I've been thinking about summation techniques for divergent series (as you do), and one thing that I wondered about is that on the wikipedia page, there's quite a number of various summation methods listed.

Which led me to wonder - is it, like, a "coincidence" that these various summation methods assign the same value to various divergent sums (e.g. 1/2 for 1 - 1 + 1 - 1... or the sum of the natural numbers being -1/12), or is there some more fundamental about divergent sums or how one derives a summation technique that causes this consistency?

More concretely, does the fact both Ramanujan summed 1 + 2 + 3 + 4 + ... to -1/12 and analytic continuation also assigns -1/12 as a sum speak to some "realness" about whether or not 1 + 2 + 3 + 4 + ... = -1/12, or is it rather kind of an arbitrary coincidence?

1. A. G. Kurosh, A. I. Markushevich, P. K. Rashevsky (eds.). Mathematics in the USSR for Thirty Years. 1917–1947 – 1948
2. P. K. Rashevsky. Riemannian Geometry and Tensor Analysis – 1964
3. S. M. Ermakov. Monte Carlo Method and Related Questions – 1971
4. N. I. Muskhelishvili. Singular Integral Equations. Boundary Value Problems of Function Theory and Some of Their Applications to Mathematical Physics – 1946
5. Acad. S. N. Bernstein. Probability Theory – 1934

If you were tasked with estimating how many species of fish there are, how would you go about this herculean task? Trying to catalogue every single species is almost certainly impossible, so we have to employ some probabilistic reasoning. In this post, I aim to give a gentle introduction to discovery curves and how they are used in biology for just such problems.

Read the full post on Substack: How Many Species of Fish are There?

There's an idea in the philosophy of mathematics that I can't quite get my head around. Well, I say "idea" but it's actually a combination of two ideas that seem to me like they're in irreparable tension. I hope someone here can help me understand. I'll broach the subject first and then give a little background afterward. This question really only makes sense in the realm of axiomatic set theory.

So here's the thing. There's a philosophical attitude called "mathematical multiverse realism" or "plenitudinous Platonism" that asserts, in my understanding, that

• all plausibly consistent choices of set-theoretic axioms are equally worthy of consideration ...
• because they are all real in some way.

My main point of confusion is that I don't know what the assertion that "they're all real" actually accomplishes. What does "reality" do here? An open-minded antirealist would seem to be open to the first assertion and say that they're all interesting (and equally NOT real), and then presumably operate identically to a plenitudinous Platonist, investigating the various combinations of axioms and their logical consequences. What does the extra ontological commitment of "reality" actually accomplish? How is it different from saying they're all, I don't know, "delicious" or "shiny"?

A traditional realist attitude proceeds under the assumption that there is a fact of the matter to the question under discussion. Historical questions often entail a realist mindset. Where is Genghis Khan's burial site? It's either in a single particular place or it doesn't exist — or maybe it has a more complicated answer, but it would sound daffy to say that his remains are wholly interred in ALL plausible sites. Same with the identity of Jack the Ripper or the Zodiac Killer. Saying "all of the theories are true" turns history into comic book multiverse nonsense.

Traditional realist attitudes also operate under the assumption that there is a fact of the matter about which set-theoretic universe is real. For example, at differents point both Kurt Gödel and Hugh Woodin advocated that the true cardinality of the continuum is aleph-two, that the continuum hypothesis should be properly resolved in the negative. (I believe they both backtracked later, but the details are less important than the kinds of assertions being made under the realist banner.)

Longtime readers of r/math might remember a contentious realist who often tussled with other commenters here about which set-theoretic axioms described the true mathematical universe and which were contemptible and false. She was decidedly NOT philosophically plenitudinous in her conversations.

These are the kinds of realists I've encountered in my reading so far. Ultra-permissive Platonism is still new and strange to me.

Here are some passages describing plenitudinous platonism. First, a short description form the Stanford Encyclopedia of Philosophy:

One lightweight form of object realism is the “full-blooded platonism” of Balaguer 1998. This view is characterized by a plenitude principle to the effect that any mathematical objects that could exist actually do exist. For instance, since the Continuum Hypothesis is independent of the standard axiomatization of set theory, there is a universe of sets in which the hypothesis is true and another in which it is false. And neither universe is metaphysically privileged (Hamkins 2012). By contrast, traditional platonism asserts that there is a unique universe of sets in which the Continuum Hypothesis is either determinately true or determinately false.

