Mathematics doesn't actually make any sense

[@SheafificationOfG]

Ngl, I forgot I scheduled this video to release yesterday. This def got a *lot* more attention than I expected! I'll try to engage, but I'm most likely going to miss most people. Anyway, some common questions: 1. *How is the typesetting being rendered?* The LaTeX is being renderred using vscode with the "GitHub markdown" preview extension. Before you judge me for using vscode, this is what happens when you sell out to coprorate. 2. *Are you French?* Non, j'suis Canadien. The "thx 4 watching" in French is to protect me from the RCMP. 3. *What's with the clicking noises?* I thought keeping the keyboard sounds was a nice touch. Perhaps I was wrong. Btw, I didn't type while talking; it took me around 45min to type that document out (after drafting up the proofs in advance), so the video is sped up dramatically. My hands hurt; I don't normally type that much in one sitting without pausing to ponder. 4. *There are mistakes in your penance!* Not a question, but thanks. The gist link in the description has a slightly different version (with fully-typed-out proofs). Please feel free to comment there with corrections, if you spot any errors! 5. *What makes the counterexample hard to draw?* Admittedly, it's not the best example for my point (I was trying to keep the mathematical prerequisites low), but branching on the rationality of the input is a lot more counterintuitive than it might seem. If you sketch two simultaneous dotted plots of the y=x^2 curve and the y=0 curve, you get some representation of this counterexample, but idk how "faithful" it is. It's certainly good-ish enough to demonstrate what the counterexample does (though the "continuity = draw with pencil" intuition doesn't really demonstrate its continuity), but I mean 100% of the points are along the x-axis, which isn't really reflected in the sketch. (Yeah, it's not important for the counterexample, but still).

[@user-sl6gn1ss8p]

For what it's worth, I really liked the typing sounds

[@ndl8722]

High-genus cow! This video was good

[@ianyap8941]

For point 5 in drawing counterexamples: In before: There exists a function (Weierstrass function) that is everywhere continuous but nowhere differentiable. 😮

[@edwardhawkins4206]

anyone with a tutorial to set up the LaTeX like in this please reply I need it so badly lol

[@SheafificationOfG]

@edwardhawkins4206 Install vscode and look for the github markdown extension. Then you can just type markdown with math as you would for github, and you can watch a live preview of it. I still prefer vanilla LaTeX but I'm very biased.

[@PrScandium]

The LaTeX subway surfers is crazy

[@AzogticMettroskik]

I get it, that’s because he’s writing that paper in LaTeX.

[@PixelSergey]

what is being used to render it? looks nice

[@dwf2606]

Looks like obsidian

[@TheWizardMyr]

I noticed Apple Pages had LaTeX capacity in my first calc course and I've been typing all my math since. Some classes have been far easier than others. Just finished a course on group & ring theory that was fun to type

[@MarkWiemer]

I kept waiting for him to show a Subway Surfers clone written in LaTeX because of this comment lol

[@aidenkoh2426]

If someone tells you math doesn’t make sense, they’ve either hate math, or they’ve studied it for 20 years.

[@dmr11235]

@aidenkoh2426 or both

[@tonibrown148]

i'm watching this twice, once to listen to what is being said, and once to read what is being written

[@Alceste_]

That's good for the algorithm (and thus for g++). 👌

[@jajaperson]

there’s a link to the document in the description

[@michaelvstemerman]

Wow I did not expect to see you on a random video I clicked on. Hello!

[@celestesimulator6539]

oh hey! it's you! you will not recognize me probably but i am a member of the conglanging community discord server

[@7oqu_ra]

r-r-rharill...

[@Rubrickety]

I took your math/legal analogy to heart; I'm now suing my topologist for a botched Dehn surgery.

[@DrDavidHerreraMath]

Wait until the judge hears that he is a "doctor" but never went to medical school.

[@fleefie]

g, could you please disable auto translations of your titles, I almost missed your vid in my feed because of it...

[@nesssel]

I completely agree. Most 3B1B video titles are completely botched in portuguese. It's silly.

[@CYXXYC]

set your youtube to the desired lang

[@fleefie]

@@CYXXYC Doesn't work as a bilingual person, since then my native language titles are also auto-translated...

[@A_doe_wasting_her_life]

I thought I had to scavenge thru youtube settings to disable that is so annoying

[@viliml2763]

Look up the extension "youtube auto-translate canceler". It has bugs but youtube's default behavior is so awful that it's worth it.

[@ingiford175]

You forgot my favorite forbidden words: "I will leave it as an exercise for the reader"

[@SheafificationOfG]

cf. the meme at 10:10

[@harsinsinquin4032]

I’m more partial to “the proof doesn’t fit within the margin”

[@ccbgaming6994]

Fermat wants a word with you

[@maythesciencebewithyou]

It's so trivial, I don't need to explain it

[@dmr11235]

Math fans: “I love math, everything just makes sense, it’s so logical” Math: “yeah so set theory is the geometry over the field with one element, an object that doesn’t exist. Also prime numbers are knots” Math fans: …. Math: “the integers are a 3-sphere”

[@thewhitefalcon8539]

The integers are a what.

[@dmr11235]

what

[@SheafificationOfG]

the 3-sphere thing is very knotty

[@sychuan3729]

Prime numbers aren't knots and integers aren't 3-sphere. there are some analogies between knots and primes. And Spec(Z) not integers have some properties of sphere. Which isn't more unusual than some set with topology having properties of sphere.

[@dmr11235]

@@sychuan3729 booooooo you’re no fun. (Yes I’m aware that the integers aren’t actually a three sphere, but you’ve got étale cohomological dimension 3 and trivial étale fundamental group. Topologically Spec(Z) “looks like” a 3 sphere viewed as a Hopf fibration over S2. I’ve got Morishita’s book kicking around here somewhere, but admittedly it’s been years since I’ve tried my hand at arithmetic topology in any serious way)

[@creativenametxt2960]

It is such good theming to intentionaly leave minor spelling mistakes in the backround document while doing this video, like "anyh" that cn be seen at 2:11 or "th eelements" at 12:17

[@Alceste_]

Nice of you to show your appreciation by following the same principle.

[@MakeAHouseAHolmes]

Oh boy can't wait to use my newly sheafified understanding of limits to figure out derivatives

[@Noname-67]

Sounds a lot like synthetic differential geometry. In this case, just plug in infinitesimal.

[@CrawliestCotter]

The problem is that people dont experience abstract mathematics unless they get their bachelors in "pure" mathematics. Abstract maths aren't required knowledge for teachers, even though they're literally the explanation of the fundamentals. I feel like the book "A Transition to Advanced Mathematics" or other books like it should be requred reading for all those who teach math. Even if they dont do the exercises, the knowledge of what the hell the fundamentals actually are, is important.

[@TheThreatenedSwan]

Ok? Good. We should care about math mostly for its applications

[@mezu-e]

​@@TheThreatenedSwan No math was ever discovered by someone looking for it's application

[@TheThreatenedSwan]

@@mezu-e Statistics. And a bunch of algebra for various things. Math was also first developed for engineering and money

[@TheThreatenedSwan]

@mezu-e That's also obviously untrue for fields like economics and physics. Developing math constructs more directly for science is more useful and the whole point is that the world is immanent and we can test how useful theories are by their predictive validity

[@catmacopter8545]

​@@TheThreatenedSwanright. Then why arent you working your ass off in the coal mine right now? Thats the easiest way for you to improve the economy or whatever. Math is art! Let it be art!

