The term "scientific method" came into popular use in the twentieth century, popping up in dictionaries and science textbooks, although there was little scientific consensus over its meaning. Although there was a growth through the middle of the twentieth century, by the 1960s and 1970s numerous influential philosophers of science such as Thomas Kuhn and Paul Feyerabend had questioned the universality of the "scientific method" and in doing so largely replaced the notion of science as a homogeneous and universal method with that of it being a heterogeneous and local practice. In particular, Paul Feyerabend, in the 1975 first edition of his book Against Method, argued against there being any universal rules of science. Later examples include physicist Lee Smolin's 2013 essay "There Is No Scientific Method" and historian of science Daniel Thurs's chapter in the 2015 book Newton's Apple and Other Myths about Science, which concluded that the scientific method is a myth or, at best, an idealization. Philosophers Robert Nola and Howard Sankey, in their 2007 book Theories of Scientific Method, said that debates over scientific method continue, and argued that Feyerabend, despite the title of Against Method, accepted certain rules of method and attempted to justify those rules with a metamethodology . The scientific method is the process by which science is carried out.: "The thesis of this book, as set forth in Chapter One, is that there are general principles applicable to all the sciences." As in other areas of inquiry, science (through the scientific method) can build on previous knowledge and develop a more sophisticated understanding of its topics of study over time.: The scientific method can function in the same way; This is the principle of noncontradiction.Francis Bacon(1629) New Organon, lists 4 types of error: Idols of the tribe (error due to the entire human race), the cave (errors due to an individual's own intellect), the marketplace (errors due to false words), and the theater (errors due to incredulous acceptance).For example, the concept of falsification (first proposed in 1934) formalizes the attempt to disprove hypotheses rather than prove them. Karl R. Popper (1963), 'The Logic of Scientific Discovery'. The Logic of Scientific Discovery pp. 17–20, 249–52, 437–38, and elsewhere. Leon Lederman, for teaching physics first, illustrates how to avoid confirmation bias: Ian Shelton, in Chile, was initially skeptical that supernova 1987a was real, but possibly an artifact of instrumentation (null hypothesis), so he went outside and disproved his null hypothesis by observing SN 1987a with the naked eye. The Kamiokande experiment, in Japan, independently observed neutrinos from SN 1987a at the same time. This model can be seen to underlie the scientific revolution.: "There is a danger that must be avoided. ... If we wish to do justice to the historical enterprise, we must take the past for what it was. And that means we must resist the temptation to scour the past for examples or precursors of modern science. ...My concern will be with the beginnings of scientific theories, the methods by which they were formulated, and the uses to which they were put; ... " The ubiquitous element in  scientific method is empiricism. This is in opposition to stringent forms of rationalism: the scientific method embodies that reason alone cannot solve a particular scientific problem. A strong formulation of the scientific method is not always aligned with a form of empiricism in which the empirical data is put forward in the form of experience or other abstracted forms of knowledge; in current scientific practice, however, the use of scientific modelling and reliance on abstract typologies and theories is normally accepted. The scientific method is of necessity also an expression of an opposition to claims that e.g. revelation, political or religious dogma, appeals to tradition, commonly held beliefs, common sense, or, importantly, currently held theories, are the only possible means of demonstrating truth. Different early expressions of empiricism and the scientific method can be found throughout history, for instance with the ancient Stoics, Epicurus,Elizabeth Asmis (1985) Epicurus' Scientific Method. Cornell University Press Alhazen,Alhazen argued the importance of forming questions and subsequently testing them: "How does light travel through transparent bodies? Light travels through transparent bodies in straight lines only... . We have explained this exhaustively in our Book of Optics. But let us now mention something to prove this convincingly: the fact that light travels in straight lines is clearly observed in the lights which enter into dark rooms through holes.... [T]he entering light will be clearly observable in the dust which fills the air. – Alhazen, Treatise on Light (رسالة في الضوء), translated into English from German by M. Schwarz, from "Abhandlung über das Licht", J. Baarmann (editor and translator from Arabic to German, 1882) Zeitschrift der Deutschen Morgenländischen Gesellschaft Vol 36 as quoted in . He demonstrated his conjecture that "light travels through transparent bodies in straight lines only" by placing a straight stick or a taut thread next to the light beam, as quoted in to prove that light travels in a straight line. David Hockney, (2001, 2006) in Secret Knowledge: rediscovering the lost techniques of the old masters (expanded edition) cites Alhazen several times as the likely source for the portraiture technique using the camera obscura, which Hockney rediscovered with the aid of an optical suggestion from Charles M. Falco. Kitab al-Manazir, which is Alhazen's Book of Optics, at that time denoted Opticae Thesaurus, Alhazen Arabis, was translated from Arabic into Latin for European use as early as 1270. Hockney cites Friedrich Risner's 1572 Basle edition of Opticae Thesaurus. Hockney quotes Alhazen as the first clear description of the camera obscura in Hockney, p. 240. "Truth is sought for its own sake. And those who are engaged upon the quest for anything for its own sake are not interested in other things. Finding the truth is difficult, and the road to it is rough." – Alhazen (Ibn Al-Haytham 965 – c. 1040) Critique of Ptolemy, translated by S. Pines, Actes X Congrès internationale d'histoire des sciences, Vol I Ithaca 1962, as quoted in . (This quotation is from Alhazen's critique of Ptolemy's books Almagest, Planetary Hypotheses, and Optics as translated into English by A. Mark Smith.) Roger Bacon, and William of Ockham. From the 16th century onwards, experiments were advocated by Francis Bacon, and performed by Giambattista della Porta, Johannes Kepler,Kepler, Johannes (1604) Ad Vitellionem paralipomena, quibus astronomiae pars opticae traditur (Supplements to Witelo, in which the optical part of astronomy is treated) as cited in The full title translation is from p. 60 of James R. Voelkel (2001) Johannes Kepler and the New Astronomy Oxford University Press. Kepler was driven to this experiment after observing the partial solar eclipse at Graz, July 10, 1600. He used Tycho Brahe's method of observation, which was to project the image of the Sun on a piece of paper through a pinhole aperture, instead of looking directly at the Sun. He disagreed with Brahe's conclusion that total eclipses of the Sun were impossible , because there were historical accounts of total eclipses. Instead  he deduced that the size of the aperture controls the sharpness of the projected image (the larger the aperture, the more accurate the image – this fact is now fundamental for optical system design). Voelkel, p. 61, notes that Kepler's experiments produced the first correct account of vision and the eye, because he realized he could not accurately write about astronomical observation by ignoring the eye. and Galileo Galilei....an approach which was advocated by Galileo in 1638 with the publication of Two New Sciences., Dover reprint of the 1914 Macmillan translation by Henry Crew and Alfonso de Salvio of Two New Sciences, Galileo Galilei Linceo (1638). Additional publication information is from the collection of first editions of the Library of Congress surveyed by . There was particular development aided by theoretical works by Francisco Sanches, John Locke, George Berkeley, and David Hume. Prediction This step involves determining the logical consequences of the hypothesis. One or more predictions are then selected for further testing. The more unlikely that a prediction would be correct simply by coincidence, then the more convincing it would be if the prediction were fulfilled;  evidence is also stronger if the answer to the prediction is not already known, due to the effects of hindsight bias (see also postdiction). Ideally, the prediction must also distinguish the hypothesis from likely alternatives; if two hypotheses make the same prediction, observing the prediction to be correct is not evidence for either one over the other. (These statements about the relative strength of evidence can be mathematically derived using Bayes' Theorem).Note: for a discussion of multiple hypotheses, see Bayesian inference#Informal Testing This is an investigation of whether the real world behaves as predicted by the hypothesis. Scientists (and other people) test hypotheses by conducting experiments. The purpose of an experiment is to determine whether observations of the real world agree with or conflict with the predictions derived from a hypothesis. If they agree, confidence in the hypothesis increases; otherwise, it decreases. Agreement does not assure that the hypothesis is true; future experiments may reveal problems. Karl Popper advised scientists to try to falsify hypotheses, i.e., to search for and test those experiments that seem most doubtful. Large numbers of successful confirmations are not convincing if they arise from experiments that avoid risk. Experiments should be designed to minimize possible errors, especially through the use of appropriate scientific controls. For example, tests of medical treatments are commonly run as double-blind tests. Test personnel, who might unwittingly reveal to test subjects which samples are the desired test drugs and which are placebos, are kept ignorant of which are which. Such hints can bias the responses of the test subjects. Furthermore, failure of an experiment does not necessarily mean the hypothesis is false. Experiments always depend on several hypotheses, e.g., that the test equipment is working properly, and a failure may be a failure of one of the auxiliary hypotheses. (See the Duhem–Quine thesis.) Experiments can be conducted in a college lab, on a kitchen table, at CERN's Large Hadron Collider, at the bottom of an ocean, on Mars (using one of the working rovers), and so on. Astronomers do experiments, searching for planets around distant stars. Finally, most individual experiments address highly specific topics for reasons of practicality. As a result, evidence about broader topics is usually accumulated gradually. Analysis This involves determining what the results of the experiment show and deciding on the next actions to take. The predictions of the hypothesis are compared to those of the null hypothesis, to determine which is better able to explain the data. In cases where an experiment is repeated many times, a statistical analysis such as a chi-squared test may be required. If the evidence has falsified the hypothesis, a new hypothesis is required; if the experiment supports the hypothesis but the evidence is not strong enough for high confidence, other predictions from the hypothesis must be tested. Once a hypothesis is strongly supported by evidence, a new question can be asked to provide further insight on the same topic. Evidence from other scientists and experience are frequently incorporated at any stage in the process. Depending on the complexity of the experiment, many iterations may be required to gather sufficient evidence to answer a question with confidence or to build up many answers to highly specific questions in order to answer a single broader question. DNA example