Inductivism

Inductivism is the traditional model of scientific method attributed to Francis Bacon, who in 1620 vowed to subvert allegedly traditional thinking. In the Baconian model, one observes nature, proposes a modest law to generalize an observed pattern, confirms it by many observations, ventures a modestly broader law, and confirms that, too, by many more observations, while discarding disconfirmed laws. The laws grow ever broader but never much exceed careful, extensive observation. Thus freed from preconceptions, scientists gradually uncover nature's causal and material structure.

At 1740, David Hume found multiple obstacles to use of experience to infer causality. Hume noted the illogicality of enumerative induction—unrestricted generalization for particular instances to all instances, and stating a universal law—since humans observe sequence of sensory events, not cause and effect. Humans thus perceive neither logical nor natural necessity or impossibility among events. Later philosophers would select, highlight, and nickname Humean principles—Hume's fork, problem of induction, and Hume's law—although Hume accepted the empirical sciences as inevitably inductive, after all.

Alarmed by Hume's seemingly radical empiricism, Immanuel Kant identified its apparent opposite, rationalism, as favored by Descartes and by Spinoza. Seeking middle ground, Kant identified that the necessity bridging the world in itself to human experience is the mind, whose innate constants thus determine space, time, and substance and determine the correct scientific theory. Though protecting both metaphysics and Newtonian physics, Kant discarded scientific realism by restricting science to tracing appearances (phenomena), not unveiling reality (noumena). Kant's transcendental idealism launched German idealism—increasingly speculative metaphysics—while philosophers continued awkward confidence in empirical sciences as inductive.

Refining Baconian inductivism, John Stuart Mill posed his own five methods of discerning causality to describe the reasoning whereby scientists exceed mere inductivism. In the 1830s, opposing metaphysics, Auguste Comte explicated positivism, which, unlike Baconian model, emphasized predictions, confirming them, and laying scientific laws irrefutable by theology or metaphysics. Finding experience to show uniformity of nature and thereby justify enumerative induction, Mill accepted positivism: the first modern philosophy of science, which, simultaneously, was a political philosophy whereby only scientific knowledge was reliable knowledge.

Nearing 1840, William Whewell thought that the inductive sciences, so called, were not so simple, after all, and asked recognition of "superinduction", an explanatory scope or principle invented by the mind to unite facts, but not present in the facts. Mill would have none of hypotheticodeductivism, posed by Whewell as science's method, which Whewell believed to sometimes, via other considerations upon the evidence, render scientific theories of known metaphysical truth. By 1880, C S Peirce had clarified the basis of deductive inference and, although recognizing induction, proposed a third type of inference that Peirce called "abduction", now otherwise termed inference to the best explanation (IBE).

Since the 1920s, although opposing all metaphysical inference via scientific theories, the logical positivists sought to understand scientific theories as provably false or true as to strictly observations. Though accepting hypotheticodeductivism to originate theories, they launched verificationism whereby Rudolf Carnap tried but never succeeded to formalize an inductive logic whereby a universal law's truth with respect to observational evidence could be quantified as "degree of confirmation". Asserting a variant of hypotheticodeductivism termed falsificationism, Karl Popper from the 1930s onward was the first especially vocal critic of inductivism and verificationism as utterly flawed models of science. In 1963, Popper declared that enumerative induction is a myth. Two years later, Gilbert Harman claimed that enumerative induction is a masked effect of IBE.

Thomas Kuhn's 1962 book—explaining that periods of normal science as but a paradigm of science are each overturned by revolutionary science whose paradigm becomes the normal science anew—dissolved logical positivism's grip in the Anglosphere, and inductivism fell. Besides Popper and Kuhn, other postpostivist philosophers of science—including Paul Feyerabend, Imre Lakatos, and Larry Laudan—have all but unanimously rejected inductivism. Among them, those who have asserted scientific realism—that scientific theory can and does offer approximately true understanding of nature's unobservable aspects—have tended to claim that scientists develop approximately true theories about nature through IBE. And yet IBE, which, so far, cannot be trained, lacks particular rules of inference. By the 21st century's turn, inductivism's heir was Bayesianism.^[1]

Scientific method

Until the 1960s, in a tradition traced to Francis Bacon at 1620, inductivism was presumed to be scientific method's manner.^[2] Even at the 21st century's turn, popular presentations of scientific discovery and progress suggest Baconian model.^[3] Until the 20th century, scientists generally well knew their own philosophies, however.^[4] Einstein remarked, "Science without Epistemology is—in so far as it is thinkable at all–primitive and muddled".^[4]

The past century was the first producing scientists, not philosopherscientists.^[4] Frequently unable to defend their works from intellectual attacks, scientists also generally cannot optimize methods and productivity.^[4] Future scientific breakthroughs ought to be produced more by scientists who have mastered both their own specialties and basics of philosophy of science, including method.^[4] Major variants upon scientific method are inductivism and hypotheticodeductivism.^[5]

Inductivism

Inductivism infers from observations of similar effects to similar causes, and generalizes unrestrictedly—that is, by enumerative induction—to a universal law.^[5] Extending inductivism, Comtean positivism explicitly aims to oppose metaphysics, shuns imaginative theorizing, emphasizes observation, then making predictions, confirming them, and stating laws.