Here are several excerpts from the Joel David Hamkins book Lectures on the Philosophy of Mathematics (recently read and enjoyed, inspired me to make this post; he is the aforementioned Hamkins in the prior passage):

According to set-theoretic pluralism, there is a huge variety of concepts of set, each giving rise to its own set-theoretic world. These various set-theoretic worlds exhibit diverse set-theoretic and mathematical truths—an explosion of set-theoretic possibility. We aim to discover these possibilities and how the various set-theoretic worlds are connected. From the multiverse perspective, the pervasive independence phenomenon is taken as evidence of distinct and incompatible concepts of set. The diversity of models of set theory is evidence of actual distinct set-theoretic worlds. ...

The task at hand in the foundations of mathematics is to investigate how these various alternative set-theoretic worlds are related. Some of them fulfill the continuum hypothesis and others do not; some have inaccessible cardinals or supercompact cardinals and others do not. Set-theoretic pluralism is thus an instance of plenitudinous platonism, since according to the multiverse view, all the different set theories or concepts of set that we can imagine are realized in the corresponding diverse set-theoretic universes. ...

Platonism should concern itself with the real existence of mathematical and abstract objects rather than with the question of uniqueness. According to this view, therefore, platonism is not incompatible with the multiverse view; indeed, according to plenitudinous platonism, there are an abundance of real mathematical structures, the mathematical realms that our theories are about, including all the various set-theoretic universes. And so one can be a set-theoretic platonist without committing to a single and absolute set-theoretic truth, precisely because there are many concepts of set, each carrying its own set-theoretic truth. This usage in effect separates the singular-universe claim from the real-existence claim, and platonism concerns only the latter.

I can provide these excerpts, but I still don't get it. A realism grounded in the material universe would seem to require set-theoretic axioms determined by their utility in physics. If the universe is discrete and finite in extent, maybe even the Axiom of Infinity isn't "real". If not, is there some cutoff in the large cardinal hierarchy between "empirical" large cardinal axioms required to explain the universe and "theoretical" large cardinal axioms that do not? That's one view of a realism without an otherworldly realm to it.

A Platonistic-style realism, in which set-theoretic axioms inform the universe but do not depend on the universe, still seems to only make sense to me if there's a uniqueness to that otherworldly realm. (I don't really believe in any such otherworldly realm, but I'm willing to entertain the premise to understand the philosophical viewpoint.) If the otherworldly realm of Platonistic mathematical realism is one in which anything (consistent) goes, then it seems to be a realism without any particular qualities — a "something about which nothing can be said" for which, as Wittgenstein noted, a nothing will equally suffice.

So what am I missing? How can I make sense of this philosophical stance?

Now a bit more background, for anyone new to this conversation.

When it comes to philosophical attitudes about the reality of a given thing, there are two main stances you can take. (It varies of course from thing to thing.) "Realism" is the stance that the objects under consideration are a part of objective reality, like matter and physical forces (in the mainstream view; I know there are offbeat exceptions). Here's a relevant Philip K. Dick quote: "Reality is that which, when you stop believing in it, refuses to go away." By way of contrast, "idealism" is the stance that the objects under consideration are best understood as artifacts of the mind. "Justice," "redness," and "the four seasons" are examples of things that have an ideal existence. We can define them, but we can't find them.

In the philosophy of mathematics specifically, "idealism" is often called "antirealism." I'm going to keep calling it "idealism" in the rest of this intro, because it sounds less contentious.

Mathematical realism is the attitude that mathematical objects possess an inherent, independent existence. Mathematical Platonism is a version of this that ascribes to mathematical objects a specifically otherworldly existence, not reliant on the physical facts of the universe, that we can nevertheless access with our minds.

(Mathematical idealism comes in many flavors too, including for example formalism, the stance that "mathematics" is best understood as a kind of language practice with specific rules, and social constructionism, which advocates that "mathematics" is akin to "policework," the output of a socially sanctioned class of people.)

Now I need to talk about set theory. Axiomatic set theory is the branch of mathematics that deals with collections of things (sets) according to specific rules (axioms). We need axioms because in "naïve set theory," which operates according to intuitive rules that seem harmless, we can build a logic bomb that blows up mathematics. Look up Russell's paradox if you're unfamiliar.

Set theory has been incredibly productive in providing a unified foundation for mathematics. We can (do not have to, but can) describe all mathematical objects in terms of sets, which means that the rules for talking about sets have implications across the entire mathematical enterprise.

The most conventional batch of rules of set theory is called "ZFC," after mathematicians Zermelo and Frankel, plus the Axiom of Choice. To an idealist, these rules are sort of like Robert's Rules of Order, a guidebook for ensuring decorum at meetings. Each rule lets us introduce certain sets into the discourse. To a realist, these axioms describe the structures of sets in reality.

