[FOM] Fictionalism About Mathematics
charlie
silver_1 at mindspring.com
Sun Mar 11 20:19:54 EDT 2012
What I wonder about is how fictionalists account for mathematical truth. For instance, physics is supposed to be true about the world, and experiments can confirm or disconfirm whether something is true. We don't resort to the world in the same way in math, yet at least for elementary math, confirmation is possible. To take a very simple example, 3 pencils plus 4 pencils = 7 pencils, and this can be verified. As math becomes more and more abstract, it becomes more difficult to correlate mathematical principles through verification. Yet there still are often means to generalize math and then verify those generalizations. The question in my mind would be when there's no possible way to independently verify mathematical principles. Of course, lots rides on the word "independently". However, even very abstract mathematical principles are founded on those that are less abstract, and the less abstract ones may be verified. Then--I'm conjecturing here--the more abstract principles might be verified because they would not work if the less abstract--and already verified--ones have not been confirmed. So, in a sense I see a ladder, with the bottom rungs confirmed and those more abstract ones above necessarily being true if the more mundane ones below are. To what extent this is possible and how high one could ascend in this manner, I do not know. This line may tie mathematics too tightly to the world for those espousing math as abstract objects (though the view of math as abstract objects may consist only from the generalization of principles being much higher up the ladder than their physicalist underpinnings).
I personally reject the essential Hersh line that math is like a game of baseball with referees making the decision of what's right and what's wrong. Mathematical referees would be, according to him, a body of experts who stamp their imprimaturs on theorems. One reason I think that's wrong is that experts can be proven wrong long after a theorem has been previously endorsed. Hersh could still maintain his view, arguing that new referees can always be later added to the referee group. This may be a good account of what we accept as mathematical theorems, but not as mathematical truth, since I believe we all agree--don't we?--that a long-lasting mathematical theorem can nevertheless be wrong no matter how many experts for how long a time have endorsed it (even when higher mathematics is based on an earlier error). Thus, to me, mathematical truth is out there beyond our fictions.
Charlie
P.S. One could maintain that there is an idealized body of referees who verify abstract mathematical theorems, but the "idealized body" would be as abstract as the theorems they evaluate.
On Mar 10, 2012, at 6:13 PM, Kevin Scharp wrote:
> The results of a recent poll of philosophers on prominent philosophical issues is here: http://philpapers.org/surveys/results.pl
> The second question is on abstract objects, and "Platonism" is the most popular answer with 39.3%. The question isn't directly on fictionalism in philosophy of mathematics, but mathematical entities are surely one of the most familiar purportedly abstract objects and one would think that Platonism is incompatible with fictionalism.
> Kevin
>
> On Sat, Mar 10, 2012 at 5:30 PM, Richard Heck <rgheck at brown.edu> wrote:
> On 03/10/2012 03:29 PM, 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.
> For what it's worth, I don't know how true this characterization is. There are philosophers of mathematics, some of them quite prominent, who defend fictionalist views, and there are others, also quite prominent, who oppose them. Then there are others who ignore the whole debate, who find the fictionalist line sufficiently implausible, or what have you, to be bothered with it. And among those, you will likely find many who would agree with Lewis's famously funny rebuttal of fictionalism and its kin in *Parts of Classes*.
>
> One would obviously have to take some kind of formal poll to find out what the percentages are, but I'm not convinced myself that fictionalism is taken seriously by anything like the majority of philosophers of mathematics.
>
> Richard
>
>
> --
> -----------------------
> Richard G Heck Jr
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> Brown University
>
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