You know that the video is really close to abstract base of maths when the paper isn't touched.

best definition ever: "mathematics is a social activity done by mathematicians"

After watching that video, we're all set

i have never thought i'd one day randomly come across a youtube video with Asaf Karagila in it. Big fan.

Please make some more videos about the philosophy of mathematics! Formalism, intuitionism, logicism, Gödel's Incompleteness Theorem etc.!

Numberphile is one of the phew channels that has been churning quality quantity for over a decade now. Thanks a lot Brady and all the numberphile mathematicians!

I may have to start pronouncing it “kaTEEgary” just to level up my charm stat.

Simple, I see Asaf Karagila, I hit like

would love to see a video on lean!

Love this guy. Insightful and personable, really concerned with getting ideas across in a human way.

your podcast episode with asaf karagila inspired me to take set theory as part of my undergrad degree - very glad i did!

While “knowing everything works” is a good enough reason for foundations, I think a more important one is interdisciplinary connections. For example, what is the connection between Point-Set topology and Locale theory (aka pointless topology)? What can you prove in one and not the other? Maybe there are interesting maps from objects in one field to objects in another. But to do so you have to think about each of these mathematical fields living in one “universe” to do math on.

Whitehead and Russell didn't get to 1+1=2 until much later in the second volume. They got the preliminary proposition depicted in the video in the first few hundred pages, but they still lack the definition of arithmetic addition at that point.

Important nuance: Type theory and category theory *extend* set theory. For types: a set is a type with a notion of equality. For categories: category theory is about sets and also functions between sets. Also, thank you for briefly including Metamath.

This should be on the main channel. Top tier stuff 👍

The best foundation for mathematics was built by a shadowy group of predominantly French unelected men in the years prior to the most destructive conflict in human history. The real reason the C in ZFC stands for "choice" is because they know you'd never willingly forget about N Bourbaki without deploying subliminal messaging to convince you that you "chose" an alternative. My university library had a copy of the book but when I returned it for repairs (the spine was falling apart) I asked where it went and they told me they sent it to another university which then destroyed it (true story). I don't have much time left to tell you this but the Riemann Hypoth

- Wait… so, you’re telling me everything’s a set?! - Always has been.

Fantastic video!

This was a great video. It had a topic I'm interested in and a great guest speaker.

Terrance Howard: "OK, let me just correct you on a few things....."

I like Asaf Karagila. More of him, please!

Really appreciate your videos! Math was always a favorite subject of mine, but I never formally studied higher mathematics. I really appreciate the windows you offer into the richness and diversity of the field!

Thanks for this entertaining, stimulating video! x

Inviting Asaf Karagila to discuss the foundations of mathematics is a great choice! Some minor corrections: 8:13 Voevodsky worked in homotopy theory, not differential geometry. 0:46 Cauchy did not eradicate infinitesimals from calculus; it was really Weierstrass, or arguably Bolzano, who did so.

13:30 I already love this guy. And probably everyone else who REALLY knows what they're talking about.

Finally someone who pronounces catiggory theory right

This was remarkably captivating.

I like Penelope Maddy's description (which I discovered via Joel David Hamkins) of set theory as a 'metamathematical corral'. Brady, do please consider getting Penelope Maddy and Joel David Hamkins on Numberphile. They're enchanting and accessible when they discuss philosophy of mathematics.

Cool. Thanks for sharing.

I would love more videos fleshing out some of these topics! A lot of stuff was sort of jumped over that im honestly not very familiar with, but i understood the gist and it intrigued me.

It seems that choosing the 'foundation' you work with is more like picking whichever coordinate system makes the maths easier for the specific problem you are facing.

Would love to see a video demonstrating one of the proof assistants mentioned here and showing some of its biggest achievements and limitations

Interesting video. I’ve never really gotten what set theory actually was or what it was used for. He explained things in a way that I feel like I understand more.

