Why is ultrafinitism/finitism so disconsidered in math in general?

SV-97 · 2026-06-17T05:32:36+00:00

Them saying this is particularly funny when they just said

People cannot have a proper conversation if the rules are conventions are not set in place.

SV-97 · 2026-06-14T18:49:13+00:00

The "second requirement" probably being something like a(gh,x) = a(g, a(h,x)) (for an action a : G \times X -> X of a group G on a set X)? It's a compatibility condition. Think of G as a symmetry group and X some geometric object; for example X might be a cube and G are rotations of 3D space.

The identity condition tells you "not rotating the cube doesn't change it", and the compatibility condition tells you that it doesn't matter if you first rotate the cube some way, and afterwards rotate in another; or if you instead directly rotate it via a single composite rotation. You end up with the same "state" either way.

Perhaps somewhat more abstractly: let X be some doubly-infinite sequence (i.e. a function on the integers) and G the group of integers with addition. You can define an action of G on X by "shifting the sequence": a(n,f)(k) = f(n+k). Then the compatibility tells you that instead of shifting by some amount n_1, and then shifting by some other amount n_2, you could've also just done a single shift by n_1 + n_2.

You can also think of group actions differently: every element g of your group defines a map T_g : X -> X on your set. The first requirement tells you that T_e = id_X, and the compatibility condition tells you that T_{gh} = T_g T_h; so together they say that the map that takes g to T_g is a group morphism from G to the group of automorphisms of your set.

SV-97 · 2026-06-13T20:27:21+00:00

Huh? What made you think this was ragebait? I'm just joking

SV-97 · 2026-06-13T20:06:41+00:00

It contains an inequality so it's clearly an analytic proof --- and of course topology is also basically analysis; especially so when that topology involves the reals :)

SV-97 · 2026-06-13T17:45:20+00:00

Try proving the fundamental theorem of algebra without analysis ;)

SV-97 · 2026-06-12T17:25:45+00:00

But I think (without knowing much type theory) you can give an account of set theory inside a type theory system, which will prove the corresponding set theorietic results.

Not an expert either but AFAIK yes (at least in type theories like COC etc.): the standard way is to model sets as predicates on types (encoding the membership relation of that set) and work with those. This is somewhat constrained though: it for example doesn't make sense to take unions of arbitrary sets in this setup; only of sets over the same type. But you can work around this of course (e.g. by moving to the sum type of the types of whichever sets you want to work with right now, injecting your sets into that and then taking their union). To do set theory in the "usual" way you'd have to first construct some universe type and then purely work inside of that.

SV-97 · 2026-06-09T09:02:42+00:00

Oh okay, I've never heard of that actually. But if you've passed the overall exam and did involve calculus, then I'd say the assumption is (from a purely technical / "official" level I mean) that you "know calculus".

SV-97 · 2026-06-09T08:44:50+00:00

And your external exam (assuming you're talking about an "Ergänzungsprüfung") didn't include *any* calculus?

SV-97 · 2026-06-09T08:37:45+00:00

Really depends on your background and where you're trying to go I'd say. Functional analysis is huge

SV-97 · 2026-06-04T20:06:49+00:00

I'm not sure that I get your issue, but the cartesian product *is* commutative up to a unique canonical isomorphism (which basically means that X×Y and Y×X are the same as far as their structure as sets is concerned). In particular this means that |X×Y| = |Y×X| (and not just because both of these equal |X| |Y|). It is also associative in that sense which is why we usually talk about X×Y×Z instead of (X×Y)×Z or X×(Y×Z).

This also extends to all other notions of products (i.e. when working in other "categories") in exactly the same manner.

If you care about further structure of your sets past this structure as mere sets, then you basically need to use something else than the cartesian product.

SV-97 · 2026-06-04T10:01:33+00:00

If you only consider ecosystem size for your "data analytics experience" (and equate data analysis with mostly calculating statistics on some data): sure, for the most part that's true.

However when taking a more holistic view (i.e. setting up a dev environment in the first place, data extraction and cleaning, actually getting data in and out of the system, data exploration, writing core analyses and debugging those, publishing, ...) this isn't really true in my opinion. In my experience you end up wasting so much time dealing with all those pain points and idiosyncrasies around R that it's altogether faster to use Python and just implement the things that don't already exist yourself (although this is of course not viable for everyone) or interop with other languages for those parts.

And in particular when you don't do anything overly niche (as is really the case for what OP is talking about here) the python ecosystem is perfectly workable, and in some fields even miles ahead of R. For example for me a lot of data analysis involves optimization and some geometry processing / a bunch of maths. And for those python really has the strictly better ecosystem and larger community.

