Maths: Discovered or Invented? Do Computer-Aided Proofs Have Value? 10K Q&A
ฝัง
- เผยแพร่เมื่อ 8 ก.ค. 2024
- Huge thank you to everyone who submitted questions -- this was a fun and relaxing video to make. Hopefully my views aren't too contentious.
00:00 - Intro
00:44 - Surreal Numbers?
01:54 - Why TH-cam?
03:59 - Negative Bases?
04:17 - Do I Make the Props?
06:08 - Axiom of Choice?
08:24 - Film, Music, Book?
11:47 - Why Bricks?
12:37 - How to Pronounce Gödel?
13:12 - Favourite Theorem?
17:17 - Do Computer-Assisted Proofs Have Value?
21:10 - What's my PhD Thesis?
23:37 - Favourite Type of Graph?
24:12 - Channel Plans?
24:57 - Favourite Number(s)?
25:54 - Is Mathematics Invented or Discovered?
29:10 - Outro - บันเทิง
Sometimes I forget how small of a TH-camr you are with how good your videos are
Spread the word!
@@AnotherRoof w o r d
I like that idea: the framework is invented, the consequences are discovered. Conway invented his Game of Life, but he discovered the Glider within it.
That past tense still stings
I have now adopted this viewpoint as well
@Dane Newland Not to detract from the truly tragic loss that was his death, but it seems to me that the past tense in that comment would have been written exactly the same if he was still alive today.
In my view "creation (or invention) is a subset of discovery". The game of life is an example of that - there are 2^(2^9) most basic 2D cellular automata rules, and the Game of Life is just one of them.
I suspect the difference in pronunciation between Möbius and Gödel comes from the fact that Möbius strips are something that young kids learn about in school, and Gödel isn't introduced until higher education.
At least from my experience being a kid in the United States, young kids aren't the greatest at making sounds that don't exist in English. (I would assume that kids who live in a household where English isn't the first language would be better at that, but if the sound also doesn't exist in their home language, they might have just as much difficulty with it.) And I wouldn't be surprised if teachers tend to simplify the pronunciation for younger students, so the kids might not even be introduced to the German pronunciation when they learn the word for the first time.
I've been rewatching your content a lot recently. One of my favorite math channels for sure.
For the 4 color theorem, iirc there's something like ~2000 different maps that can't be reduced to simpler maps. A human can't hold all those in their head at the same time and be like "ah, yes, clearly it must be one of these 2000 cases" but if there were only 3 or 4 cases, the exact same logic might produce an "intuitive" and "elegant" proof. Very unfair!
Man. The residue theorem is my favorite!! I didn't expected It to be mentioned. It's so nice to be able to solve a real integral that seems unsolvable by going into the complex realm and then coming back. Isn't that what myth is all about? You go to the other world and come back with the tools to overcome a crisis (like Eneas, Dante or Papagheno)
I love that you mentioned finding your own style by using props rather than animations, because in the SoME2 results video when Grant talked about "stylistic originality" I immediately thought of this channel
When I was in college, I had a friend who drew a lot of cartoons on the dry-erase board on his door in the dorm. I had some ideas for cartoons I wanted to draw, and I asked him if he'd draw them since I'm not good at it. He said it would be better to do it myself. I'm still not good at it, but I definitely find it more satisfying. AR's props do what they're supposed to do, so I imagine they're satisfying as well.
As an engineer with a huge curiosity for higher maths, I love the technical depth of your channel, keep up the amazing content. Also I loved PBS Infinite Series!!
It's a shame they axed that channel! I think they made a few duds and struggled to find their feet but I wish they'd continued and improved rather than cancelling. Space-Time remains excellent even after all these years!
I just found this channel, which I'm enjoying a lot. I also loved PBS Infinite Series, and it still had so much potential.
@@AnotherRoof Matt is a treasure for explaining the complex physics, definitely an S-tier channel.
I was part of the dud crowd near the end of Infinite Series. It just kind of seemed to loose its creative pizzazz (or something - idk, I'm just an engaged viewer).
I love constructionist mathematics (infinity somethings don't exist, only _arbitrarily many_ somethings, and therefore law of excluded middle doesn't poof truths into existence, unless you can prove that the original theorem truth is constructible in the first place). Is this an area you'd enjoy talking about?
