My Christmas gift: telling you about PurpleMind, which brings CS theory to the YouTube masses
Merry Christmas, everyone! Ho3!
Here’s my beloved daughter baking chocolate chip cookies, which she’ll deliver tomorrow morning with our synagogue to firemen, EMTs, and others who need to work on Christmas Day. My role was limited to taste-testing.
While (I hope you’re sitting down for this) the Aaronson-Moshkovitzes are more of a latke/dreidel family, I grew up surrounded by Christmas and am a lifelong enjoyer of the decorations, the songs and movies (well, some of them), the message of universal goodwill, and even gingerbread and fruitcake.
Therefore, as a Christmas gift to my readers, I hereby present what I now regard as one of the great serendipitous “discoveries” in my career, alongside students like Paul Christiano and Ewin Tang who later became superstars.
Ever since I was a pimply teen, I dreamed of becoming the prophet who’d finally bring the glories of theoretical computer science to the masses—who’d do for that systematically under-sung field what Martin Gardner did for math, Carl Sagan for astronomy, Richard Dawkins for evolutionary biology, Douglas Hofstadter for consciousness and Gödel. Now, with my life half over, I’ve done … well, some in that direction, but vastly less than I’d dreamed.
A month ago, I learned that maybe I can rest easier. For a young man named Aaron Gostein is doing the work I wish I’d done—and he’s doing it using tools I don’t have, and so brilliantly that I could barely improve a pixel.
Aaron recently graduated from Carnegie Mellon, majoring in CS. He’s now moved back to Austin, TX, where he grew up, and where of course I now live as well. (Before anyone confuses our names: mine is Scott Aaronson, even though I’ve gotten hundreds of emails over the years calling me “Aaron.”)
Anyway, here in Austin, Aaron is producing a YouTube channel called PurpleMind. In starting this channel, Aaron was directly inspired by Grant Sanderson’s 3Blue1Brown—a math YouTube channel that I’ve also praised to the skies on this blog—but Aaron has chosen to focus on theoretical computer science.
I first encountered Aaron a month ago, when he emailed asking to interview me about … which topic will it be this time, quantum computing and Bitcoin? quantum computing and AI? AI and watermarking? no, diagonalization as a unifying idea in mathematical logic. That got my attention.
So Aaron came to my office and we talked for 45 minutes. I didn’t expect much to come of it, but then Aaron quickly put out this video, in which I have a few unimportant cameos:
After I watched this, I brought Dana and the kids and even my parents to watch it too. The kids, whose attention spans normally leave much to be desired, were sufficiently engaged that they made me pause every 15 seconds to ask questions (“what would go wrong if you diagonalized a list of all whole numbers, where we know there are only ℵ0 of them?” “aren’t there other strategies that would work just as well as going down the diagonal?”).
Seeing this, I sat the kids down to watch more PurpleMind. Here’s the video on the P versus NP problem:
Here’s one on the famous Karatsuba algorithm, which reduced the number of steps needed to multiply two n-digit numbers from ~n2 to only ~n1.585, and thereby helped inaugurate the entire field of algorithms:
Here’s one on RSA encryption:
Here’s one on how computers quickly generate the huge random prime numbers that RSA and other modern encryption methods need:
These are the only ones we’ve watched so far. Each one strikes me as close to perfection. There are many others (for example, on Diffie-Hellman encryption, the Bernstein-Vazirani quantum algorithm, and calculating pi) that I’m guessing will be equally superb.
In my view, what makes these videos so good is their concreteness, achieved without loss of correctness. When, for example, Aaron talks about Gödel mailing a letter to the dying von Neumann posing what we now know as P vs. NP, or any other historical event, he always shows you an animated reconstruction. When he talks about an algorithm, he always shows you his own Python code, and what happened when he ran the code, and then he invites you to experiment with it too.
