Comment by practal
2 years ago
Oh, it does give you special insight, but it doesn't make you infallible. It is easier to reach for something if you feel that it must be out there, and you just haven't got it right yet. That's how most mathematicians feel. Gödel was a Platonist, and I'd say he had really special insights.
So there's no actual advantage, since you can still be wrong and grasping for something that doesn't exist or that you can't define properly. It's just adding more mysticism to mathematics.
> It's just adding more mysticism to mathematics.
No, it is actually the opposite. It is removing mysticism (which I hate) and adds clarity. There are real things out there, and we can go and use them for our purposes. These things are so flexible that we can shape them into pretty much anything we want, bar inconsistency.
But yes, you need to state clearly the properties you are talking about. These can be axioms (and must be to a certain extent), but most of your math will consist of definitions (and theorems about them). And by doing all of this in a logic, on a computer, there is really no way to be much clearer and less mystic.
Thinking purely conceptual notions are real is mysticism. It's a confusion of layers, like in an Escher print: The hand is drawing itself, the water is going around infinitely, because the background is bleeding into the foreground and there's no consistent hierarchy. Well, an idea I have doesn't get to be as real as I am, too; I can't make a sphere be just as real as I am just because I imagined a sphere, any more than I can put a million dollars into my hand just because I imagined it to exist.
> But yes, you need to state clearly the properties you are talking about.
This is normal mathematics. It isn't unique to Platonism.
> And by doing all of this in a logic, on a computer, there is really no way to be much clearer and less mystic.
A software object, as a pattern of charge in memory circuits or in a storage device, isn't a Platonic form. It is real and it has all of the limits a real, physical object does, just like a bunch of marks on a blackboard or ink on a page. It is, conceptually, just notation, albeit one which is much more useful in some ways than paper notation. Platonism, as I understand it, posits that the purely conceptual is real, and partakes of forms, and all representations of those concepts are shadows of those forms.
Except... is Number a Form? If so, it would have one consistent set of properties, yet mathematicians have a lot of different kinds of numbers, with different properties, all useful and consistent within a given axiom system. Real numbers, Integers, Natural numbers, Complex numbers, Quaternions, Octonions, and Sedenions are all numbers, but no two of them are the same, and no two of them can, therefore, be of the same Form. In fact, we can apply what's called the Cayley-Dickson construction to create as many different kinds of numbers as we wish, to infinity, no two of which are the same but all of them are consistent and have some claim on Number-ness. If the form Number can be stretched that far, it isn't very good at predicting the properties things will have, and so it isn't all that useful, even aside from the objections I have to abstract concepts being real on the same level as me.
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> So there's no actual advantage
The advantage of a robust philosophy of mathematics is that what you're doing actually makes sense. Some people care about conceptual clarity, and others don't. To each their own.