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David Sprayberry (Oct 13 2024 at 20:55):hi all,
i'm a bookworm, want to say autodidact but i learn from the authors. anyways, i have a lack of guides (more knowledgeable people than myself telling me what to do) and am here putting myself out there and calling for help/guidance from more experienced folk. suggestions for math subjects/books would be wonderful.
self-image:
recently mind-blown by math. don't know much, but i know that category theory and logic is where i want to start.
long term goals:
short term goals: to learn:
material i've covered:
material i'm learning:
material i'd like to learn once i'm ready:
Jacob Lurie's Higher Topos Theory (2009)
for the above i'd like to learn requisite math subjects such as linear algebra, abstract algebra, algebraic topology via category theory for example
linear algebra and abstract algebra by studying the associated bicategories or tricategories or operads using Niles Johnson's and Donald Yau's books
any pointers or resource suggestions appreciated, thanks!
John Baez (Oct 13 2024 at 22:32):I have tips for how to learn math, along with lists of my favorite introductory books for many subjects including category theory, here:
For category theory, after Paolo Perrone's book it might make sense to tackle Fong and Spivak's book and then the books by Leinster and Riehl - all free from the authors, and listed on my page.
David Sprayberry (Oct 14 2024 at 00:25):John Baez said:
I have tips for how to learn math, along with lists of my favorite introductory books for many subjects including category theory, here:
For category theory, after Paolo Perrone's book it might make sense to tackle Fong and Spivak's book and then the books by Leinster and Riehl - all free from the authors, and listed on my page.
thanks for your recommendations, yes i should read those, i'll add these to my list!
also Paolo Perrone's book did recommend Riehl's and Fong and Spivak's book as a continuation.
i am planning on learning physics eventually too and your page is a goldmine for book recommendations!
i'm surprised you're so active on this zulip and yet category theory represents such a small percentage of the book recommendations on your page.
btw just fyi the first 2 links are broken on this line: "...available free online at http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf. Also see their [website with videos] and my [lectures]."
i've seen you referenced quite a bit in some literature btw!
John Baez (Oct 14 2024 at 03:40):i'm surprised you're so active on this zulip and yet category theory represents such a small percentage of the book recommendations on your page.
My page is about what people should study to learn the basics of math and physics, and there's a lot to learn besides category theory!
John Baez (Oct 14 2024 at 03:41):btw just fyi the first 2 links are broken on this line:
Thanks - I fixed them. Maybe MIT finally shut down Spivak's web page? (He used to teach there.) Luckily that stuff is available elsewhere.
Madeleine Birchfield (Oct 14 2024 at 17:11):I don't think it's actually necessary to learn calculus or linear algebra before learning set theory, logic, real analysis, or abstract algebra.
Calculus and linear algebra are usually taught before the other subjects in university because they are useful for physicists and engineers and many other people, and so the university usually makes it a graduation requirement for all its students and a prerequisite for the rest of its maths.
But one can simply jump right in to set theory and logic from the end of the basic school maths curriculum since none of the subject actually requires calculus or linear algebra. Actually in some curricula around the world the basics of set theory is taught prior to the real numbers and elementary algebra.
One can think of the subjects of set theory, logic, number theory, the ring theory part of abstract algebra, real analysis, and some basic complex analysis as simply formalising and abstracting the school maths curriculum up to algebra and trigonometry.
John Baez (Oct 14 2024 at 17:38):I certainly don't think it's necessary to study calculus or linear algebra before set theory, logic, or abstract algebra. I hope my page didn't say it was necessary. I think I said it was traditional.
I do think it's bad to study real analysis before calculus: e.g. studying measure theory before you've ever done an integral seems difficult to me, because you couldn't do the calculations that provide examples and counterexamples.
