Editing Philosophy of mathematics (section)

==History==
[[File:Pythagoras in the Roman Forum, Colosseum.jpg|thumb|[[Pythagoras]] is considered the father of mathematics and geometry as he set the foundation for [[Euclid]] and [[Euclidean geometry]]. Pythagoras was the founder of [[Pythagoreanism]]: a mathematical and philosophical model to map the universe.]]
The origin of mathematics is of arguments and disagreements. Whether the birth of mathematics was by chance or induced by necessity during the development of similar subjects, such as physics, remains an area of contention.<ref>{{cite web|title=Is mathematics discovered or invented?|url=http://socialsciences.exeter.ac.uk/education/research/centres/stem/publications/pmej/pome12/article2.htm|website=University of Exeter|access-date=28 March 2018|archive-date=27 July 2018|archive-url=https://web.archive.org/web/20180727234430/http://socialsciences.exeter.ac.uk/education/research/centres/stem/publications/pmej/pome12/article2.htm|url-status=live}}</ref><ref>{{cite web|title=Math: Discovered, Invented, or Both?|url=http://www.pbs.org/wgbh/nova/blogs/physics/2015/04/great-math-mystery/|website=pbs.org|date=13 April 2015 |access-date=28 March 2018|archive-date=28 March 2018|archive-url=https://web.archive.org/web/20180328103331/http://www.pbs.org/wgbh/nova/blogs/physics/2015/04/great-math-mystery/|url-status=live}}</ref>

Many thinkers have contributed their ideas concerning the nature of mathematics. Today, some{{Who|date=July 2012}} philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis. There are traditions of mathematical philosophy in both [[History of Western philosophy|Western philosophy]] and [[Eastern philosophy]]. Western philosophies of mathematics go as far back as [[Pythagoras]], who described the theory "everything is mathematics" ([[mathematicism]]), [[Plato]], who paraphrased Pythagoras, and studied the [[ontology|ontological status]] of mathematical objects, and [[Aristotle]], who studied [[logic]] and issues related to [[infinity]] (actual versus potential).

[[Greek philosophy]] on mathematics was strongly influenced by their study of [[geometry]]. For example, at one time, the Greeks held the opinion that 1 (one) was not a [[number]], but rather a unit of arbitrary length. A number was defined as a multitude. Therefore, 3, for example, represented a certain multitude of units, and was thus "truly" a number. At another point, a similar argument was made that 2 was not a number but a fundamental notion of a pair. These views come from the heavily geometric straight-edge-and-compass viewpoint of the Greeks: just as lines drawn in a geometric problem are measured in proportion to the first arbitrarily drawn line, so too are the numbers on a number line measured in proportion to the arbitrary first "number" or "one".{{Citation needed|date=April 2011}}

These earlier Greek ideas of numbers were later upended by the discovery of the [[irrational number|irrationality]] of the square root of two. [[Hippasus]], a disciple of [[Pythagoras]], showed that the diagonal of a unit square was incommensurable with its (unit-length) edge: in other words he proved there was no existing (rational) number that accurately depicts the proportion of the diagonal of the unit square to its edge. This caused a significant re-evaluation of Greek philosophy of mathematics. According to legend, fellow Pythagoreans were so traumatized by this discovery that they murdered Hippasus to stop him from spreading his heretical idea.<ref>[[Morris Kline]] (1990), ''Mathematical Thought from Ancient to Modern Times'', page 32. Oxford University Press.</ref> [[Simon Stevin]] was one of the first in Europe to challenge Greek ideas in the 16th&nbsp;century. Beginning with [[Gottfried Wilhelm Leibniz|Leibniz]], the focus shifted strongly to the relationship between mathematics and logic. This perspective dominated the philosophy of mathematics through the time of [[George Boole|Boole]], [[Gottlob Frege|Frege]] and [[Bertrand Russell|Russell]], but was brought into question by developments in the late 19th and early 20th&nbsp;centuries.

===Contemporary philosophy===
A perennial issue in the philosophy of mathematics concerns the relationship between logic and mathematics at their joint foundations. While 20th-century philosophers continued to ask the questions mentioned at  the outset of this article, the philosophy of mathematics in the 20th century was characterized by a predominant interest in [[formal logic]], [[set theory]] (both [[naive set theory]] and [[axiomatic set theory]]), and foundational issues.

It is a profound puzzle that on the one hand mathematical truths seem to have a compelling inevitability, but on the other hand the source of their "truthfulness" remains elusive. Investigations into this issue are known as the [[foundations of mathematics]] program.

At the start of the 20th century, philosophers of mathematics were already beginning to divide into various schools of thought about all these questions, broadly distinguished by their pictures of mathematical [[epistemology]] and [[ontology]]. Three schools, [[formalism (mathematics)|formalism]], [[intuitionism]], and [[logicism]], emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and [[mathematical analysis|analysis]] in particular, did not live up to the standards of [[certainty]] and [[rigor]] that had been taken for granted. Each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.

Surprising and counter-intuitive developments in formal logic and set theory early in the 20th&nbsp;century led to new questions concerning what was traditionally called the ''foundations of mathematics''. As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of [[Euclid]] around 300&nbsp;BCE as the natural basis for mathematics. Notions of [[axiom]], [[proposition]] and [[mathematical proof|proof]], as well as the notion of a proposition being true of a mathematical object {{Crossreference|(see [[Assignment (mathematical logic)|Assignment]])}}, were formalized, allowing them to be treated mathematically. The [[Zermelo–Fraenkel set theory|Zermelo–Fraenkel]] axioms for set theory were formulated which provided a conceptual framework in which much mathematical discourse would be interpreted. In mathematics, as in physics, new and unexpected ideas had arisen and significant changes were coming. With [[Gödel numbering]], propositions could be interpreted as referring to themselves or other propositions, enabling inquiry into the [[consistency proof|consistency]] of mathematical theories. This reflective critique in which the theory under review "becomes itself the object of a mathematical study" led [[David Hilbert|Hilbert]] to call such study ''[[metamathematics]]'' or ''[[proof theory]]''.<ref>{{cite book
|last=Kleene
|first=Stephen
|author-link=Stephen Cole Kleene
|title=Introduction to Metamathematics
|page=5
|year=1971
|publisher=North-Holland Publishing Company
|location=Amsterdam, Netherlands
}}</ref>

At the middle of the century, a new mathematical theory was created by [[Samuel Eilenberg]] and [[Saunders Mac Lane]], known as [[category theory]], and it became a new contender for the natural language of mathematical thinking.<ref>[[Saunders Mac Lane|Mac Lane, Saunders]] (1998), ''[[Categories for the Working Mathematician]]'', 2nd edition, Springer-Verlag, New York, NY.</ref> As the 20th&nbsp;century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at the century's beginning. [[Hilary Putnam]] summed up one common view of the situation in the last third of the century by saying:
<blockquote>
When philosophy discovers something wrong with science, sometimes science has to be changed—[[Russell's paradox]] comes to mind, as does [[George Berkeley|Berkeley]]'s attack on the actual [[infinitesimal]]—but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that "philosophical interpretation" is just what mathematics doesn't need.<ref>*Putnam, Hilary (1967), "Mathematics Without Foundations", ''Journal of Philosophy'' 64/1, 5-22. Reprinted, pp.&nbsp;168–184 in W.D. Hart (ed., 1996).</ref>{{rp|169–170}}
</blockquote>

Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.