Noether's first isomorphism theorem is my current favorite, it's insane how one theorem completely changes the way you think of homomorphisms!
My opinions on favorite theorem and computer assisted proofs:
Favorite theorem: Idk for sure, but the first one that popped into my mind was Fermat's last theorem. When you were taking about group theory, I also remembered the monster, which is like fascinating.
Computer-assisted proofs: all proofs are helpful and computers let us get more of them, which is great. They might take some fun out of math, but it's still better to have a proof than to not have a proof.
I senator Iro seconded your motion.
Members of the House in support of the motion say aye.
I had to stop at 5:41 because you managed to address so many things, that I will surely forget to write my opinion in a comment (like it were important, hahaha). I will only write my thoughts on some things said in the video. I do not need to comment everything. Also I will not write about things I already commented on your other videos.
I find it very nice that you found your (professional) Love and can live embracing it. Not all of us are that lucky. There are "weird" but still interesting and useful topics, that are not being recognized as such by the general public (even non-mathematician general public will, if not agree, then at least respect theoretical mathematics). I do not only respect it, I also feel joy seeing people like you immersed into what you love doing. It is a pleasure.
Access to higher education: I think that there is way more out there than people might realize. But it is not (formal) education. I have watched literally hundreds of taped lectures, at Stanford or Yale. Watching all those videos, reading all those books and research papers does not make me educated. But I do all those things not to earn those letters in before or after my name. I do it for the joy of learning. At the same time, I refuse to be an expert on anything. I am merely scratching the surface, to be honest. I know, I underestimate myself (based on real-world experience), but I do not want to sound arrogant.
Your videos are not a replacement for a formal education. I do not seek a formal education. I seek the joy of learning (and forgetting, more than two decades ago, I learned how to use and compute Hamiltonian's, but I did not forget that I knew it, I forgot that I even was taught what a Hamiltonian is and how it can be used!). So much for how my memory is... :-)
Mathematics is the most abstract thing? It is the ONLY "natural science" that cannot be touched, scanned or detected by any instrument. It is utterly mental. All other sciences have, at least theoretically, some physical processes at the fundamental level (whether understood or not).
You are showing your age? Guess what my age is compared to yours? :-)
It is lovely that you are a cinephile and have other passions than mathematics. This might be an interesting tangent to write here. Two words for you: Jan Švankmajer.
Bricks! HA! Reminds me of something I did back in 2011. I wonder where that ONE particular brick ended up.
Gödel, Möbius. Learn to pronounce the letter ö. Ö! ÖÖÖÖ!!! ;-)
Computer assisted proofs will outperform any human proofs very soon. It is just that humans are not able anymore to catch up with AI. The first law of papers is "do not look where we are now, look where we will be two more paper down the line". Cheek in tongue, maybe not two papers, but ten papers down the line, AI will outperform any human.
Truth will eventually move into the domain of belief (ask me how I know). I do not know what Erdös said, but I surely know that the TH-cam closed captions messed up here. (see what I did there?)
Videos: Maybe something like Kathy loves physics does, but with Mathematics. Check out her channel! She brilliantly combined things that were were disconnected in my mental model of the world.
I disagree. I am a firm believer that chess was invented independently and infinitely across the (infinite) multiverse. There are infinite iterations of you where you are a chess grand-master. Also an infinite amount of you who do not understand the basic rules of chess.
Disclaimer: I am intoxicated by two bottles of egri bikavér. Were I not, I would not write this comment (or rant, or whatever).
Best wishes,
Erik
I like the idea of history based videos. That's exactly what I was left wanting after you defined every number. The precise way things are derived from axioms is very different from the messy history of how mathematicians came up with the axioms and proofs in the first place. Two sides of the story, both interesting and worth telling.
I generally don't have favourites, but when it comes to mathematics, I do have a favourite result, which comes from Graph Theory: "Every tree has a leaf". I count it as my favourite due to the quality wordplay.
