Ordinary mathematical claims are best interpreted at face value as making claims about objects, because typical mathematicians do not have positive intentions to be speaking nonliterally
(The ontological commitment allegedly forced by scientific realism combined with the indispensability argument)
The view that abstract mathematical entities exist independently of minds and physical reality
mathematical anti-realism(as the position being rejected in this statement)
The opposite view: the idea that mathematical objects don't really exist independently, and mathematical statements are just useful tools or ways of talking about patterns.
nonliterally(as used in philosophy of language)
Speaking in a way that's not meant to be taken as straightforward truth—like using metaphors or symbolic language instead of direct statements.
positive intentions(as used in this statement about what mathematicians intend when they speak)
Deliberate, conscious purposes or goals that someone actively chooses to pursue.
But platonists and fictionalists are not committed to the thesis that people have positive intentions to be talking about abstract objects. Rather, they can say the following: (i) ordinary mathematical claims are best interpreted at face value—and, hence, as making claims about objects—because typical mathematicians (and, indeed, typical examples of ordinary folk) do not have positive intentions to be speaking nonliterally when they utter mathematical sentences; and (ii) there are features of th