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For a more advanced book: Category Theory in Context by Emily Riehl

F. William Lawvere - Conceptual Mathematics: A First Introduction to Categories

Eugenia Cheng - The Joy of Abstraction: An Exploration of Math, Category Theory, and Life

She builds up to the category theory chapters.

Book club: https://youtube.com/playlist?list=PLhgq-BqyZ7i7tEEQVG5rlOG8y...

GRPO project is neat. Would you be willing to do a Karpathy-style explainer, breaking down the algorithm from scratch? It’s hard to understand on its own without prior background knowledge.

Understood! How much is it to enroll in your “Methods of Proof” course or in Math Academy more generally? I didn’t didn’t quite understand how it works based on the FAQ, is it lecture-problem based with an interactive testing element for course placement?

Thank you for the response and for making these resources fully available.

Do you plan to make a proofs based book available as well?

This is phenomenal. I currently don’t have a great internet connection, so the pdfs won’t load. Do these books have solutions?

What is the pricing model for this? I’m interested in enrolling.

Thanks! So, you are considering expanding the Zero to Hero series to include building a basic GPT-2 toy chatbot? I believe you mentioned in one of the early lectures that you planned to include building a toy version of Dalle. Do you still have plans for that as well?

You might have covered this topic before, but I'm curious about the main performance differences between nanoGPT and llm.c. I'm planning to take your "Zero to Hero" course, and I'd like to know how capable the nanoGPT chatbot you'll build is. Is its quality comparable to GPT-2 when used as a chatbot?

Do you plan to do any investigative pieces on AI alignment? I’d be interested in a piece that interviewed people like Paul Christino, Eliezer Yudkowsky, Chris Olah and the like. Covering opposing views from doomerism to e/acc.

Thanks for the link. I like taking these for fun, any idea where to find the advanced set for older teenagers and adults?

Has anyone ever had success on an app like Hinge, then deleted it only to see a large drop in your success rate when nothing else changed about your profile?

If so, it could be due to things like this:

https://tech.okcupid.com/evaluating-perceptual-image-hashes-...

Match Group owns OkCupid, Hinge, Tinder and so on.. so wouldn’t be surprising if they all used the same or similar algorithms for spam prevention and noob boost abuse.

They probably also collect other data like your phone number, deviceID, etc.

This is bad because if you ever delete your profile and remake it then your internal ranking could plummet due to these aggressive spam filters.

>“ There is a facebook group called "Topology Without Tears Readers" where readers of the book can communicate with each other. But this is definitely not a place to ask others to solve your homework problems. If you ask questions like "How do you solve the problem ...." your post will be removed and you will probably be blocked from this group.”

My complaint with this is the audience reading this book aren’t students in classes and likely a lot of them don’t have access to mathematicians or professors. Working with others is a great way to learn how to properly do proofs when first starting out. I think the FB groups / discord groups or whatever communities of learners should discuss their proofs and suss out difficulties or logical errors with each other.

This approach instead seems to assume the learner will know whenever they develop a correct proof but often one can just fool themselves into thinking their proofs are correct or even get utterly stuck. Or it assumes seeing solutions will “rob” the readers of learning.

Also, who really cares? Most of the people working through this book aren’t getting a grade from it. If they want to rob themselves of learning something just by seeing solutions without thinking first, then that’s on them.

Otherwise, beyond that complaint this seems like a good resource and it’s impressive it’s all free. I’d also recommend another topology book that’s free https://topology.mitpress.mit.edu/, albeit it’s more advanced.

Despite using Google maps I like having a separate GPS navigation system like this running in the background as a backup. That’s in case if something happens to my phone and it no longer works, then I have a secondary GPS device.

this video may help

https://youtu.be/iC-KVdB8pf0

yes

https://youtu.be/iC-KVdB8pf0

http://zxcalculus.com/

https://github.com/Quantomatic/pyzx

https://en.m.wikipedia.org/wiki/ZX-calculus

To be honest I care more about the high iq kids interested in math who come from poor or working class families that either won’t be identified or will be identified but not much can be done for them given the lack of resources. Plus this doesn’t negate that someone of a similar IQ may accomplish less / seem less impressive at an early age due to such disadvantages.

Not to downplay any of Daniel’s accomplishment but sometimes it isn’t a “fair” comparison when others started younger with more resources. His father is a distinguished professor of mathematics and his mother is a professor of mathematics. When you have that sort of resources available at a young age and advanced training you’ll probably accomplish more sooner than someone of similar IQ without those resources who started later

Depends what “non-expert” means. Because I actually think even if you have an undergraduate knowledge in mathematics then you still may not be able to follow some of his arguments unless you specifically studied mathematical logic. I have a more advanced background in math but not in logic and don’t understand some of his arguments. He sometimes makes loose statements without proof or without providing strong background knowledge to closely follow those things if you don’t already know them. Some of the ZFC stuff I didn’t follow either, but I never really needed to know set theory to the extent Aaronson uses it. I think the statement it’s quite approachable isn’t very accurate. If you want to gloss over it and still get the main idea then it’s good for that but not for rigorously understanding all the mathematical arguments without preexisting background knowledge. It’s between a pop sci book and textbook with a casual tone. It’s a fun book.