[@issaccoy1802]

I entered this video thinking the title "doesn't make sense" was suggesting "illogical". I was quickly reminded of my laziness with rigor about definitions as a consequence of "senority". You (likely), inadvertently proved your point, at least with me. Lastly I say "likely" as a measure of belief. Bravo, sir.

[@treborhuang233]

It's interesting to note that not too long ago - as late as the time of Poincaré - the general view of mathematics and proofs was drastically different. My proof works for all these nice closed curves that separate the plane into two, and if you want to apply my theorem with your cursed curve that doesn't satisfy this then that's your problem not mine. This difference in mindset makes a lot of definitions in, say, Poincaré's Analysis Situs, very shocking to modern readers. And indeed he made a mistake because of it: He mixed up the free rank and the generating rank of a (finitely generated) abelian group when relating Betti numbers and homology (which he just invented).

[@samb443]

Saying he made a mistake because of a difference in philosophy seems a bit disingenuous. He made a mistake because he made a mistake.

[@treborhuang233]

@@samb443 The "it" refers to the shocking definitions, not philosophy.

[@Noname-67]

What is an example of the shocking definitions? I typically don't read any math before Gödel since math was so weird back then.

[@Rudxain]

I've seen you on that one portal video. I'm sorry if I sound creepy, lol

[@mrjoe5292]

@@samb443 Please just... before writing a comment, ask yourself why.

[@AhmedIsam]

Same story with programming languages. When coming from high level to lower level language, you notice that a simple function to add two numbers is needlessly verbose, but actually, its necessary in order to be precise about what you mean by that function, what are its limitations, how to handle the edge cases. Verbosity is inevitable if you require rigour.

[@willmungas8964]

It forces you to say what you mean and set the rules for interaction between types/functions, the benefit being it’s a lot easier to figure out where you broke the rules or asked the computer for something different than what you intended to, easier enough that a compiler will catch whole classes of problems before you even run the program

[@Daniel_Zhu_a6f]

@@AhmedIsam when C and Fortran were made things like 9 bit bytes were relatively common, compilers were baremetal and CPUs usually had 1 core. a lot of verbosity is not rigor, it's pure legacy

[@AhmedIsam]

​@@Daniel_Zhu_a6f The converse is not necessarily true: Being explicity about details ==> Verbosity. At a broader scale, the endemic problem of human languages (as opposed to technical languages, e.g. math, chesmistry, programming etc) is lack of precision and vague defintions of words. As such, when you write a legal document, or a constituion, people differ on how to interpret them, hence the need for police, courts, lawyers and one authority (e.g. supreme court) to inforce their opinion on what they think x means. This is a joke, from science perspective, hence the need for all the technical languages to describe their perspective knowledge where everyone in the world agrees on what this means.

[@AhmedIsam]

​@@Daniel_Zhu_a6f The converse is not necessarily true: Being explicity about details ==> Verbosity. At a broader scale, the endemic problem of human languages (as opposed to technical languages, e.g. math, chesmistry, programming etc) is lack of precision and vague defintions of words. As such, when you write a legal document (e.g. constituion) people differ on how to interpret them, hence the need for police, courts, lawyers and one authority (e.g. supreme court) to enforce their opinion on what they think x means. This is a joke, from science perspective, hence the need for all the technical languages to describe their perspective knowledge where everyone in the world agrees on what this means.

[@TheThreatenedSwan]

In human language it can be the opposite

[@ZeroPlayerGame]

Honestly automated proof assistants are so incredibly cool. Obviously they still have a long way to go to become mainstream, but using one is a blessing, they will call you out for any wrongful "proof by triviality" (and I've often found that I'm plain wrong, i.e. thinking A iff B when actually A if B, but not the other way).

[@redoxr2815]

To me (an undergraduate who's only been doing real maths for 1.5 years), meaning and maths are two very seperated things, and maths specifically exists to be separated from meaning. In maths, there is an infinity of axioms or definitions you could come up with and build theories on, but we somehow only thoroughly study a handfull. Meaning is to me the reason we choose the axioms and definitions we will study. I believe a better approach in introducing derivative is WHY do we give those definitions, what are we trying to get out of it ? We want some sort of instanteneous speed or whatever, how can we go about it ? For that, we figured we need a notion of limit would be good, how would we define that ? But once that's all settled, we defined the mathematical objects we will be using, meaning is seperated from them. Those objects will give us definitive statements, wich we may interpret if we want. But when reasoning with the objects, they hold NO meaning, pure logical objects, that is exactly why they interest us, they must stay meaningless, separated from all intuition. That said, intuition can (and inevitably does) help us navigate through this meaningless space, finding inspiration for proofs, other objects we want to introduce. As you showed, intuition is flawed, we need maths to get definitive answers but really the answers we find are just about meaningless theoretical objects that we thought looked similar enough to the meaning we wanted.

[@Noname-67]

In my view, all choices of axioms are equally meaningful. The reason we study one more than the other is whether they're interesting or useful. The problem with using the word "meaningful" is that it's binary and absolute. We often don't just either study something or not, but study them with varying amounts of effort. What is considered worthwhile to study also changes depending on the time, the place, and the people. If "meaning" binary and absolute, then there would be a least complicated meaningless mathematical object. I would say the existence of such an object is absurd. If "meaning" is not binary or absolute, then meaning in this case is more like a synonym for usefulness rather than the other meaning of "meaning", which is a synonym of coherence. The latter meaning is one people usually think of that word in this context.

[@ronan.pellen]

Interesting points you make but it's *separated not "seperated" 😉

[@lih3391]

@@Noname-67who said meaningful is binary and absolute?

[@FreakGUY-007]

​@@lih3391 This is now philosophy at best... 🎉😂

[@Ishaan_Garud]

Once at night, when I was 16 (last year) I was pondering about the 'approximation of area under a curve' and I made a formula with sigma notation in my head (I couldn't write it down at night because my parents would've killed me for staying up)... In the morning I check that formula to check if 'I might've won the Nobel Prize' and to my discomfort it was the Integral written in Sigma Notation. We weren't taught any calculus, and it was unknown. So this really helped me this year when we were first taught calculus❤

[@Dr_Y_Doodle]

Once my physics professor told me that when he was a student and first submerged into the vast limitless (ba-dum-tss) swamp of math their professor, in turn, told them not trying to build an 'understanding' or 'intuition' about abstract topics, and rather _to get used to it_ , like, accept math entities' intrinsic rules and abandon common sense for a time. And then to build up a _new_ "common sense" now based on the things they've been simmering into for a while. And for them it worked.

[@jacobharris5894]

That is a valid way of learning mathematics that stops you from getting in your in way when things don't intuitively make sense right away. It's exactly why KZbin videos like 3blue1brown meant to build intuition have the most value to you only after you've spent a long time grappling with new mathematical concepts. They are not a proper introduction to the subject. If you watch them when your not ready, at best they serve as inspiration to go learn a new area of mathematics. At worst it is just entertainment. Very little of what you learn actually sticks and what sticks doesn't really help you learn the topic for real in the future. Because the more experienced math people that watch those videos get exciting epiphanies from those videos, they tend to assume they should have been taught that way earlier. But I doubt it would help them in most cases. Stealing the first example in the video, can you really say you understand derivatives when you've only been given visual intuition of them as a tangent line? A derivative is a limit by definition, which you either haven't been introduced to or if you saw it in a edutainment style video, you probably glossed over it without understanding what it really means or how to use it. You don't know about the simple differentiation rules typically taught in an intro calc class or if you do, you don't know how they follow from the limit definition and you probably can't apply them. For something as simple as differentiation, it's easy enough to give some visual intuition from the start but learning it later is just as valid and you may understand it better after grappling with the difficult stuff.

