The Foundations of Mathematics

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In this episode I answer a question about the foundations of mathematics. Just how important is it to fully understand the foundations? What do you think?
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Пікірлер: 115

@conjurors-prelude Жыл бұрын

I’m disinclined to say that someone new to in-depth mathematics should simply move forward. I recently earned my Bachelor’s in math and I feel like it all flew by so quickly. If there’s a topic that interests you, take your time and figure it out your own way. Yes, absolutely, there is lots and lots to discover, but if you are doing this simply for yourself then do things at your own pace. I’ve noticed that most people don’t even have a good foundation is simple algebra even though it is the literal foundation. My advice would actually be to make sure you really understand algebra, and learn to apply it to calculus (really apply it). Get good at graphing and visualizing things in your head. In so many of my upper division classes I thrived while others failed simply because my foundations in algebra and calculus were strong. I know not everyone is getting a degree in math 😊 but I did and I loved it. I would also recommend to read books on the history of mathematics. I recommend “The Math Book - From Pythagoras to the 57th Dimension 250 milestones in the History of Mathematics ” by Clifford A Pickover. It covers everything, although it is a bit pricey.

@camerongriffin7027 Жыл бұрын

Thank you! Love all your content!

@TheMathSorcerer 10 ай бұрын

You are so welcome!

@HanzoHimemiya Жыл бұрын

I thought you can talk about how every math subject evolved in history, from prehistoric counting to computus calculus

@soyoltoi Жыл бұрын

Like Wildberger but without the ultra-finitism

@PowerK1 Жыл бұрын

These videos are so chill, I can be doing something else and also watch your videos at the same time listening to you talk, please make these more often if you can

@boogerie Жыл бұрын

There's an old joke that mathematicians are formalist on Sundays and Platonists Monday thru Friday. Also the title of Velleman's book is of course How to Prove It: A Structured Approach

@topdog5252 Жыл бұрын

Apparently 2 books by Herbert Enderton A Mathematical Introduction to Logic and A Course in Mathematical Logic are good. I have not read them, I only remember hearing that the latter goes over Gödel’s Incompleteness Theorems, something that would fascinate you as a philosopher.

@shadowshadow2724 Жыл бұрын

We have a branch in high school called mathematical sciences ,we study algebra and trigonometry in first year but in second and the third year we study logic and types of reasoning ,sets ,series and sequences ,calculus and we take a little bit of abstract algebra. It is considered the hardest branch amongst students .

@techeternal8362 Жыл бұрын

Thank you for your time and efforts, your videos helps us so much.

@TheMathSorcerer Жыл бұрын

You are very welcome!!

@nkanyezitshabalala5256 9 ай бұрын

I also went down an the path of trying to understand the foundations of mathematics since I was intrigued by this subject and I can also say that for a person not within a more or less advanced level in mathematics will simply not be able to engage with questions regarding the foundations of mathematics. It is just like philosophy were one needs to engage with a bit of its areas to really know what it is. That is why a more historical look at the subject will help (that being the evolution of mathematics). But I also encountered problems going through that route as the books I had read did not fully contextualise what the nature of the discipline is all about. --- What had enabled be to understand what mathematics is and how the discipline operates was a book titled "What is Mathematics?" by Richard Courant and Herbert Robbins. This book really does speak to the philosopher or anyone who wants to analyze what characterises mathematical activity from a philosophical lense. Trust me the authors are also conscious this.

@nkanyezitshabalala5256 9 ай бұрын

I would say that after reading the fist few chapters of this book you can the move on to a more philosophical treatment of the foundations of mathematics or mathematics in general like the philosophy of mathematics. The university of Oxford starts of with Gottlob Frege's Foundations of Arithmetic.

@AbduL-wb4pz Жыл бұрын

Hey Math sorcerer love your content, just wanted to ask about what book,method or resources you would recommend for mental math strengthening?

