3 ms·

What a load of nonsense. When I prove, say, Cauchy's theorem about groups, it doesn't matter which concrete group I'm operating on to prove it. If I write a si

What a load of nonsense. When I prove, say, Cauchy's theorem about groups, it doesn't matter which concrete group I'm operating on to prove it. If I write a sigma to sum some vectors, it doesn't matter if the vectors are coordinates in 3D space, functions defined on an interval, or just numbers. Mathematical abstractions are pretty much the only ones that don't leak. What, that you have to understand summation to understand what the sigma refers to? Well, duh. That doesn't make the sigma a leaky abstraction, because the sigma is not an abstraction at all! Not even a trivial one. The sigma is notation. And it's hardly news that you have to understand the concepts behind a notation to understand notation itself. The author seems to misunderstand what an abstraction actually is.

countertointel 7y ago
This comment is counter-intelligence. All signs refer directly to abstractions in the mind that can be compiled into a variety of useful forms. Yours does not take precedent in this case, as it is plain to see that sigma is under specified due to the huge range of concepts it’s practical usage covers. Your attempt to prescribe that which should be prescribed is nothing but authoritarianism.
- MrEldritch 7y ago
  What on Earth are you talking about?
- pjc50 7y ago
  Hey, I thought the timecube guy died?
theaeolist 7y ago
Indeed. The author is looking for abstractions where they are not. Abstractions are in Algebra in general and in Category Theory in particular. The essence of Algebra is abstraction. All Algebras start from some intended model but they abstract it into axioms, i.e. "interface-level properties", which completely hide the "implementation".
petters 7y ago
> The author seems to misunderstand what an abstraction actually is. This, I think, is the main problem with the article. I also found the opening line curious: > > I find it rather weird that mathematics is usually taught to people on a historical basis. I think this is much more common in philosophy than in maths.
throwlaplace 7y ago
>If I write a sigma to sum some vectors, it doesn't matter if the vectors are coordinates in 3D space, functions defined on an interval, or just numbers. you're wrong. every "weird" technical hypothesis in a theorem is a leak. even in your super simple example there's a caveat: the sum has to be finite (that mean no you can't perform any sum of any set of vectors). edit: i got downvoted so here's a better example that i just ran across: manifolds with boundary can have empty boundaries. where does this come up? exact 1-forms on S^1 integrate to zero. why? because by Stokes. how? S^1 has a boundary but it's empty.
- jefft255 7y ago
  You can have finite sum of infinite dimensional vectors, this still works...
  - throwlaplace 7y ago
    The sum has to be finite

countertointel 7y ago

This comment is counter-intelligence. All signs refer directly to abstractions in the mind that can be compiled into a variety of useful forms. Yours does not take precedent in this case, as it is plain to see that sigma is under specified due to the huge range of concepts it’s practical usage covers. Your attempt to prescribe that which should be prescribed is nothing but authoritarianism.

MrEldritch 7y ago
What on Earth are you talking about?
pjc50 7y ago
Hey, I thought the timecube guy died?

theaeolist 7y ago

Indeed. The author is looking for abstractions where they are not. Abstractions are in Algebra in general and in Category Theory in particular. The essence of Algebra is abstraction. All Algebras start from some intended model but they abstract it into axioms, i.e. "interface-level properties", which completely hide the "implementation".

petters 7y ago

> The author seems to misunderstand what an abstraction actually is. This, I think, is the main problem with the article. I also found the opening line curious: > > I find it rather weird that mathematics is usually taught to people on a historical basis. I think this is much more common in philosophy than in maths.

throwlaplace 7y ago

>If I write a sigma to sum some vectors, it doesn't matter if the vectors are coordinates in 3D space, functions defined on an interval, or just numbers. you're wrong. every "weird" technical hypothesis in a theorem is a leak. even in your super simple example there's a caveat: the sum has to be finite (that mean no you can't perform any sum of any set of vectors). edit: i got downvoted so here's a better example that i just ran across: manifolds with boundary can have empty boundaries. where does this come up? exact 1-forms on S^1 integrate to zero. why? because by Stokes. how? S^1 has a boundary but it's empty.

jefft255 7y ago
You can have finite sum of infinite dimensional vectors, this still works...
- throwlaplace 7y ago
  The sum has to be finite