I might even say that the results singlehandedly justify the existence of YouTube, as the ten righteous men would’ve saved Sodom—with every crystal-clear animation of a CS concept canceling out a thousand unboxing videos or screamingly-narrated Minecraft play-throughs in the eyes of God.
Strangely, the comments below Aaron’s YouTube videos attack him relentlessly for his use of AI to help generate the animations. To me, it seems clear that AI is the only thing that could let one person, with no production budget to speak of, create animations of this quality and quantity. If people want so badly for the artwork to be 100% human-generated, let them volunteer to create it themselves.
Even as I admire the PurpleMind videos, or the 3Blue1Brown videos before them, a small part of me feels melancholic. From now until death, I expect that I’ll have only the same pedagogical tools that I acquired as a young’un: talking; waving my arms around; quizzing the audience; opening the floor to Q&A; cracking jokes; drawing crude diagrams on a blackboard or whiteboard until the chalk or the markers give out; typing English or LaTeX; the occasional PowerPoint graphic that might (if I’m feeling ambitious) fade in and out or fly across the screen.
Today there are vastly better tools, both human and AI, that make it feasible to create spectacular animations for each and every mathematical concept, as if transferring the imagery directly from mind to mind. In the hands of a master explainer like Grant Sanderson or Aaron Gostein, these tools are tractors to my ox-drawn plow. I’ll be unable to compete in the long term.
But then I reflect that at least I can help this new generation of math and CS popularizers, by continuing to feed them raw material. I can do cameos in their YouTube productions. Or if nothing else, I can bring their jewels to my community’s attention, as I’m doing right now.
Peace on Earth, and to all a good night.
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Comment #1 December 25th, 2025 at 3:42 am
Thank you so much for the tip on Aaron’sown blog. (Apostrophes, I mean, apostoles, are important as well). I wish I could eat one of those cookies that I do not deserve, as I am not working on Christmas. Very happy Xmas to you and your family!
Comment #2 December 25th, 2025 at 6:21 am
Those cookies look wonderful, Scott.
The videos are great for sure and I wish more people knew about them. Thanks for putting them out there.
But you are mistaken about ox-drawn plows. An important aspect of helping people learn is taking them from where they are to where you want them to go. (An excellent statement of this is the Monads are Burritos essay.) I’ve taught TOC every year for many years and much of the learning takes place in working through examples and exercises, with people getting stuck and the instructor answering questions, etc. All this stuff involves careful work, perhaps less splashy than the videos but a central contribution nonetheless. My plow is a text (in which you appear a number of times, Scott) but you do you. Thanks for the blog, and Peace on Earth.
Comment #3 December 25th, 2025 at 9:06 am
Hey now, let’s not diminish the role of taste testers! 🙂 A proud father also limited to same role most of the time 🙂 🙂
Merry Christmas and Happy Holidays to you and Aaronson family!
Comment #4 December 25th, 2025 at 3:12 pm
Ack, welcome back! Those look delicious.
Thank you for the recommendations — those videos look great and they have topics that look fascinating!
I hope you’re feeling well going into the new year!
Comment #5 December 25th, 2025 at 6:45 pm
I guess many don’t realize yet that in just a few years a single person will be able to generate an entire movie on their own, using generative AI, with nothing but their own voice guiding the AI in a very natural way – no special tools or fancy “prompt engineering” skills required.
The only limiting factor will be the patience of the creator to exactly tweak the photography, scenery, costumes, actors, script, delivery, music, … until they’re happy with the final product.
Based on how good the creator is as an artist and “film director”, the final product would be indistinguishable from “real” movies made nowadays with hundreds of people.
It’s very much how writers work alone for years to create entire universe on paper, just that now they’ll be using very advanced typewriters that create moving images.
Comment #6 December 25th, 2025 at 7:16 pm
That said, it’s also very likely that the AI will be able to generate a more compelling movie on its own, just from a few vague hints… it’s just that the end product will be somewhat unexpected, but you could tell the it to aim for a certain genre, audience demographics, etc.
