harder for an entire field to be biased in the same direction. That's because it's unlikely that all the people working in a field have the same biases. It also helps make the biases of individuals more obvious because if you repeatedly misinterpret information in the same way in multiple experiments or over a period, the other people working on the same question will notice. If you don't correct your bias when others point it out to you, you'll lose your credibility. Other people might then stop believing what you have to say.

Why Is the Scientific Method Important?

When you use the scientific method, your goal is to do research in a fair, unbiased, and repeatable way. The scientific method helps meet these goals because:

It's a systematic approach to problem-solving. It can help you figure out where you're going wrong in your thinking and research if you're not getting helpful answers to your questions. Helpful answers solve problems and keep you moving forward. So, a systematic approach helps you improve your problem-solving abilities if you get stuck.

It can help you solve your problems.  The scientific method helps you isolate problems by focusing on what's important. In addition, it can help you make your solutions better every time you go through the process.

It helps you eliminate (or become aware of) your personal biases.  It can help you limit the influence of your own personal, preconceived notions . A big part of the process is considering what other people already know and think about your question. It also involves sharing what you've learned and letting other people ask about your methods and conclusions. At the end of the process, even if you still think your answer is best, you have considered what other people know and think about the question.

The scientific method is a systematic way of conducting experiments or studies so that you can explore the world around you and answer questions using reason and evidence. It's a step-by-step problem-solving process that involves: (1) observation, (2) asking questions, (3) forming hypotheses and making predictions, (4) testing your hypotheses through experiments or more observations, (5) using what you learned through experiment or observation to guide further investigation, and (6) sharing your results.

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Solving Everyday Problems with the Scientific Method

• By (author): 
• Don K Mak , 
• Angela T Mak , and 
• Anthony B Mak
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• Description
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This book describes how one can use The Scientific Method to solve everyday problems including medical ailments, health issues, money management, traveling, shopping, cooking, household chores, etc. It illustrates how to exploit the information collected from our five senses, how to solve problems when no information is available for the present problem situation, how to increase our chances of success by redefining a problem, and how to extrapolate our capabilities by seeing a relationship among heretofore unrelated concepts.

One should formulate a hypothesis as early as possible in order to have a sense of direction regarding which path to follow. Occasionally, by making wild conjectures, creative solutions can transpire. However, hypotheses need to be well-tested. Through this way, The Scientific Method can help readers solve problems in both familiar and unfamiliar situations. Containing real-life examples of how various problems are solved — for instance, how some observant patients cure their own illnesses when medical experts have failed — this book will train readers to observe what others may have missed and conceive what others may not have contemplated. With practice, they will be able to solve more problems than they could previously imagine.

Sample Chapter(s) Chapter 1: Prelude (83 KB) Chapter 2: The Scientific Method (173 KB)

• The Scientific Method

Observation

Recognition, problem situation and problem definition, induction and deduction, alternative solutions, mathematics, probable value.

• Demonstrates how to cope with problems in both familiar and unfamiliar situations, using The Scientific Method
• Redefines problems by viewing the problem situation from different levels and perspectives
• Formulates solution methodology by borrowing ideas from economics, philosophy, logic, statistics, probability theory and scientific research
• Offers test-driven recommendations, supported with real-life examples

Type on 09/06/2008

Updated pp & pub date on 7/1/2009

Added s/c updated to systems price on 09/11/2009

Added review on 14/10/2010

Added review on 7/3/2011

FRONT MATTER

• Pages: i–xiii

https://doi.org/10.1142/9789812835109_fmatter

• Claimers and Disclaimers

https://doi.org/10.1142/9789812835109_0001

The father put down the newspaper. It had been raining for the last two hours. The rain finally stopped, and the sky looked clear. After all this raining, the negative ions in the atmosphere would have increased, and the air would feel fresh. The father suggested the family of four should go for a stroll. There was a park just about fifteen minutes walk from their house…

• Pages: 3–16

https://doi.org/10.1142/9789812835109_0002

• Edwin Smith papyrus
• Greek philosophy (4 th century BC)
• Islamic philosophy (8 th century AD–15 th century AD)
• European Science (12 th century AD–16 th century AD)
• Scientific Revolution (1543 AD–18 th century AD)
• Humanism and Empiricism
• Application of the Scientific Method to Everyday Problem
• Pages: 17–44

https://doi.org/10.1142/9789812835109_0003

• Missed information
• Misinformation
• Hidden information
• No information
• Unaware information
• Evidence-based information
• Self-denied information
• Biased information
• Unexploited information
• Peripheral information
• Pages: 45–63

https://doi.org/10.1142/9789812835109_0004

• Wild conjectures
• Albert Einstein (1879–1955)
• Pages: 65–82

https://doi.org/10.1142/9789812835109_0005

• Experiment versus hypothesis
• Platonic, Aristotelian, Baconian, and Galilean methodology
• Pages: 83–95

https://doi.org/10.1142/9789812835109_0006

• John Nash (1928– )
• Pages: 97–105

https://doi.org/10.1142/9789812835109_0007

• Perspectives on different levels
• Perspectives on the same level
• Pages: 107–117

https://doi.org/10.1142/9789812835109_0008

• Pages: 119–138

https://doi.org/10.1142/9789812835109_0009

• Lotion bottle with a pump dispenser
• Pages: 139–167

https://doi.org/10.1142/9789812835109_0010

• Ordinary thinking
• Unconscious mind
• Genetic material
• Watson and Crick at Cavendish Laboratory, Cambridge
• Rosalind Franklin at King's College, London
• The triple helix model
• The double helix model
• Creative thinking and Ordinary thinking
• Scientific Research and Scientific Method
• Can we be more creative?
• Pages: 169–194

https://doi.org/10.1142/9789812835109_0011

Mathematics, even some simple arithmetic, is so important in solving some of the everyday problems, that we think a whole chapter should be written on it.

Let us take a look at an example. When we see an advertisement which says "Buy one, get the second one at half price", we should be able to figure out what exactly does it mean, and how much discount are we actually getting. Is it a better deal than another company that advertises 30% off?…

• Pages: 195–207

https://doi.org/10.1142/9789812835109_0012

For a certain problem, we may come up with several plausible solutions. Which path should we take? Each path would only have certain chance or probability of success in resolving the problem. If each path or solution has a different reward, we can define the probable value of each path to be the multiplication of the reward by the probability. We should, most likely, choose the path that has the highest probable value. (The term "probable value" is coined by us. The idea is appropriated from the term "expected value" in Statistics. In this sense, expected value can be considered as the sum of all probable values.)…

• Pages: 209–212

https://doi.org/10.1142/9789812835109_0013

We run into problems every day. Even when we do not encounter any problems, it does not mean that they do not exist. Sometimes, we wish we could be able to recognize them earlier. The scientific method of observation, hypothesis, and experiment can help us recognize, define, and solve our problems…

BACK MATTER

• Pages: 213–220

https://doi.org/10.1142/9789812835109_bmatter

• Bibliography

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Scientific Method Example

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The scientific method is a series of steps that scientific investigators follow to answer specific questions about the natural world. Scientists use the scientific method to make observations, formulate hypotheses , and conduct scientific experiments .

A scientific inquiry starts with an observation. Then, the formulation of a question about what has been observed follows. Next, the scientist will proceed through the remaining steps of the scientific method to end at a conclusion.