link=|alt= The basic elements of the scientific method are illustrated by the following example from the discovery of the structure of DNA: Question: Previous investigation of DNA had determined its chemical composition (the four nucleotides), the structure of each individual nucleotide, and other properties. X-ray diffraction patterns of DNA by Florence Bell in her Ph.D. thesis (1939) were similar to (although not as good as) "photo 51", but this research was interrupted by the events of World War II. DNA had been identified as the carrier of genetic information by the Avery–MacLeod–McCarty experiment in 1944, but the mechanism of how genetic information was stored in DNA was unclear. Hypothesis: Linus Pauling, Francis Crick and James D. Watson hypothesized that DNA had a helical structure.October 1951, as noted in :"That's what a helix should look like!" Crick exclaimed in delight (This is the Cochran-Crick-Vand-Stokes theory of the transform of a helix). Prediction: If DNA had a helical structure, its X-ray diffraction pattern would be X-shaped.June 1952, as noted in : Watson had succeeded in getting X-ray pictures of TMV showing a diffraction pattern consistent with the transform of a helix.Watson did enough work on Tobacco mosaic virus to produce the diffraction pattern for a helix, per Crick's work on the transform of a helix. pp. 137–38, Horace Freeland Judson (1979) The Eighth Day of Creation This prediction was determined using the mathematics of the helix transform, which had been derived by Cochran, Crick  and Vand– Cochran W, Crick FHC and Vand V. (1952) "The Structure of Synthetic Polypeptides. I. The Transform of Atoms on a Helix", Acta Crystallogr., 5, 581–86. (and independently by Stokes). This prediction was a mathematical construct, completely independent from the biological problem at hand. Experiment: Rosalind Franklin used pure DNA to perform X-ray diffraction to produce photo 51. The results showed an X-shape. Analysis: When Watson saw the detailed diffraction pattern, he immediately recognized it as a helix.Friday, January 30, 1953. Tea time, as noted in : Franklin confronts Watson and his paper – "Of course it [Pauling's pre-print] is wrong. DNA is not a helix." However, Watson then visits Wilkins' office, sees photo 51, and immediately recognizes the diffraction pattern of a helical structure. But additional questions remained, requiring additional iterations of their research. For example, the number of strands in the backbone of the helix (Crick suspected 2 strands, but cautioned Watson to examine that more critically), the location of the base pairs (inside the backbone or outside the backbone), etc. One key point was that they realized that the quickest way to reach a result was not to continue a mathematical analysis, but to build a physical model."The instant I saw the picture my mouth fell open and my pulse began to race." – Page 168 shows the X-shaped pattern of the B-form of DNA, clearly indicating crucial details of its helical structure to Watson and Crick. p. 52 dates the Franklin-Watson confrontation as Friday, January 30, 1953. Later that evening, Watson urges Wilkins to begin model-building immediately. But Wilkins agrees to do so only after Franklin's departure. He and Crick then produced their model, using this information along with the previously known information about DNA's composition, especially Chargaff's rules of base pairing.Saturday, February 28, 1953, as noted in : Watson found the base pairing mechanism which explained Chargaff's rules using his cardboard models. Replication If an experiment cannot be repeated to produce the same results, this implies that the original results might have been in error. As a result, it is common for a single experiment to be performed multiple times, especially when there are uncontrolled variables or other indications of experimental error. For significant or surprising results, other scientists may also attempt to replicate the results for themselves, especially if those results would be important to their own work. Replication has become a contentious issue in social and biomedical science where treatments are administered to groups of individuals. Typically an experimental group gets the treatment, such as  drug, and the control group gets a placebo. John Ioannidis in 2005 pointed out that the method being used has led to many findings that cannot be replicated. External review The process of peer review involves evaluation of the experiment by experts, who typically give their opinions anonymously. Some journals request that the experimenter provide lists of possible peer reviewers, especially if the field is highly specialized. Peer-review does not certify the correctness of the results, only that, in the opinion of the reviewer, the experiments themselves were sound (based on the description supplied by the experimenter). If the work passes peer review, which occasionally may require new experiments requested by the reviewers, it will be published in a peer-reviewed scientific journal. The specific journal that publishes the results indicates the perceived quality of the work.In Two new sciences, there are three 'reviewers': Simplicio, Sagredo, and Salviati, who serve as foil, antagonist, and protagonist. Galileo speaks for himself only briefly. But note that Einstein's 1905 papers were not peer reviewed before their publication. Theories can also become subsumed by other theories. For example, Newton's laws explained thousands of years of scientific observations of the planets almost perfectly. However, these laws were then determined to be special cases of a more general theory (relativity), which explained both the (previously unexplained) exceptions to Newton's laws and predicted and explained other observations such as the deflection of light by gravity. Thus, in certain cases independent, unconnected, scientific observations can be connected to each other , unified by principles of increasing explanatory power. The scientific method is not a single recipe: it requires intelligence, imagination, and creativity."To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advance in science." – . In this sense, it is not a mindless set of standards and procedures to follow, but is rather an ongoing cycle, constantly developing more useful, accurate  and comprehensive models and methods. For example, when Einstein developed the Special and General Theories of Relativity, he did not in any way refute or discount Newton's Principia. On the contrary, if the astronomically massive, the feather-light, and the extremely fast are removed from Einstein's theories – all phenomena Newton could not have observed – Newton's equations are what remain. Einstein's theories are expansions and refinements of Newton's theories and, thus, increase confidence in Newton's work. Define a question Gather information and resources (observe) Form an explanatory hypothesis Test the hypothesis by performing an experiment and collecting data in a reproducible manner Analyze the data Interpret the data and draw conclusions that serve as a starting point for  new hypothesis Publish results Retest (frequently done by other scientists) While this schema outlines a typical hypothesis/testing method,See, e.g., , esp. chapters 5–8 a number of philosophers, historians, and sociologists of science, including Paul Feyerabend, claim that such descriptions of scientific method have little relation to the ways that science is actually practiced. Definition Measurements demand the use of operational definitions of relevant quantities. That is, a scientific quantity is described or defined by how it is measured, as opposed to some more vague, inexact  or "idealized" definition. For example, electric current, measured in amperes, may be operationally defined in terms of the mass of silver deposited in a certain time on an electrode in an electrochemical device that is described in some detail. The operational definition of a thing often relies on comparisons with standards: the operational definition of "mass" ultimately relies on the use of an artifact, such as a particular kilogram of platinum-iridium kept in a laboratory in France. The characterization element can require extended and extensive study, even centuries. It took thousands of years of measurements, from the Chaldean, Indian, Persian, Greek, Arabic  and European astronomers, to fully record the motion of planet Earth. Newton was able to include those measurements into  consequences of his laws of motion. But the perihelion of the planet Mercury's orbit exhibits a precession that cannot be fully explained by Newton's laws of motion (see diagram to the right), as Leverrier pointed out in 1859. The observed difference for Mercury's precession between Newtonian theory and observation was one of the things that occurred to Albert Einstein as a possible early test of his theory of General relativity. His relativistic calculations matched observation much more closely than did Newtonian theory. The difference is approximately 43 arc-seconds per century. In general scientists tend to look for theories that are "elegant" or "beautiful". Scientists often use these terms to refer to a theory that is in accordance with the known facts , but is nevertheless relatively simple and easy to handle. Occam's Razor serves as a rule of thumb for choosing the most desirable amongst a group of equally explanatory hypotheses. If the predictions are not accessible by observation or experience, the hypothesis is not yet testable and so will remain to that extent unscientific in a strict sense. A new technology or theory might make the necessary experiments feasible. For example, while a hypothesis on the existence of other intelligent species may be convincing with scientifically based speculation, there is no known experiment that can test this hypothesis. Therefore, science itself can have little to say about the possibility. In the future, a new technique may allow for an experimental test and the speculation would then become part of accepted science. Scientists assume an attitude of openness and accountability on the part of those conducting an experiment . Detailed record-keeping is essential, to aid in recording and reporting on the experimental results, and supports the effectiveness and integrity of the procedure. They will also assist in reproducing the experimental results, likely by others. Traces of this approach can be seen in the work of Hipparchus (190–120 BCE), when determining a value for the precession of the Earth, while controlled experiments can be seen in the works of al-Battanial-Battani, De Motu Stellarum translation from Arabic to Latin in 1116, as cited by "Battani, al-" (c. 858–929) Encyclopædia Britannica, 15th. ed. Al-Battani is known for his accurate observations at al-Raqqah in Syria, beginning in 877. His work includes  measurement of the annual precession of the equinoxes. (853–929) and Alhazen (965–1039). Books I-III (2001 — 91(4)) Vol 1 Commentary and Latin text via JSTOR; — 91(5) Vol 2 English translation, Book I:TOCpp.339-341, Book II:TOCpp.415-6, Book III:TOCpp.559-560, Notes 681ff, Bibl. via JSTOR DNA-experiments link=|alt= Watson and Crick showed an initial (and incorrect) proposal for the structure of DNA to a team from Kings College – Rosalind Franklin, Maurice Wilkins, and Raymond Gosling. Franklin immediately spotted the flaws which concerned the water content. Later Watson saw Franklin's detailed X-ray diffraction images which showed an X-shapePBS WBGH, NOVA: the Secret of Photo 51 X-shape and was able to confirm the structure was helical. This rekindled Watson and Crick's model building and led to the correct structure. ..1. DNA-characterizations DNA-iterations link=|alt= After considerable fruitless experimentation, being discouraged by their superior from continuing, and numerous false starts, p. 53: The weekend (January 31 – February 1) after seeing photo 51, Watson informed Bragg of the X-ray diffraction image of DNA in B form. Bragg gave them permission to restart their research on DNA (that