Editing Philosophy of mathematics (section)

===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.