The six steps of the scientific method are as follows:

Observation

The first step of the scientific method involves making an observation about something that interests you. Taking an interest in your scientific discovery is important, for example, if you are doing a science project , because you will want to work on something that holds your attention. Your observation can be of anything from plant movement to animal behavior, as long as it is something you want to know more about.​ This step is when you will come up with an idea if you are working on a science project.

Once you have made your observation, you must formulate a question about what you observed. Your question should summarize what it is you are trying to discover or accomplish in your experiment. When stating your question, be as specific as possible.​ For example, if you are doing a project on plants , you may want to know how plants interact with microbes. Your question could be: Do plant spices inhibit bacterial growth ?

The hypothesis is a key component of the scientific process. A hypothesis is an idea that is suggested as an explanation for a natural event, a particular experience, or a specific condition that can be tested through definable experimentation. It states the purpose of your experiment, the variables used, and the predicted outcome of your experiment. It is important to note that a hypothesis must be testable. That means that you should be able to test your hypothesis through experimentation .​ Your hypothesis must either be supported or falsified by your experiment. An example of a good hypothesis is: If there is a relation between listening to music and heart rate, then listening to music will cause a person's resting heart rate to either increase or decrease.

Once you have developed a hypothesis, you must design and conduct an experiment that will test it. You should develop a procedure that states clearly how you plan to conduct your experiment. It is important you include and identify a controlled variable or dependent variable in your procedure. Controls allow us to test a single variable in an experiment because they are unchanged. We can then make observations and comparisons between our controls and our independent variables (things that change in the experiment) to develop an accurate conclusion.​

The results are where you report what happened in the experiment. That includes detailing all observations and data made during your experiment. Most people find it easier to visualize the data by charting or graphing the information.​

Developing a conclusion is the final step of the scientific method. This is where you analyze the results from the experiment and reach a determination about the hypothesis. Did the experiment support or reject your hypothesis? If your hypothesis was supported, great. If not, repeat the experiment or think of ways to improve your procedure.

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Chapter 6: Scientific Problem Solving

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Science is a method to discover empirical truths and patterns. Roughly speaking, the scientific method consists of

1) Observing

2) Forming a hypothesis

3) Testing the hypothesis and

4) Interpreting the data to confirm or disconfirm the hypothesis.

The beauty of science is that any scientific claim can be tested if you have the proper knowledge and equipment.

You can also use the scientific method to solve everyday problems: 1) Observe and clearly define the problem, 2) Form a hypothesis, 3) Test it, and 4) Confirm the hypothesis... or disconfirm it and start over.

So, the next time you are cursing in traffic or emotionally reacting to a problem, take a few deep breaths and then use this rational and scientific approach. Slow down, observe, hypothesize, and test.

Explain how you would solve these problems using the four steps of the scientific process.

Example: The fire alarm is not working.

1) Observe/Define the problem: it does not beep when I push the button.

2) Hypothesis: it is caused by a dead battery.

3) Test: try a new battery.

4) Confirm/Disconfirm: the alarm now works. If it does not work, start over by testing another hypothesis like “it has a loose wire.”  

• My car will not start.
• My child is having problems reading.
• I owe $20,000, but only make $10 an hour.
• My boss is mean. I want him/her to stop using rude language towards me.
• My significant other is lazy. I want him/her to help out more.

6-8. Identify three problems where you can apply the scientific method.

*Answers will vary.

Application and Value

Science is more of a process than a body of knowledge. In our daily lives, we often emotionally react and jump to quick solutions when faced with problems, but following the four steps of the scientific process can help us slow down and discover more intelligent solutions.

In your study of philosophy, you will explore deeper questions about science. For example, are there any forms of knowledge that are nonscientific? Can science tell us what we ought to do? Can logical and mathematical truths be proven in a scientific way? Does introspection give knowledge even though I cannot scientifically observe your introspective thoughts? Is science truly objective?  These are challenging questions that should help you discover the scope of science without diminishing its awesome power.

But the first step in answering these questions is knowing what science is, and this chapter clarifies its essence. Again, Science is not so much a body of knowledge as it is a method of observing, hypothesizing, and testing. This method is what all the sciences have in common.

Perhaps too science should involve falsifiability, which is a concept explored in the next chapter.

Return to Logic Home                            Next (Chapter 7, Falsifiability)

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Redefining the scientific method: as the use of sophisticated scientific methods that extend our mind

Alexander krauss.

London School of Economics, London, UK

Institute for Economic Analysis, Spanish National Research Council, Barcelona, Spain

Associated Data

Data used for the analysis are available from the sources outlined in the text.

Scientific, medical, and technological knowledge has transformed our world, but we still poorly understand the nature of scientific methodology. Science textbooks, science dictionaries, and science institutions often state that scientists follow, and should follow, the universal scientific method of testing hypotheses using observation and experimentation. Yet, scientific methodology has not been systematically analyzed using large-scale data and scientific methods themselves as it is viewed as not easily amenable to scientific study. Using data on all major discoveries across science including all Nobel Prize and major non-Nobel Prize discoveries, we can address the question of the extent to which “the scientific method” is actually applied in making science's groundbreaking research and whether we need to expand this central concept of science. This study reveals that 25% of all discoveries since 1900 did not apply the common scientific method (all three features)—with 6% of discoveries using no observation, 23% using no experimentation, and 17% not testing a hypothesis. Empirical evidence thus challenges the common view of the scientific method. Adhering to it as a guiding principle would constrain us in developing many new scientific ideas and breakthroughs. Instead, assessing all major discoveries, we identify here a general, common feature that the method of science can be reduced to: making all major discoveries has required using sophisticated methods and instruments of science. These include statistical methods, particle accelerators, and X-ray methods. Such methods extend our mind and generally make observing, experimenting, and testing hypotheses in science possible, doing so in new ways and ensure their replicability. This provides a new perspective to the scientific method—embedded in our sophisticated methods and instruments—and suggests that we need to reform and extend the way we view the scientific method and discovery process.

Science is fascinating because discoveries like new vaccines, more efficient forms of electricity generation, and new medical therapies can spread across the globe and improve the lives of many people. Science and discoveries have enhanced our ability to understand and predict many aspects of our natural and social world. Einstein's special relativity revolutionized physics in the 20th century and how we understand the relationship between space and time. Darwin and Wallace's theory of evolution via natural selection transformed biology and how we comprehend the historical origins of our species. Franklin, Crick, and Watson's discovery of the double helix structure of DNA radically redefined genetics and how we conceive the way genetic information of living organisms is stored, copied, and passed along. These scientists fundamentally changed the way we view the world, but they did not carry out an experiment to make these path-breaking discoveries. In fact, hundreds of major scientific discoveries did not use “the scientific method”, as defined in science dictionaries as the combined process of “the collection of data through observation and experiment, and the formulation and testing of hypotheses” ( 1 ). In other words, it is “The process of observing, asking questions, and seeking answers through tests and experiments” ( 2 , cf. 3 ). Many recent science textbooks also present the scientific method as a sequence of steps or a process of observing, experimenting, and testing hypotheses, as shown in systematic studies of university-level science textbooks across science ( 4–7 ). The common scientific method is thus embedded in science dictionaries and textbooks ( 4–7 ). A study of major science institutions like the National Science Foundation and National Institutes of Health also found that they primarily endorse this scientific method focused on hypothesis testing, and generally not other exploratory research methods that do not test a predefined hypothesis ( 8 ). Researchers have not however yet used large representative data to assess the extent to which the scientific method is actually applied in science or they investigate it at an abstract level ( 9 , 10 ). In general, this universal method is commonly viewed as a unifying method of science and can be traced back at least to Francis Bacon's theory of scientific methodology in 1620 which popularized the concept ( 11 ). This seminal book in many ways has laid the foundation of philosophy of science and fundamentally influenced generations of scientists and the common conception of how science is conducted, which remains widespread and institutionalized today ( 4–8 , cf. 12 ).