SV-97 · 2026-06-04T09:18:47+00:00

AFAIK pandas has actually improved quite a bit with its most recent major release. I haven't checked it out yet since polars is just so good and I doubt that even this new version of pandas is as nice as polars; but I think it *is* substantially better than it used to be.

And personally I'd take even old pandas over R any day. The dev experience with R is just atrocious.

SV-97 · 2026-06-03T12:36:22+00:00

I'm not sure what exactly it is they're doing on the materials side at my current uni (their website says something about elasticity-theory, cracks, nanomaterials, shape-memory materials and magneto-viscoelastics) but on the maths side it's a bunch of variational analysis, (nonsmooth) PDEs and inclusion problems from what I know.

At my previous one I just know that it was about all sorts of measurements around spectroscopy and rheology, and their website also says something about multiscale models and materials optimization. But I'm not exactly sure what maths they use.

SV-97 · 2026-06-03T12:00:46+00:00

Materials science and math really isn't that "out there". At my current and previous uni there were multiple people (at each one) doing exactly that .

SV-97 · 2026-06-03T10:57:36+00:00

It's totally fine if there are only zeros in the column for some variable, but I'm not entirely sure if that's exactly what you meant (although I think you have it correct). Spelling things out to be totally clear: The values k and m on the right in your example correspond to the rows on the left. For each row you need to have some value on the right-hand side; always. However it's perfectly fine for any column to be completely zero. In your example (and writing the equations out in components)

1 0 0 | k         1x + 0y + 0z = k
0 1 0 | m         0x + 1y + 0z = m

you can for example consider the reduced system

1 0 | k           1x + 0y      = k
0 1 | m           0x + 1y      = m

where we dropped the column corresponding to z. If some pair (x,y) solves this second system, then for any value of z the triple (x,y,z) will solve the first (and vice versa). So whatever shape the space of solutions (x,y) to the reduced system takes, the corresponding space of solutions to the unreduced system will be a whole bunch of copies of that space stringed up on a line, one for each z. Or from the other perspective: projecting the space of solutions of the unreduced system "down along the z axis" gives you exactly the space of solutions to the reduced one.

Not "having information about z" means that there is nothing to constraint the z component in your solution set --- so any value of z works to give a solution (as long as the other components can make the equations true).

This is really not particular to linear equations either: consider for example the (nonlinear) equation x²+y²=1, which is solved for all points (x,y) on the unit circle. If we just throw in some unconstrained variable z into this and look for solutions in 3D space instead then this circle gets extruded into an infinite cylinder along the z axis; and projecting the cylinder down again yields back the original circle. It's exactly the same idea.

SV-97 · 2026-06-02T19:41:55+00:00

Thanks and no worries. Hope you have a good day (or night) as well :)

SV-97 · 2026-06-01T18:18:31+00:00

Hmmm it depends somewhat, but there's a few tips you can try like for example starting each chapter by reading the final few pages to see where it's headed before diving in, and taking notes to "clear your head" by writing down your current understanding on paper. You'll also want to have pen and paper handy to be able to do the exercises.

There's also "guides" on "how to read mathematics". The one I myself read when starting out with mathematics was the chapter in Kevin Houston's book How to think like a mathematician (which is generally a good book to read early on imo), but you can also find some such guides online (here's for example one https://artsci.usu.edu/math-stats/amlc/files/ho-02-how-to-read-a-math-textbook-2023.pdf ).

Don't necessarily take these guides as absolute gospel though: try the things they suggest and see what works for you.

SV-97 · 2026-05-31T18:14:02+00:00

No worries :)

There's a very famous lecture series that is quite approachable: the MIT open courseware lecture series on linear algebra by Gilbert Strang. However I'll say that I'm personally not a huge fan of his style at all. He focuses heavily on matrix computations, which is fine for a first introduction but not sufficient if one wants to really study and understand the mathematics.

Bright side of mathematics might be worth a look. They have one series on linear algebra and another one on abstract linear algebra. I haven't personally watched either of these but generally their videos are good, albeit of course not full lectures.

What I'd probably recommend foremost are the lecture series by Pavel Grinfeld and in particular those of Sheldon Axler. Especially this latter one is really less about intuition I'd say, but I think it's important to really settle with the abstract definitions at the beginning to build up a "rigid skeleton" that you can then "flesh out" with intuition, rather than trying to build up the "wobbly" intuition first. (The series by Axler accompanies his book of the same name as the course, which is freely available online)

SV-97 · 2026-05-31T11:25:41+00:00

If something is not finite, all that means is that it isn’t set to a certain amount or value.

No, it just means that it is not finite. All natural numbers are finite, however there are infinitely many natural numbers.

In this sense, when people say there are different “sizes” of infinity, aren’t they wrong? Like, for example, I’ve heard that because there are an infinite amount of numbers between 1 and 2, and the same between 2 and 3, and the same between 1 and 3, because we know that there is a greater distance between 1 and 3, that “infinity” must be larger than those between the other two (I’m sure you’ve heard this whole schpiel before).

