Editing Philosophy of mathematics (section)

===Beyond the traditional schools===
====Unreasonable effectiveness====
Rather than focus on narrow debates about the true nature of mathematical [[truth]], or even on practices unique to mathematicians such as the [[mathematical proof|proof]], a growing movement from the 1960s to the 1990s began to question the idea of seeking foundations or finding any one right answer to why mathematics works. The starting point for this was [[Eugene Wigner]]'s famous 1960 paper "[[The Unreasonable Effectiveness of Mathematics in the Natural Sciences]]", in which he argued that the happy coincidence of mathematics and physics being so well matched seemed to be unreasonable and hard to explain.

====Popper's two senses of number statements<!--Linked from 'Karl Popper'-->====
Realist and constructivist theories are normally taken to be contraries. However, [[Karl Popper]]<ref>Popper, Karl Raimund (1946) Aristotelian Society Supplementary Volume XX.</ref> argued that a number statement such as {{nowrap|"2 apples + 2 apples {{=}} 4 apples"}} can be taken in two senses. In one sense it is irrefutable and logically true. In the second sense it is factually true and falsifiable. Another way of putting this is to say that a single number statement can express two propositions: one of which can be explained on constructivist lines; the other on realist lines.<ref>Gregory, Frank Hutson (1996) "[[s:Arithmetic and Reality: A Development of Popper's Ideas|Arithmetic and Reality: A Development of Popper's Ideas]]". City University of Hong Kong. Republished in Philosophy of Mathematics Education Journal No. 26 (December 2011)</ref>

====Philosophy of language====
{{original research|section|date=February 2023}}
Innovations in the philosophy of language during the 20th century renewed interest in whether mathematics is, as is often said,{{citation needed|date=February 2023}} the ''language'' of science. Although some{{who|date=February 2023}} mathematicians and philosophers would accept the statement "mathematics is a language" (most consider that the [[language of mathematics]] is a part of mathematics to which mathematics cannot be reduced),{{citation needed|date=February 2023}} linguists{{who|date=February 2023}} believe that the implications of such a statement must be considered. For example, the tools of [[linguistics]] are not generally applied to the symbol systems of mathematics, that is, mathematics is studied in a markedly different way from other languages. If mathematics is a language, it is a different type of language from [[natural languages]]. Indeed, because of the need for clarity and specificity, the language of mathematics is far more constrained than natural languages studied by linguists. However, the methods developed by Frege and Tarski for the study of mathematical language have been extended greatly by Tarski's student [[Richard Montague]] and other linguists working in [[formal semantics (linguistics)|formal semantics]] to show that the distinction between mathematical language and natural language may not be as great as it seems.

Mohan Ganesalingam has analysed mathematical language using tools from formal linguistics.<ref name="Ganesalingam-2013">{{cite book |last1=Ganesalingam |first1=Mohan |title=The Language of Mathematics: A Linguistic and Philosophical Investigation |volume=7805 |date=2013 |publisher=Springer |isbn=978-3-642-37011-3 |doi=10.1007/978-3-642-37012-0 |series=Lecture Notes in Computer Science |s2cid=14260721 }}</ref> Ganesalingam notes that some features of natural language are not necessary when analysing mathematical language (such as [[grammatical tense|tense]]), but many of the same analytical tools can be used (such as [[context-free grammar]]s). One important difference is that mathematical objects have clearly defined [[type (mathematics)|types]], which can be explicitly defined in a text: "Effectively, we are allowed to introduce a word in one part of a sentence, and declare its [[part of speech]] in another; and this operation has no analogue in natural language."<ref name="Ganesalingam-2013" />{{rp|251}}