(Personally I think many important theorems in real analysis, like the Dominated Convergence Theorem, only become really interesting when you use them for something, e.g. Fourier analysis, which in turn only becomes interesting when you use that for something, e.g. solving PDE, which in turn only becomes interesting when you use those for something, e.g. understanding the physical world. Other people may have more ability to get interested in self-contained mathematical subjects, so I don't want to project my taste on everyone else. But I only became passionate about real analysis because I saw how it was used to understand light, heat, gravity and so on. And this is indeed a major reason it was developed. So for me, personally, I found a mixed curriculum of logic, math and physics to be very satisfying. I think some pure mathematicians find ODE and PDE boring because they aren't using these equations to understand something.)
Madeleine Birchfield (Oct 14 2024 at 20:27):John Baez said:
I do think it's bad to study real analysis before calculus: e.g. studying measure theory before you've ever done an integral seems difficult to me, because you couldn't do the calculations that provide examples and counterexamples.
I wasn't thinking about measure theory or some of the more advanced topics in real analysis, but rather basic stuff like formally constructing the real numbers (i.e. using Dedekind cuts, Cauchy filters, etc) and proving theorems about the real numbers (like its complete Archimedean ordered field structure); similarly formally constructing and proving theorems about transcendental functions like the exponential, logarithm, and trigonometric functions, as well as other stuff taught in algebra like the fundamental theorem of algebra. This would all lie in the realm of real analysis, and would be approachable by somebody straight out of secondary or high school who has some knowledge of set theory and logic.
The key here, like the case for elementary number theory, wouldn't be learning new facts about the real numbers and real-valued functions, but constructing and proving in a rigourous way the structures and theorems you had already learned previously.
John Baez (Oct 14 2024 at 20:56):I wasn't thinking about measure theory or some of the more advanced topics in real analysis, but rather basic stuff like formally constructing the real numbers (i.e. using Dedekind cuts, Cauchy filters, etc) and proving theorems about the real numbers (like its complete Archimedean ordered field structure); similarly formally constructing and proving theorems about transcendental functions like the exponential, logarithm, and trigonometric functions, as well as other stuff taught in algebra like the fundamental theorem of algebra. This would all lie in the realm of real analysis, and would be approachable by somebody straight out of secondary or high school who has some knowledge of set theory and logic.
Sure, that could be fine, for someone who has an interest in understanding what the real numbers are "really all about".
John Baez (Oct 14 2024 at 20:58):When I hear "real analysis" I think Royden or baby Rudin.
Kevin Carlson (Oct 14 2024 at 21:07):Yes, Madeleine’s description fits something more like Spivak’s Calculus, which treats both calculus and basic real analysis together. This is nice since indeed many people’s first real analysis course is just a retread of their calculus I course, with proofs this time, which can be boring.
Kevin Carlson (Oct 14 2024 at 21:08):It’s a tough book for most new students though!
Madeleine Birchfield (Oct 14 2024 at 21:40):Well, most people's first number theory course is a retread of what they had already learned about the natural numbers, the integers, and rational numbers in their schooling before university, only this time with formal definitions and rigourous proofs.
Compare that with something like elliptic curves or modular forms which are commonly used in more advanced number theory, but won't be taught in people's first number theory class.
Madeleine Birchfield (Oct 14 2024 at 21:51):I do have to say that I am somewhat biased by my background, since the real analysis I've studied and worked with is constructive real analysis. There is a strong connection between logic and real analysis and one can write multiple books on the various axioms one can assume in constructive mathematics, such as LPO or Brouwer's continuity principle, and the impact that would have on the properties of the real numbers and real-valued functions, such as the Heine-Borel theorem, the Bolsano-Weierstrass theorem, and the intermediate value theorem. There is also a lot of emphasis on computations and infinite digit representations in constructive real analysis and how those are affected under the various axioms, especially Markov's principle.
I guess from the perspective of classical mathematics this is all just reverse mathematics and thus a branch of mathematical logic rather than real analysis.
John Baez (Oct 14 2024 at 22:06):Yes, and I never heard about the use of constructive reasoning when taking a year-long course called "real analysis" as an undergrad. We studied the properties of continuous and differentiable functions , then measures and integrals, spaces, distributions, and such. The axiom of choice was used ad libitum, e.g. to prove the existence of nonmeasurable functions. This is still very common at universities.