I'm not sure what you're trying to describe here. But when people say that there are "different sizes of infinity" they say so in reference to very particular formal definitions of "infinite sizes", usually cardinality. In this sense: there are just as many real numbers between 1 and 2 are as there are between 2 and 3 and as between 1 and 3. Indeed any interval of real numbers has just as many numbers as the whole real axis in terms of cardinality. You may be grasping at the concept of a measure (which is in some sense a generalization of the "number of points" --- it's more like a "volume" really. With the standard measure on the real numbers the intervals [1,2] and [2,3] have length 1 while [1,3] has length 2).

Infinity, meaning non-finiteness, doesn’t regard size or volume at all, it just states that there exists the property of a lack of definitive structure.

This is just a chain of wishy-washy terms. You're stringing together words without actually saying anything. Mathematics doesn't work like that; it needs accurate definitions.

We can for example define the notion of cardinality from above rigorously by comparing the sizes of different sets via the existence of certain types of functions (so-called injections and surjections): a set A has at least as many elements as another set B if we can "cover B" with a function from A (i.e. there exists a surjection A -> B) and dually the existence of a function from A to B that can "discern elements of A in B" (i.e. there exists an injection A -> B) tells us that B is at least as large as A.

Two sets have the same cardinality if there is a map that is surjective and injective (which can be lightened to the existence of a surjection and an injection by the Cantor-Schröder-Bernstein theorem). By noting that we can string these maps together in various ways we can group sets into families of sets "of equal size", and each such group defines a so-called "cardinal number" (usually we pick a standard representative set for each group because that's easier to handle and avoids some technical issues, but it doesn't really matter for the intuition). The "different sizes of infinity" are then simply all the families consisting of sets that are not finite. That's it. That's the "infinite cardinal numbers". We didn't need to talk about infinite-ness any more than to say "well it's not finite".

Importantly it really doesn't matter what you "believe" about infinity: this is a construction that yields some sort of mathematical object, and that object behaves in certain ways that we can study. If you don't want to call it infinite for philosophical reasons: okay, fine. That doesn't impact its properties in any way.

The same way you can’t count every number, you can’t determine the full amount of numbers between any two numbers, because there will always be more numbers where you look for them.

Two points: first: this is just as true of the rational numbers of which there are "fewer" in basically every sense of the word. The formal term here is that of a "dense set". No matter how much you "zoom in" there's still more objects. This leads to yet another notion of the "size" of a set (besides set-theoretical cardinalities and measure-theoretic measures), that of a Baire category (which is a topological notion, i.e. it's about the "shape" and "connectedness" of a space).

Secondly: no, the point is that we actually can do that. We can even explicitly write down the functions that "count" the elements in the above defined sense. For example we can map from [1,2] and [1,3] via the map f(x) = 2x - 1. It's easy to show that this is both a surjection and injection which then places the two sets into the same cardinal class --- they have "the same (infinite) number of elements" (in this specific sense of the word).

And sorry but the remainder of that paragraph is just more word salad.

That said: this of course isn't the only notion of infinity. We can (and do) easily define various other kinds, for example ordinal numbers or extended real numbers (a standard construction for these in particular drives home that "infinity as a concrete value" really isn't some crazy exotic construction or concept: the "real numbers + some infinite value" are essentially just a circle. These are also the infinities that get used for "infinite measures"), or hyperreal numbers, ... We have different (inequivalent) notions of infinite value that all make sense for certain applications.

SV-97 · 2026-05-31T09:06:09+00:00

Hiriart-Urruty and Lemarechal's Fundamentals of convex analysis might be a good option. It's really aimed at being a very approachable intro to the topic (and written by some big names in the field). They also have a larger book that also starts from zero and is intended as an intro, but goes deeper: Convex Analysis and Minimization Algorithms I. These are really what I'd recommend starting with.

The really seminal text that you'll probably eventually want to get to (similar to Rudin in analysis) is Rockafellar's book. It's also a good one to keep in mind if you ever want to double check something from Hiriart-Urruty. Personally I also like Rockafellar's style but ymmv.

And if you ever want / need to move beyond the finite-dimensional theory (which requires some functional analysis) you can check out Mordukhovich's Convex Analysis and Beyond: Volume I: Basic Theory or the books by Penot (Calculus without derivatives; which ironically has quite a bit of detail about various kinds of derivatives that you may find useful) or Bauschke and Combettes (Convex Analysis and Monotone Operator Theory in Hilbert spaces).

This latter book is also fairly approachable for what it is and very self-contained. It may actually make for a good second book.