HD model

Hypotheticodeductivism introduces some explanation or principle from any source, such as imagination or even dreams, infers logical consequences of it—that is, deductive inferences—and compares them with observation, perhaps experimental.^[5] In simple or Whewellian hypotheticodeductivism, one might accept a theory as a true or probably true if its predictions withstand testing and it meets other considerations, such as consilience.^[6] Yet the falsificationism variant of HD model forbids ever inferring the truth of theory, whether as to axioms, laws, or principles.

Affirming

Inductivism as well as its positivist extension and Whewellian hypotheticodeductivism, too, rely on the deductive fallacy of affirming the consequent—If A, then B; indeed B; therefore A^[7]—illogical, since even if B is observed, A could be consequence instead of X or Y or Z, or XYZ combined, as A is but one possibility among potentially infinite. Or the sequence A trailed by B could be consequence of U, simply constant conjunction but not causality.

Uncertainty

No confirmations of an explanation's predictions verify the explanation true, since any phenomenon can host multiple logically possible explanations—the problem of underdetermination—leaving the move from data to theory lacking any formal, that is, logical, rules of inference. Also, we can readily find confirming instances of a theory's predictions even if most of the theory's predictions are false. As observation is laden with theory, scientific method cannot ensure that one will perform experiments inviting disconfirmations, or even notice incompatible findings. Even if they are noticed, the experimenter's regress permits one to discard, while ontological relativity permits one to reinterpret, them.

Denying

Guiding axioms as rules of inference, postulates are principles accepted without proof, themselves, and that if followed lead to conclusions upon input of information. Any number of logically invalid and even empirically false explanations can be maintained by deductive inference from postulates. A natural deductive reasoning form, rather, is logically valid without postulates, is true by simply the principle of nonselfcontradiction. Falsificationism is hyotheticodeductivism restricted to the natural deductive form denying the consequent—If A, then B; not B; thus not A—logically valid, while confirmed predictions and other considerations never justify belief in a theory as true or probably true, simply corroborate the theory.^[8]

Inductivist reign

Bacon

In 1620 in England, Francis Bacon's Novum Organum alleged that scholasticism's Aristotelian method of deductive inference via syllogistic logic upon traditional categories was impeding society's progress.^[9] Admonishing the alleged classic induction for proceeding immediately from "sense and particulars up to the most general propositions", then deducing generalizations onto new particulars without empirically verifying them,^[10][11] Bacon stated the "true and perfect Induction".^[10] In Bacon's inductivist method, a scientist—at the time, a natural philosopher—ventures an axiom of modest scope, makes many observations, accepts the axiom if it is confirmed and never disconfirmed, then ventures another axiom only modestly broader, collects many more observations, and accepts that axiom, too, only if it is confirmed, never disconfirmed.^[10]

In Novus Organum, Bacon used the term hypothesis rarely, and usually then in pejorative senses, as prevalent in Bacon's day.^[12] Yet Bacon's term axiom is more similar now to hypothesis than to law,^[12] which today is nearer to the synonym of axiom, a rule of inference. By the 20th century's close, historians and philosophers of science generally agreed that Bacon's actual counsel was far more balanced that traditionally thought, although some assessment went so far indicating that Bacon was a falsificationist, presumably as far from inductivism as one can get.^[12] Yet, Bacon was not a strict inductivist and included aspects of hypotheticodeductivism,^[12] those aspects of Bacon's model were long glossed over by others,^[12] and the "Baconian model" was regarded as true inductivism—which mostly it was.^[13]

During this repeating process of modest axiomatization confirmed by extensive and minute observations, axioms expand in scope and deepen in penetrance tightly in accordance with all the observations—opening clear and true view of nature as it exists independently of human preconceptions—as the general axioms among observables render matter's unobservable structure and nature's causal mechanisms perceptible.^[14] As Bacon provides no clear way to frame axioms, let alone develop principles or theoretical constructs universally true, researchers might observe and collect data endlessly.^[10] For this vast venture, Bacon's advised precise record keeping and collaboration among researchers—a vision resembling today's research institutes—while the true understanding of nature would permit technological innovation, heralding a New Atlantis.

Newton

Modern science arose against Aristotelian physics,^[15] Both geocentric were Aristotelian physics and Ptolemaic astronomy, which latter was a basis of astrology, a basis of medicine. Nicolaus Copernicus proposed heliocentrism, perhaps to better fit astronomy to Aristotelian physics' fifth element—universal essence, or quintessence, or aether—its intrinsic motion of perpetual, perfect circles. Yet Johannes Kepler modified Copernican orbits to ellipses, soon after Galileo Galilei's telescopic observations disputed the Moon's composition of aether, and his experiments with earthly bodies attacked Aristolelian physics. Galilean principles were subsumed by René Descartes, whose Cartesian physics structured his cosmology, modeling heliocentrism and employing mechanical philosophy—whose first principle, stated by Descartes, was No action at a distance—as termed by chemist Robert Boyle who, seeking for his own discipline a mechanical basis via corpuscularism, sought chemistry's divorce from alchemy.