Madeleine Birchfield (Oct 14 2024 at 23:02):I guess something similar happened with set theory - where if you take set theory from a university most of the time it will be ZFC or a similar theory using a single membership relation , whereas the first set theory I learned was ETCS and the second theory of sets I learned was dependent type theory with axiom K to make every single type in the theory a set.
As well as topology - the topology taught in universities is point-set topology rather than point-free approaches such as locales or formal topologies.
Madeleine Birchfield (Oct 14 2024 at 23:08):Also, about this:
John Baez said:
I certainly don't think it's necessary to study calculus or linear algebra before set theory, logic, or abstract algebra. I hope my page didn't say it was necessary. I think I said it was traditional.
Your page says
After basic schooling, the customary track through math starts with a bit of:
and
To dig deeper into math you need calculus and linear algebra, which are interconnected:
David Sprayberry (Oct 15 2024 at 00:03):Madeleine Birchfield said:
I don't think it's actually necessary to learn calculus or linear algebra before learning set theory, logic, real analysis, or abstract algebra.
Calculus and linear algebra are usually taught before the other subjects in university because they are useful for physicists and engineers and many other people, and so the university usually makes it a graduation requirement for all its students and a prerequisite for the rest of its maths.
But one can simply jump right in to set theory and logic from the end of the basic school maths curriculum since none of the subject actually requires calculus or linear algebra. Actually in some curricula around the world the basics of set theory is taught prior to the real numbers and elementary algebra.
One can think of the subjects of set theory, logic, number theory, the ring theory part of abstract algebra, real analysis, and some basic complex analysis as simply formalising and abstracting the school maths curriculum up to algebra and trigonometry.
yea i agree it's not necessary to learn in the traditional order, for example linear algebra first, then abstract algebra, then homological algebra and category theory. most linear algebra books teach how matrix multiplication works and have students follow the algorithms like robots whereas i liked how Birkhoff and Mac Lane's explanation made it more interesting from the get-go by showing matrix multiplication as a morphism between biproducts (products coinciding with coproducts) in a category of modules. i think it makes more sense to start with category theory. since you mentioned logic, i think the unity of math is very nice and i think it has logic at it's core. i now understand why the math majors and computer science majors are bundled together at a lot of universities, is because logic and computer science is where math really starts.
i agree with John Baez's point that theorems become really interesting (and i would say study motivation is generated) only when you know you're about to use them for some application or further learning closer to your interests that depends on it.
i like category theory because (i'm just a noobie, correct me if i'm wrong, but seems to me that) if you think of math subjects as landscapes, category theory's general approach is to occupy the mountain peak and have a good perspective.
for example, i wanted to learn algebra (needed for understanding a topology text for example) reading Birkhoff and Mac Lane's Algebra (1967-99) and Aluffi's Algebra: Chapter 0 (2009-16) and Anderson and Fuller's Rings and Categories of Modules (1974-92) (which I was happy to see on your page you mentioned, John Baez!), all of which are exceptional books in the fact that they teach using a categorical approach and are often mentioned to people on q&a sites looking for such books.
i'm glad i just stuck with learning categorical algebra, reading Niles Johnson and Donald Yau's 2-Dimensional Categories (2021) and was blown away by the 'small asides' of first, Example 2.1.26 (that there's a bicategory Bimod, objects: rings, 1-cells R -> S: bimodules ,...), and second Section 6.3 (i say 'aside' because: "This section describes several basic examples of duality in algebra as adjunctions in the bicategory Bimod. The results in this section are not used elsewhere in this book" pg. 256). Like that's how I want to learn linear algebra! The older books above do not mention bicategories, or a monad or monoidal category either for that matter.