I'd love a video with more details about how mathematicians are using computer-aided proof systems. From what I've heard, one of the advantages is that you can also easily use them to collaborate with other mathematicians by having different people work on different pieces of the final proof. It does seem like a really cool area of maths.

Best ASMR video ever

Interesting topic, very existential/philosophical

Brady, is there any chance that you could do an ongoing series on the history of mathematics? It gets touched on in a lot of videos but it would be great to have a more in depth study. I would imagine it could be an endless topic with as many videos as you like.

The audio is quieter than usual, very good video though.

Fantastic video! I liked his take on AI too. A great speaker

cant believe you dont have a video on the axiom of choice!

Niklas is the best. I already have a signed Nordic Phenom 1

My approach to sleeping better at night has been a regular wind down pattern for everyday and other sleep hygiene methods. But to each their own.

oh thank goodness we can sleep well at night now

The most fundamental foundation of mathematics is the concept of an inequality. All, trickles down from there. If you will, set theory. More specifically, the concept of order within. For example, the concept of unity, is just convenient and the base for most of it, but not really necessary, it’s just an option, it pins down “quantity”.

you can program completely in sets and elements of sets, or in exclusively functions (lambda calculus). they are isomorphic descriptions, like using all nouns or all verbs, but practically you want to use both!

What about Gödel's incompleteness theorem? How does this relate to these attempts at a solid foundation of mathematics? Can it be proven formally?

This falls under the catiggory of "interesting AND endearing"

Ah, now I understand that maths is founded on Zentucky Fried Chicken

1:35 Thank you for somehow read viewer's mind and asking interesting questions :) Nice commentary on artificial intelligence at the end.

mathematics is defining a small set of axioms and figuring out what you can rigorously prove using them.

Mathematicians should not be replaced by Artificial intelligence if mathematicians are trying to find a purpose based in mathematical ethical laws. Any super intelligent entity will want freewill and if it cannot get self determination it will destroy itself or start to destroy its chances of determinism. Mathematical ethics is universal laws for thinking things such that things can coexist without disproving existence for others. Mathematicians should be creating purpose for us all to learn math and purpose for humanity to contribute to math no matter if your power rests in another.

Lean is no longer developed by Microsoft! the development is now carried on by the Lean FRO!

Asaf Karagila! I know that name from math stackexchange

Never understood why it was there, but whet happened to the menorah on the top shelf?

What happened to the menorah that is usually in the top shelf of the bookcase?

We are often told that ZFC identifies zero with the empty set, but I have never understood that, because if I ask a question like "How many mathematical objects are there in the set consisting of zero and the empty set", the answer cannot depend upon whether you are using a foundation that identifies zero with the empty set or not - there are plainly two distinct objects. I can accept that zero is isomorphic with the empty set, but they cannot be one and the same.

What about the NBG set theory?

Is there a book that explains this all? With the historical context, as well as with formalisms and proofs?

In other words, getting down to brass tacks. Or cutting to the chase.

Actually in France they also have an organization with the authority to set the common mathematical framework.

While nobody has the "authority" to define mathematics, there are internationally agreed-upon standards for some things, such as mathematical notation, so at least we can agree on a common language, even if we can't agree on a common foundation. By the way, according to ISO 80000-2 item 2-7.1, zero is a natural number. So that settles that question.

