[FOM] Fictionalism About Mathematics

Vaughan Pratt pratt at cs.stanford.edu
Mon Mar 12 00:56:12 EDT 2012

On 3/10/2012 12:29 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.

Is it in fact the case that many philosophers view a proposition that is 
true "within a story," more broadly in a context, as having been 
falsified by removing it from that context?  Would not that then make 
its negation true?  It would be like saying that, except in the story of 
Santa Claus, Santa Claus does not have a beard.  One would hope 
philosophers handle context shifts better than that, in mathematics as 
in stories.

> 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.

This begs two questions, that unlike stories mathematics is not 
entertaining, and that unlike mathematics stories have no lessons for us 
(such as how, what, when, where, and why).  I have yet to see a 
convincing defense of either position.

> How do mathematical logicians and mathematicians in general react to
 > this fictionalist doctrine?

In case I was not completely clear in my previous post under this 
subject heading where I juxtaposed Harry Potter and eigenvectors in 
response to Kevin Sharp's "one would think that Platonism is 
incompatible with fictionalism," and wrote "Mathematics is an intensely 
psychological experience for its practitioners, who can follow any 
precisely specified plot however intricate as long as those of its 
entities postulated to exist remain coherent," let me state my position 
more directly.

First, the proper juxtaposition of mathematics is not with fiction but 
stories that people can relate to independently of their mathematical 
abilities.  Fiction embraces both: mathematics and stories are instances 
of fiction differing largely in style.

In the other principal respects there is no essential difference. Genres 
in stories (love, war, crime, fantasy, SF) correspond to number systems 
in arithmetic (additive groups, rings, fields, modular arithmetic, 
associative algebras).  Media in stories (plays, books, movies) 
correspond to logical frameworks (set theory, universal algebra, 
category theory).

Mathematics and fiction have in common the central purposes of 
instruction and entertainment.  Granted there are those who do not find 
mathematics entertaining, but the same can be said of stories.  And for 
those who claim not to find stories instructive I would ask, how do you 
decide whether you've been instructed, especially when by such an 
insidious means as an entertaining story?  Stories were *invented* to 
instruct, the more entertaining the more instructive.

Moreover people can become just as absorbed in a mathematical world as 
in a story world, finding in it a temporary reality on which the curtain 
falls entr'acte when it is time to put the book or notepad aside and 
focus on something else for the time being.

As to style, there is no accounting for taste.

As others have noted here, philosophers and mathematicians alike have 
found the fictionalist account of mathematics implausible.  However I 
have yet to see a logically sustainable attack on this account that does 
not boil down to any difference more essential than in minor matters of 

Vaughan Pratt

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