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> ... then that person rejects the Axiom of Choice. You're looking at it from a philosophical perspective where people care about whether the axiom is "true"
by pron 5y ago
> ... then that person rejects the Axiom of Choice.
You're looking at it from a philosophical perspective where people care about whether the axiom is "true" or not. Other approaches to mathematics don't care, because they view its value not in some metaphysical notion of truth but in utility. They might say: this proof requires the axiom of choice and that proof doesn't, without assigning either one any value beyond utility (where utility might be, in some case, just their intellectual edification). I think this is what the article is trying to drive at as well. We can't "accept" or "reject" the axiom of choice based on some philosophical grounds, because those philosophical grounds would need to involve infinities [1], which, themselves, are not philosophical but mathematical. We either choose to use it or not based not on where we come from but on where we want to go.
[1]: The view that mathematical philosophy could be derived from core basic principles outside of mathematics was called Logicism, and was favoured by Bertrand Russell. Unfortunately, mathematical systems without some mathematical (i.e. non-"logical") axioms are quite limited, and almost all of them include at least one mathematical axiom, usually one that is equivalent to the existence of the natural numbers.
- cannabis_sam 5y ago>they view its value not in some metaphysical notion of truth but in utility. Isn’t this a philosophical perspective?
- pron 5y agoIt is a perspective in the philosophy of mathematics that rejects the idea that all of mathematics is necessarily anchored in philosophical truths beyond itself.
- cannabis_sam 5y ago> It is a perspective in the philosophy of mathematics So it is a philosophical perspective…?
- pron 5y agoOf course. I think you may have misread my original comment, which compared two philosophical perspectives on mathematics, one philosophical perspective that views mathematics as an expression of some truth beyond mathematics, and another philosophical perspective that sees mathematics as a tool to achieve some ends.
- cannabis_sam 5y ago> not in some metaphysical notion of truth but in utility. So one is not metaphysical because it is a tool to achieve ends? (Sorry, if I come across as argumentative, I don’t mean to. I’m genuinely just curious.)
- jfengel 5y agoIt is a metaphysical statement about whether metaphysics is important in this instant, rather than being dispensed with in favor of something useful. It's picking a particular metaphysics rather than being bogged down in the meta-metaphysical question of the necessity of metaphysics. Or at least, that's the way I read it. The question of the value of metaphysics is an open one, which easily comes to dominate other questions if allowed to. I think the OP is saying, "I'm going to make a metaphysical commitment, and let other people worry about whether doing so is good metaphysics." (And there's good reason to think that it isn't -- but it's unclear whether that matters, or should matter.)
- cannabis_sam 5y agoYeah, I misread. Thanks for helping me understand!
- pron 5y agoFor some reason you keep skipping the qualifiers in what I write. It does not say "some metaphysical notion" but "some metaphysical notion of truth". In other words, what matters isn't whether an axiom is "true" or not in some way that is external to mathematics, but what use could be made of that axiom. You did the same above; I didn't write "from a philosophical perspective" but "from a philosophical perspective where ..."
- eeegnu 5y ago> Unfortunately, mathematical systems without some mathematical (i.e. non-"logical") axioms are quite limited What's an example without any axioms? It doesn't seem like you can do anything without say ∀x: x=x (which I believe is reciprocal identity.)
- pron 5y agoI think Russell would consider that a logical axiom, and it commonly is an axiom of predicate logic with equality. BTW, by "logic", he didn't mean just a specific set of familiar axioms, but any axiom which could be considered "obviously true" and didn't involve numbers specifically. https://en.wikipedia.org/wiki/Logicism https://en.wikipedia.org/wiki/Logicism