[FOM] Fictionalism About Mathematics
Justin D'Ambrosio
jzdpendragon at gmail.com
Sun Mar 11 17:59:19 EDT 2012
Fictionalism about mathematics seems to be a strange view in the following sense: Mathematics is, or is nearly, a paradigm case of knowledge. We are as certain of the claim that 2 + 2 = 4 as we are of anything else in our "web of belief", and we don't appear to mean this non-literally. But given that mathematical truths are so central to our thought, and we are so certain of them, it seems like they must be true strictly speaking if anything is. And yet the Fictionalist claims that mathematical statements are not true strictly speaking, they are only true in some ersatz sense.
It seems more reasonable to grant mathematical truths the status of "strictly speaking true", and then investigate what our semantic theory has to look like in order to accommodate the literal, bona-fide truth of mathematical claims. One reason that I think the fictionalist denies the literal truth of mathematical claims is because we have defined "literal truth" quite narrowly: as direct correspondence to some domain of objects under a standard referential semantics. That is to say, on a standard referential account of semantics, the literal truth of mathematical claims practically implies Platonism. But why think that this is the only reasonable account of truth in mathematics? Why think that this is the only reasonable account of truth in natural (or any) language?
Various alternatives to referential accounts of meaning and truth are available, many of them among the candidates that Colin McLarty outlined in his earlier message. Particularly relevant to the points made above is contextualism about truth, which, on some accounts, denies that the truth of a statement requires objects as the references of the terms in that statement, or as satisfying the statement's existential commitments. Terence Horgan has numerous papers on Contextualism, and he takes himself to be drawing on Lewis' claim that there are hidden contextual parameters in many ordinary types of statements. Perhaps this might be a direction for new work if one is concerned about Platonism, but unhappy with other competing philosophies of mathematics.
Justin D'Ambrosio
On Mar 10, 2012, at 10:41 PM, ARF (Richard L. Epstein) wrote:
> 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 addr!
> es!
>> 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.
>>
>> Harry Deutsch
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