Fictionalism in the Philosophy of Mathematics [pdf]

It is interesting to interpret the body of mathematics as you would a collection of fictional tales. However, the philosophy begins to unravel (to me) when it asserts that &quot;8 is larger than 5&quot; is false while &quot;Sydney is larger than San Francisco&quot; is true because the latter statement &quot;has referents&quot;.<p>What is it that makes Sydney and San Francisco real objects with meaningful sizes while 8 and 5 are not real and do not have meaningful sizes? Sydney and San Francisco are defined by political and legal &quot;stories&quot; in the same way that 8 and 5 are defined in mathematical &quot;stories&quot;. The theory only seems to be consistent if all out-of-context falsifiable statements are taken to be false.<p>This theory placates me, since it leaves the truth value of mathematical statements (in the context of the mathematical story) to mathematicians. However, it renders any conclusions meaningless to mathematics, even if it is meaningful for a philosophy dealing with human stories.

I&#x27;m having trouble understanding the significance of Hartry Field&#x27;s work. It sounds like he replaced the axiomatic system of what we call mathematics with an axiomatic system that he developed specifically for Newtonian gravity, and based on this he was able to claim that mathematics is dispensable. As an undergrad in logic class, I got the impression that mathematics is the discipline that studies axiomatic systems, so if you build an axiomatic system, you&#x27;re doing mathematics. If that&#x27;s true, then isn&#x27;t all he did is just redefining the word mathematics in a very narrow sense and then dispensing with that narrowly defined notion?

Does this represent the state of philosophy of mathematics, because I find this view quite naive. Mathematics is in the business of making a model of the real world and then making falsifiable predictions with it, just like science. Take Fermat&#x27;s last theorem: for all positive a,b,c and n&gt;2, the value a^n + b^n - c^n never comes out 0. This is an empirically falsifiable statement. You can even make it a statement about real world objects if you wish: represent a number by a jar of coins, and do addition C=A+B by filling jar C with the coins in jar A and jar B together, multiplication AB by taking a whole jar B for each coin in A, etc. Rules of mathematical deduction are just devices for making predictions. So mathematical objects are neither &quot;real&quot; as in platonism, nor meaningless as in fictionalism.

There&#x27;s a bit of a litmus test you can apply to belief systems to find out if they&#x27;re definitely objective.<p>An intelligent alien species that&#x27;s never met humans would almost certainly invent math. The syntax and organization would probably be different, but the rules would be the same.<p>On the other hand, aliens would almost certainly not write The Hobbit.

This is absurd. Clearly mathematical statements are highly predictive of the real world. They aren&#x27;t false.<p>These people seem confused over the meaning of the word &quot;exist&quot;. Regardless of whether or not numbers &quot;exist&quot;, we can show that objects in the real world can obey the same laws as abstract numbers. If I have 2 apples, and take 2 more apples, I will never have 5 apples. The properties of math are real and apply to the real world.<p>If you insist on modelling philosophy on the language we happen to use, then just treat numbers as adjectives. As if 5 is a property an object can have, rather than a physical object itself. You don&#x27;t need to worry about 2 &quot;existing&quot; any more than worrying about &quot;tallness&quot; exists, when talking about objects that are taller than other objects.

Whatever one chooses to say about the truth or falsity of mathematical statements (&amp; imo, calling them all false is ludicrous), it is hard to refute the argument that they are quite <i>effective</i>. See Hamming and Wigner[0,1]<p>[0]<a href="http:&#x2F;&#x2F;www.dartmouth.edu&#x2F;~matc&#x2F;MathDrama&#x2F;reading&#x2F;Hamming.html" rel="nofollow">http:&#x2F;&#x2F;www.dartmouth.edu&#x2F;~matc&#x2F;MathDrama&#x2F;reading&#x2F;Hamming.htm...</a><p>[1]<a href="https:&#x2F;&#x2F;www.dartmouth.edu&#x2F;~matc&#x2F;MathDrama&#x2F;reading&#x2F;Wigner.html" rel="nofollow">https:&#x2F;&#x2F;www.dartmouth.edu&#x2F;~matc&#x2F;MathDrama&#x2F;reading&#x2F;Wigner.htm...</a>

&gt; Fictionalism in the philosophy of mathematics is the view that mathematical statements, such as ‘8+5=13’ and ‘π is irrational’, are to be interpreted at face value and, thus interpreted, are false.<p>Arrrgh! So annoying! How much time will it pass until philosophers of mathematics finally understand that mathematical truth has nothing to do with philosophical truth?<p>&gt; Fictionalists are typically driven to reject the truth of such mathematical statements because these statements imply the existence of mathematical entities, and according to fictionalists there are no such entities.<p>Crash course in logic: If mathematical objects don&#x27;t exist, then statements about them aren&#x27;t “false” - they&#x27;re <i>meaningless</i>.

Fictionalism, and other formalist theories, will have to confront that problem that when we speak of mathematics as abstract rules governing strings of symbols, these rules themselves are mathematical. So it only replaces &quot;numbers are real&quot; with &quot;abstract symbols are real&quot;. There are axiomatic systems that are strong enough to express manipulations of abstract symbols, but weaker than the usual systems that mathematicians deal with (e.g. see the work of Edward Nelson on so called predicative arithmetic). But to my knowledge these have not been explored much.