However, before hypothesizing about science, what its general method is and how it should be conducted, we need to first assess the evidence on how science is actually conducted in practice. Assessing science's major discoveries across scientific fields and time provides a new systematic way to do so and enables us to evaluate how this universal concept of scientific methodology holds up. Science's major discoveries are defined here as all 533 Nobel Prize–winning discoveries in science (from the first year of the prize in 1901 to 2022) ( 13 ) and all other major discoveries that were made prior to or did not receive a Nobel Prize; these are derived from all science textbooks (a total of seven) that provide a top 100 list of the greatest scientists and their discoveries and that span across scientific fields and history ( 14–20 ) (with textbooks specific to a field or time period not included). After excluding duplicate cases within the seven textbooks, 228 other major discoveries remained. A total of 761 major discoveries, which have driven humankind's knowledge, have thus been included in the study. The main source for compiling the data in this study is the main publication of the discovery that indicates the methods used to make the breakthrough (in the case of discoveries earning a Nobel Prize, the prize-winning papers) ( 13 ). For further description of the data, see figure captions (for overall greater details on the data, see the companion study that outlines the features and characteristics of science's major discoverers) ( 21 ).

Examining science's major discoveries, we find that the common scientific method (the combined use of observation, experimentation, and hypothesis testing) is applied in making 71% of all discoveries; and the share is 75% for all discoveries in contemporary science, defined as all Nobel Prize and major non-Nobel Prize discoveries since 1900. Among all major scientific discoveries, we find that 94% have required using observation, 81% testing a hypothesis, and 75% experimentation (Fig. ​ (Fig.1)—with 1 )—with some hypotheses tested using experimental research designs and others using only observation. Science thus does not always fit the textbook definition.

Methods of science pyramid : share of each methodological approach used for making discoveries. Data reflect all 761 major discoveries.

Comparison across fields provides evidence that the common scientific method was not applied in making about half of all Nobel Prize discoveries in astronomy, economics and social sciences, and a quarter of such discoveries in physics, as highlighted in Fig. ​ Fig.2b. 2 b. Some discoveries are thus non-experimental and more theoretical in nature, while others are made in an exploratory way, without explicitly formulating and testing a preestablished hypothesis. Importantly, the common scientific method does not take into account that all Nobel Prize discoveries across fields require applying sophisticated methods (such as statistics and randomization techniques) or instruments (such as centrifuges and computers)—Fig. 2 b.

Share of discoveries made using the classic and the sophisticated scientific method, across time and fields. Data reflect all 761 major discoveries (including all Nobel Prize discoveries) (a), and all 533 Nobel Prize discoveries (b). Each of these discovery-making publications are classified as using observation if the study describes collecting observational data (using eyesight) (bar 1 in the figure), as using experimentation if the study conducted an experiment (bar 2), and as testing a hypothesis if the study formulated and assessed a proposed explanation (rather than conducted exploratory research) (bar 3). The publication is classified as using the classic scientific method if the study applied the three features (bar 4). In contrast, the publication is classified as using the sophisticated scientific method if the study applied a complex scientific method or instrument (bar 5), as defined below. The 10 most commonly used scientific methods and instruments—among all Nobel Prize discoveries—include statistical/mathematical methods, spectrometers, X-ray methods, chromatography, centrifuges, electrophoresis, lasers, (electron) microscopes, particle accelerator, and particle detector. Analysis expanding the data in (b) to include, in addition, the other major discoveries that did not earn a Nobel Prize but were made within the same time period (633 discoveries in total) illustrates comparable results (except for astronomy) and serves as a robustness check, with for example the share of discoveries made applying “the classic scientific method” at 40, 35, 75, 93, and 89% across these five fields, respectively.

When we systematically assess all major discoveries, what is the common method of science that we use to be able to do science and make discoveries? We find that one general feature of scientific methodology is applied in making science's major discoveries: the use of sophisticated methods or instruments. These are defined here as scientific methods and instruments that extend our cognitive and sensory abilities—such as statistical methods, lasers, and chromatography methods. They are external resources (material artifacts) that can be shared and used by others—whereas observing, hypothesizing, and experimenting are, in contrast, largely internal (cognitive) abilities that are not material (Fig. ​ (Fig.2). 2 ). Applying sophisticated methods or instruments is thus a necessary condition for discovery in contemporary science. We find that a number of sophisticated methods and instruments have each been used in making at least 10% of all major discoveries, such as centrifuges, X-ray diffraction, and spectrometers—and statistical methods for example have been used in making 62% of all discoveries. Without such scientific tools, discovery and scientific progress is not possible.

In fact, this sophisticated scientific method is actually more unique to science, as the most common scientific methods and instruments—such as particle accelerators, electrophoresis methods, and X-ray diffraction—are largely only used in science. In contrast, we also often make observations, test hypotheses, and experiment in business, industry, public policy, and everyday life and they are thus not just prototypical or distinctive of science. Recognizing the vast importance of such complex methods and instruments adds an essential element to understanding science and especially how science has evolved from its early origins in directly observing, hypothesizing and experimenting to now only being able to do so by using such complex tools. The classic scientific method dominated how science was done for much of history (especially when early scholars like Bacon described it) ( 11 ) but now sophisticated scientific methods dominate contemporary science by enabling us to observe, experiment, and test hypotheses in much more diverse, complex, and efficient ways. Just as science has evolved, so should the classic scientific method—which is construed in such general terms that it would be better described as a basic method of reasoning used for human activities (non-scientific and scientific).

While features of science such as observation, experimentation, and hypothesis testing are commonly used in science and making discoveries, they are thus not universal. An experimental research design was not carried out when Einstein developed the law of the photoelectric effect in 1905 or when Franklin, Crick, and Watson discovered the double helix structure of DNA in 1953 using observational images developed by Franklin. Direct observation was not made when for example Penrose developed the mathematical proof for black holes in 1965 or when Prigogine developed the theory of dissipative structures in thermodynamics in 1969. A hypothesis was not directly tested when Jerne developed the natural-selection theory of antibody formation in 1955 or when Peebles developed the theoretical framework of physical cosmology in 1965. These scientists all earned a Nobel Prize for these discoveries, but they did not directly apply or generally could not apply the “scientific method” to make their discovery. The common scientific method captures much of scientific practice but not all domains. If we were to abide by the common definition of the scientific method, Copernicus ( 22 ), Darwin ( 23 ), Einstein ( 24 ), Franklin, Crick, and Watson, and many others would not be viewed as having applied it as they did not directly carry out experiments to make their seminal breakthroughs. These scientists have however become iconic figures of science.