It's also easy to prove once you understand the meaning - a graph is a finite set V of vertices and a set E of edges which is a subset of the set of unordered (distinct) pairs from V (think of it as dots connected by lines), a tree is a connected, acyclic graph - one where you can get from any vertex to any other vertex ("connected") but the only way to get back to where you started is to backtrack - there are no loops ("acyclic"), and a leaf is a vertex with at most one edge to it - a dead-end. To prove there must be a leaf, just pick a vertex and start following edges without backtracking. At each step, you either visit a new vertex (you can't visit an old one without backtracking) or you're unable to continue, meaning you're at a leaf (since this is the first time you visited this vertex, you've only used up the edge you came in on, and if there is another edge, you can follow it, so the vertex can't have more than one edge directly attached). Since there are only finitely many vertices, you run out of new vertices to visit after only finitely many steps, so you have to reach a leaf eventually. If the tree has at least two vertices, then you can extend the argument to show there must also be a second leaf (consider the start of the path you just followed - either it's a leaf, or you can extend the path in the other direction until you reach a different leaf).
And it's central to proving many results about trees, since it allows you to perform mathematical induction on trees - taking a tree with n+1 vertices and plucking a suitable leaf leaves a tree with n vertices which, by assumption, satisfies whatever you're trying to prove, so you just need to show either that plucking the leaf can't change that property from false to true, or that adding the leaf back can't change it from true to false to prove the induction hypothesis.
I don't know much graph theory, but what about a graph without any vertices? Would that even count as a graph? And if so, is it a tree?
@@kerstinhoffmann2343 A tree, by definition, has to have a root node. So there has to be a least one and it can't be empty.
@@Yotanido You're thinking of a rooted tree, which has a specific privileged vertex. It's common in computer science to use "tree" to refer to rooted trees, but in mathematical graph theory, "tree" generally refers to a "free tree" - one without a designated root vertex.
Whether the graph with no vertices counts as a tree varies. A lot of properties that otherwise apply to all trees don't apply to the null graph (such as having one fewer edges than vertices, or having at least one leaf) so counting it as a tree means having to call it out as an exception a lot. On the other hand, there are contexts when it's a useful base case - such as when constructing tree data structures in computer science, where, for example, allowing NULL to count as a binary tree means every non-null entry of a binary tree can have two binary trees as children.
My spot research suggests that the standard approach when considering trees is to ignore the null graph unless it happens to come up (in which case, either make explicit use of it, or call it out as an exception). There's a parallel with division where, for example, it's standard to say x/x=1 without calling out the special case x=0, which is generally going to be irrelevant.
@@Yotanido ok thanks but can there be an empty (non-tree) graph?
Your videos are so unique. I’m so grateful you popped up in my recommended a few months ago, and have watched everyone through since. :)
16:45 One basic corollary to Lagrange's theorem is that groups with a prime number of elements don't have non-trivial sub-groups. When I was a child, I liked to mentally jump through patterns in a regular way. Imagine that like chess pieces moving over the board in a regular pattern. I was trying to find interesting paths that run only through certain sub-patterns. For some patterns, it worked, for others, each path was jumping through the whole pattern. So I was kinda prepared to Lagrange's theorem when I first heard about it.
More videos as soon as possible, please. Yours are the most watchable on TH-cam! 😃
18:50 this is a very fun idea to play with. i once thought "what would math look like if all proofs were required to give insight?". although i didn't get very far because of how amazingly informal this is, we could have things like 19 not being prime, or an integer between 3 and 4 (i think there was a movie about this specifically, although i haven't watched it yet, something like "the secret number").
it's sort of like breaking the law of excluded middle, but weirder. i mean, we still know every positive integer has a unique prime factorization, and we still know that there are more real numbers than integers, but we don't know whether 19 is prime.
i think that, in such a setting, the only way we could know something like the primality of a specific number is with tools like the MOG from your steiner systems video, which somehow provide insight into specific finite structures, as if they couldn't be defined by a list and could only be studied with such tools.
When you started talking about music I somehow knew you were going to name snarky puppy. They are awesome live I can tell you that.
Yeah I saw them a few years ago and they were incredible, can't wait to see them again!
@@AnotherRoof thanks for answering. As a math student myself, I have found a lot of value in your videos and I have shared them with most mathematicians wannabe friends.
