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.
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.
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.
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.
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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.