[@FreakGUY-007]

I mean how did these things even develop if institution or say common sense or logic as we know aren't followed? How did the humans thinks and know yes that's it.. It's how it's suppose to be? Do we really know if it's true? Or it's just an idea from some crazy guy who does nothing but maths?

[@jacobharris5894]

@@FreakGUY-007 Well first of all, logic and common sense are entirely different things. A statement can be perfectly logical and still defy common sense. Unlike logic, common sense isn’t universal. It’s intuition. It relies on personal experience and rules of thumb. Logic works outside of that but logic can improve our intuitions over time, which is why people often equate the two. Common sense assumptions and logically backed statements often overlap but not always which is why rigorous proofs are always sought after by mathematicians. Second of all, there really is no, “this is how it’s supposed to be” in mathematics. Because one can come up with any definitions and axioms/postulates they want and then further theorems, lemmas and corollaries can follow from that with deductive reasoning, the only pure form of logic that we know of from philosophy. So from a pure mathematics perspective there is only definitions, axioms (statements we assume to be true), correct proofs (of statements that are neither a definition nor an axiom) and faulty proofs. But that doesn’t really answer the question I think you are getting at. How do mathematicians choose their definitions and axioms? The answer to this is complicated and of course depends on what part of mathematics you’re talking about. If you’re interested in that you might want to delve more into the history of mathematics. I suspect in the beginning it depended a lot more on intuitions and what’s useful in the real world. As mathematical theory became more advanced, it became more and more about exploring what we can deduce from different logical frameworks, which may or may not be true in the real world. I imagine that, the modern axioms and definitions we see today, were refined over several years based on the most interesting and/or useful statements that could be proven with the least amount to go off of. After hundreds of years of that, you can see why modern mathematics can seem so abstract and why even axioms and definitions can seem quite complex or difficult to read. This makes mathematics in the modern era less accessible but that doesn’t mean the abstraction isn’t useful. Mathematics still continues to be remarkably useful in the physical sciences, more than ever before. But pure mathematicians care about generalizing things beyond what might be considered useful, at least initially. It’s pretty common for new things in mathematics to initially seem impractical but then find a use in the physical science much later, sometimes centuries later. Let’s start with the foundations of mathematics so I can give some concrete examples. To my knowledge, besides basic arithmetic, Euclidean geometry was the first field of mathematics and the foundations of it are very much rooted in common sense. It is also pretty useful in the real world even today and many more advanced fields of math branched off from it. So from Euclidean geometry, what are some examples of intuitive axioms/postulates that are also useful? I think some of the common postulates for congruent triangles make for good examples. You may remember some of them from high school geometry but maybe they didn’t explicitly call them postulates/axioms. Getting the definition out of the way, a congruent triangle is a 3 sided polygon where all the corresponding sides are of equal length and all corresponding angles are of equal measure. With this definition, we can state the side, side, side (SSS) postulate which says two triangles are congruent if they have congruent sides. With just the knowledge that the side lengths are equal for two triangles, it’s impossible to prove that the corresponding angles are of equal measure. There’s just not enough information to go off of and we’re going to need more than just the definition of congruent triangles to form a logical foundation we can build off of. At the same time it seems pretty obvious that the statement is in fact true. All our observations in the real world seem to support this and for any pair of triangles, with congruent sides, you draw in a flat plane, you should find this to be true. So we all collectively agree to make it a postulate. Then for similar reasons we can do the same with the side, angle, side (SAS) postulate and the angle, side, angle (ASA) postulate. The SAS postulate says that, if two triangles have a congruent angle constructed from 2 congruent sides the triangles are congruent. The ASA postulate says that if two triangles have two congruent angles, where the adjacent side to these two angles is also congruent, then the two triangles are congruent. Like before there isn’t enough information I’ve given you to prove these statements but you can easily convince yourself they’re true so we usually make them postulates. Now with this we can prove our first theorem related to congruent triangles. The angle, angle, side (AAS) theorem which says that if an angle, an adjacent angle and a side adjacent to the second angle are congruent then the triangles are congruent. With this theorem and the other postulates we can then prove the Hypothenuse Leg (HL) theorem which states that if two right triangles (triangles with right angles) have a congruent hypotheses (longest side of triangle) and congruent leg (side that isn’t the hypotenuse) then the two right triangles are congruent. Finally let’s add one more common sense postulate. Corresponding Parts of Congruent Triangles are Congruent, which is pretty self explanatory. If two triangles are congruent their corresponding sides and angles are congruent. We now have lots of ways to immediately tell two triangles are congruent which comes in handy both for physical applications and for further proofs. With some algebra and additional proofs/postulates related to either area or similar triangles, we should have everything we need to prove the Pythagorean theorem, the law of sines and the law of cosines. This puts trigonometry on a firm logical foundation which is immensely useful all over the place, everywhere from physics to construction. From my long winded exposition, it’s hopefully clear that there is no correct logical framework for mathematicians to choose from and if you don’t like the ones that exist, you can, in theory at least, make your own. It’s kind of like writing a novel (or multiple novels) except the stories never really end and everything is logically consistent. Moreover, you can start writing your own novel from scratch or continue the story someone else started. This is why many people consider mathematics an art rather than a science. There is a surprising amount of creativity that goes into it and it can literally be explored forever.

[@FreakGUY-007]

@@jacobharris5894 Great explanation dude. Appreciated.. Will look into history of maths.. I have less time because of my job.. Will take maybe months to even scratch the surface level history of maths subject. Lol But really appreciated for the time you put.. 👍

[@jacobharris5894]

@@FreakGUY-007 Yeah no problem. I’m glad you liked my explanation.

[@runnerduck4844]

I'm a computer science senior, and my hardest-earned grades are my 2 B's and a B- (determining which grades go to which classes is an exercise left to the reader) in Calculus I, II, and III. I put in a lot of work and felt like I didn't understand much of anything at all, but I got through. Later I took Discrete Math and everything made sense. I learned from this class that I do indeed love logic and that helps me to know that CS is a good fit for me.

[@Dextrostat]

Calculus 2 is what kicked me out of a CS degree in uni. I went through a foundation exam for CS1 and took Discrete Math 1 and CS2 and passed those, but Calc I could not keep up with. I just couldn't focus or get interested enough and the fact it relied on previous math that I forgot really hurt. At the time it really pissed me off and was one of the reasons for my ADHD diagnosis lol

[@StrayChoom]

As a physics undergraduate, I spent my whole life trying to do the least amount of work possible to get by. Aimed to pass. I was always afraid that I wouldn’t do as amazingly as I thought if I actually tried… Now past few months I’ve ACTUALLY been putting effort in my course, and my grades are the best they’ve ever been. Been dropping 80% avg which in England… above 70% is the highest grade you can get :o Yet… now I’m annoyed at the years I could’ve put effort in, but didn’t. Much of it, I do not regret. But… most of it… I’m mad how far behind I still am in many ways. How little I actually feel like I understand. I know radically more than I ever have, but I feel like I know less. And the more and more I learn, there’s more I feel that I don’t know. When I did nothing, at least I could egotistically overvalue my intelligence. Now I’m putting effort in to learn, I feel dumber. Logically I know, I am easily the smartest I’ve ever been. But I feel like I’m the dumbest. It’s probably that I actually have to actively deal with my limitations. At least in the past I could just go “if I tried, I could get 100%”. Dismiss it all to stroke my ego. Now I know, if I try, I know I won’t get 100%. But I guess at least, I can get happily get 90%. And you know what? Intelligence is overrated. Who cares how smart you are? Or how dumb someone is? It has nothing to do with anything. Not really. There’s dumb people in the highest places, and smart people who have nothing. So intelligence is definitely overrated.

