Editing Philosophy of mathematics (section)

===Fictionalism<!--'Mathematical fictionalism' redirects here-->===
{{See also|Fictionalism}}
'''Mathematical fictionalism'''<!--boldface per WP:R#PLA--> was brought to fame in 1980 when [[Hartry Field]] published ''Science Without Numbers'',<ref>[[Hartry Field|Field, Hartry]], ''Science Without Numbers'', Blackwell, 1980.</ref> which rejected and in fact reversed Quine's indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as a body of truths talking about independently existing entities, Field suggested that mathematics was dispensable, and therefore should be considered as a body of falsehoods not talking about anything real. He did this by giving a complete axiomatization of [[Newtonian mechanics]] with no reference to numbers or functions at all. He started with the "betweenness" of [[Hilbert's axioms]] to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by [[vector field]]s. Hilbert's geometry is mathematical, because it talks about abstract points, but in Field's theory, these points are the concrete points of physical space, so no special mathematical objects at all are needed.

Having shown how to do science without using numbers, Field proceeded to rehabilitate mathematics as a kind of [[useful fiction]]. He showed that mathematical physics is a [[conservative extension]] of his non-mathematical physics (that is, every physical fact provable in mathematical physics is already provable from Field's system), so that mathematics is a reliable process whose physical applications are all true, even though its own statements are false. Thus, when doing mathematics, we can see ourselves as telling a sort of story, talking as if numbers existed. For Field, a statement like {{nowrap|"2 + 2 {{=}} 4"}} is just as fictitious as "[[Sherlock Holmes]] lived at 221B Baker Street"—but both are true according to the relevant fictions.

Another fictionalist, [[Mary Leng]], expresses the perspective succinctly by dismissing any seeming connection between mathematics and the physical world as "a happy coincidence". This rejection separates fictionalism from other forms of anti-realism, which see mathematics itself as artificial but still bounded or fitted to reality in some way.<ref>{{cite book |last=Leng |first=Mary |date=2010 |title=Mathematics and Reality |publisher=Oxford University Press |page=239 |isbn=978-0199280797}}</ref>

By this account, there are no metaphysical or epistemological problems special to mathematics. The only worries left are the general worries about non-mathematical physics, and about [[fiction]] in general. Field's approach has been very influential, but is widely rejected. This is in part because of the requirement of strong fragments of [[second-order logic]] to carry out his reduction, and because the statement of conservativity seems to require [[quantification (logic)|quantification]] over abstract models or deductions.{{Citation needed|date=March 2023}}