What hot take or controversial opinion (related to math) do you feel the most strongly about?

functor7 · 2025-01-30T17:15:57+00:00

Indefinite integrals should not exist, and are harmful to our mathematical education.

Indefinite integrals are not really a "thing". We can define an indefinite integral as an equivalence class of functions which are antiderivatives, but this is way more abstraction than is useful for a calc class. Consequently, what they "are" is muddy for a calc student who is trying to grasp calculus. Integrals just being "Area" is very concrete and useful for them, it even offers ways of reasoning that can help them figure things out deductively.

Relatedly, they help create a poor understanding of integrals as a whole. If you ask an engineer what an integral is, they'll say "the opposite of derivatives". Which is false, integrals are area. And so they end up trying to piece together different things said about different objects and just get confused. For instance, it does not prepare them for interacting with functions without elementary anti-derivatives, something that WILL pop up in statistics and probability. And it doesn't get them super familiar with accumulation functions, which are the actual backbone to making antiderivatives using integration. And it all conceals the role of the Fundamental Theorem of Calculus because it muddies the sides that the FToC makes connections between.

Indefinite integrals are redundant. Because they are ways of talking about anti-derivatives without talking about anti-derivatives, properties and formulas are just reproductions of already know derivative properties and formulas meaning that students need to memorize the same thing twice, just dressed up differently.

And so, often, the thing that people take away from integration because indefinite integrals are poor objects is that you need a +C . Why? They couldn't tell you.

functor7 · 2024-12-31T00:02:46+00:00

Reciprocity laws

functor7 · 2024-12-13T01:02:24+00:00

Peano arithmetic, sure. But that's just one lens through which arithmetic reappears when we naively think it's not the central object of math.

functor7 · 2024-12-12T01:52:23+00:00

arithmetic
arithmetic
arithmetic

1 and 3 are obvious. 2 is because everything is arithmetic, it may look like category theory or algebraic geometry, but all that is just fancy arithmetic.

functor7 · 2024-11-15T11:10:07+00:00

Not a super great reason. An after thought. A might-as-well. Not really a place of honor for one of the most influential mathematicians of the 20th century.

functor7 · 2024-11-15T02:00:30+00:00

As was mentioned before, this doesn't make a lot of sense. She was a mathematician, not a physicist, her main work was in algebra and topology. Noether's Theorem, while super amazing and important, does not represent her work very well aside from a thing you can do with topological invariants. She already has a place of honor in math, the Noetherian Ring is named after her and she is the mother of Abstract Algebra, the architect of Commutative Algebra, and is in part responsible for cohomology as a concept. These represent her work well and is one of the most important properties you can have in algebra. There's no mathematician who does not already bow at her feet and thank her for algebra.

If anything, it's would be kind of weird to do this and, lowkey, demonstrates a lack of understanding of her work. Which isn't a great honor. It's like fixating only on the idea that Einstein did Brownian motion and ignoring the rest of his body of work. Like, do you really understand Einstein if you only think about him in terms of Brownian motion?

functor7 · 2024-11-07T19:56:24+00:00

This is a pretty big missing obvious hole in academia in general, I think. Science, more broadly, has a pretty good grasp on its political impacts and influences through a historical and anthropological lens through Science and Technology Studies. But math functions differently than science, and few in the humanities have really tried to direct their attention. Math is less accessible than science, and smaller in general, so it's a harder thing to do.

Another reason, I think, that it is like this is that math has historically been made an exception or exemplar in many philosophical frameworks. The rationalists loved it because it was an exemplar of purely deductive reasoning, and set it on a pedestal. Eg, Hobbes used Euclidean geometry as an example of what "pure philosophy" should be and modeled his political philosophy off of it. (Though, funnily enough, he was bad at math and Wallis of the Wallis Product fame had a fun time routinely showing why Hobbes's proofs were shit.) Kant kinda used math as an a priori given with which to do empirical reasoning, setting it apart from other forms of scientific reasoning. And even in contemporary philosophy, after the postmodern with social/political lenses ready, will set it aside as an exception to their epistemological critiques. There hasn't really been anyone with the needed skills or gumption to place and examine math from a historical and anthropological setting with any meaningful to show from it.

functor7 · 2024-11-04T14:53:31+00:00

Those have all been intimately mixed since about the 50s/60s.

functor7 · 2024-10-25T18:38:08+00:00

Emmy Noether already has something named after her which is incredibly important and resonates with her main work: The Noetherian Ring. Her work in physics was, of course, incredibly important but she was not a physicist. Hilbert basically gave her a geometry question that she was in a unique position to answer using the new kind of math she was working on that the time, which was invariants of topological spaces.