I disagree that it is “quite approachable for the non-expert” assuming you meant a general audience without a background in high school mathematics. They probably wouldn’t exactly understand the Löwenheim–Skolem theorem or ZFC, or consistency arguments and so forth as presented in the book.

Some parts require more math than probably the general audience will be able to grasp but if they gloss over those sections they could still get the gist of what follows.

I think I’d rephrase it as “quite approachable for a talented high schooler / someone with knowledge in undergraduate mathematics”

what math background is needed to specialize in ZKP?

for proofs i really liked two sources: susanna epps’s discrete math book and also rosen’s discrete math text (though epp’s is friendlier for the beginner)

supplement those books with these lecture videos:

https://youtube.com/playlist?list=PLl-gb0E4MII28GykmtuBXNUNo...

they appear to follow rosen but they’re the same topics you’d see in epp

if you’re the complete beginner go with epp and then use rosen as a supplement.

if you are an advanced undergraduate in math but don’t know category theory then:

https://math.jhu.edu/~eriehl/context.pdf

or

Basic Category Theory by Tom Leinster

(even though i’ll list a bunch more resources below i’d probably start with leinster and work up enough math maturity to push through it.)

i also liked this: https://arxiv.org/pdf/1912.10642.pdf

if you don’t know advanced math then there is an upcoming (not yet released book) called “the joy of abstraction”

i cannot attest for the book as i haven’t read it but may be good for true beginners:

https://www.cambridge.org/us/academic/subjects/mathematics/l...

(click “look inside”)

they say “no formal mathematical background needed”

you can read the description and see if that is something that would be of interest to you.

this guy has videos + a book

https://youtu.be/fY02LIW8fvk

there are a lot of books like “category theory for programmers”, “programming in categories” and the “seven sketches” books along with lecture recordings and videos that you may find helpful.

category theory for programmers might be the easiest of those three books. i worked a bit out of the programming with categories (http://brendanfong.com/programmingcats_files/cats4progs-DRAF...) book and ignored all the haskell sections and just focused on the math parts (this is a distinct book from “category theory for programmers”. people confuse the two due to the names and the fact they both use haskell).

if you know category theory and want to learn topology then there is “ Topology A Categorical Approach” by bradley et al.

As Jensson recommended I would learn abstract algebra first to understand concrete examples in category theory.

If abstracts algebra is too difficult then learn introduction to proofs before continuing with abstract algebra.

Try doing problems and posting them online to have them checked. Currently you can use discord, reddit, math stack exchange etc

Don’t worry about rushing it. Everything builds on itself. For instance once you rigorously study introduction to proofs you’ll see the same sort of proofs again when you begin studying abstract algebra and will already be familiar with the concepts.

The author also wrote this excellent article:

https://www.newyorker.com/magazine/2015/02/02/pursuit-beauty

Does anyone have experience with being a non-traditional PhD student and doing your PhD later in life?

I worked a bit and want to return for a PhD in CS. I took some courses at a nearby (and good) institution and did well (for fun). I want to continue taking graduate courses and and get involved in research before I apply for PhD programs (since I been out of school for a while).

Curious to hear from older students their experiences or from professors on thoughts about older students (say late 30s/early 40s) and whether there are job opportunities in industry for folks in this age range upon completion (in say applied areas).

I’m doing it because I really want to do research and enjoy hard theoretical problems outside the scope of your regular SWE job.

I’ll copy paste what I wrote before about quantum country

This doesn’t provide solutions to the problems and doesn’t show you how to do the mathematical calculations. It assumes you know proof based linear algebra and if you’re a math beginner the level that this is written at will be beyond you.

Instead I recommend

https://www.thomaswong.net/introduction-to-classical-and-qua...

which doesn’t assume much pre-req knowledge and contains solutions so you can check your work

This doesn’t provide solutions to the problems and doesn’t show you how to do the mathematical calculations. It assumes you know proof based linear algebra and if you’re a math beginner the level that this is written at will be beyond you.

Instead I recommend http://www.thomaswong.net/introduction-to-classical-and-quan... which doesn’t assume much pre-req knowledge and contains solutions so you can check your work

Grothendieck was the best in class at abstract mathematical reasoning and some regard him as the best mathematician in the 20th century. The Heron Formula or “prime” example doesn’t negate that

I wouldn’t go far as to say Von Neumann was smarter than Grothendieck. I think they’re both different types of geniuses, where their genius manifest in different ways. Grothendieck was a genius in working with extremely deep abstractions, I’d say he eclipses Von Neumann in this way, whereas Von Neumann had a different type of genius in which he eclipsed others at. In Grothendieck’s case he was a profound genius, who made profound impacts in mathematics.

Another mathematician that reminds me of Von Neumann is Euler. He also memorized long passages and could do complicated calculations in his head quickly.

A quote on Euler from wikipedia:

“He was able to, for example, repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which was the last even decades after having read it”

This is a proof assistant, not an automated theorem prover. The user has to supply* the mathematics and the proof checker formally verifies whether or not the steps are correct. It doesn’t have any creativity (that’s up to the mathematician).

*I should have clarified there is some proof generation, see the comment below by opnitro, but I meant the meat and potatoes of novel non-trivial proofs currently has to be supplied by the user.