Editing Philosophy of mathematics (section)

===Indispensability argument for realism<!--linked from 'Hilary Putnam' and 'Willard Van Orman Quine'-->===
{{Main|Quine–Putnam indispensability argument}}
This argument, associated with [[Willard Quine]] and [[Hilary Putnam]], is considered by [[Stephen Yablo]] to be one of the most challenging arguments in favor of the acceptance of the existence of abstract mathematical entities, such as numbers and sets.<ref>{{cite web |author=Yablo, S. |title=A Paradox of Existence |url=https://www.mit.edu/%7Eyablo/apex.html#fn1 |date=November 8, 1998 |access-date=August 26, 2019 |archive-date=January 7, 2020 |archive-url=https://web.archive.org/web/20200107113129/http://www.mit.edu/%7Eyablo/apex.html#fn1 |url-status=live }}</ref> The form of the argument is as follows.
#One must have [[ontological]] commitments to ''all'' entities that are indispensable to the best scientific theories, and to those entities ''only'' (commonly referred to as "all and only").
#Mathematical entities are indispensable to the best scientific theories. Therefore,
#One must have ontological commitments to mathematical entities.<ref name="Putnam">Putnam, H. ''Mathematics, Matter and Method. Philosophical Papers, vol. 1''. Cambridge: Cambridge University Press, 1975. 2nd. ed., 1985.</ref>

The justification for the first premise is the most controversial. Both Putnam and Quine invoke [[naturalism (philosophy)|naturalism]] to justify the exclusion of all non-scientific entities, and hence to defend the "only" part of "all and only". The assertion that "all" entities postulated in scientific theories, including numbers, should be accepted as real is justified by [[confirmation holism]]. Since theories are not confirmed in a piecemeal fashion, but as a whole, there is no justification for excluding any of the entities referred to in well-confirmed theories. This puts the [[nominalism|nominalist]] who wishes to exclude the existence of [[Set (mathematics)|sets]] and [[non-Euclidean geometry]], but to include the existence of [[quark]]s and other undetectable entities of physics, for example, in a difficult position.<ref name="Putnam"/>