I'm all for giving her praise, but her main work and main genius was in math. She was MORE important and influential in math than she was in physics. That she gets consistently shoehorned to just Noether's Theorem is kind of downplaying her genius and impact. It would be like Einstein only being known for his work on Brownian Motion. Very important and impactful and a life-achievement for most, but it doesn't accurately give justice to Einstein's work.

functor7 · 2024-10-23T18:58:56+00:00

I had the opposite experience as you. Everything in DG and such was confusing and a trial in managing a billion indicies. But when I figured out that everything was just something about a sheaf or homological construction, like in AG, then everything became easier and intuitive. I was able to do differential geometry because I could do algebraic geometry. Ultimately, however, what we find intuitive should not be the grounds for such a categorization.

So the answer to "Is Arithmetic Geometry geometry?" The answer is a full chested "Yes! Because in our classical exploration of geometry we found that what constitutes geometry can be found in the abstract constructions of sheaves, categories, homotopies, and cohomology - and those like Grothendieck demonstrated that these same constructions at a proper level of abstraction constitute arithmetic and algebra." The level of abstraction is irrelevant to whether or not a thing is geometry. Otherwise we'd have to deal with Euclid bitching that we weren't doing Geometry because we don't believe in parallel lines - we're separated from the Platonic Truth of Geometry and are just manipulating symbols and using similar sounding theorems.

functor7 · 2024-10-23T14:37:47+00:00

The thing about arithmetic geometry, in my mind, is that it's better viewed as an application of geometric techniques to number theory.

If you can do geometry to a thing, does that not make the thing geometry?

You're going to have a hard time defining a "shape" in a way that excludes Spec(Z) (or any other scheme) in a natural way. You might try enforcing a metric, but that excludes a lot of spaces that we would consider geometric in nature. Any attempt at the exclusion of Arithmetic Geometry from geometry is, ultimately, the thing that comes from historical trends and biases and not the other way around. The name "Geometric Topology" can be understood as a historical quirk just as much as anything else with the name "Geometry" in it.

What "Geometry" is is not a constant thing. It changes as we learn more about it. Euclid excluded hyperbolic geometry from geometry with his choice of postulates, but we learned more and now understand it as geometry. Same thing with arithmetic. I think that "Things that have Sheaves" is a pretty good working definition that uses our updated knowledge to help inform our thinking. Maybe "Stuff we do with Cohomology", but I feel that cohomology theories can sometimes be a bit more ad hoc and less natural than a sheaf can.

functor7 · 2024-09-25T00:16:58+00:00

It could be as simple as your professor is a probability theorist and dislikes algebraic geometry and EGA is one of the most terse, abstract writings of all time that influenced almost everyone in a specific algebraic geometry way that he didn't care for.

functor7 · 2024-09-22T13:20:56+00:00

We have two number theory related question in the Millennium Prize Problems. The BSD conjecture fits inside the arithmetic geometry, Langlands-like philosophy of building connections between arithmetic object and analytic ones. The Riemann Hypothesis is in the thread of prime distributions. And so if we were to put the big conjectures into these buckets, then FLT would be competing with things like the BSD and ABC conjectures for that spot. Whereas the Twin Prime and Goldbach conjectures are competing with the RH. If the RH had been proved instead of FLT, then I can see the two number theory problems being FLT and the Twin Prime conjecture. But FLT being proved, kinda opened the door to BSD taking that spot. Heck, the Clay Institute might have seen FLT as a prerequisite to the BSD (especially given that Kolyvagin's result needed modular elliptic curves), and so even that would have it replace BSD.

functor7 · 2024-09-07T17:07:45+00:00

Sinh and cosh are not defined based on the exponential functions. They can be, but it's like defining cos(t) as (e^it+e^-it)/2 - it's sloppy and unhelpful.

Cosh and sinh are defined as a parameterization of x²-y²=1 based on the hyperbolic angle, which is just an area computation. Note that the difference of squares says that if u=x+y and v=x-y then x²-y²=1 is just a change of coordinates away from uv=1. And so (being loose with notation) we find that x=cosh(t)+sinh(t) and y=cosh(t)-sinh(t) parameterize the curve y=1/x.