@root2over1 Жыл бұрын

As a philosophically-minded CS/math major, I struggle learning things without some sort road map to place more advanced knowledge later on. For example, programming didn't click at a visceral level until I studied MIT's 6.0001 that began with "knowledge exists -> we separate declarative/imperative knowledge -> algorithms via imperative knowledge -> can be stored in computers via language -> there are rules of these languages". In fact, a lot of what we study in general can be sorted somewhere on this map. The book 'A New History of Western Philosophy' by Kenny explains in the intro that the question of "which ideas are innate/acquired" eventually split into psychology and mathematics via the development of its sub-questions. My advice isn't "how" to learn the rigorous foundations since I can't say much about that, but I'd say to this person that it pays to write and think a lot about these topics. Some historical context may help (how did math originate, when were the foundations of math formalized, etc), but I'd wager that it's more comprehensive to just be patient and sort these ideas out while some sort of map begins to form in your mind. It's okay to keep digging and asking why, but at some point, I think Wolfram's idea of computational irreducibility will kick in and to keep trying to understand "why" will no long positively affect your studies. This is all just what I think right now, and I'm sure it'll change eventually.

@nkanyezitshabalala5256 9 ай бұрын

I can personally relate to you when you say you are somewhat philosophically minded, especially in the example you had given. My journey to understanding code (at this point trying to understand it) was very similar, I will needed to know the bigger picture of it all like how computers hardware works and how that ultimately gives rise to what it can do. So far I only know how to code in python and I learnt all the web dev languages for making websites but I still don't have the big picture understanding of computer science so that is why I plan on majoring in it alongside math and philosophy. (I will check out the mit course you suggested though). On the second part of what you had mentioned: I also went down an the path of trying to understand the foundations of mathematics since I was intrigued by this subject and I can also say that for a person not within a more or less advanced level in mathematics will simply not be able to engage with questions regarding the foundations of mathematics. It is just like philosophy were one needs to engage with a bit of its areas to really know what it is. That is why a more historical look at the subject will help (that being the evolution of mathematics). But I also encountered problems going through that route as the books I had read did not fully contextualise what the nature of the discipline is all about. --- What had enabled be to understand what mathematics is and how the discipline operates was a book titled "What is Mathematics?" by Richard Courant and Herbert Robbins. This book really does speak to the philosopher or anyone who wants to analyze what characterises mathematical activity from a philosophical lense. Trust me the authors are also conscious this.

@nkanyezitshabalala5256 9 ай бұрын

By the way checked the MIT 6.0001 course out and it had a great explanation of what computation was.

@martinhawrylkiewicz2025 Жыл бұрын

Study books on logic and set theory as well as constructuon of number systems like N, Z, Q, R..

@bugsdonttakefalldamage Жыл бұрын

Thank you for the recommendations!

@dirkvillarrealwittich Жыл бұрын

You have just proved that Mathematics is interesting, beautiful (and useful)!

@djisuruperera4520 Жыл бұрын

Thank you so much for this brilliant video response and everyones help in the comments. Kind regards, Isuru.

@bigstroker1300 Жыл бұрын

When will you do a video about the incompleteness theorems of Godel ?

@eddtlpz Жыл бұрын

I'd love to see your reviews on logic books! Saludos desde México :)

@mrtienphysics666 Жыл бұрын

The completeness "axiom" is very interesting (1) It is not an axiom but a theorem in Set Theory (ref: Mendelson) (2) There are many logically equivalent forms, that look on the surface totally different in meaning (3) Personally I like the version with the Dedekind Cut, upper cut , lower cut etc ref: Taylor (my favourite), even though the most useful form in analysis is the LUB, supremum version (4) Intuitively, it means that there are no "holes" in the real number line