David Sprayberry (Oct 15 2024 at 00:26):Madeleine Birchfield said:
ETCS
yes exactly! i have not yet learned it but have only seen point-free topology mentioned by authors of category theory books. very cool that you know DTT! i'll paste an edit (a question) i made above below.
i wanted to quote Yu Manin author of A Course in Mathematical Logic for Mathematicians (1977-2010): "It would be difficult to dispute nowadays that category theory as a language is replacing set theory in its traditional role as the language of mathematics." (pg. ix)
question:
there are some conjectures in a paper that came out this year by Mario Carneiro arXiv:2403.14064v1
why have these conjectures not been decided yet? i'd like to have the requisite knowledge to be able to try to prove/disprove them. what books should i read for this?
John Baez (Oct 15 2024 at 00:40):Madeleine Birchfield said:
Also, about this:
John Baez said:
I certainly don't think it's necessary to study calculus or linear algebra before set theory, logic, or abstract algebra. I hope my page didn't say it was necessary. I think I said it was traditional.
Your page says
To dig deeper into math you need calculus and linear algebra, which are interconnected:
Okay, I may water that down a bit. I still think most people who are serious about math will learn these pretty soon, and actually should, but it's true there's a lot that doesn't logically require these subjects. I have to think about who is the audience for this page, and how much I'm trying to aim them down the "usual" route.
Mike Shulman (Oct 15 2024 at 00:43):David Sprayberry said:
"It would be difficult to dispute nowadays that category theory as a language is replacing set theory in its traditional role as the language of mathematics."
I don't know the context of this quote, so it might have made sense there. But taken literally, I would dispute it. I would say this is, if you'll forgive the phrase, a category mistake in the philosophical sense. Set theory and other foundational systems (ZFC, ETCS, DTT) answer the question "what are mathematical objects built out of?", while category theory answers a different question "how are mathematical objects organized?". They are complementary; neither can replace the other.
David Sprayberry (Oct 15 2024 at 01:32):Mike Shulman said:
I don't know the context of this quote, so it might have made sense there. But taken literally, I would dispute it. I would say this is, if you'll forgive the phrase, a category mistake in the philosophical sense. Set theory and other foundational systems (ZFC, ETCS, DTT) answer the question "what are mathematical objects built out of?", while category theory answers a different question "how are mathematical objects organized?". They are complementary; neither can replace the other.
It's page ix of Preface to the Second Edition https://link.springer.com/content/pdf/bfm:978-1-4419-0615-1/1
haha very cool way to put it.
what would you recommend as a good introduction to philosophy which explains such things as category mistakes or that you've found to be great reads?
i saw that Awodey agreed with a quote of you:
"I would say -- and I think Steve and Andrej would agree -- that the HoTT notion of propositional truncation is the "modern successor" of Propositions As [Types]. IIRC the rules in that paper aren't quite right in the intensional / higher-categorical case, but now in HoTT we have the correct rules and a general justification."
i'm interested in the understanding you have of foundational systems (ZFC, ETCS, DTT).
i understand the idea of propositions as types,
i've studied Avigad's Mathematical Logic and Computation (but the end of the book seemed like he was only just getting started and not sure exactly where to go from there),
but other than that the quote goes over my head.
for example, what is the IIRC paper? i'd like to learn the intensional / higher-categorical case.
what do you recommend as next books so that i might learn these correct rules?
i've seen https://homotopytypetheory.org/book/ recommended and was planning on reading it at some point but i've been reading Clive Newstead's thesis: Algebraic models of dependent type theory (2018) to get an idea of that.
i'd like to be able to understand the difficulties in proving some conjectures in a paper that came out this year by Mario Carneiro arXiv:2403.14064v1
Mike Shulman (Oct 15 2024 at 01:40):I'm not a good person to recommend philosophy books. The paper I was referring to in that quote is the one called "Propositions as [Types]". I think the HoTT Book is a good introduction to propositional truncation in the homotopical type theory case, although Egbert Rijke's book "Introduction to homotopy type theory" is also good.
Madeleine Birchfield (Oct 15 2024 at 01:46):David Sprayberry said:
question:
there are some conjectures in a paper that came out this year by Mario Carneiro arXiv:2403.14064v1why have these conjectures not been decided yet? i'd like to have the requisite knowledge to be able to try to prove/disprove them. what books should i read for this?