In 1666, Isaac Newton fled London from the plague.^[16] Isolated, he applied rigorous experimentation and mathematics, including development of calculus, and reduced both terrestrial motion and celestial motion—both physics and astronomy—to one theory stating Newton's laws of motion, several corollary principles, and law of universal gravitation, set in a framework of postulated absolute space and absolute time. Newton's unification of celestial and terrestrial phenomena overthrew vestiges of Aristotelian physics, and disconnected physics from chemistry, which each then followed its own course.^[16] Newton became the exemplar of the modern scientist, and the Newtonian research program became the modern model of knowledge.^[16] Although absolute space, revealed by no experience, and a force acting at a distance discomforted Newton, he and physicists for some 200 years more would seldom suspect the fictional character of the Newtonian foundation, as they believed not that physical concepts and laws are "free inventions of the human mind", as Einstein in 1933 called them, but could be inferred logically from experience.^[17] Supposedly, Newton maintained that toward his gravitational theory, he had "framed" no hypotheses.

Hume

At 1740, Hume aggressively sorted truths into two, divergent categories—"relations of ideas" versus "matters of fact and real existence"—as later termed Hume's fork. "Relations of ideas", such as the abstract truths of logic and mathematics, known true without experience of particular instances, offer a priori knowledge. Yet the quests of empirical science concern "matters of fact and real existence", known true only through experience, thus a posteriori knowledge. As no number of examined instances logically entails the conformity of unexamined instances, a universal law's unrestricted generalization bears no logical basis, but one justifies it by adding the principle uniformity of nature—itself unverified, thus a major induction to justify a minor induction—an obstacle to empirical science later termed the problem of induction.^[18]

For Hume, humans experience sequences of events, not cause and effect, by pieces of sensory data whereby similar experiences might exhibit constant conjunction—first an event like A, and always an event like B—but there is no revelation of causality to reveal either necessity or impossibility.^[19][20] Although Hume apparently enjoyed the scandal that trailed his explanations, Hume did not view them as fatal,^[19] and found enumerative induction to be among the mind's unavoidable customs, required in order for one to live.^[21] Rather, Hume sought to counter Copernican displacement of humankind from the Universe's center, and to redirect intellectual attention to human nature as the central point of knowledge.^[22]

Hume proceeded with inductivism not only toward enumerative induction but toward unobservable aspects of nature, too. Not demolishing Newton's theory, Hume placed his own philosophy on par with it, then.^[23] Though skeptical at common metaphysics or theology, Hume accepted "genuine Theism and Religion" and found a rational person must believe in God to explain the structure of nature and order of the universe.^[24] Still, Hume had urged, "When we run over libraries, persuaded of these principles, what havoc must we make? If we take into our hand any volume—of divinity or school metaphysics, for instance—let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames, for it can contain nothing but sophistry and illusion".^[25]

Kant

Awakened from "dogmatic slumber" Hume's work, Immanuel Kant sought to explain how metaphysics is possible.^[25] Kant's 1781 book introduced the distinction rationalism, whereby some knowledge results not by empiricism, but instead by "pure reason". Concluding it impossible to know reality in itself, however, Kant discarded the philosopher's task of unveiling appearance to view the noumena, and limited science to organizing the phenomena.^[26] Reasoning that the mind contains categories organizing sense data into the experiences substance, space, and time,^[27] Kant thereby inferred uniformity of nature as knowledge a priori.^[28]

Kant sorted statements into two types: the analytic and the synthetic. The analytic are true by their terms' arrangement and meanings—thus are tautologies, merely logical truths, true by necessity—whereas the synthetic arrange meanings to refer to factual states, which are contingent. Yet a class of synthetic statements was contingent but, through the mind, true by necessity.^[20] Thus discovering the synthetic a priori, Kant precariously saved both physics—at the time Newtonian—and metaphysics, but incidentally discarded scientific realism and developed transcendental idealism, which triggered German idealism, including G F W Hegel's absolute idealism.^[26][29]

Positivism

Comte

In the French Revolution's aftermath, fearing Western society's ruin again, Auguste Comte was fed up with metaphysics.^[30] As suggested in 1620 by Francis Bacon,^[31] developed by Saint-Simon, and promulgated in the 1830s by his former student Comte, positivism was the first modern philosophy of science.^[32] Human knowledge had evolved from religion to metaphysics to science, said Comte, which had flowed from mathematics to astronomy to physics to chemistry to biology to sociology—in that order—describing increasingly intricate domains, all of society's knowledge having become scientific, as questions of theology and of metaphysics were unanswerable.^[33] Comte found enumerative induction reliable upon experience available, and asserted science's use as improving human society, not metaphysical truth.^[31]