@aradarbel4579 Жыл бұрын

if you're working in set theory with a certain definition of the real numbers maybe it's a theorem, but in general I'd argue it should be thought of as an axiom. recently I've been very much becoming a believer in the model theoretic approach: first pick your axioms (various abstract properties the real numbers *should* satisfy) and later you can find all sorts of *concrete* mathematical entities that satisfy the required interface. this puts a barrier between your high level math and set theory, as you really don't want to work in set theory most of the time. when we do analysis we don't care how the reals are defined, be it with dedekind cuts, or cauchy sequences, or any other definition. similarly we don't care how tuples and natural numbers are encoded using sets. these are all hacks, if you think about it. like writing directly in assembly. why do that when you can use a proper programming language and let your compiler figure out the translation? instead of focusing on the low level details what we should do is define things by their actual behavior. then it doesn't matter how you define your primitive entities, you don't even have to be in ZFC at all. that's also becoming a common trend in type theory, first you define a very high level language and then you look for a model (usually as the internal logic of some category/topos) to prove things like soundness. I believe if we do that, we can get a lot more mileage out of our axioms, and better modularity between different areas of math!

@martinepstein9826 Жыл бұрын

"[Completeness] is not an axiom but a theorem in Set Theory" You're confusing theories and models. To show that a structure S is a model of a theory T is to show that S satisfies the axioms of T. So for every axiom in the theory there is a corresponding theorem about the model. For example, completeness is an axiom in the theory of complete ordered fields and it is a theorem about Dedekind cuts of Q. Of course it is, since that's part of how we know Dedekind cuts form a model of the theory.

@mrtienphysics666 Жыл бұрын

Of course you cannot prove the Completenss "Axioms" within analysis. But you can prove it outside of analysis, in set theory. The other way would simply be appealing to proof by intuition. It has to be, because I feel it has to be.

@EducationEducation-nn2zu Жыл бұрын

Do you any videos about permutation and combination? If you have please give me the link.

@jhonsen9842 Жыл бұрын

Please make a vidio on Maths book to follow Computer Scientists and ML engineers.

@OrdenJust Жыл бұрын

Two comments on foundations: (1) In a sense, math does not (yet) have a foundation, since research is ongoing into the foundations of mathematics and the jury is still out. E.g., are sets foundational? is logic foundational? Categories? If it's turtles all the way down, there is no turtle that is the foundation. One might think a foundation is a beginning point; like Euclid's postulates, you start with the postulates, and make progress by deriving results from them. Yet Bertrand Russell wryly observed that if he were identifying all the assumptions implicit and unstated in Euclid, he would have come up with more than the five Euclid came up with. Maybe there is no beginning point. Maybe there are only starting points. (2) If physics is mathematical, shouldn't the foundations of physics and mathematics be the same? Yet the foundations of mathematics look like abstract concepts, while the foundations of physics looks like quantum field theory, or something like it. So, to quote John Wheeler, how do we get "it" from "bit", the physical universe from information?

@robertlumbangaol7491 Жыл бұрын

Thank you❤

@bendavis2234 Жыл бұрын

What are your thoughts on Gödel’s Incompleteness Theorems? Have you studied them before?

@syedmdgufranazam1260 Жыл бұрын

Please make next vedio on books recommended for Differential and Intregal Calculus books.

@nirrvana3504 Жыл бұрын

I’m grateful that I’ve found your channel

@bigstroker1300 Жыл бұрын

When will you do a video about Category theory?

@hatemalkd1633 Жыл бұрын

In your opinion what's the best logic book you have read it and after read it you said oh It's a new areas I wasn't recognize it.

@ayush2966 Жыл бұрын

Can u make a video on CSIR NET MATHEMATICS

@stevenwongso66 Жыл бұрын

This is very cool, Math Sorcerer Sir. You are now delving into Philosophy of Mathematics itself, your introduction here is very nice.

@TheMathSorcerer Жыл бұрын

thank you!

@Jacksparrow13963 Жыл бұрын

You are great teacher ☺️☺️☺️

@valor36az Жыл бұрын

Great video, beautiful thumbnail artwork

@surrealistidealist Жыл бұрын

2:37 If that's the way you could explain everything in Set Theory, then I'd LOVE to sign up for your course! For some reason, this is the only topic in math that still feels a bit too dry for me when learning from books alone.

