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John Baez (Oct 03 2024 at 15:55):On Mathstodon, Boarders wrote:
Platonism vs nominalism: an old fashioned debate in philosophy is that of universals - talking of triangle as a property commits oneself to the existence of the abstract concept of triangle or perhaps a “generic triangle”.
In this case, we can, to some extent, resolve the question mathematically in a precise way - there is a mathematical object called the moduli stack of triangles, the generic triangle is the generic point of this moduli stack. We can answer various precise questions about this generic triangle such as what automorphism group it has etc. It also specializes to various more particular triangles with a precise notion of specialize due to algebraic geometers. It seems to me this is a better formal apparatus for thinking about mathematical universals than any kind of vague waffle available, but the idea has not managed to escape the purview of mathematicians.
David Corfield (Oct 03 2024 at 16:48):There's a huge industry in philosophy in working out what's meant by when we say "Let be a natural number". A common position has it that refers to an "arbitrary natural number" which is a different thing from the individual natural numbers and has only generic properties.
Andrej Bauer at his curtest on the subject here:
What I am saying is that "a free variable is a projection" is a great deal better at explaining various perceived mysteries about free variables than fancy words from philosophy books. But your mileage may vary.
A two-pronged attack with Egbert Rijke writing:
The generalised elements of an object are morphism into . Under this point of view, the generic element of is the identity morphism on .
John Onstead (Oct 04 2024 at 02:43):Category theory actually gives us the most powerful ever version of finding "the generic X". This process is known as "externalization" and it involves going from a specific instance of X to viewing it as a functor out of a particular category, known as the "generic X" or "walking X". In a sense, the "walking X" is the platonic ideal of X, the archetypal or prototypical X. Making statements about the walking X then you can specialize to all instances of X across mathematics. There's many examples ranging from the "walking morphism" A -> B which is the archetypal morphism, to the Lawvere theory of groups which is the archetypal group object. This is also related to the notion of classifying object, which is the formal dual of the walking object since every morphism into it represents an instance of X.