According to Comte, scientific method constrains itself to observations, but frames predictions, confirms them, and states laws—positive statements—irrefutable by theology or by metaphysics, and laid as foundation for subsequent knowledge.^[31] Later, concluding science insufficient for society, Comte launched Religion of Humanity, whose churches, honoring eminent scientists, led worship of humankind.^[31][32] Comte coined the term altruism,^[32] and emphasized science's application for humankind's social welfare, which would be revealed by Comte's spearheaded science, sociology.^[31] Comte's influence is prominent in Herbert Spencer of England and Émile Durkheim of France establishing modern empirical and functionalist sociology.^[34] Influential in the latter 19th century, positivism was often linked to evolutionary theory,^[31] yet was eclipsed in the 20th century by neopositivism: logical positivism and logical empiricism.^[32]

Mill

J S Mill thought, unlike Comte, that scientific laws were susceptible to recall or revision.^[31] And Mill withheld from Comte's Religion of Humanity.^[31] Still, regarding experience to justify enumerative induction by having shown uniformity of nature,^[28] Mill was fond of Comte's positivism.^[31][34] Mill noted that within the empirical sciences, the natural sciences well surpassed Baconian model, too simplistic, whereas the human sciences, such as moral and political inquiry, had not attained even Baconian scrutiny of immediate experience and enumerative induction.^[11] Similarly, economists of the 19th century tended to pose explanations a priori, and reject disconfirmation by posing circuitous routes of reasoning to maintain their a priori laws.^[35] In 1843, Mill's A System of Logic introduced Mill's methods,^[36] explaining the five principles whereby causal laws can be identified to enhance the empirical sciences as the inductive sciences.^[34] For Mill, all explanations have the same logical structure, while society can be explained by natural laws.^[34]

Social

In the 17th century, England had taken the lead in science, which shifted to France in the 18th, then to Germany in the 19th—and to America in the 20th—transitions affecting each country's contemporary roles for science.^[34] Before Germany had taken science's lead, France led in science immediately before the French Revolution ^[34] Amid the social crisis of its aftermath, Comte inferred that society's natural condition is order, not change.^[34] As in Saint-Simon's industrial utopianism, Comte's vision—later well expressed by modernity—posed science as the only true, objective knowledge and thus as industrial society's secular spiritualism, too, whereby science is the political and ethical guide.^[34]

Positivism arrived in Britain well after science's lead had departed.^[34] British positivism, found in Victorian ethics of utilitarianism, for instance J S Mill's, and later in the social evolutionism of Herbert Spencer, associated science with moral improvement, but rejected science as political leadership.^[34] For Mill, all explanations held the same logical structure—thus, society could be explained by natural laws—yet Mill criticized "scientific politics".^[34] From its outset, then, sociology was pulled between moral reform versus administrative policy.^[34]

Spencer helped popularize the word sociology in England, and compiled vast data aiming to build general theory through empirical analysis.^[34] Spencer's 1850 book Social Statics shows Comtean as well as Victorian concern for social order.^[34] Yet whereas Comte's social science was social physics, Spencer would take biology—via Darwinism, so called, which arrived in 1859—as the model of science.^[34] Spencer's functionalist-evolutionary account identified social structures as functions that adapt, thus explaining social change.^[34]

In France, Comte's sociology influence was prominent with Émile Durkheim, whose 1895 Rules for the Sociological Method also posed natural science as sociology's model.^[34] For Durkheim, social phenomena are social functions without psychologism—that is, lacking consciousness of individuals—while Durkheim's sociology was antinaturalist, in that social facts differed from natural facts.^[34] Still, social representations were real entities to be examined, without prior theory, by assessing raw data and discovering causal laws, according to Durkheim.^[34] Durkheim's sociology was a realist and inductive science where theory would trail observations, while method proceeded from social facts to hypotheses to general laws—their priority being their causal accord—identified inductively.^[34]

Logical

World War erupted in 1914 and closed in 1919 with a treaty including a plan of reparations that British economist John Maynard Keynes immediately and vehemently predicted had the consequence of crumbling German society via hyperinflation—prediction fulfilled by 1923.^[37] Upon 29 May 1919's solar eclipse, Einstein's gravitational theory apparently overthrew Newton's—a revolution in science by astonishing prediction^[38]—a revolution bitterly resisted by many scientists but completed nearing 1930.^[39] An yet race science flourished, appearing quite scientific to many, not obviously pseudoscience,^[40] while overtaking medicine and public health^[41] with excesses of negative eugenics.^[42]

In the 1920s, philosophers and scientists in Berlin Circle and Vienna Circle were appalled by the flaring nationalism, racism, and bigotry, yet perhaps no less by countermovements toward metaphysics, intuitionism, and mysticism.^[43][44] Inspired by developments in philosophy,^[45] mathematics,^[46] logic,^[47] and physics,^[48] they sought to give the world a universal, transparent, truly scientific language whereby falsity or truth could be verified either logically or empirically, no more confusion and madness.^[44] Seeking radical reform of philosophy to convert it into scientific philosophy emulating empirical sciences to become a special science,^[49][50] they called themselves the logical positivists.

