Platonism in the Philosophy of Mathematics (2018)
The correct answer is: &quot;why give a fuck?&quot;.<p>The only practical value of having an answer to this question is giving philosophy students something to write about.<p>Mathematics is a human activity, so whether you think it&#x27;s &quot;exploring&quot; or &quot;inventing&quot; mathematics, you would agree that we aren&#x27;t pursuing the exploration or invention in <i>all</i> possible directions.<p>So let&#x27;s look at what we, the humans, actually <i>do</i>.<p>Any mathematical object ever defined is ultimately an abstraction and a model of something the humans actually experience in the physical world.<p>Number theory is abstract, numbers are not. You can count your sheep in a pasture; you can count your neighbor&#x27;s. Natural numbers model  the idea that neither of you has more than the other.<p>This is <i>just a model</i>, of course; you have a sheep named Dolly which is dear to you, and your neighbor has one named Sue which gives more wool than any sheep in the county.<p>So you really <i>don&#x27;t</i> have the same thing, because these sheep are not interchangeable. But we have a concept — <i>natural numbers</i> — which <i>models</i> the idea of sameness if we, for a second, <i>forget</i> that.<p>It&#x27;s an <i>imperfect</i> model; the concept of something that can grow without bound is an outcome of this imperfection. When you use numbers, you might as well assume that 10^(10^100) is the largest number there is. Numbers of that scale are so large, that all computers in the world taken together aren&#x27;t enough to just <i>store</i> this number. But modeling the concept of &quot;<i>not knowing what the bound is</i>&quot; is modeling an experience we run into all the time. 640KB isn&#x27;t enough for everyone.<p>We dispense with this abstraction with ease, though. Most calculations in practice are done with 64-bit &quot;numbers&quot;. These numbers don&#x27;t behave like the one in math do, but it&#x27;s OK.<p>Even simpler, our clocks have just 12 numbers, even though the time doesn&#x27;t stop running.<p>Euclidean geometry can be verified experimentally. Cut out a triangle out of paper; tear off the angles; put them together, and see that they make up a straight line.<p>It&#x27;s an imperfect model, and you don&#x27;t need Einstein or Lobachevsky to tell you that. Long-distance sailing made people run into this model breaking. The corners of a triangle you cut out of orange peel with a straight knife won&#x27;t make a straight line.<p>But that only gives rise to other kinds of geometries being interesting to us.<p>People do math because it&#x27;s <i>beautiful</i>, and something that doesn&#x27;t connect to any human experience is decidedly <i>not</i> beautiful.<p>Take a deck playing cards, or just a dozen cards, and come up with a set of rules of how a pair of cards can be traded for a single one. You can make different rules depending on which of the two cards is on top; 2-for-1 is the only restriction.<p>We could come up with an immense number of complicated systems that way. But most of them would be so arbitrary, they won&#x27;t connect to anything. So nobody cares about systems like that. Unless they come from <i>something else</i> and therefore have more than an arbitrary structure.<p>Like finite symmetry groups, e.g. rotations of a cube. You can model that with such card rules. <i>Then</i> it&#x27;s interesting.<p>So the question of <i>whether</i> mathematical objects exist is beside the point.<p>The reality is that <i>all</i> mathematical objects that <i>humans</i> care to study are, ultimately, <i>modeling</i> an experience that does.<p>And this is consistent with mathematics truly being an art, not a science.<p>Because art that doesn&#x27;t <i>abstract</i> human experience in some way is not something we care about. Strings of letters spit out by a random number generator aren&#x27;t an exciting read.<p>That&#x27;s what Vladimir Arnold means when he says, seemingly controversially, that mathematics is a branch of physics (in which experiments are cheap)[1].<p>It&#x27;s that mathematics that doesn&#x27;t connect to the world we live in is too <i>boring</i> to exist - or be invented and remembered.<p>[1] <a href="https:&#x2F;&#x2F;www.uni-muenster.de&#x2F;Physik.TP&#x2F;~munsteg&#x2F;arnold.html" rel="nofollow">https:&#x2F;&#x2F;www.uni-muenster.de&#x2F;Physik.TP&#x2F;~munsteg&#x2F;arnold.html</a>