23:00 10^74 being "very big"
17, from your video on addition: "see? How dare you call me huge, when that behemoth only gets called 'big'"
I absolutely LOVE your content. Thank you so much for making these videos. 😊
As someone really interested in philosophy, math and more specifically the metaphysics of math, its dope to hear an actual mathematician use the same reasoning I have toward the "discovery / invention" question. Like its fully true that we decide the axioms of what certain things are, and from those starting premises, deduce truths of them through logic (i.e "the discovery" or the truth hidden in the consequences of the axioms), so its cool that you gave your take on it all!
The background in this video is actually really great 😊
Also, in the context of pronunciation of German names/words: Z in German is pronounced as TZ in English, and all vowels are pronounced "as they should" (e.g. the oppoaote of how North Americans pronounce the a in "can") - so "Zahlen" should be pronounced as if you're saying "Tzullen" in NA English... ish.
(this probably belongs in the relevant number sets video, but you mentioned German pronunciation here, so 🤷🏼♀️)
Lastly, your channel is really good - and also refreshing in style, tbh.
Cool video! Nice to get to know you a bit. I for one would ABSOLUTELY love to see you make some videos about some of the lesser known, "down the rabbit hole" bits of group theory. I'm a programmer not a mathematician but group theory has always been fascinating to me. Glad you're having fun making these videos, you really are a natural at it haha.
In a way, don't all possible axioms exist latently - which can then be combined by a process of deciding - and act as entry points for discovery within the domains they elaborate? In the sense that the decision of the axioms used for exploration is artificial (or invented), while all possible individual axioms exist in the form of all possible relationships between any things. Is discovery itself the invention, created by - and possible only through - the restricting of all possible axioms down to combinations of a few?
I may not be as knowledgeable about mathematics as many people (attempting to learn it on a collegiate level has become more of a passion than a hobby at my age), but I would love to see your take on how best to teach calculus. While the concepts are all the same, at least on an elementary level, the way these concepts are presented are as unique as fingerprints when it comes to actually publishing a textbook or making a TH-cam video. I still have only a vague definition of a limit, for example, and I've been trying to find someone who will explain it in concrete terms that I can understand; it seems easier said than done. Just like the limit itself, some have come close, but nobody seems to be able to hit the target, and I may have just answered my own question, without any level of self-trust that I know what I'm talking about. Anyway, keep up the good work, and one day, I might be able to get through one of your videos without having to watch a particular section more than six times, taking copious notes along the way...
I think calculus, in general, is hard. My calculus skills hit their limit very early in my first degree and after that I focused on algebra, geometry, and logic wherever possible. Maybe one day I'll tackle the subject from the perspective of someone who finds is difficult!
I just thank you so so much for making that first video explaining Groups. Since watching that, I’ve continued watching a bunch of physics and maths related videos, but Ive found it so frequent how often those random videos are now more transparent, graspable, and interesting because of the foundational ideas of Sets and Groups you gave me in your first video.
I’ve been watching Sean Carroll’s Biggest Ideas in the Universe series, and am currently on the episode of Symmetry, and I often found myself referencing your video in my head. But it’s like, since I watched your video, the other videos I watch now have so much more depth, and so in turn there’s so much more I appreciate about other videos. In the Symmetry video (that I literally watched earlier today) Sean talks about that classification of finite Groups in 2004. And then at 16:00ish u mention it. And I don’t think I could have tried to keep up with Sean long enough to get to that if it wasn’t for your video.
I still need to watch the rest in the series, and yet it seems like I’ve already gotten so much out of it. And it’s not like I even fully understand all the fundamentals yet, but despite that, having that background knowledge has opened up other ideas for me. It’s like I can see more of how different ideas are related.
'Ppreciate it
i will call it:
a surreal numbers video is this guy's petscop 2
Thanks for the video! Very nice.
I think I completely agree with you on computer-aided proofs. There is nothing I would add or change about your answer to describe my own.
Also, I did have to have a program called Macaulay2 compute a couple free resolutions for me to finish a proof in my dissertation. While not the same extent as what you needed, the theoretical results I developed were just shy of completely answering a particular question, and some brute force computations were all I needed to answer this question in this particular case.
