Proofs and the Logical Structuring of Mathsy Stuff
Let’s contrast the two perspectives we have considered so far. On the one hand we have the Platonist idea that math is just about unchanging mathematical stuff like numbers and shapes, which our activities cannot influence in any way. On the other hand, we can see the importance of math as being bound up with the activity and creativity of the people doing it. In the second case, proofs and demonstrations can be seen as not being directed at changing the mathsy stuff, but as guiding us through a process of coming to know and understand how mathematics fits together. But on the first, Platonist point of view, what is the point of proving things?
Figure 21.1 Minifig Gottlob Frege and minifig Bertrand Russell chatting about LEGO and math. Created by the author using http://www.ldraw.org/.
Certainly, proofs are about coming to know things about the mathsy stuff, but the question is what we are coming to know exactly? Many modern Platonists tend to endorse a kind of foundational- ism. We might attribute this idea to Gottlob Frege (1848-1925) and Bertrand Russell (1872-1970), although the latter famously changed his mind a lot. The foundationalist idea is that we can build math “from the ground up.” If we want to be secure and rigorous in our reasoning and proving, then we want to make sure no bad assumptions sneak in. For this, you set out the basic principles that hold for the mathematical things you are interested in and the logical rules by which you can figure out new truths. Historically, an important reason for wanting to stop sneaky extra assumptions being used and for making everything explicit was to do with concerns about the use of infinity in math at the end of the nineteenth century and start of the twentieth century. This issue was becoming ever more crucial for analysis and calculus, both central to modern mathematics but relying at the time on potentially spurious uses of the notion of infinity.
I think one way to understand the point of proving for the Platonist is via an analogy to showing off a mighty structure made out of LEGO.
If you want your mighty LEGO structure to stand up, it needs to be built up solidly all the way from the base to the top. Each brick needs to support those above it and in turn be supported by those below it. For each brick, we can check exactly which others it is resting on and which bricks above depend on it. The full model that is built up this way is also a model of mathematics: for the foundationalist (and therefore also for the Platonist with foundationalist leanings) the
Figure 21.2 A mighty structure! Created by the author using http://www.ldraw.org/.
importance of a proof is that it shows the logical structuring of the math in much the same way the diagram above shows the structuring of our mighty LEGO build.
One big way in which the analogy between building a foundation for mathematics and building the mighty structure in LEGO breaks down is that you might well think we really need to build a model out of LEGO bricks for that model to exist. For the Platonist, the structuring of mathematics is already existent: it is simply the abstract objects and the relations between them. So the purpose of proving on this view is to see or reveal the structure, not to create it afresh.