Apparently it’s already the case that AI generated short novels are just as good as the ones written by humans.
And if you push it to the limit, one could even imagine that movies/books/music will be created on-the-fly (nothing persisted), for an audience of one, based on whatever that person wants to “consume” at that very moment and platforms that act as static content repositories (netflix, youtube) will disappear.
Comment #7 December 25th, 2025 at 11:16 pm
As someone whose youtube consumption contains both 3Blue1Brown and Minecraft, I’m pleased(?) to be able to remark that Minecraft Redstone is Turing complete. See, to paraphrase, I might even say that [redstone] singlehandedly justif[ies] the existence of [Minecraft].
Happy holidays,
Commenter #(7 + ε)
Comment #8 December 26th, 2025 at 4:22 am
Beautiful smart daughter and smart worshipful son. This must be an AI simulation!
Merry Christmas to you and your family and to all the readers here.
Comment #9 December 26th, 2025 at 7:25 am
Today there are vastly better tools, both human and AI, that make it feasible to create spectacular animations for each and every mathematical concept
What precisely are these tools and what stops you from using them too?
Comment #10 December 26th, 2025 at 9:21 am
OhMyGoodness #8: “worshipful”? ROFL
Comment #11 December 26th, 2025 at 10:36 am
Scott #10
…and grace maintained in the family hearth nourished by a father’s modesty.
Comment #12 December 26th, 2025 at 2:03 pm
Theorist Israel #9: I don’t know exactly what tools Aaron uses (I could ask him). But even with AI, it’s clearly an enormous amount of effort and craftsmanship. If I were 20 years younger, I would learn, but by now I’m too set in my ways.
Comment #13 December 26th, 2025 at 10:03 pm
Hi Scott,
Merry Christmas and may you and your family have an amazing 2026!
You surely must be teaching Lily calculus by now. If so can you please tell me how (book/website/your own unique pedagogy/..) you are teaching her?
I plan to teach my own daughter (an 8th grader) calculus to get her interested in math, so was looking for good options. Any advice for “precalculus” too would be very helpful – that trigonometric stuff etc. is perhaps more difficult to get people interested in.
Comment #14 December 26th, 2025 at 10:09 pm
Ashley #13: No, I am not teaching Lily calculus. If I’d continued to homeschool her (as I did during Covid), we surely would’ve gotten to it by now. But she’s in school, and has been content to put it off until she gets to AP Calc, most likely a few years from now when she’s 15.
Comment #15 December 27th, 2025 at 11:07 am
The first time I came across the name Scott Aaronson was when I read “Who can name the bigger number” which I still regard as an absolute classic of mathematics communication, taking an immediately accessible question that even young children have wondered about and showing how it relates to deep mysteries of the universe.
You may wish to do more than you have done so far, and I hope you do, but what you have done so far is still pretty amazing.
Comment #16 December 27th, 2025 at 6:09 pm
Scott, thank you for linking the diagonalization video. I always understood the argument logically, but not intuitively. This video goes a long way toward rectifying that. And an offhand remark in the video, that diagonalization proves that, under some reasonable assumptions, there are just more questions than answers, really hits home. “You can just construct a question that has no answer!”
Comment #17 December 27th, 2025 at 7:25 pm
kyb #15: Thanks!!
Comment #18 December 27th, 2025 at 7:27 pm
Shmi #16: Strictly speaking, if questions and answers are both finite, then there’s “only” a countable infinity of both of them. On the other hand, there’s an uncountable infinity of computational problems (since each one takes infinitely many bits to specify), and only a countable infinity of Turing machines with which to solve those problems.
Comment #19 December 28th, 2025 at 11:54 am
Aaron Gostein’s use of AI animations in the videos makes them worse, not better. Even if he managed to generate animations that look genuinely human-made (and they don’t, they look terrible), all they’d do is to make his channel look like second-rate Veritasium rather than his own thing. Archival videos, static portraits and even just geometric figures are much more respectable than this.