Vienna Circle, led by Moritz Schlick, included Otto Neurath, and was converted to logical positivism by Rudolph Carnap, introduced to Schlick by Hans Reichenbach, leading Berlin Circle, whom Carl Hempel, later of Vienna Circle, had studied with.^[51][52] Rejecting Kant's synthetic a priori, they asserted Hume's fork^[53] and staked it at the analytic/synthetic gap to dissolve confusions by freeing language from "pseudostatements", added the verifiability criterion—that only statements logically or empirically verifiable are cognitively meaningful—and presumed a semantic gulf between observational versus theoretical terms.^[54] Withholding credence from science's claims about unobservable aspects of nature,^[55] thus rejecting scientific realism,^[56] they embraced instrumentalism, whereby scientific theory is simply useful to predict human observations,^[56] while sometimes regarding talk of unobservables as either metaphorical^[57] or meaningless.^[58]

Pursuing both Bertrand Russell's logical atomism deconstructing language into supposedly elementary parts and Russell's logicism reducing swaths of mathematics to symbolic logic, logical positivists envisioned both everyday language and mathematics—thus physics, too—sharing a logical syntax in symbolic logic. To gain cognitive meaningfulness, theoretical terms would be translated, via correspondence rules, into observational terms—revealing theories' empirical claims—and then empirical operations would verify them within the observational structure, related to the theoretical structure through the logical syntax whereby a logical calculus could verify the theory's falsity of truth. With this program termed verificationism, logical positivists battled Marburg school's neoKantianism, Husserlian phenomenology, and, as the very epitome of their opposition, Heidegger's "existential hermeneutics", accused by Carnap of the most flagrant "pseudostatements".^[44][51]

Opposition

In friendly spirit, Vienna Circle's Otto Neurath nicknamed Karl Popper, a fellow philosopher in Vienna, the "Official Opposition".^[59] Popper asserted that any effort to verify a scientific theory, or even to inductively confirm a scientific law, was fundamentally misguided.^[59] Popper asserted that although exemplary science not dogmatic, science inevitably relies on "prejudices". Popper accepted Hume's criticism—the problem of induction—as making verification logically impossible. Popper accepted hypotheticodeductivism, sometimes termed it deductivism, but restricted it to denying the consequent, and thereby, refuting verificationism, reframed it as falsificationism. As to law or theory, Popper found confirmation of probable truth untenable,^[59] as any number confirmations is finite: empirical evidence approaching 0% probability of truth, as a universal law's predictive run is infinite. In fact, Popper found that a scientific theory is better if its truth is more improbable.^[60] Popper asserted that logical positivism "is defeated by its typically inductivist prejudice".^[61]

Problems

Having highlighted Hume's problem of induction, John Maynard Keynes posed logical probability as its answer—but then figured not quite.^[62] Bertrand Russell found Keynes's Treatise on Probability the best examination of induction, and if read with Jean Nicod's Le Probleme logique de l'induction as well as R B Braithwaite's review of that in the October 1925 issue of Mind, to provide "most of what is known about induction", although the "subject is technical and difficult, involving a good deal of mathematics".^[63]

Rather than validate enumerative induction—the futile task of showing it a deductive inference—Herbert Feigl as well as Hans Reichenbach, apparently independently, sought to vindicate it by showing it simply useful, either a "good" or the "best" method for the goal at hand, making predictions.^[64] Feigl posed it as a rule, thus neither a priori nor a posteriori but a fortiori.^[64] Reichenbach's treatment, similar to Pascal's wager, posed it as entailing greater predictive success versus the alternative of not using it.^[64]

In 1936, Rudolf Carnap switched the goal of verification, clearly impossible, to confirmation,^[52] while A J Ayer identified two types of verification—strong versus weak—the strong impossible, but weak attained when a statement is probable.^[65] Carnap sought to formalize inductive logic through probability as "degree of confirmation".^[52] Employing abundant logical and mathematical tools, yet never attaining the goal, Carnap's formulations of inductive logic always held a universal law's degree of confirmation at zero.^[52]

Kurt Gödel's incompleteness theorem of 1931 had made the logical positivists' logicist reduction doubtful, and Alfred Tarski's undefinability theorem of 1934 made it hopeless.^[66] Some, including Carl Hempel, still argued that logicism is possible,^[66] since, for instance, nonEuclidean geometry had shown that even the truths of geometry are via axioms among postulates. As to formalism, rather—which coverts talk to logical forms and axioms but does not reduce it to logic—neopositivists accepted hypotheticodeductivism for theory development, but held to symbolic logic as the language to justify, by verification or confirmation, its results.^[67] This was stymied by Hempel's paradox of confirmation, whereby to formalize confirmatory evidence of an hypothesized universal law, All ravens are black—logically equivalent to All nonblack things are not ravens—one could, at least in the symbolic logic, observe any nonblack thing, even a white shoe, and report a confirming instance of the law All ravens are black.^[67]