In general, scientific methods—like scientists—come in many sizes, shapes, and levels of sophistication. We use many methods to conduct science across fields: combining mathematics with measurement instruments, statistics with experimentation, X-ray diffraction, spectrometers, and particle detectors using systematic observation, and hundreds of other combinations. We may think of the diverse methods needed in immunology, oceanography, neuroscience, and astrophysics, or chemistry, agronomy, and behavioral economics. We cannot do science without our sophisticated methods and instruments which make it possible, for most phenomena in science, to observe, experiment, and test hypotheses and especially do exploratory research in the first place—and also to do so in new and innovative ways (Table ​ (Table1). 1 ). The sophisticated scientific method integrates the use of observation, experimentation, and hypothesis testing into our central methods and instruments (Fig. ​ (Fig.3). 3 ). Replicability, a central feature of science, is also tied to particular sophisticated methods, such as statistical methods and X-ray devices. Different researchers applying sophisticated methods ensures that studies, theories, and discoveries are replicable (while observation, experimentation, and hypothesis testing are too general to do so and are subject to each researcher applying them differently and thus more susceptible to researcher bias). Sophisticated methods make research more accurate and reliable and enable us to evaluate the quality of research.

Main types of methodological approaches used in science and making discoveries.

Data reflect all 761 major discoveries (including all Nobel Prize discoveries) (first row of data), and all 533 Nobel Prize discoveries (second row of data). Applying observation and a complex method or instrument, together, is decisive in producing nearly all major discoveries at 94%, illustrating the central importance of empirical sciences in driving discovery and science.

Overall, with the classic scientific method, we would not be able to label many major scientific discoveries as scientific, though they have vastly impacted science and our lives. The concept of the common scientific method, as a golden principle connecting the scientific community together, can be misunderstood as being universal. It is an idealization, embedded in university science textbooks ( 4–7 ), science dictionaries ( 1–3 ), and several major science institutions ( 8 ), that can be confusing for students and less-experienced researchers when learning about science and scientific discoveries and realizing it does not always apply. We do science and make breakthroughs using our diverse and complex methodological toolbox. We can best view the method of science as the use of our sophisticated methodological toolbox . The classic scientific method needs to be integrated into and redefined as the sophisticated scientific method that better reflects actual scientific practice:

Scientific methodology is defined as the use of sophisticated scientific methods or instruments (such as mathematics, particle accelerators, and chromatography methods), which are systematic techniques and tools that extend our cognitive and sensory abilities, are generalizable and enable better observing, hypothesis-testing, problem-solving, and experimenting and thus acquiring knowledge about the world.

A generalizable method or instrument means that it is applicable in different contexts to do science. This definition can provide a more accurate understanding of the nature of scientific methodology. It also directs our attention to refining and expanding our sophisticated methodological toolbox that is what enables us to drive science and push the scientific frontier. Other features of science’s major discoveries are outlined in a series of forthcoming papers and forthcoming book The Motor of Scientific Discovery . Ultimately, the best path to discovery is not the classic scientific method but the sophisticated scientific method.

Acknowledgments

The author is thankful for comments from Corinna Peters, Nikolas Schöll, Milan Quentel, Uwe Peters, Julia Hoefer Marti, Alina Velias, Alfonso García Lapeña, and J.P. Grodniewicz.

Contributor Information

Alexander Krauss, London School of Economics, London, UK. Institute for Economic Analysis, Spanish National Research Council, Barcelona, Spain.

This research was funded by the European Commission (grant 745447) and the Ministry of Science and Innovation of the Government of Spain (grant RYC2020-029424-I).

Contributions : Alexander Krauss is the sole author.

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How to Use the Scientific Method in Everyday Life

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The scientific method is a procedure consisting of a series of steps with the goal of problem-solving and information-gathering. The scientific method begins with the recognition of a problem and a clear elaboration or description of the problem itself. A process of experimentation and data collection then follows. The final steps consist of the formulation and testing of a hypothesis or potential solution and conclusion. For people unaccustomed to using the scientific method, the process may seem abstract and unapproachable. With a little consideration and observation, any problem encountered in daily life is a potential possibility to use the scientific method.

Locate or identify a problem to solve. Your personal environment is a good place to start, either in the workplace, the home, or your town or city.

Describe the problem in detail. Make quantifiable observations, such as number of times of occurrence, duration, specific physical measurements, and so on.

Form a hypothesis about what the possible cause of the problem might be, or what a potential solution could be. Check if the previously collected data suggests a pattern or possible cause.

Test your hypothesis either through further observation of the problem or by creating an experiment that highlights the aspect of the problem you wish to test. For example, if you suspect a faulty wire is the cause of a light not working, you must find a way to isolate and test whether or not the wire is actually the cause.

Repeat the steps of observation, hypothesis formation and testing until you reach a conclusion that is reinforced by supporting data or directly solves the problem at hand.

• The scientific method is best suited to solving problems without direct or simple answers. For example, a light bulb that burns out may simply need to be replaced. A light bulb that works intermittently is a much more suitable candidate for use of the scientific method, because of all of the potential causes of it not working.

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• Britannica Online Encyclopedia: Scientific Method; June 2011

About the Author

Alex Jakubik began his writing career in 2000 with book-cover summaries for Barnes & Noble. He has also authored concert programs and travel blogs, and worked both nationally and internationally in the arts. Jakubik holds a Bachelor of Music degree from Indiana University and a Master of Music from Yale University.

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15 Scientific Method Examples

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The scientific method is a structured and systematic approach to investigating natural phenomena using empirical evidence . 

The scientific method has been a lynchpin for rapid improvements in human development. It has been an invaluable procedure for testing and improving upon human ingenuity. It’s led to amazing scientific, technological, and medical breakthroughs.

Some common steps in a scientific approach would include:

• Observation
• Question formulation
• Hypothesis development
• Experimentation and collecting data
• Analyzing results
• Drawing conclusions

Definition of Scientific Method

The scientific method is a structured and systematic approach to investigating natural phenomena or events through empirical evidence. 

Empirical evidence can be gathered from experimentation, observation, analysis, and interpretation of data that allows one to create generalizations about probable reasons behind those happenings. 

As mentioned in the article published in the journal  Nature,

“ As schoolchildren, we are taught that the scientific method involves a question and suggested explanation (hypothesis) based on observation, followed by the careful design and execution of controlled experiments, and finally validation, refinement or rejection of this hypothesis” (p. 237).

The use of scientific methods permits replication and validation of other people’s scientific analyses, leading toward improvement upon previous results, and solid empirical conclusions. 

Voit (2019) adds that:

“…it not only prescribes the order and types of activities that give a scientific study validity and a stamp of approval but also has substantially shaped how we collectively think about the endeavor of investigating nature” (p. 1).

This method aims to minimize subjective biases while maximizing objectivity helping researchers gather factual data. 

It follows set procedures and guidelines for testing hypotheses using controlled conditions, assuring optimum accuracy and relevance in concluding by assessing a range of aspects (Blystone & Blodgett, 2006).

Overall, the scientific method provides researchers with a structured way of inquiry that seeks insightful explanations regarding evidence-based investigation grounded in facts acquired from an array of fields.