I also completely agree with your answer to the question of whether mathematics is invented or discovered. I think this perspective is more common than you think, but maybe I'm biased here.
1. Congratualions 2. Looking forward to seeing where you take things next :)
I love the Cornetto trilogy ^^ And I love mathematics and your videos!
2:05 that is the feeling my friend
thanks! keep up the Free Open-Source Education!
Genuinely, honest to god love your videos... They really do a lot for me, math-wise! But I have to know! When will you continue your investigations in set theory?
The generalization of the classification of finite simple groups to Moufang Loops is kinda neat. The non-associative loops end up being really closely related to the octonions, which isn't surprising, but kinda nifty
With regards to the axiom of choice, I think there will always be more to say about it! Especially in the context of examining "weakenings" of AC that are still consistent with other principles, for example full AC is not compatible with Determinacy, but weakened forms of AC are. Also there's an entire book about AC by Jech, appropriately enough called "The Axiom of Choice." As an example of a particularly fun exercise in Finite Choice that I've never seen anywhere outside of the aforementioned book and "Set Theory" by Stoll: Suppose we have choice for any family of 2-element sets. Prove that choice for any family of 4-element sets holds. (Jech p 107)
Could you please talk about the Hilbert Axioms of Geometry? From an undergrad math major.
Your favorite theorem is a great choice. I forgot most about complex analysis, but I loved it at the time. I guess I would have answered 1. Yoneda lemma (just for lolz) and 2. borsuk ulam
Borsuk Ulam is a good one! The other two that sprang to mind were the Hairy Ball theorem for funny name reasons (but I feel it's a bit overdone at this point) and Noether's theorem which I think it's such a profound reframing of our fundamental understanding of reality. Wish I'd talked about them!
I've been doing a little digging into λ-calculus recently, and the definition of the Church encoding there reminds me a lot of the set theory definition of numbers that you gave in your video on what numbers are. Would you maybe consider doing a video on that?
Awww yeah - love you watch the Fuzz given that back poster. It was great British comedy/action movie imo.
All three are excellent! World's End is the best imo 👌
I would love to see you make a cinema channel as well.
Second this. I'd listen to you talk about films, 100%.
"i don't own any other t-shirts" rip yellow t-shirt
That t-shirt is the property of Yellow T-Shirt Alex, and we're nothing like each other.
I am working in health care but I enjoy maths. I love your videos. Please make a video if you can on
Axiom of Choice (AC)
Axiom of Countable Choice (AC_w)
Axiom of Dependent Choice (DC)
Axiom of Determinacy (AD)
I also connect with you on Fantasy (though I liked Scott Lynch, Mistborn is still too good😁)
Mistborn is the series that rekindled my love of reading, it's near and dear to my heart! I may yet make a video on AC.
@@AnotherRoof That would be welcome. A lot of people have said that if math was developed after computers then AD is ZF+AD is what anyone would have opted for. We know how AC leads to Vitali Sets & can cause chaos. AD makes everything measurable.
I was surprised to hear that after u gushed on about how much of a perfect world maths is, that you aren’t a Platonist. Same!
I think, in a way, the universe of mathematics is so perfect *because* it isn't real
About pronunciation: if you want to say Gödel‘s name in german (as he was from Austria) you are pronouncing the Ö correctly :) In his first name though, Kurt, the u should sound more like the oo in „school“ or ough in „through“ :)
What's your favourite movie, and why is it Primer ? 😅
I do like Primer. And I love the Star One song that's based on it called "Prescient". My favourite is actually Fight Club! But I'm also very fond of Lord of the Rings, the Cornetto Trilogy, Arrival, Donnie Darko, Mad Max: Fury Road, Wall-E, Aliens, and many more...
12:36 you should have him lay them face-up, in a mathematically sensible order as you walk up the path...
Your take about discovered or invented is the same as mine, but I was never able to fully explain my thoughts I think this chess analogy is great I gonna use in the future. But changing the subject: Luca is great! not midiocre!