That’s a shame, because his videos really are fascinating otherwise.
Comment #20 December 28th, 2025 at 10:50 pm
There are other channels that have been trying to do this, e.g., https://www.youtube.com/@Reducible
I am not making an endorsement and would say that the choice of topics may not appeal to people here.
My point here is that there are perhaps a few more channels like this. manim, the tool used for production of 3b1b videos, has been publicly available for ages (i.e., before this AI boom).
ps: I am not making any statement about use of AI in this context.
Comment #21 December 29th, 2025 at 6:01 am
Max Chaplin #19: Sounds like you’ve just volunteered to help! 😀 Again, I don’t see any way to make similar videos without AI but also without a production team and a budget — is there one?
Comment #22 December 29th, 2025 at 8:12 am
Scott, did either of your kids complain about the AI animations or express they wished archival footage or other “respectable” media was used instead?
Comment #23 December 29th, 2025 at 8:29 am
What has always bothered me with this diagonalization argument is that it relies on an iterative progression in an infinite set, but at some point it casually jumps from finite index numbers to infinity, which is a huge leap of faith that isn’t obvious at all since the entire argument was to prove you can or can’t do that sort of things in the first place.
On the other hand, why can’t you just start with a finite set, like a space of length N, that has N+1 ints in it (0,1,…N) and clearly a bigger number of floats (0.0, 0.5,1.0,1.5,…), and by the pigeon hole principle you can’t have a one to one mapping, and moving N to infinity (or just adding more and more finite intervals) isn’t going to help?
Comment #24 December 29th, 2025 at 9:28 am
Adam Treat #22: What do you think? 😀
Comment #25 December 29th, 2025 at 9:33 am
QMAnon #23: These things were once confusing, but were figured out in great detail the early 20th century.
If you’re applying Cantor’s argument within the context of ZFC, then you’ve already assumed both the Axiom of Infinity and the Power Set Axiom — so, you have both the set N of natural numbers and the set 2N of subsets of N. The non-obvious part, which you then prove, is that among all the sets in your ZFC universe, whatever they might be, not one of those sets can encode a one-to-one correspondence between N and 2N.
Comment #26 December 29th, 2025 at 9:43 am
Scott #21: That’s the thing – Gostein can make his videos easier on the eyes and more appealing to the public by doing a little bit *less* work, not more. He already has enough visualization material without needing to pad it with uncanny twitching figures. Math/simulation channels like 3Blue1Brown, Pezzza’s Work and 2swap rake in millions of views while sticking to simple vector/raster animations, and other channels have gotten famous with nothing but crude doodles.
Comment #27 December 29th, 2025 at 10:42 am
Max Chaplin #26: Ok, thanks for clarifying. I thought you were making a transparently ridiculous demand on someone else’s time and resources, when you were merely expressing a debatable aesthetic preference. I actually really enjoyed Aaron’s animated reconstructions of Cantor, Gödel, Von Neumann, etc, but if you don’t, then to each his own and let a thousand TCS YouTube videos bloom.
Comment #28 December 29th, 2025 at 11:54 am
Scott #25
thank you Scott.
maybe a more explicit way to express my point about diagonalization.
The diagonal number that’s supposedly different from all the others: after k bits, there are 2^k possible bit sequences starting with k bits (followed by an infinite trail of bits), and the diagonal number (no matter how you pick its k first bits) is always going to match one those 2^k sequences. And as you move from k to k+1, you can drop the first k bits and keep the infinite matching subset and you’re just back where you started, so it doesn’t matter how far you get, for every new bit you pick you still have as much work to do – you don’t seem to progress in building a sequence that’s actually different from all the others.