[@StrayChoom]

As a physics undergraduate, I spent my whole life trying to do the least amount of work possible to get by. Aimed to pass. I was always afraid that I wouldn’t do as amazingly as I thought if I actually tried… Now past few months I’ve ACTUALLY been putting effort in my course, and my grades are the best they’ve ever been. Been dropping 80% avg which in England… above 70% is the highest grade you can get :o Yet… now I’m annoyed at the years I could’ve put effort in, but didn’t. Much of it, I do not regret. But… most of it… I’m mad how far behind I still am in many ways. How little I actually feel like I understand. I know radically more than I ever have, but I feel like I know less. And the more and more I learn, there’s more I feel that I don’t know. When I did nothing, at least I could egotistically overvalue my intelligence. Now I’m putting effort in to learn, I feel dumber. Logically I know, I am easily the smartest I’ve ever been. But I feel like I’m the dumbest. It’s probably that I actually have to actively deal with my limitations. At least in the past I could just go “if I tried, I could get 100%”. Dismiss it all to stroke my ego. Now I know, if I try, I know I won’t get 100%. But I guess at least, I can get happily get 90%. And you know what? Intelligence is overrated. Who cares how smart you are? Or how dumb someone is? It has nothing to do with anything. Not really. There’s dumb people in the highest places, and smart people who have nothing. So intelligence is definitely overrated.

[@volbla]

I think it was James Grime who said "Most people who like maths do so _because_ it's hard." Most people enjoy challenges, be they with maths, crossword puzzles, video games or go-karts. So what gravitates different people to different kinds of challenges? Beats me. That sounds like a very deep and complex psychological question.

[@alexsere3061]

A challenge needs to be the right amount and type of stimulating. Too easy is boring, too hard is draining. To make interesting math questions you need to have a basic understanding

[@janmartense]

we don't understand math, we get used to it

[@astonishinghypothesis]

John von Neumann. Yeah, fits well.

[@notsojharedtroll23]

Unironically, this how I learned Calc; i got used to it because it felt intuitive to me

[@Coppermeshman]

I've uet to ever see a scenarios that is not throught this.

[@isomeme]

When i was an undergrad, a college-wide convention was that if you couldn't figure out one step in a multistep problem, you could take a guess at that intermediate result in order to move on to the remaining steps. Such a guessed result was labeled IOTTMCO -- "Immediately obvious to the most casual observer." 😁

[@corbincanavarros2655]

This was amazing. Life changing, perhaps. First yt video I’ve watched until the end in memory. First time commenting on a stranger’s video. Seriously spoke to me. Thank you.

[@SheafificationOfG]

I'm glad to hear it left such an impression on you!

[@Rojuvid]

I love the example of differentiation. I still remember very well in the first year of my bachelor's that we redefined the derivative as an operator that gave the best linear approximation to a function using the Fréchet derivative as opposed to the Gateaux derivative, which is the high-school definition you often see. Now I'm doing a PhD in numerical analysis (e.g. the field of numerically solving partial differential equations, also the field that highjacked the term 'curse of dimensionality' as the amount of variables in a classical numerical method (e.g. finite elements) scales exponential in dimension, something which machine learning is supposed to do better (e.g. Physics Informed Neural Networks (PINN)) and I use this higher order abstraction of a derivative to calculate derivatives of mappings between Banach spaces, something I would have never imagined to be a thing. It just goes to show how deep the rabbit hole of mathematics can go and I still often find myself with the same bewilderment as in my first year when doing research now. You get confused, you struggle, and you learn; it is all part of the progress. Love the video!

[@vectorsahel5420]

I understood about 40% of this comment😂

[@HumanSektor]

As a math teacher who will teach this year in prep nursing school, therefore needs to teach math to people who normally doesn't care much for math, this video was mindblowing. I'm really re-evaluating my teaching methods now.

[@SheafificationOfG]

I'm glad this video has made an impression on you, especially given that you're a maths educator (for non-maths students)!

[@parrotkoi4048]

I will definitely have to use the phrase “Only an absolute buffoon would think” in my proofs from now on

[@gabitheancient7664]

I was wating for the part where you said "in category theory: [bizarre concepts and symbols]" but this was such a good video, amazing

[@voidvector]

Low self-confidence is not a bad thing in STEM, as it means they are capable at questioning their own judgments/opinions/intuitions, and as long as they are capable, they can achieve results by "doing/showing the work". Would you rather your civil engineer be naively self-confident about the bridges you are driving over everyday? Or would you rather they derive such confidence via work? Of course this trait sucks if they were to pursue social aspects of the field (e.g. education, sales).

[@ElusiveEel]

people who are self-confident can do that too. I would rather the civil engineer just know their stuff. A lot to expect nowadays I know.

[@gamedepths4792]

You are one of my favorite math content creators on yt. Thank you for making such good quality videos. You have motivated me to start making math videos myself someday. Love your work, kudos!

[@SheafificationOfG]

So glad to hear you enjoy my stuff, and you should totally go for making maths videos!

[@crafty_zombie_minecraft]

High school student here. I thought your examples with people ignoring the definition of the derivative very interesting and relatable. For a long time, the mathematical rules that we've memorized in class have always intuitively made sense to me, and if they didn't, I'd research them until they made sense. The concern I've come to is that many of the other "gifted" people around me don't experience any of this. They simply memorize and carry out rules and procedures, or "computations", as you called them, with not a care or desire to understand what they're doing. This shocked me because many of these things I either visualized in my head to better understand or wrote down and experimented with, but most people never give a deep concentration to math. As for what you said about intuition, I think I partially agree. Mathematical rigor is, of course, necessary because unifies and formalizes ideas and is necessary for the consistency of mathematics. That being said, rigorous definitions aren't wrong. For example, even though the trigonometric functions via power series, teaching Geometry students (who are only dealing with their real definitions) about representing them via the ratios of triangle sides shouldn't be looked at as inaccurate or incorrect, even if it's not the full definition. Same things with derivatives being called a velocity function. Forgive me if I misinterpreted what you were saying. Anyway, cool video. :)

[@toby9999]

Sometimes it can be the opposite way. The gifted ones intuativelly understand the concepts, and therefore, the rules seem obvious. That's how it was for me at high school. It was all easy and intuitive. 20 years later, I returned to education and struggled somewhat with calculus while studying for my CS degree. I never really reached a point where any of it felt intuative, and now, I struggle with those original proofs and definitions, e.g. 0:33 We had to show the full working based on that definition. It always hurt my brain.

[@kathryncollings9421]

So I’m wondering how you made sense of a negative times a negative equals a positive. Because I could never get this to make sense to me.

[@crafty_zombie_minecraft]

@@kathryncollings9421 That’s actually something I hadn’t thought about until recently. One way to look at it is the concept that a/a equals 1, except for 0/0, as that is often a representation of an indeterminate or undefined result. Since the reciprocal of -1 is -1, division and multiplication by -1 are the same. If you factor out the -1 from both factors and multiply, you get +1, which joins the other positive factors.

