Mathematics and Reality by Mary Leng

By Mary Leng

Mary Leng deals a safeguard of mathematical fictionalism, in response to which we haven't any cause to think that there are any mathematical gadgets. might be the main urgent problem to mathematical fictionalism is the indispensability argument for the reality of our mathematical theories (and for that reason for the life of the mathematical items posited through these theories). in accordance with this argument, if we now have cause to think whatever, we have now cause to think that the claims of our greatest empirical theories are (at least nearly) real. yet considering that claims whose fact will require the lifestyles of mathematical gadgets are integral in formulating our greatest empirical theories, it follows that we've got stable cause to think within the mathematical gadgets posited through these mathematical theories utilized in empirical technology, and for that reason to think that the mathematical theories used in empirical technological know-how are actual. prior responses to the indispensability argument have focussed on arguing that mathematical assumptions could be allotted with in formulating our empirical theories. Leng, in contrast, deals an account of the function of arithmetic in empirical technology based on which the winning use of arithmetic in formulating our empirical theories don't need to depend upon the reality of the maths applied.

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It is usually hopeless and pointless to determine just what microphysical states lapsed and what ones supervened in the event, but some reshuffling at that level there had to be; physics can settle for no less. If the physicist suspected there was any event that did not consist in a redistribution of the elementary states allowed for by his physical theory, he would seek a way of supplementing his theory. Full coverage in this sense is the very business of physics, and only of physics. Whether or not Quine is right about this presumption of supervenience is not particularly important as regards our own application of the naturalistic approach to answer ontological questions about whether there are mathematical objects.

In holistically viewing all the utterances used to express our best physical theories (when properly regimented) as equally confirmed by our theoretical successes, Quine’s claim is that in our ultimate, naturalism and ontology 39 cleaned up, and carefully regimented best theoretical efforts to describe and organize our experience, the distinction between merely practical and genuinely evidential reasons to speak as if there are φs cannot be made. It is only against the backdrop of this assumption that Quine’s response to Carnap (that practical reasons can also be evidential) allows him to hold that, in our best science, practical reasons are always evidential, such that a reason to utter a sentence ‘there are φs’ in the context of the theoretical framework of our best science is automatically viewed as a reason to believe that there are φs.

But, on Carnap’s (1950: 208) view, practical reasons to speak as if there are φs do not count as reasons to believe in the reality of φs, for ‘there is no such belief or assertion or assumption’. As a result, on Carnap’s view we cannot hope to answer the philosophical question of what (we ought to believe that) there is, for there is no meaningful question for us to answer. Carnap presents this worry in his (1950) paper ‘Empiricism, semantics, and ontology’, where he puts forward his famous distinction between internal and external questions regarding the existence claims of a given discourse, or linguistic framework.

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