Early criticism

During the 1830s and 1840s, the French Auguste Comte and the British J S Mill were the leading philosophers of science.^[68] Debating in the 1840s, J S Mill claimed that science proceeds by inductivism, whereas William Whewell, also British, claimed it proceeds by hypotheticodeductivism.^[5]

Whewell

William Whewell found the "inductive sciences" not so simple, but, amid the dominance of inductivism, described "superinduction".^[69] Whewell proposed recognition of "the peculiar import of the term Induction", as "there is some Conception superinduced upon the facts", that is, "the Invention of a new Conception in every inductive inference". Rarely spotted by Whewell's predecessors, such mental inventions rapidly evade notice.^[69] Whewell explained,

"Although we bind together facts by superinducing upon them a new Conception, this Conception, once introduced and applied, is looked upon as inseparably connected with the facts, and necessarily implied in them. Having once had the phenomena bound together in their minds in virtue of the Conception, men can no longer easily restore them back to detached and incoherent condition in which they were before they were thus combined".^[69]

These "superinduced" explanations may well be flawed, but their accuracy is suggested when they exhibit what Whewell termed consilience—that is, simultaneously predicting the inductive generalizations in multiple areas—a feat that, according to Whewell, can establish their truth. Perhaps to accommodate prevailing view of science as inductivist method, Whewell devoted several chapters to "methods of induction" and sometimes said "logic of induction", and yet stressed it lacks rules and cannot be trained.^[69] Whewell also pointed out that Bacon himself was not a strict inductivist, for Bacon had actually, said Whewell, "held the balance, with no partial of feeble hand, between phenomena and ideas".^[12]

In Whewell's hypotheticodeductivism, one discovers from any source imaginable—perhaps even a dream—a model or principle of explanatory power, and then deductively infers logical consequences of it, and tests those consequences versus observation, which can include experimental outcomes.^[6] If they are disconfirmed, then the theory is deductively inferred to be false, yet if they are confirmed, then perhaps, upon other explanatory considerations, the theory will be accepted as true or approximately or probably.^[6]

Peirce

As had Kant noted in 1787, theory of deductive inference had not progressed since antiquity.^[70] In the 1870s, C S Peirce and Gottlob Frege, unbeknownst to one another, revolutionized deductive logic through vast efforts identifying it with mathematical proof.^[70] Originator of pragmatism—or, since 1905, pragmaticism, distinguished from more recent appropriations of Peirce's original term—the American Peirce recognized induction, too, but continuously insisted on a third type of inference that Pierce variously termed abduction or retroduction or hypothesis or presumption.^[70] Later philosophers gave Peirce's abduction, etc, the synonym inference to the best explanation (IBE).^[71] Many philosophers of science espousing scientific realism have maintained that IBE is how scientists develop approximately true scientific theories about nature.^[72]

Inductivist fall

After defeat of National Socialism with the close of World War II in 1945, logical positivists lost their revolutionary zeal and led emergence of philosophy of science as a devoted subdiscipline within the Anglosphere's philosophy academia to research such questions and aspects of scientific theory and knowledge.^[51] The movement shifted, thus, into a milder variant bettered termed logical empiricism or, in any case, neopositivism, led principally by Rudolf Carnap, Hans Reichenbach, and Carl Hempel.^[51] Amid apparent contradictions in its central tenets—verifiability principle, analytic/synthetic distinction, observation/theory gap—Hempel in 1965 abandoned ship for a far wider conception of "degrees of significance", signaling neopositivism's official demise.^[73] Neopositivism became mostly maligned,^[74][75] while credit for its fall generally has gone to W V O Quine and to T S Kuhn,^[51] although its "murder" was first confessed to, quite prematurely, in the 1930s by K Popper.^[76]

Fuzziness

Willard Van Orman Quine's 1951 paper "Two dogmas of empiricism"—explaining semantic holism, how any term's meaning is networked to one's beliefs about the entire world—attacked Hume's fork whereby the analytic/synthetic division was supposedly unbridgeable, a principle apparently untenable.^[73] Among verificationism's greatest internal critics, Carl Hempel had recently concluded the same as to the verifiability criterion, which would cast not only religious assertions and metaphysical statements, but even scientific laws of universal type, too, as meaningless.^[77] In 1958, Norwood Hanson's book Patterns of Discovery subverted the supposed gap between observation and theory—how direct observation would permit neutral comparison from theory to theory—by Hanson explaining that even direct observations, the scientific facts, so called, are laden with theory guiding collection, sorting, prioritization, and interpretation of direct observations as well as the ability to apprehend a phenomenon to start with.^[78] Meanwhile, as to all knowledge generally, Quine's thesis eroded foundationalism, which retreated to modesty.^[79]