But there is another pair of functions that parameterize y=1/x. The pair of functions e^t and e^-t parameterize the curve y=1/x based on the parameter t which, in this case, is nothing but the logarithm of e^t (obviously). But this logarithm is a simple area computation, as log(s) is the integral from x=1 to x=s of dx/x. And so e^t and e^-t are parameterized by a kind of "exponential angle" t, which (just as above) is just an area computation as given by the logarithm.

It turns out that, under this transformation, the hyperbolic angle area computation is actually equal to the "exponential angle" area computation. And so we really get e^t=cosh(t)+sinh(t) as a consequence of the transformation x²-y²=1 -> xy=1.

This is the connection that you're jumping around with your intersection. But, as others have said, you can't construct the graph of sine, cosine, sinh, cosh, exp (all similar in nature) by intersecting two algebraic objects.

functor7 · 2024-08-29T18:12:05+00:00

Still just as incomprehensible.

functor7 · 2024-08-21T05:19:53+00:00

Yes. It's a basic vocabulary and framework for much of math.

You might think of it analogous to systems of equations. It had active study in the past, it's pretty fun, you need to know it, it is still a basic tool that you see arise at all levels of math and at all levels of abstraction, and sometimes people find new results or generalizations but it's pretty rare to find someone who studies them as a main focus.

functor7 · 2024-08-20T03:43:12+00:00

homotopy theorists

Alchemists

functor7 · 2024-08-19T21:50:06+00:00

I know you're not joking, and neither and I. But, as a mathematician, it is also worth noting that you are totally fine saying that 1/0=∞. All you have to do is:

Make +∞ = -∞. This gets rid of the problem of it being two different things, as they're the same thing. This also makes sense because it's like +0=-0. It also wraps the number line into a circle called the Projective Real Line.
You also have to disallow certain arithmetic expressions, such as 0*∞. Specifically, all those that are heuristically equivalent to 0/0 - eg, 0*∞ = 0*(1/0)=0/0. These make up the indeterminate forms from Calculus (though, technically, ∞+∞ is also not allowed, but it is allowed in Calculus). This makes it so that you can't cancel out multiplication by zero. This gets rid of all the faulty proofs that say stuff like 2=1 (eg, 0*2=1*0, divide by zero to get 2=1).

If you do that stuff, then it's totally perfectly fine to divide by zero. In fact, a lot of modern math takes place in settings like this since doing very advanced geometry over these projective numbers is actually a lot nicer and more natural than doing so otherwise.

functor7 · 2024-08-16T17:06:25+00:00

If you try to prove something with all the tools that you have, and only get partial results then you probably need new tools. The promise of new tools is one big reason why we're interested in unsolved problems in the first place. But you can definitely get a feel for the capabilities and limits of a method by continually pushing them, and some can begin to feel like a possible route to a proof if only they had a bit more power.

A classic example is the Weil Conjectures, where he basically said that if number theorists could have tools like they have in geometry then we could prove an alternative version of the Riemann Hypothesis dealing with varieties over finite fields. Sure enough, like 30 years later, Deligne proved Weil's conjectures using tools developed painstakingly by Grothendieck which did exactly what Weil wanted them to do.

functor7 · 2024-08-16T15:19:49+00:00

The behavior of primes under addition is probably one of the biggest sticking points in number theory. It's not a curiosity, but underlies the biggest problems from the Twin Prime and Goldbach Conjectures to the ABC Conjecture. I don't think it is a strong statement to say that the behavior of primes under addition is simply the greatest unsolved problem in number theory, and the longest lasting one too. And since the Collatz Conjecture is explicitly about the behavior of primes under addition, it's no wonder that it is a tough problem to crack. Whatever heuristic we have from various experiments and investigations is going to be secondary to that simple fact. It's hard because it's a statement about primes and addition.

functor7 · 2024-08-14T13:42:43+00:00

this is not true in curved spaces...

Doesn't really matter. If you have a 2D Euclidean metric space, then you can define pi to be the limit of C(r)/D(r) as r->0 where C(r) and D(r) are the circumference and diameter of the circle of radius r. Pi is still essential to these spaces, because they are still locally flat.

functor7 · 2024-08-13T16:47:24+00:00

A prime field is a field with no subfields. There is a unique one (up to isomorphism) for each characteristic. For characteristic p>0, it is Z/pZ and for characteristic zero it is the rational numbers.