19:30 why are charges equal and opposite. I believe it's group theory. It's a property of su(1) IIRC and the way gauge symmetry falls out of that. I'm not an expert but it's funny, you were raving about groups and then that group theory property pops up 😂
I'll have to look into that! But I think my point remains -- why is the universe such a way that such symmetries fall out? It might be that it just... is.
@@AnotherRoof it's a fascinating question, and noethers theorem about symmetries and conservation laws makes it even more fascinating..
I agree with your stance of philosophy of numbers. But, if axioms are inventions, can we posit different axioms to get completely different, "unnatural" mathematics? And if so, how would that maths look like? Would that be kind of metamathematics? (Maybe it even already exists, haven't even checked.)
Mathematics as a discipline is somehow "mimetic," in the sense that its axioms tries to mimetically represent out intuitive sense of the world. I think that if it can be shown that there are infinite amout of mathematics that stems from different initial axioms, if can be "proven" that mathematics cannot be platonic, i.e. numbers, axioms, theorems, etc. do not exists out there as a seperate objective entity.
The same kind of conundrum is prevalent in physics, where mathematics is usually interpreted as "language of nature." Given this premise (which also presupposes that "language of nature", i.e. maths and it's axioms, is somehow "objective"), it's usually uncritically assumed that physical laws, theories, equations, and units are not only good representations that just works but also the laws which governs reality itself. By that I mean that it is usually assumed that equations determines how matter and energy should behave and not the other way around. I think, that the problem of defining what energy even is, is indicative that all physics is descriptive and "mathematical construction." In this sense, "energy" does not exist. The same way as we can question energy (what it is), we can say for other physical phenomena: power, work, force, pressure, current, etc. etc. To cut this short: I think and I am sure of this, that the universe does not compute this equations in any sense, but it is determined by underling symmetries and invariant topologies that cannot be simply expressed in one quantity (like energy), even if this describes it perfectly (for humans).
And to come back to my original point: can new maths that is based on different axioms (which are not mimetic, but even more abstract) be more adequately used in physic or in other areas?
Good video, like always! Keep it up!
When I took my first geometry class at a level for math majors, the professor said that geometry was the study of those properties that are invariant under certain transformations. I later broadened this, and I say that mathematics is the study of all invariant properties. (Even in sports, we have the phrase, "mathematically eliminated," meaning that there is no possible situation, short of overturning a prior result, in which the outcome would change.) To the extent that there are invariant truths about the universe, those would be expressed mathematically.
@@JayTemple That's true, and thanks for reply. But I think you can create any number of systems that are in themselves consistent but are contradicting each other if we compare them outside of their frames. What I am trying to say is, that invariant properties are properties within the tautological systems, what math actually is; it it precisely effect of tautology that it creates properties that are invariant.
My question is, if we can create other tautological systems, based on different axioms, that would be something similar to math, but not exactly how maths is right now. And if so, would this in any way be useful or it would be just a logical exercise.
The open question is, why does math seems to be so useful in describing the properties that we can observe in the world. Of course, there could be a simple answer: because we create it precisely to describe the world and understand it better (anything that wouldn't do so or anything that would not be consistent, we would throw out; that's why I claimed that mathematics is actually "mimetic"). Another short answer would also be, that the universe itself is tautological in a logical sense and could not produce inconsistencies; thus, math can somehow align with adequate description of the world. But maybe this is also bias that is introduces when we see the world through the lens of the math-as-tautology.
Since you are already a Prog Metal fan and a Maths guru: are you into Math Rock? If you are not I’m afraid you’re into a rabbit hole. Try listening to “Giraffes? Giraffes!” (Yeah, that’s the band name, punctuation and all)
You deserve a million subs but in a way, I'm not looking forward to it. I've been an early viewer of a few clearly good channels in their infancy and I'm going to miss this early stage where we viewers get to easily interact with you. It sucks how it just can't really happen once you have like, oh, 50k subs.
Anyway, you're on a nice road Another Roof. I see a nice future for you.
One of the unexpected joys has been interacting with my fans and supporters. I thought I'd just put videos up and be done with it but I really enjoy the community side and I'm not in a hurry to abandon that. I want my channel to grow and I know that when that happens my attention will thin out, but I'll always endeavour to read and respond to as many comments as I can, as well as respond to emails, on my subreddit, and on Patreon.