I'm afraid I won't be able to help you here. While I am familiar with dependent type theory, I am not all that familiar with either the metatheory of dependent type theory, or the Lean project, or formal verification of programming languages. However, the author Mario Carneiro is on the Lean Zulipchat, so you might be able to ask him there what he thinks are the prerequisites to understanding his paper.
David Sprayberry (Oct 15 2024 at 01:50):Madeleine Birchfield said:
the author Mario Carneiro is on the Lean Zulipchat, so you might be able to ask him there what he thinks are the prerequisites to understanding his paper.
i shall do this, thanks!
Madeleine Birchfield (Oct 15 2024 at 02:16):David Sprayberry said:
most linear algebra books teach how matrix multiplication works and have students follow the algorithms like robots
This is because most linear algebra books and most linear algebra courses at universities are geared towards physicists and engineers and computer scientists where the goal is to teach them to use these algorithms in helping them construct various models and solve various problems in their domain, not necessarily to help them understand the properties of vector spaces/modules and linear maps in a deeper level.
There is usually a separate course called "abstract linear algebra" or something similar at universities which is aimed at mathematicians and does go more deeply into the theory of vector spaces and modules and linear maps between vector spaces.
John Baez (Oct 15 2024 at 02:27):Yeah, the math majors at my university have to take a course on linear algebra that's not just mindless matrix manipulation. You can see that part 1 of this course covers the concepts of vector space, linear map, dimension, isomorphism, basis, subspace, quotient space and dual space.
Then part 2 covers eigenvectors, Gram-Schmidt orthonormalization, projections, change of basis for matrices, determinants, diagonalization, and Jordan normal form.
This is different from the applied linear algebra course.
John Baez (Oct 15 2024 at 02:29):Part 1 is a decent warmup to the theory of modules over rings that you'd learn in abstract algebra, though it exploits all the simplications of working over a field. Part 2 involves the inner product more, so part of it is a warmup for studying Hilbert spaces.
Madeleine Birchfield (Oct 15 2024 at 02:35):Something I do not get about universities and American universities in particular is why they teach calculus with multiple variables before linear algebra. You would think that learning about real vectors and their linear maps first would be useful in differentiating non-linear maps with multiple variables.
John Baez (Oct 15 2024 at 02:41):So much of a typical elementary multivariable calculus class is really just 2- and 3-variable calculus - div, grad, curl, Gauss' theorem, Stokes' theorem - in short, the stuff that people need to understand Maxwell's equations - that I believe universities prefer not to force their physicists and engineers to have studied linear algebra first, before they can study electromagnetism. Every prerequisite costs time, and the physics and engineering departments would complain bitterly if they were increased. They're always threatening to teach their own math courses - a threat that math departments are constantly guarding against.
John Baez (Oct 15 2024 at 02:43):All this div grad curl stuff was basically invented for physics, roughly speaking.
Alex Kreitzberg (Oct 15 2024 at 02:59):Kind of a funny story, at my undergrad college a professor dropped a multi variable course so a grad student was asked to cover it. The grad student said they would, under the condition they were allowed to teach differential forms.
Their thought process was that it'll be fine provided he simply explains the calculations like any other applied course.
The students complained their professors wouldn't be able to grade their homework, and that the methods were incongruous with their physics notes and textbooks. But, the grad student was so passionate about differential forms, they offered to grade all of their physics homework if the physics professors couldn't or wouldn't.
The physics students didn't like this! I wish I knew whether any of them took him up on that offer :rolling_on_the_floor_laughing:
David Sprayberry (Oct 15 2024 at 03:02):Madeleine Birchfield said:
Calculus and linear algebra are usually taught before the other subjects in university because they are useful for physicists and engineers and many other people, and so the university usually makes it a graduation requirement for all its students and a prerequisite for the rest of its maths.
yes this makes sense
David Sprayberry (Oct 15 2024 at 03:04):John Baez said:
Yeah, the math majors at my university have to take a course on linear algebra that's not just mindless matrix manipulation.
forgive my exaggeration. i like how Mike above said category theory is about organization. my opinion is that category theory should be taught before linear algebra because it's more useful regardless, as Fong and Spivak's book (which i have not yet read) i assume shows, and that one should not wait to give a breakdown from the abstract to the particular when teaching math. seems current curricula have their students blindfolded to the larger structure. they could have studied those courses knowing bicategories and tricategories beforehand, and gotten much more out of it, seems to me, but idk.