is, model building). p. 54: On Sunday, February 8, 1953, Maurice Wilkes gave Watson and Crick permission to work on models, as Wilkes would not be building models until Franklin left DNA research. p. 56: Jerry Donohue, on sabbatical from Pauling's lab and visiting Cambridge, advises Watson that textbook form of the base pairs was incorrect for DNA base pairs; rather, the keto form of the base pairs should be used instead. This form allowed the bases' hydrogen bonds to pair 'unlike' with 'unlike', rather than to pair 'like' with 'like', as Watson was inclined to model, on the basis of the textbook statements. On February 27, 1953, Watson was convinced enough to make cardboard models of the nucleotides in their keto form. Watson and Crick were able to infer the essential structure of DNA by concrete modeling of the physical shapes of the nucleotides which comprise it."Suddenly I became aware that an adenine-thymine pair held together by two hydrogen bonds was identical in shape to a guanine-cytosine pair held together by at least two hydrogen bonds. ..." – . p. 57 Saturday, February 28, 1953, Watson tried 'like with like' and admitted these base pairs didn't have hydrogen bonds that line up. But after trying 'unlike with unlike', and getting Jerry Donohue's approval, the base pairs turned out to be identical in shape (as Watson stated above in his 1968 Double Helix memoir quoted above). Watson now felt confident enough to inform Crick. (Of course, 'unlike with unlike' increases the number of possible codons, if this scheme were a genetic code.) They were guided by the bond lengths which had been deduced by Linus Pauling and by Rosalind Franklin's X-ray diffraction images. ..DNA Example To protect against bad science and fraudulent data, government research-granting agencies such as the National Science Foundation, and science journals, including Nature and Science, have a policy that researchers must archive their data and methods so that other researchers can test the data and methods and build on the research that has gone before. Scientific data archiving can be done at a number of national archives in the U.S. or in the World Data Center. Hypothetico-deductive model The hypothetico-deductive model or method is a proposed description of  scientific method. Here, predictions from the hypothesis are central: if you assume the hypothesis to be true, what consequences follow? If  subsequent empirical investigation does not demonstrate that these consequences or predictions correspond to the observable world, the hypothesis can be concluded to be false. In 1877,. Charles Sanders Peirce (1839–1914) characterized inquiry in general not as the pursuit of truth per se but as the struggle to move from irritating, inhibitory doubts born of surprises, disagreements, and the like, and to reach a secure belief,  belief being that on which one is prepared to act. He framed scientific inquiry as part of a broader spectrum and as spurred, like inquiry generally, by actual doubt, not mere verbal or hyperbolic doubt, which he held to be fruitless."What one does not in the least doubt one should not pretend to doubt; but a man should train himself to doubt," said Peirce in a brief intellectual autobiography; see Ketner, Kenneth Laine (2009) "Charles Sanders Peirce: Interdisciplinary Scientist" in The Logic of Interdisciplinarity). Peirce held that actual, genuine doubt originates externally, usually in surprise, but also that it is to be sought and cultivated, "provided only that it be the weighty and noble metal itself, and no counterfeit nor paper substitute"; in "Issues of Pragmaticism", The Monist, v. XV, n. 4, pp. 481–99, see p. 484, and p. 491. (Reprinted in Collected Papers v. 5, paragraphs 438–63, see 443 and 451). He outlined four methods of settling opinion, ordered from least to most successful: The method of tenacity (policy of sticking to initial belief) – which brings comforts and decisiveness but leads to trying to ignore contrary information and others' views as if truth were intrinsically private, not public. It goes against the social impulse and easily falters since one may well notice when another's opinion is as good as one's own initial opinion. Its successes can shine but tend to be transitory.But see Scientific method and religion. The method of authority – which overcomes disagreements but sometimes brutally. Its successes can be majestic and long-lived, but it cannot operate thoroughly enough to suppress doubts indefinitely, especially when people learn of other societies  present and past. The method of the a priori – which promotes conformity less brutally but fosters opinions as something like tastes, arising in conversation and comparisons of perspectives in terms of "what is agreeable to reason." Thereby it depends on fashion in paradigms and goes in circles over time. It is more intellectual and respectable but, like the first two methods, sustains accidental and capricious beliefs, destining some minds to doubt it. The scientific method – the method wherein inquiry regards itself as fallible and purposely tests itself and criticizes, corrects, and improves itself. Peirce held that slow, stumbling ratiocination can be dangerously inferior to instinct and traditional sentiment in practical matters, and that the scientific method is best suited to theoretical research,Peirce (1898), "Philosophy and the Conduct of Life", Lecture 1 of the Cambridge (MA) Conferences Lectures, published in Collected Papers v. 1, paragraphs 616–48 in part and in Reasoning and the Logic of Things, Ketner (ed., intro.) and Putnam (intro., comm.), pp. 105–22, reprinted in Essential Peirce v. 2, pp. 27–41. which in turn should not be trammeled by the other methods and practical ends; reason's "first rule" is that, in order to learn, one must desire to learn and, as a corollary, must not block the way of inquiry."... in order to learn, one must desire to learn ..." – Peirce (1899), "F.R.L." [First Rule of Logic], Collected Papers v. 1, paragraphs 135–40, The scientific method excels the others by being deliberately designed to arrive – eventually – at the most secure beliefs, upon which the most successful practices can be based. Starting from the idea that people seek not truth per se but instead to subdue irritating, inhibitory doubt, Peirce showed how, through the struggle, some can come to submit to  truth for the sake of belief's integrity, seek as truth the guidance of potential practice correctly to its given goal, and wed themselves to the scientific method.Peirce, C.S., Collected Papers v. 5, in paragraph 582, from 1898: For Peirce, rational inquiry implies presuppositions about truth and the real; to reason is to presuppose (and at least to hope), as a principle of the reasoner's self-regulation, that the real is discoverable and independent of our vagaries of opinion. In that vein  he defined truth as the correspondence of a sign (in particular, a proposition) to its object and, pragmatically, not as  actual consensus of some definite, finite community (such that to inquire would be to poll the experts), but instead as that final opinion which all investigators would reach sooner or later but still inevitably, if they were to push investigation far enough, even when they start from different points. In tandem he defined the real as a true sign's object (be that object a possibility or quality, or an actuality or brute fact, or a necessity or norm or law), which is what it is independently of any finite community's opinion and, pragmatically, depends only on the final opinion destined in a sufficient investigation. That is a destination as far, or near, as the truth itself to you or me or the given finite community. Thus, his theory of inquiry boils down to "Do the science." Those conceptions of truth and the real involve the idea of a community both without definite limits (and thus potentially self-correcting as far as needed) and capable of definite increase of knowledge.Peirce (1868), "Some Consequences of Four Incapacities", Journal of Speculative Philosophy v. 2, n. 3, pp. 140–57. Reprinted Collected Papers v. 5, paragraphs 264–317, The Essential Peirce v. 1, pp. 28–55, and elsewhere. Arisbe Eprint As inference, "logic is rooted in the social principle" since it depends on a standpoint that is, in a sense, unlimited.Peirce (1878), "The Doctrine of Chances", Popular Science Monthly v. 12, pp. 604–15, see pp. 610–11 via Internet Archive. Reprinted Collected Papers v. 2, paragraphs 645–68, Essential Peirce v. 1, pp. 142–54. "... death makes the number of our risks, the number of our inferences, finite, and so makes their mean result uncertain. The very idea of probability and of reasoning rests on the assumption that this number is indefinitely great. ... logicality inexorably requires that our interests shall not be limited. ... Logic is rooted in the social principle." Paying special attention to the generation of explanations, Peirce outlined the scientific method as a coordination of three kinds of inference in a purposeful cycle aimed at settling doubts, as follows (in §III–IV in "A Neglected Argument" with added notes. Reprinted with previously unpublished part, Collected Papers v. 6, paragraphs 452–85, The Essential Peirce v. 2, pp. 434–50, and elsewhere. except as otherwise noted): Abduction (or retroduction). Guessing, inference to explanatory hypotheses for selection of those best worth trying. From abduction, Peirce distinguishes induction as inferring, on the basis of tests, the proportion of truth in the hypothesis. Every inquiry, whether into ideas, brute facts, or norms and laws, arises from surprising observations in one or more of those realms (and for example at any stage of an inquiry already underway). All explanatory content of theories comes from abduction, which guesses a new or outside idea so as to account in a simple, economical way for a surprising or complicative phenomenon. Oftenest, even a well-prepared mind guesses wrong. But the modicum of success of our guesses far exceeds that of sheer luck and seems born of attunement to nature by instincts developed or inherent, especially insofar as best guesses are optimally plausible and simple in the sense, said Peirce, of the "facile and natural", as by Galileo's natural light of reason and as distinct from "logical simplicity". Abduction is the most fertile but least secure mode of inference. Its general rationale is inductive: it succeeds often enough and, without it, there is no hope of sufficiently expediting inquiry (often multi-generational) toward new truths.Peirce (c. 1906), "PAP (Prolegomena for an Apology to Pragmatism)" (Manuscript 293, not the like-named article), The New Elements of Mathematics (NEM) 4:319–20, see first quote under "Abduction" at Commens Dictionary of Peirce's Terms. Coordinative method leads from abducing a plausible hypothesis to judging it for its testabilityPeirce, Carnegie application (L75, 1902), New Elements of Mathematics v. 4, pp. 37–38: and for how its trial would economize inquiry itself.Peirce (1902), Carnegie application, see MS L75.329330, from Draft D of Memoir 27: Peirce calls his pragmatism "the logic of abduction".Peirce (1903), "Pragmatism – The Logic of Abduction", Collected Papers v. 5, paragraphs 195–205, especially 196. Eprint. His pragmatic maxim is: "Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object". His pragmatism is a method of reducing conceptual confusions fruitfully by equating the meaning of any conception with the conceivable practical implications of its object's conceived effects – a method of experimentational mental reflection hospitable to forming hypotheses and conducive to testing them. It favors efficiency. The hypothesis, being insecure, needs to have practical implications leading at least to mental tests and, in science, lending themselves to scientific tests. A simple but unlikely guess, if uncostly to test for falsity, may belong first in line for testing. A guess is intrinsically worth testing if it has instinctive plausibility or reasoned objective probability, while subjective