Inductive reasoning  

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"Contemporary science is founded upon the principle of induction: most people have seen a certain phenomenon precede or follow some other phenomenon most often, and conclude therefrom that it will ever be thus." --Exploits and Opinions of Dr. Faustroll, Pataphysician (1911) by Alfred Jarry

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Train wreck at Montparnasse (October 22, 1895) by Studio Lévy and Sons.
Train wreck at Montparnasse (October 22, 1895) by Studio Lévy and Sons.

Inductive reasoning is the process of making inferences based upon observed patterns, or simple repetition. Often used in reference to predictions about will happen or does happen, based upon what has happened.

The problem of induction is one of considerable debate and importance in the philosophy of science: is induction indeed justified, and if so, how?

It is used to ascribe properties or relations to types based on an observation instance (i.e., on a number of observations or experiences); or to formulate laws based on limited observations of recurring phenomenal patterns. Induction is employed, for example, in using specific propositions such as:

This ice is cold. (or: All ice I have ever touched was cold.)
This billiard ball moves when struck with a cue. (or: 100/100 billiard balls struck with a cue moved.) infer general propositions such as:

All ice is cold.
All billiard balls move when struck with a cue.



Ancient philosophy

For a move from particular to universal, Aristotle in the 300s BCE used the Greek word epagogé, which Cicero translated into the Latin word inductio. In the 300s CE, Sextus Empiricus maintained that all knowledge derives from sensory experience—concluded in his Outlines of Pyrrhonism that acceptance of universal statements as true cannot be justified by induction.

Early modern philosophy

In 1620, early modern philosopher Francis Bacon repudiated mere experience and enumerative induction, and sought to couple those with neutral and minute and many varied observations before to uncover the natural world's structure and causal relations beyond the present scope of experience via his method of inductivism, which nonetheless required enumerative induction as a component.

The supposedly radical empiricist David Hume's 1740 stance found enumerative induction to have no rational, let alone logical, basis but to be a custom of the mind and an everyday requirement to live, although observations could be coupled with the principle uniformity of nature—another logically invalid conclusion, thus the problem of induction—to seemingly justify enumerative induction and reason toward unobservables, including causality counterfactually, simply that modifying such an aspect prevents or produces such outcome.

Awakened from "dogmatic slumber" by a German translation of Hume's work, Kant sought to explain the possibility of metaphysics. In 1781, Kant's Critique of Pure Reason introduced the distinction rationalism, a path toward knowledge distinct from empiricism. Kant sorted statements into two types. The analytic are true by virtue of their terms' arrangement and meanings—thus are tautologies, merely logical truths, true by necessity—whereas the synthetic arrange meanings to refer to states of facts, contingencies. Finding it impossible to know objects as they truly are in themselves, however, Kant found the philosopher's task not peering behind the veil of appearance to view the noumena, but simply handling phenomena.

Reasoning that the mind must contain its own categories organizing sense data, making experience of space and time possible, Kant concluded uniformity of nature a priori. A class of synthetic statements was not contingent but true by necessity, then, the synthetic a priori. Kant thus saved both metaphysics and Newton's law of universal gravitation, but incidentally discarded scientific realism and developed transcendental idealism. Kant's transcendental idealism prompted the trend German idealism. G F W Hegel's absolute idealism flourished across continental Europe and fueled nationalism.

Late modern philosophy

Developed by Saint-Simon, and promulgated in the 1830s by his former student Comte was positivism, the first late modern philosophy of science. In the French Revolution's aftermath, fearing society's ruin again, Comte opposed metaphysics. 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. Comte found enumerative induction reliable by its grounding on experience available, and asserted science's use as improving human society, not metaphysical truth.

According to Comte, scientific method frames predictions, confirms them, and states laws—positive statements—irrefutable by theology or by metaphysics. Regarding experience to justify enumerative induction by having shown uniformity of nature, Mill welcomed Comte's positivism, but thought laws susceptible to recall or revision, and withheld from Comte's Religion of Humanity. Comte was confident to lay laws as irrefutable foundation of other knowledge, and the churches, honoring eminent scientists, sought to focus public mindset on altruism—a term Comte coined—to apply science for humankind's social welfare via Comte's spearheaded science, sociology.

During the 1830s and 1840s, while Comte and Mill were the leading philosophers of science, William Whewell found enumerative induction not nearly so simple, but, amid the dominance of inductivism, described "superinduction". 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. 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".

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.

Originator of pragmatism, C S Peirce who, as did Gottlob Frege independently, in the 1870s performed vast investigations that clarified the basis of deductive inference as mathematical proof, recognized induction but continuously insisted on a third type of inference that Peirce variously termed abduction or retroduction or hypothesis or presumption. Later philosophers gave Peirce's abduction, etc, the synonym inference to the best explanation (IBE).

Contemporary philosophy

Bertrand Russell

Having highlighted Hume's problem of induction, John Maynard Keynes posed logical probability as its answer—but then figured not quite. 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 it 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". Two decades later, Russell proposed enumerative induction as an "independent logical principle". Russell found,

"Hume's skepticism rests entirely upon his rejection of the principle of induction. The principle of induction, as applied to causation, says that, if A has been found very often accompanied or followed by B, then it is probable that on the next occasion on which A is observed, it will be accompanied or followed by B. If the principle is to be adequate, a sufficient number of instances must make the probability not far short of certainty. If this principle, or any other from which it can be deduced, is true, then the casual inferences which Hume rejects are valid, not indeed as giving certainty, but as giving a sufficient probability for practical purposes. If this principle is not true, every attempt to arrive at general scientific laws from particular observations is fallacious, and Hume's skepticism is inescapable for an empiricist. The principle itself cannot, of course, without circularity, be inferred from observed uniformities, since it is required to justify any such inference. It must therefore be, or be deduced from, an independent principle not based on experience. To this extent, Hume has proved that pure empiricism is not a sufficient basis for science. But if this one principle is admitted, everything else can proceed in accordance with the theory that all our knowledge is based on experience. It must be granted that this is a serious departure from pure empiricism, and that those who are not empiricists may ask why, if one departure is allowed, others are forbidden. These, however, are not questions directly raised by Hume's arguments. What these arguments prove—and I do not think the proof can be controverted—is that the induction is an independent logical principle, incapable of being inferred either from experience or from other logical principles, and that without this principle, science is impossible".

Gilbert Harman

In a 1965 paper, Gilbert Harman explained that enumerative induction is not an autonomous phenomenon, but is simply a masked consequence of inference to the best explanation (IBE). IBE is otherwise synonym to C S Peirce's abduction. Many philosophers of science espousing scientific realism have maintained that IBE is the way that scientists develop approximately true scientific theories about nature.

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