@@AnotherRoof I know you will, because I can tell you're a good human. It will just get hard to do when there are 100k of us, ya know? 😅
Again, and sincerely... Big fan. You're doing really great work. As someone with an (undergrad) math degree. Loving it. You're hitting all the notes that I love in math.
@@kruksog I appreciate that. Let's hope I actually get to 100k subscribers though first 😅
Gödel pronunciation is good.
At 3:30, you talk about tuition fees in universities. There is a HUGE change which has happened here, and no-one seems to have focused on it, and few have even noticed. We grumble, but this should have been a big stand-up-and-shout event, like if someone decided to... I daren't say, because they'd take it as a challenge and do it anyway.
But: I was a child of the fifties and sixties, and when I turned 17 in about 1967 I applied for Uni and was GIVEN a GRANT to go there. Accommodation provided, and a maintenance grant upon which you could survive. No tuition fees. You could even run a cheap car on a student grant. (Not, unfortunately, a car AND a student; but then you can't have everything.)
Nowadays we have many more students and the privilege of funding student life on that basis is probably unsustainable; but I cannot think that saddling anyone who chooses to go to university with a millstone of debt is a good idea.
How did that happen without anyone noticing? What effects is it having?
In 1950, most families were single-income, because mum stayed home, did housework, looked after the pre-school children. Now, scarcely ANY young married women are without at least a job, and preferably a career. Nothing wrong with that in itself, but - the workforce has doubled. Has our quality of life doubled? In seventy years, just two generations, this colossal change has taken place. Has anyone noticed?
It's lucky we don't do sociology in schools any more, otherwise it couldn't have happened unnoticed.
Speaking of animation with a unique style, have you seen Cartoon Saloon's stuff?
On whether mathematics is invented or discovered - I don't know what it means for something to exist, but I think that results are not true in themselves, e.g. 1 + 1 = 2 is not necessarily true. It is obviously true given the axioms and the way we define equality, addition, 1, and 2, but that statement has no meaning by itself; it only comes from the axioms and definitions. So I would agree that the axioms are invented, theresults discovered.
Can we raise a petition to save the bricks? i'm open to donating to buy new bricks to avoid losing these!
i am totally the opposite of you re: favorites. how can you have a favorite film? they're so different from one another. maybe today i'm in a Blazing Saddles kind of mood, maybe tomorrow I want to watch The Matrix, the next day 12 Angry Men. And that's three movies that are about as different as can be; how could I possibly say which one is better? When someone just goes "yeah my favorite movie is [whatever]" i just think they're basic as hell. And don't even get me started on favorite colors - adults should not have them.
Well yeah I have a favourite food but that doesn't mean I want to eat only that food all the time!
Ah I see you have the exact same music taste as me
When you say that closure is a property of a binary operation, why is that? I ask because you could alternatively omit identity if you keep closure by showing that there was always a group element that was its own inverse. And sometimes you want to talk about binary operations that aren't closed. Like with a binary relations, which can be thought of as binary operations that return a boolean value.
A binary operation on a set A, by definition, is a function AxA -> A, so a binary operation is closed by definition (because if x*y is not in A then it isn't a well-defined function).
Also, identity needs to exist as an axiom, because there are groups which don't have a self-inverse (involutory) element, like Z3.
12:58 the issue with correct pronunciation is anachronism and this idea that standards are somehow correct forms of speech. Language pronunciation evolves and depends on where you are. Even if you nail a pronunciation according to the current correct standard of the political entity with the largest influence on the linguistic group with which the person would have identified you can still be way of based on how they would've pronounced their name. Forexample this same name with a Swiss German Or Low German pronunciation would be different even today let alone according to the standard from the day or the way it would have been pronounced where they're from back then
Consider is your name incorrect when pronounced with an American accent? I think an effort is good enough
I too am of the opinion that axioms are invented, but the consequences of those axioms are discovered.
I had no idea that you were Canadian!
Huh?
@@AnotherRoof I’m so sorry! I was making a joke about how much you were saying sorry at the start of the video, but clearly it fell flat.
@@yumnuska Ooooh I get it... very sorry that I misunderstood.