[@crafty_zombie_minecraft]

@@toby9999 Yes, I get what you mean. I was trying to say that though some people have an intuitive understanding, other people who are seen as gifted in mathematics actually don’t and just end up going through the motions.

[@kathryncollings9421]

@@crafty_zombie_minecraft thank you. Something to think about… Happy New Year.

[@blacklion79]

Terminal rigor before computers: Principia Mathematica, 100+ pages for proofing 1+1=2 After computers: Lean proof assistant find hole in proof from 1950s which is used in housands of works.

[@thewhitefalcon8539]

You can start from axioms of addition or you can start from even lower axioms and then prove yours contain addition. There wasn't a NEED to prove 1+1=2 as it's normally an axiom.

[@dewah7775]

The proof itself is like a pargraph long and it's not like that was the end goal of principia mathematica.

[@halftab]

good way to end my all night math / physics study session. thank u g++!!!

[@bluegamer4210]

6:48 I really want to understand what you mean by the only barriers are social.

[@m4gh3]

Speaking of intuition however one might argue that a formal proof written for humans is an intuition for a proof written using proof assistants

[@Noname-67]

A "formal proof" in the ordinary sense is a proof that can easily be translated to formal language by someone who knows the language in question. An "intuition proof" cannot be translated so easily.

[@thewhitefalcon8539]

​@@Noname-67 elaboration: "formal" means something like "in form, alone". A formal proof is one that is true because of the form of the proof. For example, all proofs of the form "A and B, therefore A, therefore A or C" are valid (given that "A and B" is a premise) and this can be checked without understanding A, B or C, only by understanding proofs. If the proof requires you to understand one of those, it's an "intuitive proof" as you said. It's relatively easy to write a formal proof in a different language if you understand proofs and both languages. However, they are extremely long.

[@DrDavidHerreraMath]

9:26 You can sketch counter-examples that are "remotely faithful". For instance, consider f(x) = x^2*isOdd( roundUp(1/x) ) This function has discontinuities at 1/n for every integer n but is differentiable. On half of the intervals (1/n, 1/(n+1)) the function equals 0 and on the other half it equals x^2. The function is not continuous on any interval containing 0 but it differentiable at 0. If you paste this: x^{2}\operatorname{mod}\left(\operatorname{round}\left(\frac{1}{x} ight),2 ight) into Desmos then it plots a graph. (I tried to post a link but the comment may have been hidden.) Great video btw.

[@SheafificationOfG]

Yes, very nice! There was another comment with a very similar countercounterexample as well.

[@adityakhanna113]

That analogy with the legal text is great! I'll use that with my students

[@sya8002]

Tbf, I strongly think that although majority points made in the video are really interesting and worked through, some assertions would really benefit from getting more into philosophy/epistemology of mathematics, because they are inherently tied to these fields (e.g. definition of "understanding", the evolution of the concept of "proof"), some are maybe even related to pedagogical/didactic side of mathematics. Really nice video overall though, always happy to see fellow mathematicians be interested by the "meta" side of things in the field.

[@ppppppppppppppppppppppp7]

you’re the greatest youtuber of all time, g. happy new year!

[@louisrobitaille5810]

5:30 I associate "understanding mathematics" with "I can use it to solve a given problem without ever feeling unsure of the path to follow". It doesn't matter if it takes one line or 10 pages, if there're one or 10 possible methods and/or solutions. If I can get from A to B in a single continuous train of thought (considering I have shit short-term memory), I consider it easy.

[@ApostleOfApostol]

I’ve never been more excited to see sigma algebra and measure theory. EDIT: Really enjoyed this video. I also agree that studying pure maths is almost like having chronic impostor syndrome.

[@-taehyun]

Is that obsidian?

[@crix_h3eadshotgg992]

VSCode with the latex preview addon, according to the pinned comment, if you didn't check back already.

[@Xrtd62]

This video is so good and reminds me of so much of the conversations I have with my classmates. Continue sur cette lancée !!

[@apc8536]

Thanks for giving a talk on measures and (complete) boolean algebras!

[@SuperRobieboy]

What I hate about math is that every math teacher or professor is so incredibly boring, monotonous and does 0 effort to create stories or analogies to things outside raw theory. Seriously, physics and biology teachers have a spark in their eye that gets you hooked on their subject. The only math source that came close to delivering this for me is 3blue1brown. I cannot sit through this monotonous video without losing attention and feeling lost & anxious when I snap back. Reminds me of every math course I've had...

[@myb701]

Fell for the "programming is just like math!" trap, and now I've found myself studying CS, disgusted by almost every field of programming I've tried, and more interested in any of the math subjects learnt in (and out) of class. What are the most math heavy programming fields? Or atleast anything where the complexity is not just reading documentation of 10 layers of abstraction (*cough* Gamedev *cough*), or super low level, linux and memory management autism.

[@averageumbrella3424]

I totally understand where you are coming from. Imo, the study of programming languages themselves have some aspects that are very similar to pure math. Also theory of computation. The mathematical basis for a computer is a Turing machine, or equivalently, Alonzo Church’s lambda calculus. If you want pure math from computer science, fall down the rabbit hole of lambda calculus, functional programming, intuitionistic logic, type theory, and the automated proof verification systems mentioned in the video (Lean I believe is the mathmaticians proof assistant, but I would also recommend checking out Agda as it’s more of a fun little type theory playground).

[@toby9999]

Well, I went crazy and studied both. I loved mathematics at high school. In fact, that and physics were the only things I was any good at. I was terrible at everything else and I loathed school in general. After dropping out from school, computing became my hobby. I eventually went back and completed two Bachelors degrees... Computer Science, the other Mathematics. Computer Science won out and I've worked as a C and C++ developer for 27 years. LOL. Now retiring.

[@rivershen8199]

Have you tried Shader programming? You're not gonna find new theoretical breakthroughs but it requires extremely solid understanding of practical linear algebra and a little physics. But only as long as you actually want to create new effects with your Shaders. If all you're gonna do is implement standard effects you can just copy paste algorithms from some guys blog post like everyone else is doing. Also do crazy game physics or something idk. It's crazy to me how you're mentioning game dev as a negative example when it's the field with the most math potential other than machine learning shenanigans and quantum computing. Of course you're not gonna do math if you never work with a custom game engine but only use Unity or Unreal who do all the math for you in particularly abstract and low-performant ways.

[@thewhitefalcon8539]

Computer science is applied mathematics and you don't like the application...

[@DrMcCrady]

The “somewhat ironically I’m not going to define what understanding means…” was gold.

[@JAMath-314]

Nice video! I think you are delving into a topic that much needs to be discussed i.e. what it means to understand something (and to what degree we should want to understand something to use it) that has implications for other disciplines besides Math as as well. But for more on the Math side of things, I highly recommend the book :Mathematica, A Secret World of Intuition and Curiosity: by David Bessis that could probably be included in the Citations Needed Section. It's actually just a very great book that talks honestly about learning math like you do here; and it will probably be very refreshing for anyone with any sort of higher education in math.

[@Lantalia]

You also need the limit definition to _derive_ the rules of differentiation (plus, the limit differentiation is super useful for generating numerical methods so you can simulate things)

[@alonsoviton8278]

About the error's of the famous mathematician. There is a conference titled "Materiales no impresos en la historia de las matemáticas: el caso de Dedekind" which was published by the UCM, an Spanish University, that goes around the countless hours and calculations (errors as well) that are behind the published work of mathematicians like Dedekind and Riemann which advocated about the importance of logical reasoning versus the mecanichal calculations

[@mhadzovic]

The best video on “mathematics” I’ve ever seen. You’ve captured what it feels like to be a math student: a perpetual learner. Great job illustrating that in a succinct, concise way. I especially love that what you said about professors making analogies vs. the sitting with and digesting of the material on your own. Keep doing this, g! You (g)ot this!

