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My favourite way of thinking about these operations comes from category theory: * addition: the cardinality of the disjoint union of two sets * multiplication
by Garlef 6y ago
My favourite way of thinking about these operations comes from category theory:
* addition: the cardinality of the disjoint union of two sets
* multiplication: the cardinality of the direct product of two sets
* exponentiation: a^b is the cardinality of the set of maps from B to A where B is a set of cardinality b and A is a set of cardinality a
- Rerarom 6y agoThis is just set theory, not category theory.
- Garlef 6y agoNeither is more foundational than the other. So I think the use of "just" is not justified here. https://mathoverflow.net/questions/360578/category-theory-and-set-theory-just-a-different-language-or-different-foundati https://mathoverflow.net/questions/360578/category-theory-an...
- deleted 6y ago[deleted]
- nicky0 6y agoBut talking about category theory makes you sound smarter.
- Garlef 6y agoI'm personally more scared of set theory.
- eigenket 6y agoYeah but your comment didn't use the concept of a "category" at all, whereas you used "set" three times. Its definitely more natural to say these definitions come from set theory.
- Garlef 6y agoTo me, "disjoint union", "direct product" and "set of maps" (as a special case of "internal hom") are the central keywords in what I wrote: Concepts from category theory.
- eigenket 6y agoThose keywords all appeared in the context of set theory long before category theory was thought of. Even ignoring that as a historical accident (we invented set theory before category theory), those ideas are all natural, meaningful and interesting to examine in the context of sets even if one has never heard of a category.
- spekcular 6y agoYes, there's no need to invoke any category theory concepts here. Everything is happening in the category of sets.
- spekcular 6y agoFirst, that's not what your link says. (See, e.g., the comments to the first answer.) Second, I find that claim highly dubious. To do metamathematics and talk about the relative strength of various axiomatic systems, you need to talk about large cardinals. So that, at least to me, seems like a good argument that set theory is foundational to mathematics in a way that category theory is not, since the latter has no non-eliminable place in the (study of) contemporary foundations of mathematics.
- t0r0nat0r 6y agoThis seems a bit restricted. How do you go about extending it to real and complex numbers?
- Garlef 6y agoFrom what I know: This is still an open question. Here's a list of some approaches. https://mathoverflow.net/questions/310004/categorifications-of-the-real-numbers https://mathoverflow.net/questions/310004/categorifications-...
- AzzieElbab 6y agoThat is exactly how I explained multiplication to my 5 years old, right after getting him into programming by playing with profunctor optics
- Garlef 6y agoI learned adding from counting disjoint unions of dots. I learned multiplication from counting squares of dots. But agreed: I learned about exponentiation first via repeated multiplication.
- AzzieElbab 6y agoThat is very cool. I prefer geometric/visual approaches. Like move that many steps along the line, figure out area of a rectangle and so on
- YeGoblynQueenne 6y agoYou waited 5 years to teach programming to your 5 year old?
- deleted 6y ago[deleted]
- threatofrain 6y agoCan this perspective accommodate fractional exponents, like 2^(0.5)?
- Garlef 6y agoNo (At least not yet: open research question). But neither can "repeated multiplication". How would you multiply 2 with itself "a half" times?
- prionassembly 6y agoOperationally, unsure. But if there was an answer, it would be something that multiplied with itself twice would have be the same as 2 multiplied with itself once. I think it was Gerolamo Cardano who in a book of worked exercises came up with a problem like "obtain two numbers that add to 10 and multiply to 40" and arrived at the solution had to involve square roots of -1. Then he goes "this makes no sense, but let's calculate nevertheless..."
- Garlef 6y ago> it would be something that multiplied with itself twice would have be the same as 2 multiplied with itself once from what i recall that's the way to go: You can use this to extend the definition of exponentials as "repeated addition" from allowing only natural numbers expoonents to allowing positive rational numbers. Afterwards you can "complete" this to allow real exponents. But this requires some hard maths to do rigorously.
- prionassembly 6y agoYou're absolutely right about rigor: I'm not claiming that Cardano discovered complex analysis or anything...
- johndoe42377 6y agoYes, very clever social signaling of assumed smartness. However, both set theory and category theory are canonical examples of synthetic purely abstract theories, which include paradoxes, while repeated addition (which is just merging together) is deeply grounded in reality. Putting an abstract theories first is the same flaw as with Froidian psychology, Marxism, and all the other abstract bullshitting, from Hegel onwards.