Madeleine Birchfield (Oct 15 2024 at 03:21):I disagree, it is better to begin with more specific examples and then generalise to the abstract case. For example, in ring theory, we give various examples of rings out there, such as integers, real numbers, polynomial algebras, matrix algebras, Clifford algebras, integers modulo a natural number, etc, so that people have some intuition of what a ring could be. Then we abstract away from all that and define rings.
It's the same with category theory. One would study set theory and Cartesian products, disjoint unions, and function sets, then one would study abelian group theory and direct sums, direct products, and tensor products of abelian groups and the abelian group of homomorphisms, then one would study linear algebra and direct sums, direct product, tensor product of vector spaces and the vector spaces of linear maps, and then one would study topological spaces and product topological spaces, disjoint union topological spaces, and function spaces. Finally, one has enough examples to then say okay in general categories these are products and coproducts and tensor products and internal homs/exponential objects, so that people can understand why these concepts in categories are so important.
Madeleine Birchfield (Oct 15 2024 at 03:26):One can teach subjects in a categorical manner, defining direct sums of abelian groups or product topological spaces in terms of its universal properties and then using the universal properties to prove theorems in the subject, without ever mentioning the word "category". Tom Leinster is doing that right now with a set theory course he is teaching.
Madeleine Birchfield (Oct 15 2024 at 03:33):There is also the issue that courses on category theory usually also teach about functors, which I do not really feel is all that useful in many areas of mathematics until one gets to much more advanced topics in that field, such as homological algebra, condensed mathematics in complex and functional analysis, or sheaves in algebraic and differential geometry.
Joe Moeller (Oct 15 2024 at 18:35):I think it’s easy to find functors, especially adjunctions, fairly early in many branches: in/discrete topological spaces, the topology generated by a metric, free groups, etc.
Alex Kreitzberg (Oct 15 2024 at 18:53):I'm partial to group and monoid homomorphisms as functors as well.
Todd Trimble (Oct 17 2024 at 20:05):Madeleine Birchfield said:
Well, most people's first number theory course is a retread of what they had already learned about the natural numbers, the integers, and rational numbers in their schooling before university, only this time with formal definitions and rigourous proofs.
Compare that with something like elliptic curves or modular forms which are commonly used in more advanced number theory, but won't be taught in people's first number theory class.
I would strongly dispute that first paragraph. Here are course notes for a course in elementary number theory theory, randomly plucked off the web. You will see that it's hardly a retread of familiar topics but with more rigor. The selection of topics is very consistent, in my experience, with a typical such course. A student in such a course will learn some very cool theorems and techniques.
Any student who takes a course in number theory and finds it's a retread of pre-university stuff should ask for their money back -- they've been robbed.
John Baez (Oct 17 2024 at 20:25):Yeah, one good feature of number theory as a first "intro to proofs" course is that in number theory there are a lot of statements that are easy to understand yet surprising, which therefore seem to require proof. I've taught a quarter-long course of that sort twice, which went up to quadratic reciprocity. I found the proof of quadratic reciprocity in the book to be too long and complicated to be fun. (There are nicer proofs that use Fourier theory, but I couldn't really get into that.) If I taught the course again I'd stick to more "bite-sized" theorems.
Madeleine Birchfield (Oct 17 2024 at 20:48):Well I guess i stand corrected about number theory; I might have gotten that course confused with a different course at university which is aimed at future maths teachers.