[@SheafificationOfG]

Glad you enjoyed the video!

[@EthanWood-tf6eu]

Great video Halfway, I used it at white noise because you convinced me to retry a problem I had some difficulties on

[@schow176]

As a final year math undergraduate student, I still find it hard to learn in different areas of mathematics because I think they are too abstract/ "not intuitively obvious". I often lose self confidence whenever I cannot get to the correct answer or thinking in the wrong direction hence leading me to nowhere. Your words are inspiring and I hope someday I can get over the thought of I'm incapable of understanding math and embrace them. Great video and thank you so much!!

[@christopherthomas484]

I really appreciate this video. Math education and people’s perception of it (from within or without the field) is a very interesting idea to me. Also at least this video was relatable and understandable to me lol (I don’t know what a limit is). Overreaching my understanding of math topics has been something that I’m getting better at avoiding (despite my ego warring with the fact that I’m behind in my own math education after a couple horrible semesters). I took my first abstract algebra course this fall, and I realized I had developed misconceptions from trying to learn about homomorphisms on my own (I basically didn’t understand that there was a radical difference between isomorphisms and general bijections). I’m trying to go through Baby Rudin for my real analysis experience, with the intention of working through every proof and exercise that I can. I am partway through the Dedekind cut construction of R, and this is all a clear example of needing to take as little for granted as I can, and to follow through everything with rigor; working through the definition of < in R had me making a couple horrendous blunders, but in the end I got much needed practice working with these structures. Rigor, man. You hate that you need it.

[@christrombley4590]

I think the best way to think about learning math is still the way Socrates talked about it in Meno. Socrates proposes a math problem to a boy (doubling the area of a square), who responds with simplistic answers (double the side length). Socrates shows the student how the boy's own understanding of his language implies that the "answers" are not correct. Then he asks leading questions to walk the boy to how his own language contains the true answer (construct a side length of root(2)). Socrates points out that though the boy can now do something he could not before but does not feel as though he has learned something. Rather, he has discovered something that was already implicit in his understanding of language.

[@SmileyMPV]

9:28 ironically… here is a counterexample to your claim that counterexamples cant be faithfully plotted: f(x)=x^2{1/x} here {1/x}=1/x-floor(1/x) is the fractional part of 1/x and f(0)=0. then f’(0)=0, but f is not continuous at 1/n for all nonzero integers n. yet you can easily plot f in eg. desmos. there is still kind of infinite precision around 0 but it’s not that bad…

[@AstaryuuGaming]

I usually notate the "fractional part" of a number x as mod(x,1) (or x%1 because I learned the modulus function from computer science lmao)

[@sonicmaths8285]

@@AstaryuuGaming yeah, discrete math classes are very diverse and very hard as well since they encapsulate proof writing, some abstract algebra, set theory, relations, combinatorics, sometimes even real analysis and linear algebra (yes, it can be that compresed) as a cs student you tend to learn some abstract algebra early on due to discrete math classes

[@SheafificationOfG]

Right, that is a good countercounterexample. My mind was dead-set on an example of a function that is only differentiable and continuous at exactly one point; I forgot that the statement was just "neighbourhood".

[@SmileyMPV]

@@AstaryuuGaming problem with the notation “mod” is that this technically gives an equivalence class and with “%” is that it is defined differently for different programming language. For example -0.5%1 could be either -0.5 or 0.5. Im not sure how standard the notation {} is but ive seen it sporadically and it seems unambiguous.

[@MagicGonads]

@@SmileyMPV most unfortunately, due to truncation being more efficient than rounding, % has the representatives inverted for negative values in old languages like C.

[@chapol2020]

This was calming brother, please do this more. I'm gonna enable your notifications.

[@lolmanthecat]

the name of your channel is amazing.

[@CasualLifeExperiencer]

9:29 I was thinking: "this can't be right, as you approach the point you jump indefinitely never reaching the point, there's no limiting behavior", but then it strucked me: "that's not the definition of a derivative; for every neighborhood of the limiting, output value y there's a neighborhood of the input value x. True because density of q, and the existence of the limit in the continuous version, even though there are infinitely many discontinuities" The function didn't have a definite integral beccate of the infinite number of discontinuities.

[@MusingsAndIdeas]

The definition of a derivative actually shows up in Control Systems, due to the plus sign. Because it's defined as a function of time, it would imply you knew the future

[@mistersir3020]

so you do the derivative from the past then? ( -f(x-h) + f(x) )/ h?

[@Katieushka]

Mfw h isnt defined as positive

[@kikivoorburg]

​@@mistersir3020 I think this "derivative from the past" is equivalent to the usual one, since we take the limit as h → 0 (which can also be taken in any direction). I have no proof right now, but I'd be surprised if it wasn't. In control theory, though, I expect you would only ever have _discrete data_ and therefore you should use the 'discrete equivalent' of the derivative, the "forward difference" (Δ^+) and "backward difference" (Δ^-) : Δ^+ f(t) := [ f(t + Δt) - f(t) ] / Δt Δ^- f(t) := [ f(t) - f(t - Δt) ] / Δt Since these are discrete (no limit), the two are not necessarily the same, and the backward difference corresponds to the "derivative from the past" you suggested

[@mistersir3020]

@@kikivoorburg Great explanation :)

[@kevincsellak296]

@@kikivoorburgThere's a difference between the derivative from above and derivative from below in functions that aren't continuously differentiable (from above or from below; the definition for continuous differentiability is equivalent for both if the function is continuous). A function may have a derivative from above at every point and a derivative from below at every point, but that doesn't mean that those two values have to be the same; just think of the function that sends each number x to max{0, x}. At x = 0, the function behaves like its slope is 1 if approached from the right, but like its slope is 0 if approached from the left. Luckily, all twice-differentiable functions are continuously differentiable, which also includes all smooth functions, so this isn't a problem in applied math.

[@robin1826]

Love your videos, g. Keep up the great work. I wish there were more like them (friendly commenters, feel free to drop recommendations).

[@alexsere3061]

As someone who compteted in math competitions and did really great in uni, its crazy how many people want a secret formula for understainding mathematics. I had the privilege of asking 3b1b in an online conference where het gets the understanding, and he said you just need to sit down and study it. Study how it works, study where and why it was created, study the extwnsions and so on. That is the way to understanding.

[@gchinmayvarma9030]

where do you even find symbols for latex? how would i type ¬x while in the flow? do we enjoy writing latex because it is in the pursuit for finding the unicodes and the syntaxes that lets us digest what we are typing that we understand the subject?

[@antonf.9278]

The symbol you described is a logical not, it's called \lnot. It fits in with \land and \lor for and and or respectively. On Wikipedia there is a page for mathematical symbols that shows and describes them. Sadly only the German and French pages give latex commands.

[@SheafificationOfG]

9+ years of LaTeX makes it easy to remember which symbols are what. But in the case where I have no clue, there's also good sites like detexify

[@user-oc8jp2bk2y]

Constantly thinking about edge cases, being rigorous.. yeah that's programming I like thinking and having a brain struggling to grasp things, but when I went onto real physics and maths.... You know, I also like to feel confident and doing easy tasks, and I prefer it to be like that most of the time. I gave up, I am going to be a coder now, not an engineer. I don't want to constantly suffer and learning real math feels like it. Especially hurts when you are on the bottom of your class. I immensely respect people who went for this route, who could take it.

