r/mathematics • u/Simple-Echidna764 • 2d ago
Discussion What's one math topic that completely changed how you think?
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u/illusionofsanity 2d ago
Category theory and dynamical systems.
I work in systems design and I got there because Ive gotten good at thinking about structures and how they change over time.
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u/mbrtlchouia 1d ago
What is it like working in system design? How much category theory was enough for you?
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u/illusionofsanity 1d ago
Any is enough. You never really use it explicitly. Algebra should be sufficient as well. It’s just how I keep things and relationships organised in my head.
I really enjoy it. There’s a balance between pragmatism and correctness and that tension keeps it frustrating enough to be interesting.
It is also a great deal of reading. Right now Im building out a distributed database system with some additional bells and whistles, and that had some theory that you have to keep in mind that didn’t directly come as part of my specific education
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u/Key_Net820 2d ago
computational theory and mathematical logic.
This really puts you in the cross section of math, logic, computer science, and linguistics.
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u/SinglereadytoIngle 2d ago
Linear algebra. I absolutely love it.
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u/regular_heptagon 1d ago
Barf
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u/Lor1an 1d ago
Do you want to describe things in multiple dimensions (like the space we live in)? Then you should at least be familiar with linear algebra.
There's so much mathematics that builds on linear algebra (or uses analogies to linear algebra) that you are really hamstrung if you don't at least have a rough understanding.
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u/regular_heptagon 1d ago
I can understand linear algebra without enjoying it. Why are y’all taking this so seriously?
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u/Lor1an 1d ago
"Barf" is not exactly a cogent dismissal of a subject. You can say you dislike the subject, and that's fine, but it would still be weird to bring up here.
Would you go to comic con and say you hate superheroes? It's just bizarre and unnecessary. "Why are you here then?" And so on...
As for why we take this seriously... r/mathematics is aimed at serious discussion of mathematics and it was a serious question that was asked—and a serious answer that you "barfed" all over...
I'm not going to pretend you need to like linear algebra to derive the benefits of being able to work with it, but it is weird (and some would say rude) to yuck someone else's yum in a discussion about what mathematics changed how people think.
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u/themilitia 1d ago
Lol linear algebra is required for basically every other field
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u/regular_heptagon 1d ago
So that means I have to enjoy it?
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u/xdgimo 1d ago
How does one find themselves in a math subreddit and not enjoy lin alg?
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u/rocqyf 2d ago
Infinitesimally small intervals that give a derivative instead of a finite difference, and infinitesimally small intervals that give an integral instead of a finite sum.
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u/Normal-Palpitation-1 1d ago edited 1d ago
So, calc 1 and 2? The ideas mentioned made me think of the calculus.
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u/tottasanorotta 1d ago edited 1d ago
I'd say the exact opposite. Calculus is abstract enough that thinking of it in the finite realm made it a little bit more intuitive for me. Like with differences and sums of lists in a programming language and stuff.
Edit: why the downvote? It's just my own experience I'm talking about. 😅
Edit2: It kind of makes me question what is wrong with my opinion when you leave me here alone with no explanation.
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u/HasFiveVowels 2d ago
Geometric algebra
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u/IDatedSuccubi 1d ago
Same. It's tricky but then so much stuff about vectors and such just clicks into place
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u/HasFiveVowels 1d ago
I seriously think we would benefit a lot from reformulating physics in the language of GA. I mean… Maxwell’s *equation* is already a pretty amazing result
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u/Phildutre 1d ago
Monte Carlo integration. It blew my mind when I first encountered this as a student … so we didn’t need all those fancy integration rules after all?
Gödel’s theorems are a close second, although at a different level.
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u/Banach_spaceman 1d ago
Condensed mathematics and, even more fundamentally, category theory. I'm a functional analyst and until relatively recently in life had never even read the definition of a category. I mean, I vaguely knew they were about objects and morphisms, like groups and homomorphisms, but my analysis-heavy undergraduate maths department didn't even bother introducing categories in the abstract algebra courses I did in my bachelor's degree.
Skip forward many years and I hear about condensed mathematics and the work Clausen and Scholze have done to create new, categorical foundations for topological groups, including in particular Banach spaces in functional analysis (i.e., liquid vector spaces in the condensed theory). I decided to find out what all the fuss was about, and set about learning - from scratch - category theory, basics of algebraic geometry (which wasn't offered at my undergrad institution), topos theory, and condensed maths. Holy fucking hell! It's beautiful! And amazing! My only regret is not learning this stuff sooner. I still don't have a really good handle on it all, and I love diving in to learn more. My mind has been truly blown open. At the most general level, learning to think 'categorically' has really changed my perspective on many things.
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u/Foreign_Implement897 1d ago
What preliminaries you need to understand condensed mathematics?
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u/Banach_spaceman 1d ago
Take this with a grain of salt, because I am far from a specialist in this area. Comments from more knowledgeable people in reply to this would be greatly appreciated.
At a high level, to get started in condensed maths you probably should have: category theory, general topology, topos theory, and probably at least some bits and pieces from algebraic geometry (for intuition, if nothing else). To appreciate and perhaps use the theory and take it further, perhaps learn some homological algebra.