likelihood, though reasoned, can be misleadingly seductive. Guesses can be chosen for trial strategically, for their caution (for which Peirce gave as an example the game of Twenty Questions), breadth, and incomplexity.Peirce, "On the Logic of Drawing Ancient History from Documents", Essential Peirce v. 2, see pp. 107–09. On Twenty Questions, p. 109: One can hope to discover only that which time would reveal through a learner's sufficient experience anyway, so the point is to expedite it; the economy of research is what demands the leap, so to speak, of abduction and governs its art. Induction. The long-run validity of the rule of induction is deducible from the principle (presuppositional to reasoning in general ) that the real is only the object of the final opinion to which adequate investigation would lead;Peirce (1878), "The Probability of Induction", Popular Science Monthly, v. 12, pp. 705–18, see 718 Google Books; 718 via Internet Archive. Reprinted often, including (Collected Papers v. 2, paragraphs 669–93), (The Essential Peirce v. 1, pp. 155–69). anything to which no such process would ever lead would not be real. Induction involving ongoing tests or observations follows a method which, sufficiently persisted in, will diminish its error below any predesignate degree. Three stages: Classification. Unclearly premised, but inductive, classing of objects of experience under general ideas. Probation: direct inductive argumentation. Crude (the enumeration of instances) or gradual (new estimate of  proportion of truth in the hypothesis after each test). Gradual induction is qualitative or quantitative; if qualitative, then dependent on weightings of qualities or characters;Peirce (1905 draft "G" of "A Neglected Argument"), "Crude, Quantitative, and Qualitative Induction", Collected Papers v. 2, paragraphs 755–60, see 759. Find under "Induction" at Commens Dictionary of Peirce's Terms. if quantitative, then dependent on measurements, or on statistics, or on countings. Sentential Induction. "... which, by inductive reasonings, appraises the different probations singly, then their combinations, then makes self-appraisal of these very appraisals themselves, and passes final judgment on the whole result". Invariant explanation Model of DNA with David Deutsch, proponent of invariant scientific explanations (2009) Peer review evaluation Scientific journals use a process of peer review, in which scientists' manuscripts are submitted by editors of scientific journals to (usually one to three, and usually anonymous) fellow scientists familiar with the field for evaluation. In certain journals, the journal itself selects the referees; while in others (especially journals that are extremely specialized), the manuscript author might recommend referees. The referees may or may not recommend publication, or they might recommend publication with suggested modifications, or sometimes, publication in another journal. This standard is practiced to various degrees by different journals , and can have the effect of keeping the literature free of obvious errors and to generally improve the quality of the material, especially in the journals who use the standard most rigorously. The peer-review process can have limitations when considering research outside the conventional scientific paradigm: problems of "groupthink" can interfere with open and fair deliberation of some new research.Brown, C. (2005) Overcoming Barriers to Use of Promising Research Among Elite Middle East Policy Groups, Journal of Social Behaviour and Personality, Select Press. Sometimes experimenters may make systematic errors during their experiments, veer from standard methods and practices (Pathological science) for various reasons, or, in rare cases, deliberately report false results. Occasionally because of this then, other scientists might attempt to repeat the experiments in order to duplicate the results. Archiving Researchers sometimes practice scientific data archiving, such as in compliance with the policies of government funding agencies and scientific journals. In these cases, detailed records of their experimental procedures, raw data, statistical analyses  and source code can be preserved in order to provide evidence of the methodology and practice of the procedure and assist in any potential future attempts to reproduce the result. These procedural records may also assist in the conception of new experiments to test the hypothesis, and may prove useful to engineers who might examine the potential practical applications of a discovery. Limitations Since it is impossible for a scientist to record everything that took place in an experiment, facts selected for their apparent relevance are reported. This may lead, unavoidably, to problems later if some supposedly irrelevant feature is questioned. For example, Heinrich Hertz did not report the size of the room used to test Maxwell's equations, which later turned out to account for a small deviation in the results. The problem is that parts of the theory itself need to be assumed in order to select and report the experimental conditions. The observations are hence sometimes described as being 'theory-laden'. Norwood Russell Hanson, Imre Lakatos and Thomas Kuhn have done extensive work on the "theory-laden" character of observation. Hanson (1958) first coined the term for the idea that all observation is dependent on the conceptual framework of the observer, using the concept of gestalt to show how preconceptions can affect both observation and description. He opens Chapter 1 with a discussion of the Golgi bodies and their initial rejection as an artefact of staining technique, and a discussion of Brahe and Kepler observing the dawn and seeing a "different" sun rise despite the same physiological phenomenon. Kuhn and FeyerabendFeyerabend, Paul K (1960) "Patterns of Discovery" The Philosophical Review (1960) vol. 69 (2) pp. 247–52 acknowledge the pioneering significance of his work. Kuhn (1961) said the scientist generally has a theory in mind before designing and undertaking experiments so as to make empirical observations, and that the "route from theory to measurement can almost never be traveled backward". This implies that the way in which theory is tested is dictated by the nature of the theory itself, which led Kuhn (1961, p. 166) to argue that "once it has been adopted by a profession ... no theory is recognized to be testable by any quantitative tests that it has not already passed".Kuhn, Thomas S., "The Function of Measurement in Modern Physical Science", ISIS 52(2), 161–93, 1961. Post-modernism and science wars Paul Feyerabend similarly examined the history of science, and was led to deny that science is genuinely a methodological process. In his book Against Method he argues that scientific progress is not the result of applying any particular method. In essence, he says that for any specific method or norm of science, one can find a historic episode where violating it has contributed to the progress of science. Thus, if believers in  scientific method wish to express a single universally valid rule, Feyerabend jokingly suggests, it should be 'anything goes'.Feyerabend, Paul K., Against Method, Outline of an Anarchistic Theory of Knowledge, 1st published, 1975. Reprinted, Verso, London, 1978. Criticisms such as his led to the strong programme, a radical approach to the sociology of science. Psychologist Kevin Dunbar says the process of discovery often starts with researchers finding bugs in their experiments. These unexpected results lead researchers to try to fix what they think is an error in their method. Eventually, the researcher decides the error is too persistent and systematic to be a coincidence. The highly controlled, cautious  and curious aspects of the scientific method are thus what make it well suited for identifying such persistent systematic errors. At this point, the researcher will begin to think of theoretical explanations for the error, often seeking the help of colleagues across different domains of expertise. Nevertheless, the connection between mathematics and reality (and so science to the extent it describes reality) remains obscure. Eugene Wigner's paper, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, is a very well known account of the issue from a Nobel Prize-winning physicist. In fact, some observers (including some well-known mathematicians such as Gregory Chaitin, and others such as Lakoff and Núñez) have suggested that mathematics is the result of practitioner bias and human limitation (including cultural ones), somewhat like the post-modernist view of science. Imre Lakatos argued that mathematicians actually use contradiction, criticism  and revision as principles for improving their work.See the development, by generations of mathematicians, of Euler's formula for polyhedra as documented by In like manner to science, where truth is sought, but certainty is not found, in Proofs and refutations (1976), what Lakatos tried to establish was that no theorem of informal mathematics is final or perfect. This means that we should not think that a theorem is ultimately true, only that no counterexample has yet been found. Once a counterexample, i.e. an entity contradicting/not explained by the theorem is found, we adjust the theorem, possibly extending the domain of its validity. This is a continuous way our knowledge accumulates, through the logic and process of proofs and refutations. (If axioms are given for a branch of mathematics, however, Lakatos claimed that proofs from those axioms were tautological, i.e. logically true, by rewriting them, as did Poincaré (Proofs and Refutations, 1976).) The particular points raised are statistical ("The smaller the studies conducted in a scientific field, the less likely the research findings are to be true" and "The greater the flexibility in designs, definitions, outcomes, and analytical modes in a scientific field, the less likely the research findings are to be true.") and economical ("The greater the financial and other interests and prejudices in a scientific field, the less likely the research findings are to be true" and "The hotter a scientific field (with more scientific teams involved), the less likely the research findings are to be true.") Hence: "Most research findings are false for most research designs and for most fields" and "As shown, the majority of modern biomedical research is operating in areas with very low pre- and poststudy probability for true findings." However: "Nevertheless, most new discoveries will continue to stem from hypothesis-generating research with low or very low pre-study odds," which means that *new* discoveries will come from research that, when that research started, had low or very low odds (a low or very low chance) of succeeding. Hence, if the scientific method is used to expand the frontiers of knowledge, research into areas that are outside the mainstream will yield most new discoveries. Bauer, Henry H., Scientific Literacy and the Myth of the Scientific Method, University of Illinois Press, Champaign, IL, 1992 Beveridge, William I.B., The Art of Scientific Investigation, Heinemann, Melbourne, Australia, 1950. Bernstein, Richard J., Beyond Objectivism and Relativism: Science, Hermeneutics, and Praxis, University of Pennsylvania Press, Philadelphia, PA, 1983. Brody, Baruch A. and Capaldi, Nicholas, Science: Men, Methods, Goals: A Reader: Methods of Physical Science, W.A. Benjamin, 1968 Brody, Baruch A., and Grandy, Richard E., Readings in the Philosophy of Science, 2nd edition, Prentice Hall , Englewood Cliffs, NJ, 1989. Burks, Arthur W., Chance, Cause, Reason: An Inquiry into the Nature of Scientific Evidence, University of Chicago Press, Chicago, IL, 1977. Chalmers, Alan, What Is This Thing Called Science?. Queensland University Press and Open University Press, 1976. Cowles, Henry M., The Scientific Method: An Evolution of Thinking from Darwin to Dewey, Harvard University Press, Cambridge, MA, 2020. Reviewed in: Riskin, Jessica, "Just Use Your Thinking Pump!", The New York Review of Books, vol. LXVII, no. 11 (2 July 2020), pp. 48–50. .