[@aimsmathmatrix]

I would even go so far as to claim that proofs 100-200 years ago were much less rigorous than they are for (under)grads now, in lectures like real or functional analysis. Proofs done by mathematicians of the 18th century are a lot more handwavy, as another commenter suggested here already. Nice video too! It brings up a very central point. Mathematics modus operandi is pain and suffering. It's masochism. But there's also joy in that. I don't know if you did a video on it, but shedding light on motivation for mathematical research and curiosity would be nice. I.e. mathematics is like art, in a way. There is no inherent reason to, say, pursue research into the theory of operator algebras or some really deep niche of multivariable complex analysis lodged inbetween some other fields of maths beyond some quantum information application 30 years down the line. The point is, it is done out of interest, curiosity, and because it brings joy. Just like my professors, for example, I derive joy from the field(s) I am working in and the research coming out of it, because it's a bit like a baby (a puzzle) you put into the world, that you get to watch grow up. (That latter part turned really unhinged, but it should get the point across. The rigour isn't necessarily what most mathematicians enjoy, it's abstraction and generalization and growing cumulative knowledge, imo.)

[@SheafificationOfG]

Hmm, a video on motivation for research and curiosity could be another topic to explore for a video (if I get in the sentimental mood again). But I do resonate with what you're saying here :) made leaving academia all the more painful tbh, but I enjoyed the journey, pain and all.

[@fizhouz328]

7:35 the negation of "Built different" would be something along the lines of "Built stupid", from this arises the grammatically perfect sentence of "Most people in mathematics are built stupid", which I think fits perfectly into this script discussing the problems in mathematics and its education.

[@kingplunger1]

I don't get that version of continuity without talking about the topology on domain and range of the function

[@ianyap8941]

Thank you for making me relief all the stress that I had as an undergraduate maths student studying pure mathematics. Now that i am fully focused on a project that doesn't use advanced maths, I fear that my intuition has declined. :/ But thank you for this video. Btw, I love how you typeset in Latex, the foundational building blocks of measure theory for no good reason at all xD.

[@SheafificationOfG]

The building blocks of measure theory were part of the penance of being (more) self-contained with my counterexample of a CBA that isn't atomic.

[@jackozeehakkjuz]

Thanks, G. Very nice video. I wish there were more videos that represented the mathematical experience as what it is: a human endeavour.

[@strangeWaters]

My private opinion is that most of the time you work in mathematics you have, like, 2 pictures in your head that make all the supposedly "abstract" symbols very concrete. And if people would merely sketch their fuzzy pictures next to their proofs we would all have a much easier time.

[@Alex-nm8xn]

Yeah i definitely have some visualisations corresponding to nice cases for more abstract generalisations. I find that that drawing out these pictures tends to help a bit and makes a proof a bit more intuitive

[@the_neto06]

why do i feel like i've watched this already, even though it just came out...?

[@Louise-j3p]

2:04 Getting jumpscared by fire emblem in a maths video, it's more likely than you'd think!

[@andychen7016]

I learned about the "difference quotient" years before learning about derivatives.

[@isavenewspapers8890]

This is really one of those tough pill videos. Thank you.

[@dawmd55]

I'm glad you made this video! I often ponder things like this and I'm extremely happy you shared your opinion on this topic (I hope for more). The only thing I did NOT like about the video is the typing sound that made it harder to focus on what was being said.

[@SheafificationOfG]

Thanks for the feedback! I thought the clicking was relaxing, but I obviously listen to my keyboard a lot.

[@gabitheancient7664]

7:32 now I'm curious, what is the curse of dimensionality on geometry? and what article are you talking about?

[@SpinDip42069]

You are quickly becoming one of my favourite channels

[@sinewavey]

Good morning fellas this is a great ass thing to wake up to

[@PigeonSwag]

as a first year student taking calculus, this was nice to hear

[@weirdo911aw]

6:45 would love to deep dive into that rabbit hole! very curious if by "social", you meant the inner politics of academia. Your about page implies you used to teach in a university.

[@SheafificationOfG]

Given my own background, I would say I'm not the most qualified to speak of the social obstacles in mathematics, but mathematics doesn't have a great inclusivity track record, and academia is fraught with annoying politics for sure.

[@shevsky]

i feel like " "differentiable in neighborhood = continuous in neighborhood" isn't true" is itself a counterexample to the formalism and it makes me run away screaming to the comforting arms of constructivism

[@MagicGonads]

yeah I feel like if you just assume that any measurement driven from observation must have an underlying computable (i.e. piecewise continuous) source, then that intuition makes perfect sense.

[@deathracoffee]

Hate the clickbait, this is about how math is badly taught not how it is somehow inherently bad. The former is important, true and talked about by many, the latter seems so impossible it interested me.

[@SheafificationOfG]

This is not about either of the things you mentioned, though; it's about how "understanding" shouldn't be a measure of success in mathematics (i.e., you'll never be "done" learning it). I used diff calc as a reference point since that would be a bit more universally relatable, but I only personally came to this conclusion during grad school, when it helped me overcome my obstacles to research.

[@academyofuselessideas]

Pretty awesome! Random thought: The definition of the derivative is NOT the derivative. But being able to come up with a definition of the derivative shows some degree of understanding of that object. This explains why some people can do calculus without caring for that particular definition. They have an intuition of the derivative that works for their purposes. Someone with a deeper understanding could then ask them something like, but what about the derivative of the function you show @ 09:30 . Now, some people would say that such a thing is not a function (depending on their own understanding of what.a function is), or maybe they'll consider it as a function and then realize that they need to be more careful in defining the derivative... IMO, one of the reasons why math doesn't makes sense, and the reason why your meme @ 09:59 hits true is that many modern math definitions are so good at representing what math is, that it is easy to forget that they are not what they are defining (a map is not the territory)... the definitions are so good at usurping real math, that professors often teach the definitions instead of teaching math. This makes it harder for students to learn concepts. In my experience the best mathematicians are those who are able to see the math behind the definitions (and the worst are those who cannot or the ones that don't even realize that there is a difference)

[@isavenewspapers8890]

You are correct. The definition of the derivative is not the derivative; it is a statement telling us what the derivative is. The reason people can do calculus calculations without the definition of the derivative is that other people have already worked it out and wrapped things up into nice formulas. Anyone can shuffle symbols around without understanding what they mean. Okay, I am really curious on what grounds one could possibly consider the object described at 9:30 not to be a function. Genuinely, why not? Is it because it's super-duper weird and might make someone uncomfortable? See, this is why definitions are important. The mathematical definition of a function as accepted by the mathematical community is fully clear, and the f we see here definitely counts. I know it's weird to think about what exactly a mathematical object is, as interrogating the ontology of abstract concepts tends to be, but I think you're confusing the set of intuitions you like to use to make a concept feel good with that concept itself. If what you wanted to say is that the object described by the definition of the derivative is not the derivative, then that's just incorrect. Even still, what I believe you're trying to say is that handing out definitions without explanation is bad math teaching, and that's a great point. I completely agree. It's just that it feels like you didn't know how to express that in the right way.