15 Examples of Scientific Method

• Medicine Delivery : Scientists use scientific method to determine the most effective way of delivering a medicine to its target location in the body. They perform experiments and gather data on the different methods of medicine delivery, monitoring factors such as dosage and time release.
• Agricultural Research : Scientific method is frequently used in agricultural research to determine the most effective way to grow crops or raise livestock. This may involve testing different fertilizers, irrigation methods, or animal feed, measuring yield, and analyzing data.
• Food Science and Nutrition : Nutritionists and food scientists use the scientific method to study the effects of different food types and diet on health. They design experiments to understand the impact of dietary changes on weight, disease risk, and overall health outcomes.
• Environmental Studies : Researchers use scientific method to study natural ecosystems and how human activities impact them. They collect data on things like biodiversity, water quality, and pollution levels, analyzing changes over time.
• Psychological Studies : Psychologists use the scientific method to understand human behavior and cognition. They conduct experiments under controlled conditions to test theories about learning, memory, social interaction, and more.
• Climate Change Research : Climate scientists use the scientific method to study the Earth’s changing climate. They collect and analyze data on temperature, CO2 levels, and ice coverage to understand trends and make predictions about future changes.
• Geology Exploration : Geologists use scientific method to analyze rock samples from deep in the earth’s crust and gather information about geological processes over millions of years. They evaluate data by studying patterns left behind by these processes.
• Space Exploration : Scientists use scientific methods in designing space missions so that they can explore other planets or learn more about our solar system. They employ experiments like landing craft exploration missions as well as remote sensing techniques that allow them to examine far-off planets without having physically land on their surfaces.
• Archaeology : Archaeologists use the scientific method to understand past human cultures. They formulate hypotheses about a site or artifact, conduct excavations or analyses, and then interpret the data to test their hypotheses.
• Clinical Trials : Medical researchers use scientific method to test new treatments and therapies for various diseases. They design controlled studies that track patients’ outcomes while varying variables like dosage or treatment frequency.
• Industrial Research & Development : Many companies use scientific methods in their R&D departments. For example, automakers may assess the effectiveness of anti-lock brakes before releasing them into the marketplace through tests with dummy targets.
• Material Science Experiments : Engineers have extensively used scientific method experimentation efforts when designing new materials and testing which options could be flexible enough for certain applications. These experiments might include casting molten material into molds and then subjecting it to high heat to expose vulnerabilities
• Chemical Engineering Investigations : Chemical engineers also abide by scientific method principles to create new chemical compounds & technologies designed to be valuable in the industry. They may experiment with different substances, changing materials’ concentration and heating conditions to ensure the final end-product safety and reliability of the material.
• Biotechnology : Biotechnologists use the scientific method to develop new products or processes. For instance, they may experiment with genetic modification techniques to enhance crop resistance to pests or disease.
• Physics Research : Scientists use scientific method in their work to study fundamental principles of the universe. They seek answers for how atoms and molecules are breaking down and related events that unfold naturally by running many simulations using computer models or designing sophisticated experiments to test hypotheses.

Origins of the Scientific Method

The scientific method can be traced back to ancient times when philosophers like Aristotle used observation and logic to understand the natural world. 

These early philosophers were focused on understanding the world around them and sought explanations for natural phenomena through direct observation (Betz, 2010).

In the Middle Ages, Muslim scholars played a key role in developing scientific inquiry by emphasizing empirical observations. 

Alhazen (a.k.a Ibn al-Haytham), for example, introduced experimental methods that helped establish optics as a modern science. He emphasized investigation through experimentation with controlled conditions (De Brouwer, 2021).

During the Scientific Revolution of the 17th century in Europe, scientists such as Francis Bacon and René Descartes began to develop what we now know as the scientific method observation (Betz, 2010).

Bacon argued that knowledge must be based on empirical evidence obtained through observation and experimentation rather than relying solely upon tradition or authority. 

Descartes emphasized mathematical methods as tools in experimentation and rigorous thinking processes (Fukuyama, 2021).

These ideas later developed into systematic research designs , including hypothesis testing, controlled experiments, and statistical analysis – all of which are still fundamental aspects of modern-day scientific research.

Since then, technological advancements have allowed for more sophisticated instruments and measurements, yielding far more precise data sets scientists use today in fields ranging from Medicine & Chemistry to Astrophysics or Genetics.

So, while early Greek philosophers laid much groundwork toward an observational-based approach to explaining nature, Islam scholars furthered our understanding of logical reasoning techniques and gave rise to a more formalized methodology.

Steps in the Scientific Method

While there may be variations in the specific steps scientists follow, the general process has six key steps (Blystone & Blodgett, 2006).

Here is a brief overview of each of these steps:

1. Observation

The first step in the scientific method is to identify and observe a phenomenon that requires explanation. 

This can involve asking open-ended questions, making detailed observations using our senses or tools, or exploring natural patterns, which are sources to develop hypotheses. 

2. Formulation of a Hypothesis

A hypothesis is an educated guess or proposed explanation for the observed phenomenon based on previous observations & experiences or working assumptions derived from a valid literature review . 

The hypothesis should be testable and falsifiable through experimentation and subsequent analysis.

3. Testing of the Hypothesis

In this step, scientists perform experiments to test their hypothesis while ensuring that all variables are controlled besides the one being observed.

The data collected in these experiments must be measurable, repeatable, and consistent.

4. Data Analysis

Researchers carefully scrutinize data gathered from experiments – typically using inferential statistics techniques to analyze whether results support their hypotheses or not.

This helps them gain important insights into what previously unknown mechanisms might exist based on statistical evidence gained about their system.

See: 15 Examples of Data Analysis

5. Drawing Conclusions 

Based on their data analyses, scientists reach conclusions about whether their original hypotheses were supported by evidence obtained from testing.

If there is insufficient supporting evidence for their ideas – trying again with modified iterations of the initial idea sometimes happens.

6. Communicating Results

Once results have been analyzed and interpreted under accepted principles within the scientific community, scientists publish findings in respected peer-reviewed journals.

These publications help knowledge-driven communities establish trends within respective fields while indirectly subjecting papers reviews requests boosting research quality across the scientific discipline.

Importance of the Scientific Method

The scientific method is important because it helps us to collect reliable data and develop testable hypotheses that can be used to explain natural phenomena (Haig, 2018).

Here are some reasons why the scientific method is so essential:

• Objectivity : The scientific method requires researchers to conduct unbiased experiments and analyses, which leads to more impartial conclusions. In this way, replication of findings by peers also ensures results can be relied upon as founded on sound principles allowing others confidence in building further knowledge on top of existing research.
• Precision & Predictive Power : Scientific methods usually include techniques for obtaining highly precise measurements, ensuring that data collected is more meaningful with fewer uncertainties caused by limited measuring errors leading to statistically significant results having firm logical foundations. If predictions develop scientifically tested generalized defined conditions factored into the analysis, it helps in delivering realistic expectations
• Validation : By following established scientific principles defined within the community – independent scholars can replicate observation data without being influenced by subjective biases or prejudices. It assures general acceptance among scientific communities who follow similar protocols when researching within respective fields.
• Application & Innovation : Scientific concept advancements that occur based on correct hypothesis testing commonly lead scientists toward new discoveries, identifying potential breakthroughs in research. They pave the way for technological innovations often seen as game changers, like mapping human genome DNA onto creating novel therapies against genetic diseases or unlocking secrets of today’s universe through discoveries at LHC.
• Impactful Decision-Making : Policymakers can draw from these scientific findings investing resources into informed decisions leading us toward a sustainable future. For example, research gathered about carbon pollution’s impact on climate change informs debate making policy action decisions about our planet’s environment, providing valuable knowledge-useful information benefiting societies (Haig, 2018).