The term "scientific method" came into popular use in the twentieth century, popping up in dictionaries and science textbooks, although there was little scientific consensus over its meaning. Although there was growth through the middle of the twentieth century, by the 1960s and 1970s numerous influential philosophers of science such as Thomas Kuhn and Paul Feyerabend had questioned the universality of the "scientific method" and in doing so largely replaced the notion of science as a homogeneous and universal method with that of it being a heterogeneous and local practice. In particular, Paul Feyerabend, in the 1975 first edition of his book Against Method, argued against there being any universal rules of science. Later examples include physicist Lee Smolin's 2013 essay "There Is No Scientific Method" and historian of science Daniel Thurs's chapter in the 2015 book Newton's Apple and Other Myths about Science, which concluded that the scientific method is a myth or, at best, an idealization. Philosophers Robert Nola and Howard Sankey, in their 2007 book Theories of Scientific Method, said that debates over scientific method continue, and argued that Feyerabend, despite the title of Against Method, accepted certain rules of method and attempted to justify those rules with a meta methodology . The scientific method is the process by which science is carried out.: "The thesis of this book, as outlined in Chapter One, is that there are general principles applicable to all the sciences." As in other areas of inquiry, science (through the scientific method) can build on previous knowledge and develop a more sophisticated understanding of its topics of study over time.: The scientific method can function in the same way; This is the principle of noncontradiction.Francis Bacon(1629) New Organon, lists 4 types of error: Idols of the tribe (error due to the entire human race), the cave (errors due to an individual's own intellect), the marketplace (errors due to false words), and the theater (errors due to incredulous acceptance).For example, the concept of falsification (first proposed in 1934) formalizes the attempt to disprove hypotheses rather than prove them. Karl R. Popper (1963), 'The Logic of Scientific Discovery'. The Logic of Scientific Discovery pp. 17–20, 249–52, 437–38, and elsewhere. Leon Lederman, for teaching physics first, illustrates how to avoid confirmation bias: Ian Shelton, in Chile, was initially skeptical that supernova 1987a was real, but possibly an artifact of instrumentation (null hypothesis), so he went outside and disproved his null hypothesis by observing SN 1987a with the naked eye. The Kamiokande experiment, in Japan, independently observed neutrinos from SN 1987a at the same time. This model can be seen to underlie the scientific revolution.: "There is a danger that must be avoided. ... If we wish to do justice to the historical enterprise, we must take the past for what it was. And that means we must resist the temptation to scour the past for examples or precursors of modern science. ...My concern will be with the beginnings of scientific theories, the methods by which they were formulated, and the uses to which they were put; ... " The ubiquitous element in the scientific method is empiricism. This is in opposition to stringent forms of rationalism: the scientific method embodies that reason alone cannot solve a particular scientific problem. A strong formulation of the scientific method is not always aligned with a form of empiricism in which the empirical data is put forward in the form of experience or other abstracted forms of knowledge; in current scientific practice, however, the use of scientific modelling and reliance on abstract typologies and theories is normally accepted. The scientific method is of necessity also an expression of an opposition to claims that e.g. revelation, political or religious dogma, appeals to tradition, commonly held beliefs, common sense, or, importantly, currently held theories, are the only possible means of demonstrating truth. Different early expressions of empiricism and the scientific method can be found throughout history, for instance with the ancient Stoics, Epicurus,Elizabeth Asmis (1985) Epicurus' Scientific Method. Cornell University Press Alhazen,Alhazen argued the importance of forming questions and subsequently testing them: "How does light travel through transparent bodies? Light travels through transparent bodies in straight lines only... We have explained this exhaustively in our Book of Optics. But let us now mention something to prove this convincingly: the fact that light travels in straight lines is clearly observed in the lights which enter into dark rooms through holes.... [T]he entering light will be clearly observable in the dust which fills the air. – Alhazen, Treatise on Light (رسالة في الضوء), translated into English from German by M. Schwarz, from "Abhandlung über das Licht", J. Baarmann (editor and translator from Arabic to German, 1882) Zeitschrift der Deutschen Morgenländischen Gesellschaft Vol 36 as quoted in . He demonstrated his conjecture that "light travels through transparent bodies in straight lines only" by placing a straight stick or a taut thread next to the light beam, as quoted in to prove that light travels in a straight line. David Hockney, (2001, 2006) in Secret Knowledge: rediscovering the lost techniques of the old masters (expanded edition) cites Alhazen several times as the likely source for the portraiture technique using the camera obscura, which Hockney rediscovered with the aid of an optical suggestion from Charles M. Falco. Kitab al-Manazir, which is Alhazen's Book of Optics, at that time denoted Opticae Thesaurus, Alhazen Arabis, was translated from Arabic into Latin for European use as early as 1270. Hockney cites Friedrich Risner's 1572 Basle edition of Opticae Thesaurus. Hockney quotes Alhazen as the first clear description of the camera obscura in Hockney, p. 240. "Truth is sought for its own sake. And those who are engaged upon the quest for anything for its own sake are not interested in other things. Finding the truth is difficult, and the road to it is rough." – Alhazen (Ibn Al-Haytham 965 – c. 1040) Critique of Ptolemy, translated by S. Pines, Actes X Congrès internationale d'histoire des sciences, Vol I Ithaca 1962, as quoted in . (This quotation is from Alhazen's critique of Ptolemy's books Almagest, Planetary Hypotheses, and Optics as translated into English by A. Mark Smith.) Roger Bacon, and William of Ockham. From the 16th century onwards, experiments were advocated by Francis Bacon, and performed by Giambattista della Porta, Johannes Kepler,Kepler, Johannes (1604) Ad Vitellionem paralipomena, quibus astronomiae pars opticae traditur (Supplements to Witelo, in which the optical part of astronomy is treated) as cited in The full title translation is from p. 60 of James R. Voelkel (2001) Johannes Kepler and the New Astronomy Oxford University Press. Kepler was driven to this experiment after observing the partial solar eclipse at Graz, July 10, 1600. He used Tycho Brahe's method of observation, which was to project the image of the Sun on a piece of paper through a pinhole aperture, instead of looking directly at the Sun. He disagreed with Brahe's conclusion that total eclipses of the Sun were impossible because there were historical accounts of total eclipses. Instead , he deduced that the size of the aperture controls the sharpness of the projected image (the larger the aperture, the more accurate the image – this fact is now fundamental for optical system design). Voelkel, p. 61, notes that Kepler's experiments produced the first correct account of vision and the eye, because he realized he could not accurately write about astronomical observation by ignoring the eye. and Galileo Galilei....an approach which was advocated by Galileo in 1638 with the publication of Two New Sciences., Dover reprint of the 1914 Macmillan translation by Henry Crew and Alfonso de Salvio of Two New Sciences, Galileo Galilei Linceo (1638). Additional publication information is from the collection of first editions of the Library of Congress surveyed by . There was particular development aided by theoretical works by Francisco Sanches, John Locke, George Berkeley, and David Hume. Prediction This step involves determining the logical consequences of the hypothesis. One or more predictions are then selected for further testing. The more unlikely that a prediction would be correct simply by coincidence, then the more convincing it would be if the prediction were fulfilled; the evidence is also stronger if the answer to the prediction is not already known, due to the effects of hindsight bias (see also postdiction). Ideally, the prediction must also distinguish the hypothesis from likely alternatives; if two hypotheses make the same prediction, observing the prediction to be correct is not evidence for either one over the other. (These statements about the relative strength of evidence can be mathematically derived using Bayes' Theorem).Note: for a discussion of multiple hypotheses, see Bayesian inference#Informal Testing This is an investigation of whether the real world behaves as predicted by the hypothesis. Scientists (and other people) test hypotheses by conducting experiments. The purpose of an experiment is to determine whether observations of the real world agree with or conflict with the predictions derived from a hypothesis. If they agree, confidence in the hypothesis increases; otherwise, it decreases. The agreement does not assure that the hypothesis is true; future experiments may reveal problems. Karl Popper advised scientists to try to falsify hypotheses, i.e., to search for and test those experiments that seem most doubtful. Large numbers of successful confirmations are not convincing if they arise from experiments that avoid risk. Experiments should be designed to minimize possible errors, especially through the use of appropriate scientific controls. For example, tests of medical treatments are commonly run as double-blind tests. Test personnel, who might unwittingly reveal to test subjects which samples are the desired test drugs and which are placebos, are kept ignorant of which are which. Such hints can bias the responses of the test subjects. Furthermore, failure of an experiment does not necessarily mean the hypothesis is false. Experiments always depend on several hypotheses, e.g., that the test equipment is working properly, and a failure may be a failure of one of the auxiliary hypotheses. (See the Duhem–Quine thesis.) Experiments can be conducted in a college lab, on a kitchen table, at CERN's Large Hadron Collider, at the bottom of an ocean, on Mars (using one of the working rovers), and so on. Astronomers do experiments, searching for planets around distant stars. Finally, most individual experiments address highly specific topics for reasons of practicality. As a result, evidence about broader topics is usually accumulated gradually. Analysis This involves determining what the results of the experiment show and deciding on the next actions to take. The predictions of the hypothesis are compared to those of the null hypothesis, to determine which is better able to explain the data. In cases where an experiment is repeated many times, a statistical analysis such as a chi-squared test may be required. If the evidence has falsified the hypothesis, a new hypothesis is required; if the experiment supports the hypothesis but the evidence is not strong enough for high confidence, other predictions from the hypothesis must be tested. Once a hypothesis is strongly supported by evidence, a new question can be asked to provide further insight on the same topic. Evidence from other scientists and experience are frequently incorporated at any stage in the process. Depending on the complexity of the experiment, many iterations may be required to gather sufficient evidence to answer a question with confidence or to build up many answers to highly specific questions to answer a