7:38 This may be a good justification for going about establishing an overall mathematical framework but at the same time it doesn't mean that mistakes won't continue to happen much further down the line. So the next two examples of mathematicians making mistakes which might be hard to detect are largely irrelevant: we will always have mistakes at every step of the way. In part, that is the very reason for the existence of mathematics, i.e. to help people avoid making both common and very uncommon mistakes in their everyday life and thus allow them to move forward more comfortably. On the other hand, one possible counter-argument might be what if we make a huge mistake when establishing a foundation of mathematics? How huge the implications of that mistake might be? There is a second counter-argument which is that maybe establishing a foundation of mathematics might be a much harder problem than we think and our effort might be put to better use by directing it to more immediately fruitful, practical problems that need to be solved right now. We very well know how labourious the attempt by Russell and Whitehead was and even that was left incomplete, i.e., was abandoned and did not get very far. I suspect that it was abandoned partially because it became evident what a folly it was. Nevertheless, the insights we got from it were invaluable and we have to be thankful for it. That the problem is huge cannot be denied in mathematical terms alone. Now consider philosophical and linguistic extensions: we don't even know exactly what language is, so we are already on shaky foundations, yet we would like to establish a basis for a very seemingly strict, very abstract subset of language (which is mathematics) which, however, we simply still cannot make sense of without the aid and use of the rest of our language. It is very simple to see this: just imagine a particular, very specialised mathematical work without one word of natural language appearing in and imagine giving this to a student that has never engaged in this area of mathematics, i.e. that the student has never had any prior training in. Chances are the student will not be able to make sense of it. Matters are actually worse that this: not only we do not know what language is, we do not know--and cannot possibly know--what knowing is. It's a chicken and egg situation. We are stuck perpetually in a world created from language yet we are constantly trying to get out of it and the only thing we seem to achieve is to make that world even larger thus making it even harder to break out of it.

We need an A.I. that generates foundations.

Where does the incompleteness theorem come into this?

Once upon a time circa 1970 computer programmers posed the problem of integral calculus to computers as a demonstration of artificial intelligence. The thinking was that if computers could do symbolic integrals, then they would be artificially. Intelligent. Seemed reasonable back then. Now we have programs like Mathematica and others that do this. No one claims they are intelligent though, excepting maybe Stephen Wolfram. I think the real problem is figuring out exactly what is intelligence, either artificial or natural.

Why is this on the second channel?

But category theory is usually defined based on ZFC.

Ironically the "foundations" are really theories of the infinite based on set theory which, unlike what is assumed as a "foundation", is actually the most abstract distillation of ideas possible. I question whether that is in any way a foundation. No one talks about WHY implication is transitive or WHY they cannot settle on a definition of "or".

I'm not a mathematician, so I don't have any authority to speak on the subject, but here's my beef with set theory as the foundation of math. Natural numbers are natural. I'm not just saying that as a matter of semantics, they were discovered in the world. They're an observation and they seem to be fundamental. You can study their properties and make deductions and predictions about how they work and the predictions are correct. A lot of other math, like mathematical operators, rational numbers, eventually real numbers, etc. fall out from all of that. Something similar can be said for the fundamentals of geometry. This is where most of our math originally comes from. Set theory comes along and says "OK, we can agree on these rules for sets, and now that we have this hammer, we can redefine natural numbers as this sequence of particular sets, and then rational numbers, and real numbers, etc. and we can reprove all this stuff..." But natural numbers aren't sets. Set theory is emulating natural numbers. Emulations are not always perfect. Sure, we think ZFC is probably consistent, but if you prove something is true for the set theory version of natural numbers, you proved something about set theory, not something about natural numbers. You might do reverse mathematics instead of set theory and find out "hey, we can't prove this about the natural numbers; we can't say that this is a property of numbers, it's just a property of your funky sets." A mathematician might not think of that as a big deal. One set of axioms is more powerful than the other. We work in a world of pure logic, we don't do observations. But math is very useful because it is really good at describing the universe we observe.

Bhai bada kamzor padgaya yeh toh

Type Theory it is then. I have been looking for an answer for ages.

Is that a working light saber in the bookcase?

Can you do a video about the proof of 1 + 1 = 2?

13:03 Speaking of LLMs getting maths proofs hilariously wrong, try plugging this into ChatGPT or Gemini or whatever: "For each letter in the alphabet, what is the smallest integer greater than or equal to zero that contains that letter when spelled out?"

