[FOM] Fictionalism About Mathematics

[ARF (Richard L. Epstein)]

In my recent book *Reasoning in Science and Mathematics* (available from the Advanced Reasoning Forum) I present a view of mathematics as a science like physics or biology, proceeding by abstraction from experience, except that in mathematics all inferences within the system are meant to be valid rather than valid or strong.  In that view of science, a law of science is not true or false but only true or false in application.  Similarly, a claim such as 1 + 1 = 2 is not true or false, but only true or false in application.  It fails, for example, in the case of one drop of water plus one drop of water = 2 drops of water, so that such an application falls outside the scope of the theory of arithmetic.

On this view numbers are not real but are abstractions from counting and measuring, just as lines in Euclidean geometry are not real but only abstractions from our experience of drawing or sighting lines.  The theory is applicable in a particular case if what we ignore in abstracting does not matter there.

I have lectured on this to three mathematics departments, where most of those in attendance were applied mathematicians.  Each time the mathematicians were unanimous that this gave a better idea of what they were doing, while providing some guidance for how they might devise new mathematical theories.  Many said that my views were commonplace.  Further, some of them said they were quite relieved to know that they needn't be platonists to do mathematics.

Harry Deutsch wrote:

> The view that mathematical objects are fictitious and that "strictly speaking"  seemingly true mathematical statements such as 5 + 6 = 11 are false, though they are true in the "story" of mathematics, is currently a very popular philosophy of mathematics among philosophers.  The claim is that such fictionalism solves the epistemological problem of how mathematical knowledge is possible, and it solves the semantical problem of providing a uniform semantics for both mathematical and non-mathematical discourse.  Fictionalist have also tried to address the obvious question of how, if mathematics is pure fiction, it nonetheless manages to be so useful in the sciences and in daily life. But I won't go into that here.  My question is this:  How do mathematical logicians and mathematicians in general react to this fictionalist doctrine?  I realize that it may not be clear whether or how the doctrine might affect foundations or one's view of foundations.  But I thought I would addres!
>  s this question to the FOM group since work in foundations and work in the philosophy of mathematics are intertwined.  Let me put it this way:  This fictionalism about mathematics is taken very seriously by philosophers of mathematics, but I  doubt that mathematicians would find it at all appealing.
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> Harry Deutsch
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