The scientific method is an essential tool that has revolutionized our understanding of the natural world.

By emphasizing rigorous experimentation, objective measurement, and logical analysis- scientists can obtain more unbiased evidence with empirical validity . 

Utilizing this methodology has led to groundbreaking discoveries & knowledge expansion that have shaped our modern world from medicine to technology. 

The scientific method plays a crucial role in advancing research and our overall societal consensus on reliable information by providing reliable results, ensuring we can make more informed decisions toward a sustainable future. 

As scientific advancements continue rapidly, ensuring we’re applying core principles of this process enables objectives to progress, paving new ways for interdisciplinary research across all fields, thereby fuelling ever-driving human curiosity.

Betz, F. (2010). Origin of scientific method.  Managing Science , 21–41. https://doi.org/10.1007/978-1-4419-7488-4_2

Blystone, R. V., & Blodgett, K. (2006). WWW: The scientific method.  CBE—Life Sciences Education ,  5 (1), 7–11. https://doi.org/10.1187/cbe.05-12-0134

De Brouwer , P. J. S. (2021).  The big r-book: From data science to learning machines and big data . John Wiley & Sons, Inc.

Defining the scientific method. (2009).  Nature Methods ,  6 (4), 237–237. https://doi.org/10.1038/nmeth0409-237

Fukuyama, F. (2012).  The end of history and the last man . New York: Penguin.

Haig, B. D. (2018). The importance of scientific method for psychological science.  Psychology, Crime & Law ,  25 (6), 527–541. https://doi.org/10.1080/1068316x.2018.1557181

Voit, E. O. (2019). Perspective: Dimensions of the scientific method.  PLOS Computational Biology ,  15 (9), e1007279. https://doi.org/10.1371/journal.pcbi.1007279

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Limitations of the Scientific Method

Clearly, the scientific method is a powerful tool, but it does have its limitations. These limitations are based on the fact that a hypothesis must be testable and falsifiable and that experiments and observations be repeatable. This places certain topics beyond the reach of the scientific method.

Science cannot prove or refute the existence of God or any other supernatural entity. Sometimes, scientific principles are used to try to lend credibility to certain nonscientific ideas, such as intelligent design . Intelligent design is the assertion that certain aspects of the origin of the universe and life can be explained only in the context of an intelligent, divine power. Proponents of intelligent design try to pass this concept off as a scientific theory to make it more palatable to developers of public school curriculums. But intelligent design is not science because the existence of a divine being cannot be tested with an experiment.

Science is also incapable of making value judgments. It cannot say global warming is bad, for example. It can study the causes and effects of global warming and report on those results, but it cannot assert that driving SUVs is wrong or that people who haven't replaced their regular light bulbs with LED bulbs are irresponsible.

Occasionally, certain organizations use scientific data to advance their causes. This blurs the line between science and morality and encourages the creation of "pseudo-science," which tries to legitimize a product or idea with a claim that has not been subjected to rigorous testing.

And yet, used properly, the scientific method is one of the most valuable tools humans have ever created. It helps us solve everyday problems around the house and, at the same time, helps us understand profound questions about the world and universe in which we live.

Most of the time, two competing theories can't exist to describe one phenomenon. But in the case of light , one theory is not enough. Many experiments support the notion that light behaves like a longitudinal wave. Taken collectively, these experiments have given rise to the wave theory of light. Other experiments, however, support the notion that light behaves as a particle. Instead of throwing out one theory and keeping the other, physicists maintain a wave/particle duality to describe the behavior of light.

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• D'Agnese, Joseph. "Scientific Method Man." Wired, September 2004. http://www.wired.com/wired/archive/12.09/rugg.html.
• Introduction to the Scientific Method on Web Site of Frank Wolfs, Department of Physics and Astronomy, University of Rochester. http://teacher.pas.rochester.edu/phy_labs/AppendixE/AppendixE.html
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• Published: 30 August 2024

Analysis of multi-wave solitary solutions of (2+1)-dimensional coupled system of Boiti–Leon–Pempinelli

• Sidra Ghazanfar 1 ,
• Nauman Ahmed 1 , 6 ,
• Muhammad Sajid Iqbal 2 , 3 ,
• Syed Mansoor Ali 4 ,
• Ali Akgül 5 , 10 , 11 ,
• Shah Muhammad 7 ,
• Mubasher Ali 8 &
• Murad Khan Hassani 9  

Scientific Reports volume  14 , Article number:  20234 ( 2024 ) Cite this article

Metrics details

• Engineering
• Mathematics and computing

This work examines the (2+1)-dimensional Boiti–Leon–Pempinelli model, which finds its use in hydrodynamics. This model explains how water waves vary over time in hydrodynamics. We provide new explicit solutions to the generalized (2+1)-dimensional Boiti–Leon–Pempinelli equation by applying the Sardar sub-equation technique. This method is shown to be a reliable and practical tool for solving nonlinear wave equations. Furthermore, different types of solitary wave solutions are constructed: w-shaped, breather waved, chirped, dark, bright, kink, unique, periodic, and more. The results obtained with the variable coefficient Boiti–Leon–Pempinelli equation are stable and different from previous methods. As compared to their constant-coefficient counterparts, the variable-coefficient models are more general here. In the current work, the problem is solved using the Sardar Sub-problem Technique to produce distinct soliton solutions with parameters. Plotting these graphs of the solutions will help you better comprehend the model. The outcomes demonstrate how well the method works to solve nonlinear partial differential equations, which are common in mathematical physics.With the help of this method, we may examine a variety of solutions from significant physical perspectives.

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Introduction.

In the past few decades, the research of traveling wave solutions explored by researchers has gained considerable attention. It includes the solutions of non-linear partial differential equations (NPDEs) which play a key role in the study of non-linear physical phenomena arising in many fields of engineering and sciences i.e., mathematical physics 1 , technical arena 2 , plasma physics 3 , ocean engineering 4 , tsunami waves 5 , etc. NPDEs have great potential for applications in various fields, therefore, these equations have the advantage of getting the special attention of researchers to find their analytical and numerical solutions. In recent times, many researchers in mathematics and physics have established various methods of constructing and analyzing exact traveling wave solutions of different non-linear problems such as Hirota’s bilinear transformation method 6 , 7 , 8 , the extended Exp-expansion method 9 , the new extended direct algebraic method 10 , the variational iteration method 11 , the semi-inverse variational principle 12 , the generalized Kudryashov technique 13 , the sine-Gordon method 14 , the Cole-Hopf transformation method 15 , the Adomian decomposition method 16 , the traveling wave scheme 17 , A special kind of distributive product 18 , the B \(\ddot{a}\) cklund transformation method 19 .

Solitons are the fascinating aspect of nonlinear physical events. The solitonic concept is accessible due to ethical balance and nonlinearity of concentration. Many scholars have conducted studies on solitary wave solutions as mentioned above.

Recently, an effective method has been introduced, called the Sardar sub-equation technique (SSET) 20 . Our primary emphasis is developing various wave soliton solutions, such as bright, singular, dark-bright, kink, dark, w-shaped, chirped, breather wave, and periodic wave solitons. The method under consideration is more universal than the others listed above. Similarly, these findings help us recognize the dynamic performance of various physical configurations. Furthermore, these results are positive, unique, and precise, and they may help illuminate particular non-linear natural phenomena in non-linear mathematical models.