On the other hand one can make the point that each new bit cuts the entire set in half, so at each bit there’s progress. That’s the problem of infinity, depending on what end you hold it, the cup appears half full or half empty :-/
Comment #29 December 29th, 2025 at 1:11 pm
QMAnon #28: But you did make progress. In the first k steps, you made sure your sequence didn’t match the first k sequences in this specific list. After ∞ steps, you’ll have made sure it matches no sequence in the list, thereby proving that the list didn’t contain all possible sequences. You just need to focus on what the goal is, and forget about other goals that you can’t achieve but don’t need to achieve anyway.
Comment #30 December 29th, 2025 at 2:06 pm
Nitpicking on the “without loss of correctness” part:
The diagonalization argument to show there are more real numbers between 0 and 1 than counting numbers is flawed, since it fails to account for multiple representations of the same real number.
Suppose your list of real numbers starts with 0.100…, and suppose moreover that the n-th entry on your list has n-th digit equal to 0. Then the diagonal sequence will be 0.1000…, and the flipped sequence is 0.0111…, which unfortunately represents the same real number as 0.1000… (in binary).
I think the lazy fix to ensure no problems like this can occur is to switch to quaternary and to `flip’ any non-one to one and any one to two, which ensures your flipped diagonal sequence only contains real numbers with a unique representation. Alternatively, you can use Cantor’s argument that uses the diagonal argument to show there are uncountably many infinite binary sequences (essentially as shown in the video), and then constructs an injection from the set of infinite binary sequences to the real numbers by mapping the binary sequences to decimal representations.
Comment #31 December 30th, 2025 at 7:39 am
Scott #29
got it, thanks!!
Comment #32 December 31st, 2025 at 4:31 pm
“Even as I admire the PurpleMind videos, or the 3Blue1Brown videos before them, a small part of me feels melancholic. From now until death, I expect that I’ll have only the same pedagogical tools that I acquired as a young’un…”
That would be a tragedy. An entirely unforced deprivation of enrichment for yourself in learning the new tools and everyone else for getting to experience the output.
AI tools have hit the threshold where it is faster and easier to simply sit and ask the LLM how to do something and to help you along than it is to “learn” them through traditional methods. They are tools that can continuously adapt to where you are and what you need to get where you want to go – for as long as you’re willing to try.
It’s a New Year. Don’t bring a false assumption from the old one into it.
Comment #33 December 31st, 2025 at 9:54 pm
Well Scott you’ve brought me a tremendous amount of happiness over the years. Listening to your interviews and watching your videos over the years have helped me come to a layman’s understanding of topics I find endlessly fascinating. I hope you’re not too melancholy about your contributions, im certainly grateful for all you’ve done on that front!
Comment #34 January 1st, 2026 at 3:04 pm
David Skelton #33: Thank you so much, and happy new year!
Comment #35 January 2nd, 2026 at 9:12 pm
QMAnon #23:
Scott addressed why diagonalization works, but to address why your alternate construction *doesn’t* work, it turns out that “moving N to infinity” breaks the pigeon hole principle! The infinite sets {0,1,…} and {0,0.5,1.0,1.5,…} are equinumerous because the function f(x)=x/2 is a bijection (something that is not true for any finite versions of the sets).
Even if you take for your second set all rational numbers between 0 and N, which absolutely dominates the puny set {0,1,…,N}, the infinite versions are equinumerous like before. (Describing a specific bijective function is a little more complicated in this case though).
Comment #36 January 5th, 2026 at 6:16 pm
QMAnon #6 “That said, it’s also very likely that the AI will be able to generate a more compelling movie on its own, just from a few vague hints… ”
The hint doesn’t have to be vague. Just say “Han shoots first” and wham, the movie is instantly more compelling. 😉
Comment #37 January 8th, 2026 at 2:12 pm
Nice videos, thanks for sharing.
My qualms about diagonalization (and Busy Beaver and the likes) is the following. And I say that as a non-mathematician so maybe there is something obvious that I miss.