@elizabethharper9081 Жыл бұрын

try Kunen

@surrealistidealist Жыл бұрын

@@elizabethharper9081 Thank you!!

@elizabethharper9081 Жыл бұрын

@@surrealistidealist but i warn You, it is graduate level book. You should know the basics of writing proofs and mathematical language for it.

@surrealistidealist Жыл бұрын

@@elizabethharper9081 Seems like a worthwhile challenge! Thanks again!!!

@lukehowley5138 Жыл бұрын

Math Sorcerer, I am going to college soon and would like to review Precalculus and Algebra 2 before studying engineering to strengthen my background. Do you know any good resources to do so?

@TheMathSorcerer Жыл бұрын

Yeah just get one of the big pre-calculus books. The one by Stewart is a solid choice, or Sullivan. There are others that are good but those two are very good. Try to find used copies:)

@lukehowley5138 Жыл бұрын

@@TheMathSorcerer thank you!

@cunningham.s_law Жыл бұрын

we move on after learning it what are the consequences if there are errors in the foundations? is it kay to build math on shaky grounds?

@declanfarber Жыл бұрын

Find them, it could make you famous. That’s one way how progress is made.

@keikiicake6370 Жыл бұрын

...Where do you find bookcases that don't bow out incredibly under the weight of that many textbooks?

@danielmrosser Жыл бұрын

Number by Tobias Dantzig is a great place to start

@tethyn Жыл бұрын

I have sympathies with the question that was asked because understanding the foundations of mathematics allows a person to expand the breadth of the discipline of mathematics. Think about the proof of Euclid 5th postulate as an example of the development of non Euclidean geometries or understanding the extent in which mathematics can find all truths such as Gödel second incompleteness theorem. Surely it is not for everyone and many are not Interested. As an illustration let’s talk about the materials and the form of the house. Some are interested in what ways you can build a house given the materials while others are more interested in what materials you can build with given that a house is being built. Good luck to either program and there is quite of bit of interesting stuff to learn and each will help the other to improve their craft.

@martinepstein9826 Жыл бұрын

"Think about the proof of Euclid 5th postulate" What proof? The existence of non-Euclidean geometry shows that you *can't* prove (or disprove) Euclid's 5th postulate from the other 4.

@tethyn Жыл бұрын

@@martinepstein9826 that was the point. In the attempt to prove Euclid postulate a new area of mathematics was developed. Once a foundation was questioned new areas of knowledge opened up. Take care.

@jizert Жыл бұрын

hey math sorcerer! hope ur doing good. i'm micah, a big fan of your math vids. im 15 now, i wrote an email a couple months ago but it didnt go thru, anyways. im freshman in high school, and honestly im really obsessed with higher math. so far, i've read "book of proof," "linear algebra done right," "set theory" by pinter, "calculus" by james stewart, a college algebra book, a game theory book, and i'm currently reading "baby rudin" now. i've nailed down the basics of proofs and set theory. anyways, my issue is that i never learned how to study well. school was boring and easy so i just never did any hw through elementary or middle school. i still ace tests but struggle with homework. my geometry class is super understimulating and i just have no motivation. next year im going into algebra 2 research honors (its a program my school does which is a bit harder than honors.) besides that, i want to be able to balance higher math with the high school curriculum and studying which is another issue🤷‍♂️ do you have any advice for me because i just have no idea what to do. anywaysc have a good one, and thank you! ✌🏼

@TheMathSorcerer Жыл бұрын

WOW it sounds like you are doing AMAZING!!! The fact you are 15 and have accomplished so much w.r.t. mathematics is amazing. I say keep doing what you are doing, few can do what you did. Good work my friend!