Revolutions

Thomas Kuhn's landmark book of 1962, The Structure of Scientific Revolutions, was first published, ironically, in a volume of the International Encyclopedia of Unified Science—a project begun by logical positivists—and somehow, at last, unified the empirical sciences by freeing them from the physics model, and calling them for assessment in history and sociology.^[80] Lacking such heavy use of mathematics and logic's formal language—an approach introduced in 1920s by Rudolph Carnap—Kuhn's book, powerful and persuasive, was written in natural language open to laypersons.^[80]

Structure finds science to be puzzlesolving toward a vision projected by the "ruling class" of a scientific specialty's community, whose "unwritten rulebook" dictates proper scientific problems and solutions, altogether normal science.^[81] The scientists reinterpret ambiguous data, discard anomalous data, and try to stuff nature into the box of their shared paradigm—a theoretical matrix or fundamental view of nature.^[81] At last, compatible data become scarce, anomalies accumulate, and "crisis" ensues.^[81] Some young scientists, newly training, defect to revolutionary science—what arose by explaining both normal and anomalous data simultaneously—whose success solving the scientific problems makes it a new "exemplar", which, however, contradicts normal science.^[81]

Of incompatible languages, rival paradigms are incommensurable.^[81] Trying to resolve conflict, scientists talk past each other, as even direct observations—like the Sun "rising"—get differing interpretations. Some working scientists convert by a perspectival shift that—to their astonishment—snaps the new paradigm, suddenly obvious, into sight. Others, never attaining such gestalt switch, remain holdouts, committed for life to the old paradigm, but one by one die, while the new exemplar—the new, unwritten rulebook—settles in as normal science.^[81] Thus, a revolution in science is fulfilled. The old theoretical matrix becomes so shrouded by the meanings of terms in the new theoretical matrix that even philosophers of science misinterpret the old one.^[81]

Kuhn critically destabilized confidence in foundationalism, which was generally presumed—though erroneously—to be a key tenet of logical empiricism.^[82][83] As logical empiricism was extremely influential in the social sciences,^[84] Kuhn's ideas were rapidly adopted by scholars in disciplines well outside natural sciences.^[80] Kuhn's thesis in turn was attacked even by opponents of positivism.^[85] In Structure's 1970 postscript, Kuhn asserted that science at least has no algorithm—and on that even most of Kuhn's critics agreed.^[85] Reinforcing Quine's assault, Kuhn ushered the Anglosphere's academia into postpositivism or postempiricism.^[51][80]

Falsificationism

Karl Popper's 1959 book proposing falsificationism, originally published in German in 1934, reached the Anglosphere and was soon mistaken for a new type of verificationism,^[76][86] yet refuted it.^[76][86][87] Falsificationism's demarcation falsifiable grants a theory the status scientific—simply, empirically testable—not the status meaningful, a status that Popper did not aim to arbiter.^[86] Popper found no scientific theory either verifiable or, as in Carnap's "liberalization of empiricism", confirmable,^[86][88] and found unscientific, metaphysical, ethical, and aesthetic statements often rich in meaning while also underpinning or fueling science as the origin of scientific theories.^[86] The only confirmations particularly relevant are those of risky predictions,^[89] such as ones conventionally predicted to fail.

Postpositivism

At 1967, historian of philosophy John Passmore concluded, "Logical positivism is dead, or as dead as a philosophical movement ever becomes",^[90] and it became philosophy of science's bogeyman.^[75] Kuhn's thesis was attacked for portraying science is irrational, mere cultural relativism even similar to religious experience. Postpositivism's poster became Popper's view of human knowledge as hypothetical, continually growing, always tentative, open to criticism and revision.^[76]

A myth?

In 1945, Bertrand Russell had proposed enumerative induction as an "independent logical principle",^[91] one "incapable of being inferred either from experience or from other logical principles, and that without this principle, science is impossible".^[92] And yet in 1963, Karl Popper declared, "Induction, i.e. inference based on many observations, is a myth. It is neither a psychological fact, nor a fact of ordinary life, nor one of scientific procedure".^[93] Popper's 1972 book Objective Knowledge opens, "I think I have solved a major philosophical problem: the problem of induction".^[93]

Within Popper's schema—Problem₁ → Tentative Solution → Critical Test → Error Elimination → Problem₂—enumerative induction is "a kind of optical illusion" shrouded by steps of conjecture and refutation during a problem shift.^[93] The tentative solution is improvised, an imaginative leap unguided by inductive rules, and the resulting universal law is deductive, an entailed consequence of all, included explanatory considerations.^[93] Controversy continued over whether there is any way to justify—or, as by Popper, simply dissolve—enumerative induction.^[94]

Some have claimed that although inductive inference is often obscured by language, as especially common in news reporting, perhaps announcing scientists' supposed experimental proof, and that such enumeraive induction ought to be tempered by proper clarification, but that inductive inference is used liberally in science—and that science even requires.^[95] Actually, strong arguments on both sides^[94]