This work has a few obvious limitations, such as the that it is usually only appropriate for a specific class of nonlinear PDEs,it might not be able to solve more complex equations, it might struggle with highly nonlinear terms,the solution it does provide may take particular forms, it frequently yields solutions under specific conditions, it frequently calls for the use of symbolic computational tools, it can be difficult to incorporate initial and boundary conditions into this method, the solutions it produces sometimes be non-trivial, and there may be easier solutions available.

In this work, the following coupled system of (2+1)-Dimensional Boiti–Leon–Pempinelli (BLP) equations 21 has been taken into consideration.

which was initially introduced by Boiti et al. 22 . Many mathematicians studied this system and developed precise explicit solutions using various methods. The Boiti–Leon–Pempinelli equation has drawn a lot of attention from researchers in the past ten years since it is used to explain the wave propagation of incompressible fluids in plasma physics, fluid dynamics, ocean engineering, astrophysics, and aerodynamics.More relevant material can also be studied in 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 .

Wazwaz and Mehanna’s precise traveling wave system suggestions of the system ( 1 ). System ( 1 )’s lump-type solutions, Lie point symmetries, and some precise answers to some other algebraic equations with new optical, lump wave, breather, periodic, and other multi-wave solutions can also be found with some additional precise solutions to the Eq. ( 1 ). There are some novel traveling wave system solutions of ( 1 ) that are provided in this article. Some other novel exact traveling wave solutions were presented in 31 , 32 , 33 , 34 .

It is a crucial system for describing how the horizontal velocity component of waves in an infinitely narrow channel with constant depth changes over time. The horizontal velocity and height of the water wave are related to the velocity components U ( x ,  y ,  t ) and V ( x ,  y ,  t ), respectively. Eq. ( 1 ) is a member of a group of equations that explain how water waves move through channels with a constant depth. The Sinh-Gordon equation can be generalized as the (2+1)-Dimensional equation in Eq. ( 1 ), and it can be transformed into the (anti)-Burgers equations, for example, in Mu et al. 35 , the model was taken into account using Hamiltonians, the Bäcklund transform, Lax Pair, and the Painlevé integrability.

More related research work can also be studied in this regard e.g., 36 , 37 , 38 , 39 , 40 , 41 .

The terms “horizontal velocity” and “height” relate to two essential characteristics that characterize the motion of water waves. The pace at which individual water particles move horizontally during the propagation of a wave is known as the horizontal wave velocity. The motion of the water particles in a wave is either elliptical or circular. This motion in the direction of wave propagation is composed of the horizontal velocity. It shows the rate at which the wave is propagating laterally. Both the vertical and horizontal components of the particle motion affect the wave’s real speed. While, the vertical distance between a water wave’s highest point, the crest, and its lowest point, the trough, determines the height of the wave. An indicator of a water wave’s energy is its height. Greater heights and energy are carried by larger waves. The amplitude of the wave, or the maximum displacement of the water particles from their undisturbed position, is correlated with the wave’s height. The wave’s energy is determined by its amplitude. To comprehend and forecast wave behavior, a water wave’s height and horizontal velocity work together. Wind speed, water depth, and the distance at which the wind has blown are some variables that affect them. An understanding of these factors is essential in disciplines like marine science, coastal engineering, and oceanography. This work is novel in itself that has not been done before. We can take this work so far till we are able to present these specific solitons/solutions in the form of physical interpretations.

The soliton-type solutions provided in this paper are beneficial for those who are physically related to it. These solutions can be applied in a variety of situations. Regarding its all-purpose applications, nevertheless, a few of these solutions can be useful to all physicists working in the field of soliton solutions for PDEs. Even though we have offered many solutions. Furthermore, since we have found the exact solutions in the absence of auxiliary data, there are infinitely many solutions. Since we haven’t connected the problem to any initial or boundary conditions, the physicist must determine which solution best fits the available information.

Problem statement

Using SSET, the following traveling wave transformation is used to create strong and authentic solitons of the BLP system

Here c is the real constant to be determined. Applying Eq. ( 2 ) in Eq. ( 1 ) to get the following form of ODE (ordinary differential equation) of the given system

By integrating and assembling the above system, we can write the following equation as

This will lead us to the exact solutions of the given BLP system. Moreover, some general insights or constraints of the equations which are worth mentioning here are its boundary conditions, nature of the equation and, its stability analysis.

Mathematical details of Sardar sub-equation technique

This thorough and straightforward method is used by many experts to discover solitons and other wave solutions to the given issue. This technique can provide precise responses for a class of NPDEs. The given system of equations can be included by following the procedures below.

Step I: Considering the NPDE as follows

where \(P=P(x,t)\) is the unknown function, O is a polynomial of P ( x ,  t ) and its derivatives with respect to x and t . Now applying the traveling wave transformation

where \(\alpha\) , and \(\beta\) are the unknown constants to be determined later.

Using the above transformation, Eq. ( 5 ) is converted to the following ODE (ordinary differential equation),

where Q is the function of \(\Psi (\eta )\) and its derivatives and its superscripts designate ordinary derivatives w.r.t \(\eta\)

Step II: Solution of Eq. ( 7 ) is then formulated as

where \(c_n(0\le n\le N)\) are real constants and \(M(\eta )\) satisfies the ODE of the following form

Here \(\mu\) and \(\nu\) are real constants and Eq. ( 9 ) presents the following solutions:

If \(\nu >0\) and \(\mu =0\) , then

If \(\nu <0\) and \(\mu =0\) , then

If \(\nu <0\) and \(\mu =\frac{\nu ^2}{4}\) , then

If \(\nu >0\) and \(\mu =\frac{\nu ^2}{4}\) , then

The above listed functions are the generalized forms of trigonometric and hyperbolic functions with parameters p and q . If we take the values of p and q to be 1, then the above functions become known functions.

Step III: By balancing the capital, we can determine the number N . Using this value of N , we get an algebraic equation in the shape of \(M^n(\eta )\) by substituting Eq. ( 8 ) into Eq. ( 7 ), which we balance by setting the powers of \(M^n(\eta ),\,n=(0,1,2,\ldots )\) to zero, resulting in a set of algebraic equations.

Step IV: This system of equations provides the necessary inputs and the precise answer to the provided equation.

Execution of the technique

The traveling wave solution to the Boiti–Leon–Pempinelli System is created in this part using SSET. Using homogeneous balance principle, we balance the equations \(U''\) and \(U^3\) to get the value of N and found to be 1.

Equation ( 8 ) is reduced by the equilibrium formula into

where \(a_0\) and \(a_1\) are the constants to determine. Substituting Eq. ( 14 ), Eq. ( 4 ) into Eq. ( 9 ) to get a polynomial in the form of \(T^n(\eta )\) . Equating the powers of \(T^n(\eta ),\,(n=0,1,2,3)\) to zero to get the algebraic equations in the form of \(a_0, a_1, \nu\) and \(\mu\) .

The system of equations is as

We discovered the following results by analyzing the above system of equations

Using these values in Eqs. ( 9 ),( 14 ) and ( 17 ) along with Eq. ( 2 ), we summarized the results for functions U along with their corresponding V as follows:

If \(u>0\) and \(\xi =0\) , then

In summary, the constraints of the above equations include:

\(u>0\) and \(\xi =0\) (parameter u must be a positive real number).