single broader question. DNA example link=|alt= The basic elements of the scientific method are illustrated by the following example from the discovery of the structure of DNA: Question: Previous investigation of DNA had determined its chemical composition (the four nucleotides), the structure of each individual nucleotide, and other properties. X-ray diffraction patterns of DNA by Florence Bell in her Ph.D. thesis (1939) were similar to (although not as good as) "photo 51", but this research was interrupted by the events of World War II. DNA had been identified as the carrier of genetic information by the Avery–MacLeod–McCarty experiment in 1944, but the mechanism of how genetic information was stored in DNA was unclear. Hypothesis: Linus Pauling, Francis Crick and James D. Watson hypothesized that DNA had a helical structure.October 1951, as noted in :"That's what a helix should look like!" Crick exclaimed in delight (This is the Cochran-Crick-Vand-Stokes theory of the transform of a helix). Prediction: If DNA had a helical structure, its X-ray diffraction pattern would be X-shaped.June 1952, as noted in : Watson had succeeded in getting X-ray pictures of TMV showing a diffraction pattern consistent with the transform of a helix.Watson did enough work on Tobacco mosaic virus to produce the diffraction pattern for a helix, per Crick's work on the transform of a helix. pp. 137–38, Horace Freeland Judson (1979) The Eighth Day of Creation This prediction was determined using the mathematics of the helix transform, which had been derived by Cochran, Crick , and Vand– Cochran W, Crick FHC and Vand V. (1952) "The Structure of Synthetic Polypeptides. I. The Transform of Atoms on a Helix", Acta Crystallogr., 5, 581–86. (and independently by Stokes). This prediction was a mathematical construct, completely independent from the biological problem at hand. Experiment: Rosalind Franklin used pure DNA to perform X-ray diffraction to produce photo 51. The results showed an X-shape. Analysis: When Watson saw the detailed diffraction pattern, he immediately recognized it as a helix.Friday, January 30, 1953. Tea time, as noted in : Franklin confronts Watson and his paper – "Of course it [Pauling's pre-print] is wrong. DNA is not a helix." However, Watson then visits Wilkins' office, sees photo 51, and immediately recognizes the diffraction pattern of a helical structure. But additional questions remained, requiring additional iterations of their research. For example, the number of strands in the backbone of the helix (Crick suspected 2 strands, but cautioned Watson to examine that more critically), the location of the base pairs (inside the backbone or outside the backbone), etc. One key point was that they realized that the quickest way to reach a result was not to continue a mathematical analysis, but to build a physical model."The instant I saw the picture my mouth fell open and my pulse began to race." – Page 168 shows the X-shaped pattern of the B-form of DNA, clearly indicating crucial details of its helical structure to Watson and Crick. p. 52 dates the Franklin-Watson confrontation as Friday, January 30, 1953. Later that evening, Watson urges Wilkins to begin model-building immediately. But Wilkins agrees to do so only after Franklin's departure. He and Crick then produced their model, using this information along with the previously known information about DNA's composition, especially Chargaff's rules of base pairing.Saturday, February 28, 1953, as noted in : Watson found the base-pairing mechanism which explained Chargaff's rules using his cardboard models. Replication If an experiment cannot be repeated to produce the same results, this implies that the original results might have been in error. As a result, it is common for a single experiment to be performed multiple times, especially when there are uncontrolled variables or other indications of experimental error. For significant or surprising results, other scientists may also attempt to replicate the results for themselves, especially if those results would be important to their own work. Replication has become a contentious issue in social and biomedical science where treatments are administered to groups of individuals. Typically an experimental group gets the treatment, such as a drug, and the control group gets a placebo. John Ioannidis in 2005 pointed out that the method being used has led to many findings that cannot be replicated. External review The process of peer review involves evaluation of the experiment by experts, who typically give their opinions anonymously. Some journals request that the experimenter provide lists of possible peer reviewers, especially if the field is highly specialized. Peer review does not certify the correctness of the results, only that, in the opinion of the reviewer, the experiments themselves were sound (based on the description supplied by the experimenter). If the work passes peer review, which occasionally may require new experiments requested by the reviewers, it will be published in a peer-reviewed scientific journal. The specific journal that publishes the results indicates the perceived quality of the work.In Two new sciences, there are three 'reviewers': Simplicio, Sagredo, and Salviati, who serve as foil, antagonist, and protagonist. Galileo speaks for himself only briefly. But note that Einstein's 1905 papers were not peer-reviewed before their publication. Theories can also become subsumed by other theories. For example, Newton's laws explained thousands of years of scientific observations of the planets almost perfectly. However, these laws were then determined to be special cases of a more general theory (relativity), which explained both the (previously unexplained) exceptions to Newton's laws and predicted and explained other observations such as the deflection of light by gravity. Thus, in certain cases independent, unconnected, scientific observations can be connected , unified by principles of increasing explanatory power. The scientific method is not a single recipe: it requires intelligence, imagination, and creativity."To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advance in science." – . In this sense, it is not a mindless set of standards and procedures to follow, but is rather an ongoing cycle, constantly developing more useful, accurate , and comprehensive models and methods. For example, when Einstein developed the Special and General Theories of Relativity, he did not in any way refute or discount Newton's Principia. On the contrary, if the astronomically massive, the feather-light, and the extremely fast are removed from Einstein's theories – all phenomena Newton could not have observed – Newton's equations are what remain. Einstein's theories are expansions and refinements of Newton's theories and, thus, increase confidence in Newton's work. Define a question Gather information and resources (observe) Form an explanatory hypothesis Test the hypothesis by performing an experiment and collecting data in a reproducible manner Analyze the data Interpret the data and draw conclusions that serve as a starting point for a new hypothesis Publish results Retest (frequently done by other scientists) While this schema outlines a typical hypothesis/testing method,See, e.g., , esp. chapters 5–8 many philosophers, historians, and sociologists of science, including Paul Feyerabend, claim that such descriptions of scientific method have little relation to the ways that science is actually practiced. Definition Measurements demand the use of operational definitions of relevant quantities. That is, a scientific quantity is described or defined by how it is measured, as opposed to some more vague, inexact , or "idealized" definition. For example, electric current, measured in amperes, may be operationally defined in terms of the mass of silver deposited in a certain time on an electrode in an electrochemical device that is described in some detail. The operational definition of a thing often relies on comparisons with standards: the operational definition of "mass" ultimately relies on the use of an artifact, such as a particular kilogram of platinum-iridium kept in a laboratory in France. The characterization element can require extended and extensive study, even centuries. It took thousands of years of measurements, from the Chaldean, Indian, Persian, Greek, Arabic , and European astronomers, to fully record the motion of planet Earth. Newton was able to include those measurements into the consequences of his laws of motion. But the perihelion of the planet Mercury's orbit exhibits a precession that cannot be fully explained by Newton's laws of motion (see diagram to the right), as Leverrier pointed out in 1859. The observed difference for Mercury's precession between Newtonian theory and observation was one of the things that occurred to Albert Einstein as a possible early test of his theory of General relativity. His relativistic calculations matched observation much more closely than did Newtonian theory. The difference is approximately 43 arc-seconds per century. In general scientists tend to look for theories that are "elegant" or "beautiful". Scientists often use these terms to refer to a theory that is following the known facts but is nevertheless relatively simple and easy to handle. Occam's Razor serves as a rule of thumb for choosing the most desirable amongst a group of equally explanatory hypotheses. If the predictions are not accessible by observation or experience, the hypothesis is not yet testable and so will remain to that extent unscientific in a strict sense. A new technology or theory might make the necessary experiments feasible. For example, while a hypothesis on the existence of other intelligent species may be convincing with scientifically based speculation, no known experiment can test this hypothesis. Therefore, science itself can have little to say about the possibility. In the future, a new technique may allow for an experimental test and the speculation would then become part of accepted science. Scientists assume an attitude of openness and accountability on the part of those experimenting . Detailed record-keeping is essential, to aid in recording and reporting on the experimental results, and supports the effectiveness and integrity of the procedure. They will also assist in reproducing the experimental results, likely by others. Traces of this approach can be seen in the work of Hipparchus (190–120 BCE), when determining a value for the precession of the Earth, while controlled experiments can be seen in the works of al-Battanial-Battani, De Motu Stellarum translation from Arabic to Latin in 1116, as cited by "Battani, al-" (c. 858–929) Encyclopædia Britannica, 15th. ed. Al-Battani is known for his accurate observations at al-Raqqah in Syria, beginning in 877. His work includes the measurement of the annual precession of the equinoxes. (853–929) and Alhazen (965–1039). Books I-III (2001 — 91(4)) Vol 1 Commentary and Latin text via JSTOR; — 91(5) Vol 2 English translation, Book I:TOCpp.339-341, Book II:TOCpp.415-6, Book III:TOCpp.559-560, Notes 681ff, Bibl. via JSTOR DNA-experiments link=|alt= Watson and Crick showed an initial (and incorrect) proposal for the structure of DNA to a team from Kings College – Rosalind Franklin, Maurice Wilkins, and Raymond Gosling. Franklin immediately spotted the flaws which concerned the water content. Later Watson saw Franklin's detailed X-ray diffraction images which showed an X-shapePBS WBGH, NOVA: the Secret of Photo 51 X-shape and was able to confirm the structure was helical. This rekindled Watson and Crick's model building and led to the correct structure. ..1. DNA-characterizations DNA-iterations link=|alt= After considerable fruitless experimentation, being discouraged by their superior from continuing, and numerous false starts, p. 53: The weekend (January 31 – February 1) after seeing photo 51, Watson informed Bragg of the X-ray diffraction image of DNA in B form. Bragg permitted them to restart their research on DNA (that is, model building). p. 