[@academyofuselessideas]

@@isavenewspapers8890 your clarifications are all great... thanks for sharing them! What math is really depends on your philosophical standpoint. If you are a platonist for example, then all those math ideas exist somewhere and we are just talking about those ideas. for some people the definitions in itself are mathematics... but my own philosophy is that the definitions are just talking about the mathematical object and they are not the object themselves... Our current definitions just happen to be really good at describing the objects. With respect to functions... well, in this point of history, every mathematician would accept a definition based on set theory... but this is fairly modern. In fact, much of the work in analysis arose precisely from attempting to understand what a function should be (a simpler example would be how the dirichlet function was not accepted by all mathematicians at the time of its conception)... What i meant is that the fact that we understand this objects very well allows us to give the modern definitions... But that someone who is not as versed in mathematics, might give a "weaker" definition (which is an indication of the fact that they don't understand the concept that well). Some of this ideas are hard to verbalize, and i am not putting enough tough in verbalizing them very precisely... so thanks for your effort on understanding my perspective! I actually made a video on what is a function (i don't even remember what i said in there. But some of those ideas might be better verbalized there) Thanks for your feedback!

[@isavenewspapers8890]

@@academyofuselessideas Excellent response! Yeah, I don't have much to add. I'll just say that I think I have a problem with controlling my tone when disagreeing with someone, and I'm sorry if that was a problem here. Regardless, I hope you have a nice day.

[@academyofuselessideas]

@@isavenewspapers8890 oh, not really! i didn't notice anything. The internet is so vast that some cultural differences can come across as rudeness to others... so i have found it healthier to not assume malice when you can explain the difference in tone by other reasons! Discussing ideas helps us, so I am always happy to hear others!

[@simdimdim]

a vid about the Langlands program would be nice :D P.S. it'd be nice if the typing in the background was at least 60% quieter, the noise was just tiring to ignore.

[@burrahobbithalf]

I happened to disagree with nearly everything said in the first few minutes of introduction. Students are generally not this stupid.

[@farkas5572]

Practical problems and real life usually requires you to be confident and "wing" stuff. In mathematics, you can have the feeling of doing something that is a 100% correct. Once you dig deeper into calculus, you realize that it is quite difficult to be 100% sure, that what you are doing is "correct". A ton of mathematicians are motivated by this feeling in my opinion. They found confort in math for giving them a 100% certainty. That's why there are also a ton of autistic mathematicians, because I think these people find more confort in math as they cannot really "wing" other situations in life like most people. Mathematicians have spent more than a 100 years to make calculus rigorous, because most of them got extremely bothered by the fact that the only placed that gave them confort of correctness was corrupted. And most of mathematics I believe is still done out of this passionate search for 100%, proven truth. WHich is kinda unattainable but it at least keeps you occupied

[@SeeingRed-e8d]

If nothing can be 100% proven true then nothing exists period.

[@computerKwastaken]

KZbin absolutely refuses to get this video out of my recommendations, so I'm watching it :P

[@SheafificationOfG]

KZbin algorithm is being a real one (for once) ✊

[@MagicGonads]

I think this is very important when it comes to our understanding of social situations (etiquette, rhetoric, awareness, intention etc.). Most people take for granted the 'rules of engagement' under a sense of intuition about 'common sense', but for some people (perhaps broadly categorised as 'autistic') there simply is no such thing, and we must derive our understanding of society from principles that are often not discussed (by analogy to be given no definition of a limit, but just the power laws). Probably that is why I was drawn to mathematics only after the introduction of 'letters', finally I could have a language with which to express the underlying connections of the seemingly arbitrary facts collected over the course of basic education. This was all the more interesting as the complexity of the language of mathematics approaches category theory and is increasingly able to connect different advanced fields of study. At risk of 'getting political' (try to stick to the point relevant to my above paragraphs with this example): Recently, I had an argument with someone about welfare economics, and they claimed that my use of the term 'pareto optimal' was just to declare myself as more intelligent by using fancy jargon (I don't know the term for it, some combination of 'gishgalloping' and 'gatekeeping'?), but that term is incredibly relevant to the topic of discussion to the point that to not know it is to basically admit you have not observed or participated in sufficiently advanced discussions of the topic, let alone read any studies on it. Their justification for not knowing this term (and having no intention to learn it) was basically (paraphrasing) "we have intuition and can use that to solve logic problems, so even if I don't know the word I can know about the issue" and I thought that was absurd, there are innumerable 'counterintuitive' facts about not only reality but logic itself, if you are at all careful with the decisions you make (based on the opinions you form) there should be extreme doubt about intuition. If no matter how concrete and well supported our theories are, they can just be dismissed by failing to match 'intuition', what hope do we have to actually change anything for the better? Can any relevant 'understanding' be boiled down to something a 'layman' could grasp, or is there a fundamental level of personal inquiry, honesty, and humility that we should expect of one another, given that the problems are just actually hard?

[@brendanchamberlain9388]

What note-taking software are you using in the background?

[@xaf15001]

LaTex. It's for formatted documents like Word, but specialized for equations and maths.

[@Vaaaaadim]

@@xaf15001 I don't think that's a sufficient answer. There are different things that allow usage of LaTeX like Overleaf, Texmaker, Obsidian(which is what I think he may be using).

[@kikivoorburg]

​@@Vaaaaadim Agreed, Obsidian definitely seems most likely, given the use of '$$' to denote equations and markdown for the rest.

[@JrgenMonkerud-go5lg]

To me, understanding is about comprehending the context in which a certain pie e of logic is embedded. Relationships between statements and applications, much like knowing how to use a tool for what it was meant for but also understanding the tool well enough to apply it in novel ways, without misunderstanding the new application. Understanding something is understanding what it can do and what is implicated in it. Difficulty is an illusion, mental strain that has nothing to do with the logic, but the illogic and poor memory wraped in anxiety that people feel.

[@trwn87]

Very cool video that illustrates the common misconceptions of mathematics. You shall know that I am grateful! 😎

[@sh23943]

(Not so) hot take, but doing "informal" formal mathematics is just a big diagram chase in your topos of choice. I'm glad a math channel made by a category theorist is going mainstream

[@charlietian4023]

based and very true. it always seems that there will be some intuition defying counterexample

[@adityakhanna113]

Damn. These arguments are so impressive.

[@geoinmot]

I feel like the derivative formula makes sense. It's the change of Y over the change of X (gradient). Except Y is written as f(x) and h is the distance between X¹ and X².

[@samicalvo4560]

Why is the example at 9:30 a counterexample? The function is not continuous at any point, right?

[@SheafificationOfG]

It's differentiable at x=0 (and in particular continuous!)

[@reehansaeed5453]

Favourite channel for a reason ❤

[@robfielding8566]

I started learning theorem proving like Lean, and getting into formal methods like TLA+. I like being able to state theorems, but want better tactics to just in the 90% case just make sure that the left and right side of equals are in fact connected. Instead of having to find some crazy set of tactics to DO the combination, I often want it to EXTRACT a path from a to b if it found one. It would be better to just make programming languages better support theorem proving. But mostly, you just want to ensure that the algebra does lead from a to b; and you sometimes want a path extracted as evidence. It is good to leave out steps, but be able to mechanically pull the steps out when they are needed.

[@calibratingform]

What's the idea behind the spongebob meme at 7:23? I guess it's that differential geometers allow "smart" people in their field to publish proofs that aren't fully rigorous? Is this a widely-held belief? Also, if it's true, is differential geometry more guilty of this than other fields?

[@danielesantospirito5743]

Very beautiful discussion. I'm in a time in my life when I'm struggling to find the connection between mathematical knowledge and the rest of culture/humanism. This is a great doubt for me, because I admire both areas and can't make a precise sense of their place in the world and their relation. Thank you. I'll try to use your video to understand better...