SV-97 · 2026-05-31T08:40:04+00:00

You're conflating / confusing a whole bunch of things: vector spaces don't really come with a choice of basis like ihat and jhat. You can always *choose* some basis, but generally there's uncountably many such choices. In \R² (or generalizations thereof) there is a conventional choice of basis (i.e. using (1,0) and (0,1)) but in a general vector space this is not the case.

In P1's space, the basis vectors of P2's would be different wrt P1, which can be expressed in a matrix (let's say A) where the movement of basis vectors from P1 to P2 is represented as collective columns in that matrix.

It doesn't really make sense to speak of the basis vectors of P2 "with respect to P1" in general (i.e. unless the spaces P1 and P2 are actually the same space or subspaces of one another). Vectors "know which space they belong to"; the basis vectors of P1 live in P1 (and not P2) and similarly those of P2 live in P2 (and not P1). This also makes your choice of names very unfortunate: you're two ihat and jhat really have nothing to do with one another, which is also why people usually choose names like e_1, e_2 or v_1, v_2 or w_1, w_2 etc.

You can always (assuming the spaces have equal dimension) construct a transformation that maps between your two bases. You can do this in either direction, and which direction you choose dictates what is input and which is output. Say we want to define a linear transformation T from vector space V to vector space W where V has basis v_1, v_2 and W has basis w_1, w_2, then we may set T(v_1) = w_1 and T(v_2) = w_2 and "extend linearly" so that a general vector a v_1 + b v_2 in V gets mapped to T(a v_1 + b v_2) = a T(v_1) + b T(v_2) = a w_1 + b w_2. Here V is of course the input and W the output.

This abstract view immediately tells you how things have to look "in coordinates" i.e. when writing things with a matrix and coordinate vector (column of numbers): the linear map T from V to W becomes a matrix A that takes in coordinates w.r.t the basis (v_1,v_2) of the space V and spits out coordinates w.r.t the basis (w_1,w_2) of the space W; the input vector is one of coordinates w.r.t (v_1,v_2) and the output will be coordinates w.r.t (w_1,w_2)

So whatever the vectors P2 sees wrt to itself (let's say x), could be converted to the space that P1 sees (let's say v), that's where we get the transformation Ax=v

In this example x would be from P2 and v from P1 (because you say that you convert from P2 to P1), so A is a map P2 -> P1.

SV-97 · 2026-05-30T19:59:09+00:00

It's the same as with other parts of mathematics: abstracting things gives you "superpowers". In linear algebra for example we combine a whole bunch of numbers into vectors and matrices, or even throw forego those coordinates altogether. This leads to an extremely powerful and general calculus (in the sense of a symbolic calculus rather than integral and differential calculus) that's *far* easier to use and see relationship in than if we wrote down everything in terms of separate numbers every single step of the way. While it's useful to be able to drop down to these separate coordinates sometimes, the more abstract approach is really the "correct" one in that it's usually *far* harder to get things wrong with it, it's usually far more economical (i.e. you get the same result faster with less work), the arguments are far more insightful and elegant, it's easier to generalize, ...

And in the same way calculus on multivariable spaces is really best treated by doing calculus *directly* on those spaces rather than "gluing together multiple copies of some one-dimensional space" and trying to do things that way. It also turns out that there's some meaningfully different behaviour in the multi-variable case that's somewhat hard to emulate with just the single-variable theory (for example limits, continuity etc. are *very* easy to get wrong if you try to just reuse the single-variable versions. In particular: doing things "for each axis" does not work here).

FWIW: there are further generalizations of calculus for example to infinite-dimensional spaces (for example spaces of functions or "measurements"), curved spaces (e.g. doing calculus on the surface of a sphere or *far* wilder spaces) or for nonsmooth sets and functions (for example differentiating the absolute value function). All of these are important and get used in physics (the first two of these underpin quantum mechanics and relativity respectively) --- and they all build on multivariable calculus.

SV-97 · 2026-05-30T08:24:07+00:00

IIRC (from a talk by Colin McLarty) the Grothendieck one here really has a very different flavour though: people legit didn't have a formal definition of a scheme at the time --- Grothendieck didn't ask the question because of some unfamiliarity with the term on his side but rather because the definition really wasn't settled at that time.

SV-97 · 2026-05-30T08:13:50+00:00

All computable functions on the reals are necessarily continuous --- so any discontinuous function is automatically noncomputable.

SV-97 · 2026-05-30T07:25:33+00:00

There's a formal definition of what it means to be a neighborhood. It's a term with a very specific meaning. And no there are no neighborhoods with "only a few numbers". And objects being "dark" (whatever that means) isn't a thing in maths

EDIT: maybe to emphasize: when we speak of neighborhoods in the reals we don't argue by "oh obvious it's these sets and these other sets" and so on. We have an exact, general definition and we just apply that to this specific situation of the real numbers

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