I'm at @0:43, please stop apologizing. And don't reply to this comment by apologizing that you apologized too many times. As someone that was also an over-apologizer, just stop. :)
Q (n + 1) : are mathematicians, say, PhD-level and above, 'born or made'?
Oooohh. Now this is a question I think I would waffle on about. I think I'd spend a good five minutes explaining why the answer is "I don't know" 😂
@@AnotherRoof I taught myself algebra in the fifth grade from a textbook I got at a garage sale. (Our school didn't teach it, at least formally, until the 9th grade.) I still have a copy of the calculus book we used in high school. It's been 30 years since I last went through the formality of a class that required enrollment, but when I see a math problem in a suggested video, I have to at least work out how I would solve it. The pull is that strong--for me. Some of us, at least, were "born" mathematicians. I'd be interested in hearing the journey of one who was "made."
Me: I don't have a favorite movie
Anther Roof: No
What is your favourite number sequence? Mine is the Catalan Numbers.
So... no bricks for sale? Even for charity?
My opinion on the "is math discovered or invented" differs slightly. Yes, the axioms are obviously invented. But theorems are neither discovered nor invented - they are proven. The way I see it, "to discover" means you obtain new information. "to invent" means you create new information. "to prove" means you transform information (ie. the axioms) to reveal a pattern in it (ie. the theorem).
It mirrors the science, engineering, mathematics trio. Neither can be adequately expressed in terms of the remaining two, because they produce qualitatively different kind of "product". The confusion comes from the fact that each of them uses the products of each other's work in its own workflow.
I don't think it's a good idea to elaborate on math arguments by bringing up physics situation. The "19 as prime" discussion is interesting, but the electron assertion is loaded. It silently assumes that electrons and protons are a thing and not merely a handy too to explain things. It brings up the antrophic principle, begging the question of whether - in a world where electrons and protons don't have oppositive charges - there could even be q coherent notion of physical beings like humans which ask the question, and so if other worlds are merely voids of question askers. The math question is purer, but also here the antrophic principle could be taken to kick in, if we take the Kantian view that the reasonable (like that math humans could do) is enabled and bound by the structure of the mind.
Also there's of course book on the axiom of choice and we won't run out of topics. I think as long as people naively blunder into adopting it, it's worth making videos explaining it's issues.
Your pronunciation of Gödel is surely better than that of other Anglos. Not sure about the sea-love theorems though.
I've actually always said "See-low" until very recently but didn't check before shooting! After checking, I can't find much agreement on the correct pronunciation. How do you say it?
@@AnotherRoof I'm also just Viennes and so I'd not know for sure, but I'd assume it's more like the cy in cyborg and a quiet w at the end.
It's ok that you don't care if mathematics discovers absolute truth. It does.
Mathematics are by default invented, since mental constructions don't exist in an a priori state before someone stumbles upon them, they aren't physical objects.
Hahsjaha first I feel so powerfull
This is the first time someone has commented "first" on one of my uploads. Now I feel like an actual TH-camr.
Well congratulations!
I guess
given or you lie
atten cake
saikguot
p
grouing
Whaaaaat u no Platonist? :o *unsubscribe*
(Just kidding, love your work)
I really like your videos but I wish they were shorter. It's hard to find the time to watch them between my CS lectures and other stuff I have to do. Usually I watch long videos at 2x speed but you can't really do that with Math stuff since you don't have time to comprehend what's being said.
You don't need to watch the video in one go.
The topics he covers are pretty advanced. Break them into pieces rather than suggest his vids be shorter. The complexity of the topics pretty much necessitates their length. If they got much shorter they'd be entirely hand wavey and without any real mathematical maturity or rigour.
I'll echo what people have already said and explain a bit about my writing choices. I break my videos into chapters to help viewers watch in multiple sittings, and I'm always striving to strike the right balance between mathematical rigour and general interest. Making them shorter would mean cutting too much and reducing the level of rigour. I did consider breaking my last video into multiple videos, but then it's not a very exciting video to go from just the naturals, go through all the preamble of setting up Cartesian products and relations, then just end at the integers. I thought it was more complete journey to go all the way to the complex numbers. Still new to this though and will probably experiment with different lengths of video going forward!