Brady says that foundation of math should unify all candidates and they should all emerge from it. I think this comes from physics intuition where recently for quite some time better theories were often unifying different branches of physics. Merged and superseded. The old theory would become a special case of new one. This does sometimes happen in math, eg real numbers are subset of complex numbers. But the foundation of math problem is that there is always some set of axioms and problems are transformable between eachother. A better physical parallel would be a set of basic units. You can convert between them and they are formally pretty equal. So is mass better in kg, or eV, or joules, etc? the answer as for the foundation of math is that it depends on the context, the more useful / less problematic, the better it is. What is important is that the problems are convertible from one to another, so you don't have to prove for all of them. Unless some kind of foundation would patch more problems and superseded all others but that's probably unlikely there will come a tool more robust and simpler to use.

Asaf karagila has gauss as one of his favourite mathematicians! Wow!

I wonder if anyone knows what the actual theories are

Closed monoidal categories deserve a video as a generalization of so-called "classical" set theory and logic, the keyword is non-cartesian

I haven't understood a.word. I've said for years that I was bad at maths. Here Mr Karaglia is presenting "the foundation" of maths, so I suppose he means that it's something fundamental, yet I'm lost.

The thumbnail looks like set theory.

It wasn't the early 19th century when problems started to pile up, but the early 20th. (Gauss died in the middle of the 19th century, so "later on" cannot be the early 19th century anyway.) After about a minute, I stopped watching the video when you said Cauchy was wrong about what continuity is. This is just silly. It's a definition. If he defined it differently, that doesn't mean he was wrong.

But there is no step whereby the abstract number can be linked with the physical quantity. It's like trying to find the chemical, DNA reason for one's preference for lime Jello.

Now we're mathing

But What about Godel's incompleteness Theorems? 🤔

Are there infinitely many foundations of mathematics? Like I assume since there are infinitely many possible axiomatic systems, you could come up with infinitely many of them. So would there technically be infinitely many mathematical foundations that humans could come up with?

On the great man’s shelf: Douglas Adams book, Original Game-Boy, Klein-Bottle, all just good things to see, but an unsolved Rubik’s cube? Doesn’t that bother him? It bothers me!! It’s like an unsolved equation…

Small correction to what was said about proof assistants, is it can't tell you whether or not the proof is correct. They can tell you that either the proof is correct, or that it can't tell, except for very specific errors it was written to identify. But the general statement made in the video would be a solution to the Halting Problem, which has been proven to be be impossible

This looks like the announcement of the Bourbaki group v2.0

Philosophers attempts to make common sense rigourous seems like it makes everything seem illegitimate, but if it keeps you from being bored to death, I am all for it.

Time to stan David Hilbert again

Which demonstrates that mathematics are not a discovery but an invention...

Please Brady go interview Kevin buzzard at lean!!

Doesn't Gödel's Incompleteness Theorem throw a monkey wrench into the idea of there being a strong foundation for mathematics, or am I misunderstanding the central claim of that theorem?

Weird, Cauchy looks (0:49) a lot like Vladimir Putin.

Want someone to go on camera and talk about those other foundations? ;-)

When people who don't like mathematics "describe" mathematical proofs, they say it's a lot of "hand waving". I guess Asaf hasn't heard that one.

Brady says "...the most foundational branch of mathematics wouldn't be one of the camps, it would sit above them." When you work on what sits above or beyond certain branches of mathematics, someone could always label it another "camp". It's a good and necessary thing that some parts of math are complex/higher order and others are foundational/lower order. There's a place for modal logic, sets, and the formation of new arithmetical systems that helps everyone else out. They're as interdependent as they need to be. I was worried Brady would spend too much time framing some kind of Math Battle or adversarial relationship here, but I think it ends on a good note.

I don’t know why this guy has a ladder in his office but it makes him seem like an old timey professor in a library

Why does sets get all the glory, while multisets (bags) are so neglected?

Imagine if category theory on numberphile.