54: On Sunday, February 8, 1953, Maurice Wilkes gave Watson and Crick permission to work on models, as Wilkes would not be building models until Franklin left DNA research. p. 56: Jerry Donohue, on sabbatical from Pauling's lab and visiting Cambridge, advises Watson that textbook form of the base pairs was incorrect for DNA base pairs; rather, the keto form of the base pairs should be used instead. This form allowed the bases' hydrogen bonds to pair 'unlike' with 'unlike', rather than to pair 'like' with 'like', as Watson was inclined to model, based on the textbook statements. On February 27, 1953, Watson was convinced enough to make cardboard models of the nucleotides in their keto form. Watson and Crick were able to infer the essential structure of DNA by concrete modeling of the physical shapes of the nucleotides which comprise it."Suddenly I became aware that an adenine-thymine pair held together by two hydrogen bonds was identical in shape to a guanine-cytosine pair held together by at least two hydrogen bonds. ..." – . p. 57 Saturday, February 28, 1953, Watson tried 'like with like' and admitted these base pairs didn't have hydrogen bonds that line up. But after trying 'unlike with unlike', and getting Jerry Donohue's approval, the base pairs turned out to be identical in shape (as Watson stated above in his 1968 Double Helix memoir quoted above). Watson now felt confident enough to inform Crick. (Of course, 'unlike with unlike' increases the number of possible codons, if this scheme were a genetic code.) They were guided by the bond lengths which had been deduced by Linus Pauling and by Rosalind Franklin's X-ray diffraction images. ..DNA Example To protect against bad science and fraudulent data, government research-granting agencies such as the National Science Foundation, and science journals, including Nature and Science, have a policy that researchers must archive their data and methods so that other researchers can test the data and methods and build on the research that has gone before. Scientific data archiving can be done at several national archives in the U.S. or the World Data Center. Hypothetico-deductive model The hypothetico-deductive model or method is a proposed description of the scientific method. Here, predictions from the hypothesis are central: if you assume the hypothesis to be true, what consequences follow? If a subsequent empirical investigation does not demonstrate that these consequences or predictions correspond to the observable world, the hypothesis can be concluded to be false. In 1877,. Charles Sanders Peirce (1839–1914) characterized inquiry in general not as the pursuit of truth per se but as the struggle to move from irritating, inhibitory doubts born of surprises, disagreements, and the like, and to reach a secure belief, the belief being that on which one is prepared to act. He framed scientific inquiry as part of a broader spectrum and as spurred, like inquiry generally, by actual doubt, not mere verbal or hyperbolic doubt, which he held to be fruitless."What one does not in the least doubt one should not pretend to doubt; but a man should train himself to doubt," said Peirce in a brief intellectual autobiography; see Ketner, Kenneth Laine (2009) "Charles Sanders Peirce: Interdisciplinary Scientist" in The Logic of Interdisciplinarity). Peirce held that actual, genuine doubt originates externally, usually in surprise, but also that it is to be sought and cultivated, "provided only that it be the weighty and noble metal itself, and no counterfeit nor paper substitute"; in "Issues of Pragmaticism", The Monist, v. XV, n. 4, pp. 481–99, see p. 484, and p. 491. (Reprinted in Collected Papers v. 5, paragraphs 438–63, see 443 and 451). He outlined four methods of settling opinion, ordered from least to most successful: The method of tenacity (policy of sticking to initial belief) – which brings comforts and decisiveness but leads to trying to ignore contrary information and others' views as if truth were intrinsically private, not public. It goes against the social impulse and easily falters since one may well notice when another's opinion is as good as one's own initial opinion. Its successes can shine but tend to be transitory.But see Scientific method and religion. The method of authority – which overcomes disagreements but sometimes brutally. Its successes can be majestic and long-lived, but it cannot operate thoroughly enough to suppress doubts indefinitely, especially when people learn of other societies ' present and past. The method of the a priori – which promotes conformity less brutally but fosters opinions as something like tastes, arising in conversation and comparisons of perspectives in terms of "what is agreeable to reason." Thereby it depends on fashion in paradigms and goes in circles over time. It is more intellectual and respectable but, like the first two methods, sustains accidental and capricious beliefs, destining some minds to doubt it. The scientific method – the method wherein inquiry regards itself as fallible and purposely tests itself and criticizes, corrects, and improves itself. Peirce held that slow, stumbling ratiocination can be dangerously inferior to instinct and traditional sentiment in practical matters, and that the scientific method is best suited to theoretical research,Peirce (1898), "Philosophy and the Conduct of Life", Lecture 1 of the Cambridge (MA) Conferences Lectures, published in Collected Papers v. 1, paragraphs 616–48 in part and in Reasoning and the Logic of Things, Ketner (ed., intro.) and Putnam (intro., comm.), pp. 105–22, reprinted in Essential Peirce v. 2, pp. 27–41. which in turn should not be trammeled by the other methods and practical ends; reason's "first rule" is that, in order to learn, one must desire to learn and, as a corollary, must not block the way of inquiry."... in order to learn, one must desire to learn ..." – Peirce (1899), "F.R.L." [First Rule of Logic], Collected Papers v. 1, paragraphs 135–40, The scientific method excels the others by being deliberately designed to arrive – eventually – at the most secure beliefs, upon which the most successful practices can be based. Starting from the idea that people seek not truth per se but instead to subdue irritating, inhibitory doubt, Peirce showed how, through the struggle, some can come to submit to the truth for the sake of belief's integrity, seek as truth the guidance of potential practice correctly to its given goal, and wed themselves to the scientific method.Peirce, C.S., Collected Papers v. 5, in paragraph 582, from 1898: For Peirce, rational inquiry implies presuppositions about truth and the real; to reason is to presuppose (and at least to hope), as a principle of the reasoner's self-regulation, that the real is discoverable and independent of our vagaries of opinion. In that vein , he defined truth as the correspondence of a sign (in particular, a proposition) to its object and, pragmatically, not as the actual consensus of some definite, finite community (such that to inquire would be to poll the experts), but instead as that final opinion which all investigators would reach sooner or later but still inevitably, if they were to push investigation far enough, even when they start from different points. In tandem he defined the real as a true sign's object (be that object a possibility or quality, or an actuality or brute fact, or a necessity or norm or law), which is what it is independently of any finite community's opinion and, pragmatically, depends only on the final opinion destined in a sufficient investigation. That is a destination as far, or near, as the truth itself to you or me or the given finite community. Thus, his theory of inquiry boils down to "Do the science." Those conceptions of truth and the real involve the idea of a community both without definite limits (and thus potentially self-correcting as far as needed) and capable of definite increase of knowledge.Peirce (1868), "Some Consequences of Four Incapacities", Journal of Speculative Philosophy v. 2, n. 3, pp. 140–57. Reprinted Collected Papers v. 5, paragraphs 264–317, The Essential Peirce v. 1, pp. 28–55, and elsewhere. Arisbe Eprint As inference, "logic is rooted in the social principle" since it depends on a standpoint that is, in a sense, unlimited.Peirce (1878), "The Doctrine of Chances", Popular Science Monthly v. 12, pp. 604–15, see pp. 610–11 via Internet Archive. Reprinted Collected Papers v. 2, paragraphs 645–68, Essential Peirce v. 1, pp. 142–54. "... death makes the number of our risks, the number of our inferences, finite, and so makes their mean result uncertain. The very idea of probability and of reasoning rests on the assumption that this number is indefinitely great. ... logicality inexorably requires that our interests shall not be limited. ... Logic is rooted in the social principle." Paying special attention to the generation of explanations, Peirce outlined the scientific method as coordination of three kinds of inference in a purposeful cycle aimed at settling doubts, as follows (in §III–IV in "A Neglected Argument" with added notes. Reprinted with previously unpublished part, Collected Papers v. 6, paragraphs 452–85, The Essential Peirce v. 2, pp. 434–50, and elsewhere. except as otherwise noted): Abduction (or retroduction). Guessing, inference to explanatory hypotheses for selection of those best worth trying. From abduction, Peirce distinguishes induction as inferring, based on tests, the proportion of truth in the hypothesis. Every inquiry, whether into ideas, brute facts, or norms and laws, arises from surprising observations in one or more of those realms (and for example at any stage of an inquiry already underway). All explanatory content of theories comes from abduction, which guesses a new or outside idea to account in a simple, economical way for a surprising or complicative phenomenon. Oftenest, even a well-prepared mind guesses wrong. But the modicum of success of our guesses far exceeds that of sheer luck and seems born of attunement to nature by instincts developed or inherent, especially insofar as best guesses are optimally plausible and simple in the sense, said Peirce, of the "facile and natural", as by Galileo's natural light of reason and as distinct from "logical simplicity". Abduction is the most fertile but least secure mode of inference. Its general rationale is inductive: it succeeds often enough and, without it, there is no hope of sufficiently expediting inquiry (often multi-generational) toward new truths.Peirce (c. 1906), "PAP (Prolegomena for an Apology to Pragmatism)" (Manuscript 293, not the like-named article), The New Elements of Mathematics (NEM) 4:319–20, see first quote under "Abduction" at Commens Dictionary of Peirce's Terms. Coordinative method leads from abducing a plausible hypothesis to judging it for its testabilityPeirce, Carnegie application (L75, 1902), New Elements of Mathematics v. 4, pp. 37–38: and for how its trial would economize inquiry itself.Peirce (1902), Carnegie application, see MS L75.329330, from Draft D of Memoir 27: Peirce calls his pragmatism "the logic of abduction".Peirce (1903), "Pragmatism – The Logic of Abduction", Collected Papers v. 5, paragraphs 195–205, especially 196. Eprint. His pragmatic maxim is: "Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object". His pragmatism is a method of reducing conceptual confusions fruitfully by equating the meaning of any conception with the conceivable practical implications of its object's conceived effects – a method of experimentational mental reflection hospitable to forming hypotheses and conducive to testing them. It favors efficiency. The hypothesis, being insecure, needs to have practical implications leading at least to mental tests and, in science, lending themselves to scientific tests. A simple but unlikely guess, if uncostly to test for falsity, may belong first in line for testing. A guess is intrinsically worth testing if it has instinctive plausibility or reasoned objective probability, while subjective likelihood, though reasoned, can be