\(pq<0\) to ensure the square root term is real for \(U_1, V_1\) and, \(V_2\) and, \(pq>0\) to ensure the square root term is real for \(U_2\) .

If \(u<0\) , and \(\xi =0\) , then

In summary, the constraints of the above equations are:

\(u<0\) and \(\xi =0\) (parameter u must be a negative real number).

\(pq>0\) to ensure the square root term is real for all \(U_3, U_4, V_3\) and \(V_4\) .

\(\cos _{pq}(\sqrt{-u}\,\eta )\ne 0\) (to avoid division by zero in the denominator of \(V_3\) ).

If \(u<0\) and \(\xi =\frac{u^2}{4}\) , then

In summary, the constraints of the above equations are given below:

\(u<0\) and \(\xi =\frac{u^2}{4}\) (parameter u must be a negative real number).

pq should be defined appropriately for the hyperbolic tangent, cotangent, secant and cosecant functions.

The expression should be well-defined for the given values of \(\sqrt{\frac{-\mu }{2}}\,\eta\) and, \(\xi =\frac{u^2}{4}\) .

If \(u>0\) and \(\xi =\frac{u^2}{4}\) , then

\(u>0\) and \(\xi =\frac{u^2}{4}\) (parameter u must be a positive real number).

pq should be defined appropriately for the tangent, cotangent, secant and cosecant functions.

The expression should be well-defined for the given values of \(\sqrt{\frac{\mu }{2}}\,\eta\) and, \(\xi =\frac{u^2}{4}\) .

Graphical behavior

Various types of solitons are shown below, each displaying the graphical behavior of the solution to the issue mentioned above.

Above plottings are associated to Case I function U.

Graphical representation of function V Case I.

For Case II

Plots associated to Case II function U.

Graphing of exact solutions linked with Case II function V.

For Case III

Plots representing Case III function U.

Surface and contour plots representing Case III function V.

For Case IV

Representation of exact solutions of Case IV function U.

Plots signifying Case IV function V.

The parameters used to generate these figures are listed below.

Results and discussion

The surface and contour plots of each solution using each condition stated in the technique are shown in the graphs above where Fig.  1 shows breather wave singular solitonic behavior of U of the given coupled system using the conditions in case I represented in Eq. ( 19 ), Fig.  2 shows the surface and contour plots of V like rogue wave (singular) solitons represented in Eq. ( 19 ) of the same case I, Fig.  3 represents the surface and contour plots showing w-shaped dark-bright solitons in the form of U using the condition given in case II represented in Eq. ( 21 ), Fig.  4 signifies the surface and contour plots representing periodic function solitons in the form of V represented in Eq. ( 21 ) of the same case II, Fig.  5 shows the plots as kink soliton type behavior of U of the given system of equations represented in Eq. ( 23 ) under conditions of case III, Fig.  6 displays the surface and contour plots signifying kink solitons with non-topological (bright) background in the form of V of Eq. ( 23 ) of the same case III, Fig.  7 represents the surface and contour plots of chirped periodic solitons of U represented in Eq. ( 25 ) under condition of case IV, Fig.  8 signifies the surface and contour plots representing dark-bright solitons in the form of V of the given system represented in Eq. ( 25 ) of the same case IV.

Physical interpretation

Discovering NPDE solutions is critical for comprehending the underlying physical processes. Solitons are important in mathematics and physics because they keep their shape and velocity constant while propagating. The modulation instability of the carrier wave train requires distinguishing between topological (dark) and non-topological (bright) solitons. Topological solitons occur when the carrier wave is unsustainable due to long-wave modulations, whereas non-topological solitons occur when the carrier wave is modulationally consistent.

Rogue waves, often known as freakish or killer waves, have progressed from maritime folklore to a recognized phenomenon. These waves, which are twice the magnitude of the surrounding waves, are unpredictable and frequently appear from other directions than the current wind and waves. The study of rogue waves advances our understanding of extraordinary phenomena in fluid dynamics.

Conclusions

This article presents new and interesting optical soliton solutions to the (2+1)-Dimensional Coupled System of the BLP equations using the analytical method of SSET. Our main goal in writing this article is to assess the BLP system using this methodology for the first time. This is a relatively new method that yields several new soliton solutions for the system being studied. The method is incredibly effective and easy to use. The results are given as hyperbolic, rational, and trigonometric functions. As we can see, this method provides a powerful, efficient, and simple tool for solving a range of nonlinear PDEs that are included in many models in the fields of natural science and engineering. The results may have practical applications and explain water waves in domains such as optics, linked circuits, elastic rods, shallow water with long wavelengths, and marine engineering. Lastly, 3D and contour plots of these solutions are produced using Maple. Bright, dark, periodic, chirped, breather-waved, singular, w-shaped, and breather-waved solitons are the results of this approach. We have investigated the forms and directions of the different solitons using the generated graphs.

As we all know, in the field of integrable systems, there is no general method to solve the analytical solution of NPDEs. The symbol calculation method based on neural networks proposed by Zhang et al. (see, 42 , 43 , 44 , 45 ) open up a general symbolic computing path for the analytic solution of NPDEs, and lays the foundation for the universal method of symbolic calculation of analytical expression. The problems studied in this paper can be solved by using this method in future work.

This work is novel in itself. Within the framework of analytical solutions, researchers can choose our solution for numerical analysis. This methodology distinguishes itself from other ways by providing a systematic approach, comprehensive applicability, efficiency and brevity, the generation of many solutions, the reduction of equations to simpler ones, and integration with other techniques.

Graphing parameters

The graphs in this article were created using the settings listed below.

Figure  1 : \(c=2; u=2.5; p=1; q=-0.09; y=1.\)

Figure  2 : \(c=2; u=-2.5; p=1; q=-0.09; y=1.\)

Figure  3 : \(c=2; u=-1.5; p=2; q=1; y=2.\)

Figure  4 : \(c=-5; u=2.5; p=1; q=-0.09; y=5.\)

Figure  5 : \(c=2; u=2.5; p=0.8; q=-0.09; y=1.\)

Figure  6 : \(c=2; u=-2.5; p=1; q=-0.09; y=1.\)

Figure  7 : \(c=2; u=-5.5; p=2; q=1; y=2.\)

Figure  8 : \(c=-5; u=2.5; p=1; q=0.09; y=5.\)

Data availibility

Data will be provided by corresponding author on reasonable request.

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Acknowledgements

This research is funded by “Researchers Supporting Project number (RSPD2024R733), King Saud University, Riyadh, Saudi Arabia.

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Sidra Ghazanfar & Nauman Ahmed

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Muhammad Sajid Iqbal

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Syed Mansoor Ali

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Nauman Ahmed

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Shah Muhammad

Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield, SYK, England

Mubasher Ali

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Murad Khan Hassani

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S.G. and N.A. and M.S.I. and S.M.A. wrote the main manuscript text and A.A. and S.M. and M.A. and M.K.H. prepared Figs. 1-8. All authors reviewed the manuscript.

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Ghazanfar, S., Ahmed, N., Iqbal, M.S. et al. Analysis of multi-wave solitary solutions of (2+1)-dimensional coupled system of Boiti–Leon–Pempinelli. Sci Rep 14 , 20234 (2024). https://doi.org/10.1038/s41598-024-67698-z

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Received : 28 May 2024

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Published : 30 August 2024

DOI : https://doi.org/10.1038/s41598-024-67698-z

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