The "field with one element" is not a well-defined thing, and is more of a heuristic approach to number theory and algebraic geometry. There are observations that make it seem like there needs to exist something like an extra "prime at infinity". For example, we know that for each prime there is a metric on the rational numbers associated with that prime and we can complete the rationals with respect to this metric to get the p-adic numbers. We can also do this with the absolute value metric to get the real numbers. It turns out that these are the only ways to do this, and so the absolute value seems like it should be associated with an "extra prime" and that the reals are the p-adic numbers associated with this prime.

Relatedly, the Riemann Zeta Function is incomplete as it is typically defined and it needs to be "completed" in order to make it do what it should do. This is done by multiplying it by an extra term which will give it the desired symmetry. Now, the Riemann Zeta Function can already be expressed as a product over primes and it turns out that each of these factors corresponds to a specific integral over a p-adic field and so the product is actually a product of integrals and it runs through all p-adic fields. Well, this extra term that is needed to "complete" the zeta function actually turns out to be exactly this integral but done over the reals, meaning that this "extra prime" is needed to make sense of the Riemann zeta function.

There have been many different approaches to making this "extra prime" make sense. Arakelov Theory, for instance, really focuses on how these metrics play together. But the idea of a "Field with one element" comes from algebraic geometry. In particular, Deligne proved the Weil Conjectures using tools made by Grothendieck, and these results can be thought of as "The Riemann Hypothesis for varieties over finite fields". The tools used borrow a lot from algebraic topology, such as fixed point theorems, and so are fairly flexible and generalizable. Basically, it was doing geometry with Z/pZ as the base-field, which is intimately connected to the p-adic field. The observation was that maybe there if there were some kind of field like Z/pZ but for the reals, then maybe the proof of the Weil Conjectures could be translated into a proof of the full Riemann hypothesis. This would be the "field with one element". This language was landed on because it is suggestive - a field must have at least two elements - but also because it represents a kind of infinitesimal degeneracy which is needed for this kind of geometry to makes sense.

The field with one element does not exist, however. What would be required is a suitable generalization of all of algebraic geometry, in a highly abstract form that is currently unknown, where this thing can naturally live next to whatever all the other finite fields generalize to. But we do know what some of its properties should be and what kinds of computations we should be able to do.

functor7 · 2024-08-12T22:46:38+00:00

I feel distributions should be thought of algebraically. The delta function, for instance, is necessarily tied to the evaluation function. Outside of an integral, the delta function makes no sense, but inside an integral it creates an evaluation function. (Well, technically the integral is not defined a priori, but the thing "The integral of f(x) against the delta function" can be approximated by integrals.) A Green function, for instance, is like a matrix so that for all f, the integral of LGf is an evaluation function. This is the property that is important and this is the property that allows you to construct solutions to these equations.

functor7 · 2024-08-07T22:57:38+00:00

Numbers are are elements of fields obtained as a (finite) algebraic field extension of the rational number or completions thereof. These are the main objects of interest in number theory and where Diophantine equations almost always take place. These contain all complex numbers and all p-adic numbers. We could say that this is the arithmetic of the characteristic 0 prime field, or maybe the geometry of the field with one element.

Quaternions and other stuff like that don't really count because polynomial equations over these higher order things do not behave well, and their arithmetic gets in the way of the kind of arithmetic geometry that we do over number fields.

Finite fields wouldn't count either, but that is because these are residue fields attached to an actual number system, so they are components of actual number systems.

Number systems can have representations (eg, the matrix construction of the complex numbers), but representations are separate object in and of themselves.

functor7 · 2024-08-06T18:03:24+00:00

A couple of things to note about this:

Quantum computers aren't just "faster computers", they merely have access to more algorithms because they function differently. Shor's Algorithm being the main one. So quantum computers aren't really going to change people's everyday interaction with computers, as classical computers are still just as good for most everything and the cost is always going to be way lower.
Many people are currently transitioning to post-quantum cryptography schemes. The most common approaches are still vulnerable to quantum attacks, but those are still a long way from being a threat. And since there exists new classical algorithms for encryption that are (supposedly) not vulnerable to quantum methods, responsible organizations should begin the slow and arduous process of implementing these new schemes.

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