misleadingly seductive. Guesses can be chosen for trial strategically, for their caution (for which Peirce gave as an example the game of Twenty Questions), breadth, and incomplexity.Peirce, "On the Logic of Drawing Ancient History from Documents", Essential Peirce v. 2, see pp. 107–09. On Twenty Questions, p. 109: One can hope to discover only that which time would reveal through a learner's sufficient experience anyway, so the point is to expedite it; the economy of research is what demands the leap, so to speak, of abduction and governs its art. Induction. The long-run validity of the rule of induction is deducible from the principle (presuppositional to reasoning , in general, ) that the real is only the object of the final opinion to which adequate investigation would lead;Peirce (1878), "The Probability of Induction", Popular Science Monthly, v. 12, pp. 705–18, see 718 Google Books; 718 via Internet Archive. Reprinted often, including (Collected Papers v. 2, paragraphs 669–93), (The Essential Peirce v. 1, pp. 155–69). anything to which no such process would ever lead would not be real. Induction involving ongoing tests or observations follows a method which, sufficiently persisted in, will diminish its error below any predesignate degree. Three stages: Classification. Unclearly premised, but inductive, classing of objects of experience under general ideas. Probation: direct inductive argumentation. Crude (the enumeration of instances) or gradual (new estimate of the proportion of truth in the hypothesis after each test). Gradual induction is qualitative or quantitative; if qualitative, then dependent on weightings of qualities or characters;Peirce (1905 draft "G" of "A Neglected Argument"), "Crude, Quantitative, and Qualitative Induction", Collected Papers v. 2, paragraphs 755–60, see 759. Find under "Induction" at Commens Dictionary of Peirce's Terms. if quantitative, then dependent on measurements, or on statistics, or on countings. Sentential Induction. "... which, by inductive reasonings, appraises the different probations singly, then their combinations, then makes self-appraisal of these very appraisals themselves, and passes final judgment on the whole result". Invariant explanation Model of DNA with David Deutsch, proponent of invariant scientific explanations (2009) Peer review evaluation Scientific journals use a process of peer review, in which scientists' manuscripts are submitted by editors of scientific journals to (usually one to three, and usually anonymous) fellow scientists familiar with the field for evaluation. In certain journals, the journal itself selects the referees; while in others (especially journals that are extremely specialized), the manuscript author might recommend referees. The referees may or may not recommend publication, or they might recommend publication with suggested modifications, or sometimes, publication in another journal. This standard is practiced to various degrees by different journals and can have the effect of keeping the literature free of obvious errors and generally improve the quality of the material, especially in the journals that use the standard most rigorously. The peer-review process can have limitations when considering research outside the conventional scientific paradigm: problems of "groupthink" can interfere with open and fair deliberation of some new research.Brown, C. (2005) Overcoming Barriers to Use of Promising Research Among Elite Middle East Policy Groups, Journal of Social Behaviour and Personality, Select Press. Sometimes experimenters may make systematic errors during their experiments, veer from standard methods and practices (Pathological science) for various reasons, or, in rare cases, deliberately report false results. Occasionally because of this then, other scientists might attempt to repeat the experiments to duplicate the results. Archiving Researchers sometimes practice scientific data archiving, such as in compliance with the policies of government funding agencies and scientific journals. In these cases, detailed records of their experimental procedures, raw data, statistical analyses , and source code can be preserved to provide evidence of the methodology and practice of the procedure and assist in any potential future attempts to reproduce the result. These procedural records may also assist in the conception of new experiments to test the hypothesis, and may prove useful to engineers who might examine the potential practical applications of a discovery. Limitations Since a scientist can't record everything that took place in an experiment, facts selected for their apparent relevance are reported. This may lead, unavoidably, to problems later if some supposedly irrelevant feature is questioned. For example, Heinrich Hertz did not report the size of the room used to test Maxwell's equations, which later turned out to account for a small deviation in the results. The problem is that parts of the theory itself need to be assumed to select and report the experimental conditions. The observations are hence sometimes described as being 'theory-laden'. Norwood Russell Hanson, Imre Lakatos and Thomas Kuhn have done extensive work on the "theory-laden" character of observation. Hanson (1958) first coined the term for the idea that all observation is dependent on the conceptual framework of the observer, using the concept of gestalt to show how preconceptions can affect both observation and description. He opens Chapter 1 with a discussion of the Golgi bodies and their initial rejection as an artefact of staining technique, and a discussion of Brahe and Kepler observing the dawn and seeing a "different" sunrise despite the same physiological phenomenon. Kuhn and FeyerabendFeyerabend, Paul K (1960) "Patterns of Discovery" The Philosophical Review (1960) vol. 69 (2) pp. 247–52 acknowledge the pioneering significance of his work. Kuhn (1961) said the scientist generally has a theory in mind before designing and undertaking experiments to make empirical observations, and that the "route from theory to measurement can almost never be traveled backward". This implies that how theory is tested is dictated by the nature of the theory itself, which led Kuhn (1961, p. 166) to argue that "once it has been adopted by a profession ... no theory is recognized to be testable by any quantitative tests that it has not already passed".Kuhn, Thomas S., "The Function of Measurement in Modern Physical Science", ISIS 52(2), 161–93, 1961. Post-modernism and science wars Paul Feyerabend similarly examined the history of science, and was led to deny that science is genuinely a methodological process. In his book Against Method he argues that scientific progress is not the result of applying any particular method. In essence, he says that for any specific method or norm of science, one can find a historic episode where violating it has contributed to the progress of science. Thus, if believers in the scientific method wish to express a single universally valid rule, Feyerabend jokingly suggests, it should be 'anything goes'.Feyerabend, Paul K., Against Method, Outline of an Anarchistic Theory of Knowledge, 1st published, 1975. Reprinted, Verso, London, 1978. Criticisms such as his led to the strong programme, a radical approach to the sociology of science. Psychologist Kevin Dunbar says the process of discovery often starts with researchers finding bugs in their experiments. These unexpected results lead researchers to try to fix what they think is an error in their method. Eventually, the researcher decides the error is too persistent and systematic to be a coincidence. The highly controlled, cautious , and curious aspects of the scientific method are thus what make it well suited for identifying such persistent systematic errors. At this point, the researcher will begin to think of theoretical explanations for the error, often seeking the help of colleagues across different domains of expertise. Nevertheless, the connection between mathematics and reality (and so science to the extent it describes reality) remains obscure. Eugene Wigner's paper, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, is a very well-known account of the issue from a Nobel Prize-winning physicist. In fact, some observers (including some well-known mathematicians such as Gregory Chaitin, and others such as Lakoff and Núñez) have suggested that mathematics is the result of practitioner bias and human limitation (including cultural ones), somewhat like the post-modernist view of science. Imre Lakatos argued that mathematicians actually use contradiction, criticism , and revision as principles for improving their work.See the development, by generations of mathematicians, of Euler's formula for polyhedra as documented by In like manner to science, where truth is sought, but certainty is not found, in Proofs and refutations (1976), what Lakatos tried to establish was that no theorem of informal mathematics is final or perfect. This means that we should not think that a theorem is ultimately true, only that no counterexample has yet been found. Once a counterexample, i.e. an entity contradicting/not explained by the theorem is found, we adjust the theorem, possibly extending the domain of its validity. This is a continuous way our knowledge accumulates, through the logic and process of proofs and refutations. (If axioms are given for a branch of mathematics, however, Lakatos claimed that proofs from those axioms were tautological, i.e. logically true, by rewriting them, as did Poincaré (Proofs and Refutations, 1976).) The particular points raised are statistical ("The smaller the studies conducted in a scientific field, the less likely the research findings are to be true" and "The greater the flexibility in designs, definitions, outcomes, and analytical modes in a scientific field, the less likely the research findings are to be true.") and economical ("The greater the financial and other interests and prejudices in a scientific field, the less likely the research findings are to be true" and "The hotter a scientific field (with more scientific teams involved), the less likely the research findings are to be true.") Hence: "Most research findings are false for most research designs and for most fields" and "As shown, the majority of modern biomedical research is operating in areas with very low pre- and poststudy probability for true findings." However: "Nevertheless, most new discoveries will continue to stem from hypothesis-generating research with low or very low pre-study odds," which means that *new* discoveries will come from research that, when that research started, had low or very low odds (a low or very low chance) of succeeding. Hence, if the scientific method is used to expand the frontiers of knowledge, research into areas that are outside the mainstream will yield the newest discoveries. Bauer, Henry H., Scientific Literacy and the Myth of the Scientific Method, University of Illinois Press, Champaign, IL, 1992 Beveridge, William I.B., The Art of Scientific Investigation, Heinemann, Melbourne, Australia, 1950. Bernstein, Richard J., Beyond Objectivism and Relativism: Science, Hermeneutics, and Praxis, University of Pennsylvania Press, Philadelphia, PA, 1983. Brody, Baruch A. and Capaldi, Nicholas, Science: Men, Methods, Goals: A Reader: Methods of Physical Science, W.A. Benjamin, 1968 Brody, Baruch A., and Grandy, Richard E., Readings in the Philosophy of Science, 2nd edition, Prentice-Hall , Englewood Cliffs, NJ, 1989. Burks, Arthur W., Chance, Cause, Reason: An Inquiry into the Nature of Scientific Evidence, University of Chicago Press, Chicago, IL, 1977. Chalmers, Alan, What Is This Thing Called Science?. Queensland University Press and Open University Press, 1976. Cowles, Henry M., The Scientific Method: An Evolution of Thinking from Darwin to Dewey, Harvard University Press, Cambridge, MA, 2020. Reviewed in: Riskin, Jessica, "Just Use Your Thinking Pump!", The New York Review of Books, vol. LXVII, no. 11 (2 July 2020), pp. 48–50. .
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