Todd Trimble (Oct 17 2024 at 23:47):When I taught it (once, at SUNY Brockport, a very modest institution), I got up to the two squares theorem, that primes congruent to 1 mod 4 are sums of two squares. This involves the arithmetic of Gaussian integers, for which there is a division lemma (or whatever it's called: the statement that is a [[Euclidean domain]]). This has some pleasant geometry underling it.
On a different occasion, this time at Northwestern, I taught a course where the textbook I used was Concrete Mathematics by Ronald Graham, Donald Knuth, and Oren Patashnik, and part of this course involved number-theoretic functions such as the Euler -function and the Moebius function. (Of all the undergraduate courses I've taught, this was probably my favorite.)
There's a beautiful "structural" proof of [[quadratic reciprocity]], ascribed to Zolotarev and more or less explained by Conway and Fung in The Sensual (Quadratic) Form, but which unlike Fourier theory (i.e., Gauss sum) type proofs, hasn't suggested further deep number-theoretic extensions along class field theory lines. Which I've always felt was a shame.
David Sprayberry (Nov 03 2024 at 23:08):Sorry about the poor etiquette from me like no caps, I haven't presented myself very professionally, and I appreciate getting feedback from professionals.
The feedback unanimously seems to be that I should learn math more or less as a university would teach it, other than some exceptions mentioned, such as the multi-variable calculus before linear algebra.
The purpose of abstracting is
not to be vague, but to create
a new semantic level in which
one can be absolutely precise.
E. W. Dijkstra
Madeleine Birchfield said:
I disagree...
I can see how starting with small examples makes a lot of sense in handling complexity. Like the definition of a tricategory might make someone want to run away (me). Having said that
John Baez said:
...not just mindless matrix manipulation...
some presentations of beginner material seem purposefully meant to 'weed' people out of their subject, such as Algebra by Michael Artin, which is an example of an introduction to linear algebra by way of unenlightening matrix manipulation algorithms.
Thanks for the references to those course materials @John Baez , I'll focus on first learning the equivalent of Peter Peterson's 2012 book, which seems to be what those course materials are based off of.
Madeleine Birchfield said:
the first set theory ... was ETCS and the second ... was dependent type theory with axiom K.... [T]opology ... point-free approaches such as locales or formal topologies.
@Madeleine Birchfield Where did you learn these subjects/approaches from? For ETCS I'm only aware of this source to learn from, and I've seen The point of pointless topology(1983) by Peter T. Johnstone cited.
@John Baez I saw the first part of the talk you gave at the Oktoberfest and found the discussion very interesting. I've been reading From Groups to Categorical Algebra by Dominique Bourne to learn about the simpler of the categories you mentioned such as linear categories and plan to learn about symmetric rig categories once I get the basics down!
I had some observations that I was hoping to get help putting into place where they stand in relation to one another:
I'm also interested in logic and model theory, which is why I'd like to study logic in connection with algebra eventually, and why I'd like an understanding of where the subjects lie in relation to each other, and the order they should be studied in.
Putting down category theory and instead looking for an overview of math subjects I looked at the contents of Mathematical Concepts by Jürgen Jost which seems to have taken me in a circle as I saw that the last chapter is on topos theory.
John Baez (Nov 04 2024 at 01:29):@David Sprayberry wrote:
The feedback unanimously seems to be that I should learn math more or less as a university would teach it, other than some exceptions mentioned, such as the multi-variable calculus before linear algebra.
I think a good approach is to learn some of whatever your heart desires, while in parallel making sure you master the usual university courses expected of math majors, so that at end you can talk to mathematicians and be accepted by them (which is extremely morale-boosting and fun) while still having your own unique style. That's more or less what I did. And it sounds like that's what you're aiming to do.
John Baez (Nov 04 2024 at 01:39):
- Theory in the vicinity of operads and generalized PROPs seems to have to do with classifying algebras. The contents of A Foundation for PROPs, Algebras, and Modules by Donald Yau and Mark W. Johnson seems to be an extensive and relatively all-encompassing way of classifying algebras.
