> The move from ‘two numbers have the same same sum whichever way round you add them’ to e.g. ‘\forall x\forall y (x + y = y + x)’ changes the medium but not the message. And the fact that you can and should temporally ignore the meaning of non-logical predicates and functions while checking that a formally set-out proof obeys the logical rules (because the logical rules are formalized in syntactic terms!), doesn’t mean that non-logical predicates and functions don’t any longer have a meaning!

Being able to use informal language to loosely describe something which also has a formal description is not Platonism. In fact, the only reason the statement "a sum is the same regardless of which way you order the arguments" functions as a language or math is because most people learn mathematics on the same set of real numbers and they take the domain for granted. In fact, I'd hazard a guess that most people don't even know there are other domains. This is not the secret operation of a world of "true" ideals about adding things. It's simply implicit information.

In my journey through academia I would say I encountered many Platonist type thinkers in math and CS departments. There was definitely a popular notion that math contained some capital T, Truth. I have always been a bit suspicious of this notion. I encountered my first real critique of it in a work of pop philosophy of all places, 'Zen and the Art of Motorcycle Maintenance'. A theme of the book is criticism of the kind of "cult of reason" which operates in most universities. Particularly there is a section about mathematics. Pirsig's contention is that math is useful and it happens to be useful because we have invented it to be so. Math appears without contradiction because we invented it to not contain contradiction. We create these base assumptions and then reject anything that creates a contradiction.

Pirsig points to the non-Eucludian geometry of Poincare as an example where we can drastically change the nature of the math itself by simply changing the base assumptions. Suddenly, what was contradictory, is no longer contradictory. For me this really solidified in my mind when learning Linear Algebra. Linear Algebra is all about laying out these 6 odd rules about vectors and scalars and then seeing where they take you and where they don't for that matter.

Poincare's math turned out to have applications but it didn't have to. There are many dead ends in the world of high mathematics. And so Pirsig's ultimate observation can be boiled down to, Math is not true but it can be useful.

This would later be echoed to me when I first heard the now popular saying amongst data scientists, "all models are false but some models are useful." This is how I've come to view mathematics. It's a giant pile of really well thought out and organized, often interconnecting models. Some of them match certain natural phenomena well. Others not at all. Some of them are obviously useful. Some of them are only discovered to be useful long after their invention (quaternions). Some of them may never prove useful.

Ultimately I'd answer any contentions of math as model building and the author's tribulations about language here with this same observation. What happens when you want to take seemingly unrelated branches of math and connect them? What do you reach for? Set builder notation. x(weird E thing)(two legged R) blah blah blah for all of your constructs and then you start operating.

You start by laying out all your assumptions in symbolic language. Then you let loose with the consequences of those definitions. Which sure sounds a lot like what modellers do in English when they're first proposing a model. The grand unifying layer of mathematics is a kind of symbolic modeling.