Comment by jalapenos
21 hours ago
> Take R as an ordered field with its usual topology and ask for a finite-dimensional, commutative, unital R-algebra that is algebraically closed and admits a compatible notion of differentiation with reasonable spectral behavior.
No thank you, you can keep your R.
Damn... does this paragraph mean something in the real world?
Probably I've the brain of a gnat compared to you, but do all the things you just said have a clear meaning that you relate to the world around you?
I'm sure you don't have the brain of a gnat, and, even if you did, it probably wouldn't prevent you from understanding this.
As for whether these definitions have a clear meaning that one can relate to 'the world': I think so. To take just one example (I could do more), finite-dimensional means exactly what you think it means, and you certainly understand what I mean when I say our world is finite (or three, or four, or n) dimensional.
Commutative also means something very down to earth: if you understand why a*b = b*a or why putting your socks on and then your shoes and putting your shoes on and then your socks lead to different outcomes, you understand what it means for some set of actions to be commutative.
And so on.
These notions, like all others, have their origin in common sense and everyday intuition. They're not cooked up in a vacuum by some group of pretentious mathematicians, as much as that may seem to be the case.
> does this paragraph mean something in the real world?
It's actually both surprisingly meaningful and quite precise in its meaning which also makes it completely unintelligible if you don't know the words it uses.
Ordered field: satisfying the properties of an algebraic field - so a set, an addition and a multiplication with the proper properties for these operations - with a total order, a binary relation with the proper properties.
Usual topology: we will use the most common metric (a function with a set of properties) on R so the absolute value of the difference
Finite-dimentional: can be generated using only a finite number of elements
Commutative: the operation will give the same result for (a x b) and (b x a)
Unital: as an element which acts like 1 and return the same element when applied so (1 x a) = a
R-algebra: a formally defined algebraic object involving a set and three operations following multiple rules
Algebraically closed: a property on the polynomial of this algebra to be respected. They must always have a root. Untrue in R because of negative. That's basically introducing i as a structural necessity.
Admits a notion of differentiation with reasonable spectral behaviour: This is the most fuzzy part. Differentiation means we can build a notion of derivatives for functions on it which is essential for calculus to work. The part about spectral behavior is probably to disqualify weird algebra isomorphic to C but where differentiation behaves differently. It seems redondant to me if you already have a finite-dimentional algebra.
It's not really complicated. It's more about being familiar with what the expression means. It's basically a fancy way to say that if you ask for something looking like R with a calculus acting like the one of functions on R but in higher dimensions, you get C.
Math and reality are, in general completely distinct. Some math is originally developed to model reality, but nowadays (and for a long time) that's not the typical starting point, and mathematicians pushing boundaries in academia generally don't even think about how it relates to reality.
However, it is true (and an absolutely fascinating phenomenon) that we keep encountering phenomena in reality and then realize that an existing but previously purely academic branch of math is useful for modeling it.
To the best of our knowledge, such cases are basically coincidence.
Opposing view (that I happen to hold, at least if I had to choose one side or the other): not only is mathematics 'reality'; it is arguably the only thing that has a reasonable claim to being 'reality' itself.
After all, facts (whatever that means) about the physical world can only be obtained by proxy (through measurement), whereas mathematical facts are just... evident. They're nakedly apparent. Nothing is being modelled. What you call the 'model' is the object of study itself.
A denial of the 'reality' of pure mathematics would imply the claim that an alien civilisation given enough time would not discover the same facts or would even discover different – perhaps contradictory – facts. This seems implausible, excluding very technical foundational issues. And even then it's hard to believe.
> To the best of our knowledge, such cases are basically coincidence.
This couldn't be further from the truth. It's not coincidence at all. The reason that mathematics inevitably ends up being 'useful' (whatever that means; it heavily depends on who you ask!) is because it's very much real. It might be somewhat 'theoretical', but that doesn't mean it's made up. It really shouldn't surprise anyone that an understanding of the most basic principles of reality turns out to be somewhat useful.
"that we keep encountering phenomena in reality and then realize that an existing but previously purely academic branch of math is useful for modeling it."
Would you have some examples?
(Only example that I know that might fit are quaternions, who were apparently not so useful when they were found/invented but nowdays are very useful for many 3D application/computergraphics)
Group theory entering quantum physics is a particularly funny example, because some established physicists at the time really hated the purely academic nature of group theory that made it difficult to learn.[1]
If you include practical applications inside computers and not just the physical reality, then Galois theory is the most often cited example. Galois himself was long dead when people figured out that his mathematical framework was useful for cryptography.
[1] https://hsm.stackexchange.com/questions/170/how-did-group-th...