Inductive reasoning in the context of "Mathematical proof"

Reason is the capacity of consciously applying logic by drawing valid conclusions from new or existing information, with the aim of seeking truth. It is associated with such characteristically human activities as philosophy, religion, science, language, and mathematics, and is normally considered to be a distinguishing ability possessed by humans. Reason is sometimes referred to as rationality, although the latter is more about its application.

Reasoning involves using more-or-less rational processes of thinking and cognition to extrapolate from one's existing knowledge to generate new knowledge, and involves the use of one's intellect. The field of logic studies the ways in which humans can use formal reasoning to produce logically valid arguments and true conclusions. Reasoning may be subdivided into forms of logical reasoning, such as deductive reasoning, inductive reasoning, and abductive reasoning.

David Hume (/hjuːm/; born David Home; 7 May 1711 – 25 August 1776) was a Scottish philosopher, historian, economist and essayist who is known for his highly influential system of empiricism, philosophical scepticism and metaphysical naturalism. Beginning with A Treatise of Human Nature (1739–40), Hume strove to create a naturalistic science of man that examined the psychological basis of human nature. Hume followed John Locke in rejecting the existence of innate ideas, concluding that all human knowledge derives solely from experience; this places him amongst such empiricists as Francis Bacon, Thomas Hobbes, John Locke, and George Berkeley.

Hume argued that inductive reasoning and belief in causality cannot be justified rationally; instead, they result from custom and mental habit. People never actually perceive that one event causes another but only experience the "constant conjunction" of events. This problem of induction means that to draw any causal inferences from past experience, it is necessary to presuppose that the future will resemble the past; this metaphysical presupposition cannot itself be grounded in prior experience.

Abductive reasoning (also called abduction, abductive inference, or retroduction) is a form of logical inference that seeks the simplest and most likely conclusion from a set of observations. It was formulated and advanced by the American philosopher and logician Charles Sanders Peirce beginning in the latter half of the 19th century.

Abductive reasoning, unlike deductive reasoning, yields a plausible conclusion but does not definitively verify it. Abductive conclusions do not eliminate uncertainty or doubt, which is expressed in terms such as "best available" or "most likely". While inductive reasoning draws general conclusions that apply to many situations, abductive conclusions are confined to the particular observations in question.

The scientific method is an empirical method for acquiring knowledge that has been referred to while doing science since at least the 17th century. Historically, it was developed through the centuries from the ancient and medieval world. The scientific method involves careful observation coupled with rigorous skepticism, because cognitive assumptions can distort the interpretation of the observation. Scientific inquiry includes creating a testable hypothesis through inductive reasoning, testing it through experiments and statistical analysis, and adjusting or discarding the hypothesis based on the results.

Although procedures vary across fields, the underlying process is often similar. In more detail: the scientific method involves making conjectures (hypothetical explanations), predicting the logical consequences of hypothesis, then carrying out experiments or empirical observations based on those predictions. A hypothesis is a conjecture based on knowledge obtained while seeking answers to the question. Hypotheses can be very specific or broad but must be falsifiable, implying that it is possible to identify a possible outcome of an experiment or observation that conflicts with predictions deduced from the hypothesis; otherwise, the hypothesis cannot be meaningfully tested.

Logos (UK: /ˈloʊɡɒs, ˈlɒɡɒs/, US: /ˈloʊɡoʊs/; Ancient Greek: λόγος, romanized: lógos, lit. 'word, discourse, or reason') is a term used in Western philosophy, psychology and rhetoric, as well as religion (notably Christianity), that most broadly means reason, logic, order, or understanding. Among its connotations is that of a rational form of discourse that relies on inductive and deductive reasoning.

Aristotle first systematized the usage of the word, making it one of the three principles of rhetoric alongside ethos and pathos. This original use identifies the word closely to the structure and content of language or text. Both Plato and Aristotle used the term logos (along with rhema) to refer to sentences and propositions.

Francis Bacon, 1st Viscount St Alban PC (/ˈbeɪkən/; 22 January 1561 – 9 April 1626) was an English philosopher and statesman who served as Attorney General and Lord Chancellor of England under King James I. Bacon argued for the importance of natural philosophy, guided by the scientific method, and his works remained influential throughout the Scientific Revolution.

Bacon has been called the father of empiricism. He argued for the possibility of scientific knowledge based only upon inductive reasoning and careful observation of events in nature. He believed that science could be achieved by the use of a sceptical and methodical approach whereby scientists aim to avoid misleading themselves. Although his most specific proposals about such a method, the Baconian method, did not have long-lasting influence, the general idea of the importance and possibility of a sceptical methodology makes Bacon one of the founders of the scientific method. His portion of the method based in scepticism was a new rhetorical and theoretical framework for science, whose practical details are still central to debates on science and methodology. He is famous for his role in the scientific revolution, promoting scientific experimentation as a way of glorifying God and fulfilling scripture.

Analogy is a comparison or correspondence between two things (or two groups of things) because of a third element that they are considered to share.

Logically, it is an inference or an argument from one particular to another particular, as opposed to deduction, induction, and abduction. It is also used where at least one of the premises, or the conclusion, is general rather than particular in nature. It has the general form A is to B as C is to D.