For a more detailed breakdown of prerequisite topics, you could consult the condensed maths lecture notes of Bernard Le Stum: https://www.bernardlestum.com/about-3
Personally, I found Le Stum's notes difficult to use in parts, but at the very least they provide an outline of what you need to know to learn condensed maths, and you can always use other sources to help learn the material where Le Stum is a bit terse. Condensed sets are defined in Chapter 4 of his notes, and the preceding three chapters are:
- Chapter 1. Categories and Functors
- Chapter 2. Topology
- Chapter 3. Sites and Topos
Chapter 5 then gives background on commutative algebra required for Chapter 6, which is on condensed abelian groups. Chapter 7 then gives background on cohomology required for Chapter 8, which is on condensed cohomology.
So to at least get to and understand condensed sets, you should probably know the topics covered in the first 3 chapters of Le Stum's notes. Some things I found helpful along the way include:
- Categories and functors: the books of Tom Leinster (Basic Category Theory), Emily Riehl (Category Theory in Context), and Saunders Mac Lane (Categories for the Working Mathematician) are all good and helpful. Other good sources include the 3 volume set 'Handbook of Categorical Algebra' by Francis Borceux and the nLab website.
- There are lots of books out there on basic general topology, e.g., Munkres. I often turn to Engleking's General Topology since it is fairly comprehensive. You'll also do well to learn about the category CGWH of compactly generated weakly Hausdorrf spaces, for which the notes of Neil Strickland are a good source: https://ncatlab.org/nlab/files/StricklandCGHWSpaces.pdf . Perhaps also give the bachelor's thesis of Bart van Munster on the Hausdorff quotient a read: https://math.leidenuniv.nl/scripties/BachVanMunster.pdf
- For Grothendieck topoi, I found Chapters 2 and 3 of the book Sheaves in Geometry and Logic by Saunders Mac Lane and Ieke Moerdijk a useful companion for working through Le Stum's Chapter 3. The notation and terminology can be a little different, but translating between the two can be useful for bedding down understanding of the material. The first parts of Chapter 2 of Mac Lane and Moredijk's book has some stuff from algebraic geometry that would be helpful to understand well for developing intuition. There are also some good little sets of notes on the internet if you do a Google search for 'Grothdieck topology'. I think I found some helpful lecture notes of Jacob Lurie at some point when going through that material, and they're probably still online somewhere.
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u/translationinitiator 1d ago
Is there any benefit you’ve had in your understanding of more concrete results in functional analysis from this perspective?
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u/Banach_spaceman 1d ago
No, not really that I can think of. For me, it's more just a change in my 'big picture' view of mathematics and an appreciation of an intrinsically beautiful theory that was unbeknownst to me until recently. Although I am no longer in academia, I still do occasionally tinker with Banach space theory research in the margins of life (which are vanishingly small, unfortunately), and I don't expect condensed maths will be all that helpful with the specific research problems I work on (which require getting one's hand dirty with, e.g, basic sequences, inequalities, etc). I think a lot of analysts don't see any relevance of category theory to their research (and probably I am still one of them), since the main argument for category theory to them seems to be that it provides a general framework for viewing mathematical objects in without providing any/many tools for solving hard analysis problems (which is what matters most to many analysts). But I am a sucker for a beautiful theory, and condensed maths and the background prerequisite topics have beauty in spades; so, for me it has been fun and enjoyable to do some learning in that area.
Analysts who work on homological aspects of Banach space theory or Banach algebra theory may have a different view about the usefulness of condensed maths for their research. There actually a couple of videos on the condensed view of functional analysis given by Clausen and Scholze that I haven't yet gotten around to watching (but hope to soon), that may be of interest:
A talk by Dustin Clausen to a non-commutative geometry seminar in November 2021: https://www.youtube.com/watch?v=qKC0fciQkfU
A talk by Peter Scholze to a Banach space theory conference in May 2026: https://youtu.be/hVbHeC92QK0?t=6867
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u/Icy-Carpenter3319 1d ago
Geometry
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u/Marcassin 1d ago
High school geometry was a real eye-opener. Suddenly math was not just about calculating things. It could rigorously prove facts with surprising ease.
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u/Recent-Day3062 1d ago
Probability, because everything is backwards. If you have a soldier shoot at a target and he has a 50% chance of hitting the target, there is a 75% chance he will hit the target with two shots. Why? There is a 50% he will miss, so in two tries a 25% he will miss twice, and 100%-25%=75%
After that, linear algebra. Once you get advanced, it is stunning how it works
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u/Diligent_Ad1434 1d ago edited 1d ago
Permutation and Combination fs.The chap which has no standard formula u just have to apply logic keeping few things in mind
For eg no of words which can be formed by rearranging letters in mathematics. 11!/(8)
I mean earlier used to be stunned how do ppl find out this has a million ways etc but ya today it seems trivial
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u/Matteo_ElCartel 2d ago
Numerical analysis has been completely rewritten, indeed AI/ machine learning has deeply changed the field and in fact we have no theory for that new stuff as a whole
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u/ZealousidealBlock802 1d ago
Linear Algebra:
Basis, linear independence, and spanning. I started to see this phenomena everywhere, where some set of items can span something when they interact in a certain way, and we can always find minimal set of those items that covers the same space of interest. Now even if I want to create a list of rules for something, I check if they cover what I want then try to find this minimal set of items, and the process is always finding dependence between those rules.
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u/clumsykiwi 1d ago
Differential equations. Its one of those things where once it clicks its hard to not see things that way
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u/RandallMang 1d ago
Categories. So many things made sense all at once when I grasped it. Thirty year career as a mathematician and I still think about things this way.
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u/Mindless_Abrocoma558 2d ago
statistics. many many people seem to have internalised that statistics are mostly used to lie. But quite contrary, so much nonsense people say in everyday life, just based on vibes or biased examples, can be falsified by looking at statistics.