Enumerative induction obviously occurs as a conclusion, but its independence is unclear, as some interpret that it derives as a deductive consequence of an underlying explanation of the observations.^[96] In a 1965 paper, now classic, Gilbert Harman had explained enumerative induction as simply a masked effect of IBE,^[71] which philosophers of science espousing scientific realism have usually maintained is how scientists develop, about the putative mind-independent world, theories approximately true.^[72] Thus, the view that Popper was obviously wrong^[95] is structured by conflicting semantics.^[97]

By now, enumerative induction has been shown to exist, but is found rarely, as in programs of machine learning in Artificial Intelligence (AI).^[98] Likewise, machines can be programmed to operate on probabilistic inference of near certainty.^[99] Yet sheer enumerative induction is overwhelmingly absent from science conducted by humans.^[98] Although much talked of is abduction or IBE, it proceeds by humans' imaginations and creativity without rules of inference, which IBE's discussants provide nothing resembling.^[96][98]

Bogeyman

Popperian falsificationism, too, became widely criticized and eventually became unpopular,^{[88][100][101]} yet Popper has been the only philosopher of science often praised by scientists.^[88] Likened to economists of the 19th century who took circuitous, protracted measures refused falsification of their preconceived principles,^[35] the verificationists—that is, the logical positivists—became identified as pillars of scientism^[102] holding strict inductivism^[103] and foundationalism^[82][83] to ground all empirical sciences to a foundation of direct sensory experience.^[50] It became fashionable among philosophers to rehash neopositivism's alleged failures before launching argument for their own views,^[50] often built atop misrepresentations and outright falsehoods about neopositivism.^[50] Not seeking to overhaul empirical sciences, neopositivists sought to understand them and to overhaul philosophy to be scientific, finding a place among special sciences.^[50]

Logical empiricists indeed posed unity of science to network all special sciences and reduce their laws—upong stating boundary conditions and supplying bridge laws within deductivenomological model—to the fundamental science, that is, fundamental physics.^[104] And Carnap sought to formalize inductive logic in order to confirm universal laws through probability as "degree of confirmation".^[52] Yet Vienna Circle pioneered nonfoundationalism, a legacy especially of Neurath, whose coherentism—the main alternative to foundationalism—likened science to a boat that scientists must rebuild at sea without ever touching shore.^[83][105] And neopositivists did not seek rules of inductive logic to regulate scientific discovery or theorizing, but to verify or confirm laws and theories once they are stated.^[106] Practicing what Popper preached—conjectures and refutations—logical positivism ran its course and catapulted Popper, initially a contentious misfit, to carry the richest philosophy out of interwar Vienna.^[76]

Anarchy

In early 1950s, studying philosophy of quantum mechanics under Popper at London School of Economics, Paul Feyerabend found falsificationism not a breakthrough but rather obvious—the controversy over it suggesting philosophy's poverty, rather.^[107] And yet, there witnessing attacks on inductivism as "the idea that theories can be derived from, or established on the basis of, facts", Feyerabend was impressed by a talk that Popper gave at British Society for the Philosophy of Science. Popper showed that higher-level laws often conflict with, and cannot be reduced to, supposedly more fundamental laws.^[107] The prime example was Kepler's laws of planetary motion, long famed to be—but not actually—reduced by Newton to the law of universal gravitation.^[107][108] Having found falsificationism trivial, Feyerabend found the utter sham of inductivism to be pivotal.^[107] Investigating, eventually Feyerabend found that among the diverse sciences, the unifying approach is Anything goes—often rhetoric, circular argumentation, even subterfuge—altogether methodological lawlessness, scientific anarchy.^[109] At persistence of claims that faith in induction is a necessary condition of reason, Feyerabend sardonically bid Farewell to Reason.^[110]

Programmes

Imre Lakatos found Popper's falsificationism not actually practiced by scientists and unrealistically impractical, but found Kuhn's paradigms of science more monopolistic than actual. Lakatos found multiple, vying research programmes to coexist. Each has a hard core of theories shielded from falsification, while a protective belt of malleable theories sustains revisions in order to advance the hard core via theoretical progress that extends the hard core into new empirical territories, whereupon empirical progress corroborates the theoretical claims, how a research programme becomes progressive. Lakatos found inductivism rather farcical, never in history of science ever practiced. Lakatos alleged that Newton had fallaciously posed his own research programme as inductivist to publicly legitimize itself.^[111]

Traditions

Lakatos's putatative methodology of scientific research programmes was criticized by sociologists of science and by some philosophers of science, too, as being too idealized and omitting scientific communinities' interplay with the wider society's social configurations and dynamics. Philosopher of science Larry Laudan identified the stable to be not the research programmes, but rather research traditions.

Inductivist heir

By the 21st century's turn, Bayesianism had become the heir of inductivism.^[1]

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Notes