- Universal algebra seems to be a way of classifying algebras and using model theory for applications in algebra, from what I read of the forward by F. W. Lawvere of Algebraic Theories by Adámek, Rosický and Vitale.
This is all roughly correct; let me just say some stuff....
When I hear "model theory" I think of hard-core classical logic and set theory, which is quite different in flavor from universal algebra. In universal algebra you greatly limit the expressive power of your logic, in order to get more powerful results about the structures you can still describe. (This is always the tradeoff.) Basically, in universal algebra you only describe structures that have a bunch of operations obeying equational laws: this leaves out a lot of math, but covers groups and rings and lots of other "algebraic" things.
The category theorist's approach to universal algebra involves Lawvere theories and algebraic theories. There's also a more old-fashioned approach due to Birkhoff, which is also worth learning.
Operads and props are similar enough to Lawvere theories that in the long run you'll see them as different parts of a spectrum.
John Baez (Nov 04 2024 at 01:42):Topos theory seems to always be mentioned in connection with logic, and in your Oktoberfest talk you mentioned topos theory is essentially the algebraic theory of rigs with products distributing over coproducts.
I hope I said that topoi are nice examples of rig categories where the multiplication is given by products, the addition is given coproducts, and the products distribute over the coproducts. There are also lots of such rig categories that aren't topoi. And it doesn't even parse to say "topos theory is the algebraic theory of rigs with products distributing over coproducts" - there are 2 separate ways this sentence fails to parse.
John Baez (Nov 04 2024 at 01:42):I've also heard about Quillen model categories being used. Are they also used in studying algebra?
Mainly "homotopical" algebra - the kind of algebra that algebraic topologists use.
John Baez (Nov 04 2024 at 01:46):I think you're talking about too many big ideas at once here for me to say anything truly helpful about any one of them. Each one takes quite a while to explain. I'm almost glad I grew up before the internet, so that I didn't meet so many big ideas at the same time! Especially before I went to college, books were precious and I would tend to get to know them very well, one at a time. Nowadays everyone has to work harder to keep from getting hopelessly distracted by the endless sea of links. Props, operads, Lavwere theories, algebraic theories, model categories, topoi.... aargh!
David Sprayberry (Nov 04 2024 at 01:51):Yes, I'm interested in math so I'd like to educate myself well in it. I'll keep these things in consideration.
Thank you, this is very informative.
Madeleine Birchfield (Nov 04 2024 at 03:49):David Sprayberry said:
@Madeleine Birchfield Where did you learn these subjects/approaches from? For ETCS I'm only aware of this source to learn from, and I've seen The point of pointless topology(1983) by Peter T. Johnstone cited.
There's Tom Leinster's Rethinking Set Theory and then there are Todd Trimble's blog posts on ETCS:
More recently, Tom Leinster is teaching a course on ETCS at university and has posted his lecture notes here.
Madeleine Birchfield (Nov 04 2024 at 04:12):As for point-free topology, I learned it piecemeal by reading research papers in that field, such as Generalized spaces for constructive algebra by Ingo Blechschmidt or Uniform locales and their constructive aspects by Graham Manuell, and then taking a look at the references inside the articles for more information.
More generally, reading new research papers, usually preprints on the arXiv, is how I learned a lot of the mathematics I know. I tend not to read textbooks or old papers from 50 years ago, since the material usually aren't available to those outside university unless they are willing to fork over a lot of money to publishers and journals. People inside universities usually have free journal subscriptions through their institution and textbooks in their university libraries.
Also the nLab has a good repository of information on mathematics, though some of the stuff on the nLab is incorrect so beware.
John Baez (Nov 04 2024 at 04:36):I tend not to read textbooks or old papers from 50 years ago, since the material usually aren't available to those outside university unless they are willing to fork over a lot of money to publishers and journals.
A lot of important math monographs and textbooks are available for free electronically from LibGen. I often find it easier to systematically learn stuff by repeatedly reading books rather than reading papers - first skimming, then digging deeper. But I also spend a lot of time mining the arXiv, and Wikipedia, and the nLab.