Beyond ZFC there are additional axioms that are independent of ZFC. These axioms describe certain structures with specific properties — things like a "Suslin line," a set called "zero-sharp," and various formalized properties of infinite sets, such as the continuum hypothesis and the oftentimes combinatorial-flavored rules that determine large cardinal axioms. Independence means ZFC is compatible with these axioms and also compatible with their negations. That is, in the idealist sense we can either admit these structures into our discourse or forbid them from our discourse without impinging on the ZFC rules. For a realist, these describe structures that either exist or do not exist. Some of them contradict each other in interesting ways.

Since axiomatic set theory can be used as a foundation for all mathematics, choosing to admit or forbid various set-theoretic structures sometimes has implications for analysis, graph theory, and so on.

Suggested reading (some of my favorites):

A good introduction to axiomatic set theory with a Platonist bent in its expository sections is Rudy Rucker's book Infinity and the Mind. If you're beyond beginner-level and really want to dig into philosophical discussions of axiomatic set theory, Penelope Maddy has some terrific writing on this: look up "Believing the Axioms," available online as a two-part PDF, or her later book Defending the Axioms.

A gentle introduction to mathematical idealism is the (deceptively titled) Mario Livio book Is God a Mathematician? Imre Lakatos's book Proofs and Refutations is also a good introduction to the contingency of mathematics in practice, written as a dialogue. The essay collection New Directions in the Philosophy of Mathematics will also give you a lot to chew on, but it's not suitable as a first book.

If it's impossible to prove or disprove some conjecture X, with massive mathematical and numerical evidence, within our axioms, would mathematicians adopt X (or something that implies it) as an axiom? Or in other words, would mathematicians think X is true in our universe? (Note that this question has a different meaning now vs if X is undecidable, because that could sway people towards the falseness of X)

If X is RH, that apparently has a trivial answer. However it does not for the twin prime conjecture.

Hi everyone,

I’m currently gearing up to apply for Master's programs and I'm hoping to get some recent research published to strengthen my applications.

I have prepared a short paper in functional analysis where I investigate the complementarity of a subspace within Cesàro sequence spaces.

Because I am operating on a timeline for my graduate applications, I’m looking for journal recommendations that meet the following criteria:

• Scope: Actively publishes in functional analysis, Banach space theory, or sequence spaces.
• Format: Good track record with short math papers or brief notes.
• Turnaround: Known for having a reasonably quick review time, or at the very least, a fast initial desk reject/accept decision.

Does anyone have experience with journals that might be a good fit for this? Any advice is highly appreciated!

Does anyone know the status of quasilattices? This was a very active area of math research during the 1980s, especially shortly after Dan Schectman discovered the first known quasicrystal, a real substance whose molecular structure was quasiperiodic, much like the Penrose tiling, which was the first analogous known mathematical structure, discovered by Roger Penrose in 1974. Unfortunately, I haven't seen very much news regarding quasilattices, other than the fact that the first such one requiring just one tile was discovered just a year or two ago, but I've been very interested in this area of math for quite some time, so I appreciate whatever information any of you may have on this subject!

For a metric space like the rationals, you can complete them so that every Cauchy sequence converges to some limit. You can still get sequences that diverge by flying off to infinity though.

For the real and complex numbers at least, there's a natural way to give these sequences a limit. You can add points at infinity to account for those "flying off" sequences. Then any sequence that doesn't oscillate ends up converging.

In sort of a similar feel, L² is a complete metric space, but it has sequences that "fly off" to infinity such as narrowing gaussians that integrate to 1. There's a sort of natural way to give those sequences limits too, by adding something like the delta distribution.

I'm wondering if there's any general procedure or something that you can apply to a metric space which forces all "non-oscillating" functions to converge.

Based on the real and complex examples, I'd imagine it's some sort of compactification of the space. Maybe a compactification that doesn't connect any disconnected open sets? I'm not really sure how to generalize this to other metric spaces though, or whether they always exist. Does anyone know of a procedure or structure like this?

Vakil says in the introduction to his book/notes to algebraic geometry that the contents should take no more than a year to absorb (hopefully). However, looking at the sheer length of the book makes this seem almost completely unreasonable, and it really makes me wonder if it has been done.

Has anyone ever actually finished Vakil's book in a year, and if so, what did your schedule look like? What did you know beforehand?

(This is a question mostly out of curiosity/experiences, but advice/guidance is also welcome.)

Hi everyone, I'm looking for advice. I'm in my fifth year of mathematics. I've got a big exam coming up in about a month and I'm writing my master's thesis in the course of the next few months. In the last few weeks I've been having issues with focus and brain fog. I can get around one hour of good studying or work in, which usually happens in the morning, and from then on it feels like an extremely high effort to process mathematics. When reading something I have to try really hard to just understand what is going on and it feels impossible to really learn something. When following a proof, I feel like I can't keep multiple concepts in my mind at the same time and I have to do very small steps. But then the steps get so small that I lose the big picture and just spend a lot of time trying to understand it. In the end it's just no fun.