While from a practical perspective is clear that we need to deal with “countable infinite” (and actually more practically even with finite numbers only, given that our machines are limited in size), it seems silly to do so while dealing with the foundation of math.
For example, in the diagonalization leading to Godel’s incompleteness theorem, the guy lists all proofs on the left, including n < 3. It seems obvious to me that he can't really do that, because there is an uncountable number of "n < x" for any value of x, including pi, e, sqrt(2) and many more real numbers than we have symbols to represent. I suspect the actual proofs of Godel's theorems don't have such problem (I haven't read them), but I am puzzled by the tiny number of mathematicians who have tried to "escape" the limitations of "countable proofs".
The original argument about real number vs integers (and hence rational) numbers taught us that there are many more real than rational numbers, despite the latter being dense and hence arbitrarily close to any of the former. I suspect (but again, not a mathematician) some if not many things we do with reals (e.g. calculus) won't work correctly with rational (even though we invent tricks that do, such as Runge-Kutta algorithm).
As such, to me these theorems about the foundation of math only say that there must be a larger framework with uncountable more proofs and that we are leaving it off the table for no reason. Okay, I do know (a little bit) about https://en.wikipedia.org/wiki/Infinitary_logic but it appears to be that very few mathematicians are interested into it, and that is as shocking to me as if very few mathematicians were interested in real numbers and arbitrarily restricted themselves to rational numbers instead (like we do in computer science, but only because we are forced to do so)
Comment #38 January 15th, 2026 at 2:14 am
Regarding “Strangely, the comments below Aaron’s YouTube videos attack him relentlessly for his use of AI to help generate the animations.”
For me it is the other way around: I find it bewildering that it is not clear why people are opposed to AI generated images and videos. The ethical problems with generative AI for image/video creation have been discussed a thousand times by now and I would be surprised if these discussions have gone past you.
Generative AI like this is (right now) inherently built on the exploitation of artists. Taking their work as training data without any form of reimbursement, payment or credit. Not even asking for consent, just “going fast and breaking things”. The whole lives work of millions of artists simply taken and abused.
Ask yourself how you would feel if someone would take one of your papers, books or whatever else and republish it in their name. I guess you would not be happy about it. Of course, this analogy is not perfect, because that’s not how generative AI works.
As someone who is both a computer scientist and an artist, I’m personally happy that at least some parts of the general public are pushing back against this.
Of course, there are other problems and misunderstandings that come on top (artists loosing work, supposed environmental impact of gen AI, quality etc.). While these things might be debatable, the ethical problem is clear cut. People using gen AI seem to have no problem with this exploitation. Which, honestly, is very sad to see.
“To me, it seems clear that AI is the only thing that could let one person, with no production budget to speak of, create animations of this quality and quantity.”
There’s of course some truth to this, although there are plenty of YouTubers doing stuff like this on their own. To be fair: I have thought about trying to start a YouTube channel on cryptography (explaining it for the general public, not experts) quite often, but I simply don’t have the skills to create nice and fitting animations, nor the time to learn all of this. It’s not for a lack of interest in these topics, quite the opposite, but there’s simply not enough time in the day to follow all interests. Of course, generative AI would be a technical solution here and might make creating a channel like this feasible to me.
But the ends don’t always justify the means.
Now, of course, the YouTube channel you’re presenting/discussing here is a special case. It’s not a Coca Cola advert, an AI song on Spotify trying to make a quick buck or something like that, it tries to educate people on a niche topic. Seemingly in good quality as well (based on your description – I haven’t watched a video and am frankly not interested after you mentioned that the animations are AI generated).
I think an example like this could however lead to some interesting discussions: Seeing the many ethical problems of generative AI, are there maybe use cases that are “more ethical” than others and which should be accepted?
Or is there maybe room for an “ethical” generative AI company that trains a model based works of artists that willingly gave their consent, because they knew the model’s goal is to be used for such “ethical” applications and all other uses will be restricted? (which poses the question how such a restriction could be accomplished)