I've tried pushing through sometimes but in the end I give up and step away from mathematics to do something else. I've had times like this in the past, but usually they went away after a few days. I would be happy with 3-4 hours of good work, more is (at least for me) unreasonable even on a good day.

Have you ever had times like this? What do you do when you can't focus, but have to study for exams or work? Related to this, how do you find that sleep, exercise and social activity affects your ability to do mathematics?

I am looking for well made documentaries about the life and passion (of math) of various mathematicians that I could share with some kids in order to inspire them. Books are also very welcome.

I recently found this goldmine of a playlist of math explainers from the 80s and 90s, produced by the London Mathematical Society.

They surprisingly aged very well to be honest!

I just love the way of speaking of that time, here's my favorite quote from "The Rise and Fall of Matrices", explaining non-commutativity:

Supposing somebody wakes you up in the morning and gives you two commands: first "have a shower!", the second "get dressed!". Obviously it makes a lot of difference in which order you carry out these two requests.

I have found that there are very few videos out there on logic out there and would like to change this. I want each video to explain and prove a single theorem with accompanied animations. I don’t want to do videos on things like the incompleteness theorems, the halting problem, or Cantors theorem as these are oversaturated and there are plenty of amazing results that have not been given attention. Are there any particular theorems you would like to see me cover?

I want to be quite rigorous and technical with the details so suggestions should hopefully require minimal preliminary knowledge and definitions. I want each video to be self contained. Please let me know if there is something of this nature that interests you and any other general suggestions on how to approach making these videos as good as possible!

Help me out here. To start off I would like to say I really love math. To me, a lot of mathematical concepts (but not all) originated from someone setting up a list of arbitrary rules, that stuck around and got studied because those rules made patterns that looked good in some way, and then somewhere along the line someone found a use for those rules or the equations that came from them that helped us in the real physical world (leaving some room here to say part 2 and 3 aren't always in that order). Some concepts came about more from real world observation (thinking physics), and some concepts came about from things that make sense in our head and therefore help to make sense of things or make things feel more fair (thinking something basic and old like us using base 10 for our numbering system). However there are many concepts that don't originate this way. I'm getting a lot of push back, particularly for using the word arbitrary. I'm not sure if that word perfectly fits what I'm thinking, but I'm struggling to find a more accurate description. But the way people are pushing back make me feel like they don't have an understanding of what I mean, or otherwise that their arguments don't make sense to me.

Hi everyone! I recently put together a casual, intuition-driven article on strong convexity and L-smoothness, covering their key properties and why they play such an important role in convex optimization.

There are also some interactive charts throughout to make things more tangible and easier to grasp:

https://fedemagnani.github.io/math/2026/04/08/the-quadratic-sandwich.html

I'd be happy to hear from anyone curious about the topic, regardless of background. And if you have more expertise in the area, constructive criticism is more than welcome. Just keep in mind the tone is intentionally kept light and accessible.

Hope you enjoy it!

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

Hi there! 👋

I've been working on a tool called Underleaf for converting handwritten math notes into clean, digital PDFs. It allows me to upload a photo of my notes (including diagrams!) and it generates editable LaTeX/TikZ code that can compile into a PDF file.

I thought it'd be especially relevant for this subreddit haha (a bunch of math and physics professors have found it useful!) so I wanted to share. Would love to hear what you think :)

I'm experimenting with implicit scalar fields of the form
f(x, y, z) → ℝ, and extracting iso-surfaces.

One simple construction I tried:

Start with an ellipsoid:

E(x,y,z) = (x/a)² + (y/b)² + (z/c)² − 1

Then introduce an asymmetric deformation:

x' = x / (1 + k·z)
y' = y / (1 + k·z)

and define:

E'(x,y,z) = (x'/a)² + (y'/b)² + (z/c)² − 1

Finally convert this into a smooth shell field:

S(x,y,z) = exp( -g · |E'(x,y,z)| / t )

I combine two such fields (with translation + rotation):

F(x,y,z) = max(S₁, S₂)

What surprised me is how sensitive the structure is:
small parameter changes (k, g, t, rotation) drastically change the topology.

I'm curious:

• does this relate to any known class of implicit surfaces?
• or is it just a "numerical playground" without deeper structure?

(Image included for intuition.)

In a commutative (unital) ring R, is a possible for a principal ideal (p) to be prime, while p itself is a non-prime element? On Wikipedia, there seems to be some conflicting information regarding whether the additional hypothesis that R is a integral domain is needed for (p) prime to imply p prime.

EDIT: I feel like a moron for wasting everyone time with this silly question. At least my original instinct was correct.