@jizert Жыл бұрын

@@TheMathSorcerer woah the legend himself replied 😆! thank you! i guess my issue is like, i find it near impossible to do things that are easy (or like busywork specifically), and i find the high school math curriculum to be busy work. i want to be able to have that sort of motivation, how do i build it? (again thanks so much! do you have like a discord server or something like that btw)

@jizert Жыл бұрын

@@TheMathSorcerer do u have any advice for that

@toyshopenterprises Жыл бұрын

I thought this was a maths channel. Now it has become a channel for motivation, book reviews, life advice 😞

@toyshopenterprises Жыл бұрын

Please make some valuable content so we can remember you after you leave.

@Jacksparrow13963 Жыл бұрын

Sir please make more video about Physics 🙂🙂🙂

@Victor-Soria Жыл бұрын

In my off topic opinion, the ai thumbnails are off-putting for a few reasons; i much prefer the real, simple photographs. Great content as always.

@whitb6111 Жыл бұрын

I find it odd that people typically think the subjects of philosophy and mathematics are polar opposites. There is huge overlap between the two from rigorous reasoning to thinking abstractly. Both rely on and integrate logic heavily. Philosophy, especially formal and symbolic logic really helped me understand mathematics better while math helped me sharpen my philosophical reasoning. It’s not a coincidence that so many philosophers over the centuries were also brilliant mathematicians. The line between pure math and metaphysics is a lot thinner than most think.

@schrodingcheshirecat Жыл бұрын

Any math you're learning is foundational. New mathematical discoveries are also foundational. Always under construction. That zone where the safety cones are numbers, truth, and existence. Watch out for unexpected detours.

@seraph127 Жыл бұрын

_Foundations and Fundamental Concepts of Mathematics_ is a Dover book that could be a god starting place. To get into a little more technically you could look at Robert Stoll’s _Set Theory and Logic_. Stephen Kleene’s _Mathematical Logic_ is advanced but well-written. Charles Pinter, of _A Book of Abstract Algebra_ fame has also written _A Book of Set Theory. Of I can think of others, I’ll post them later.

@bitesizedmath9686 Жыл бұрын

I am concerned, as I think it's clear you did not answer this person's question. This is someone interested in set theory as a modern subject. Things like Archimedes' property are not relevant to this. Foundations are not what you study when you flip open to the first page of a math textbook. Things related to set theory as a first-order theory, and those results of philosophical interest (e.g. Skolem's paradox arising from the Löwenheim-Skolem theorem(s), V=L, independence results, and results about ZF without choice) are probably what this person who emailed you is looking for. Telling someone interested in learning more about these topics to "read a proofs book" instead of pointing them in the direction of more genuine resources seems counterproductive, and I think you should be more cautious regarding the questions you answer from your email list. When you tell someone with an interest in a particular subject to just "move on" from that interest because you think they are interested in just studying the preliminary content of math texts over and over, I think you are sending a message that (I hope) you don't intend to put out. For that student who sent this email: if you are interested in any of the things I listed above, there are a few good books and other math KZfaq channels which will have the kind of content you're looking for. UCLA's Artem Chernikov has a playlist from a class 220A, called mathematical logic, which goes over these things at an introductory (albeit graduate) level, which would be perfect for someone trying to get into the more interesting details fairly quickly. I have linked it below. kzfaq.info/sun/PL54Pt_mZzBqibWHgesgEICeQHnwHom8xz

@faraminesmailiseraji7777 Жыл бұрын

I was a geometry student and all foundational questions was "irrelevant" and i had to "move on" and study elliptic curves instead :)) Tnx for sharing that link; These days I'm trying to study more foundational stuff, currently Category theory using "Topoi - Robert Goldblatt" book. but i picked up that book randomly :)) I'm looking for some justifications for ZFC despite Godel's incompleteness theorems. i have ordinary undergrad level set theory and logic knowledge. Do you recommend any specific book? any suggestion appreciated~ Ty

@shawnruby7011 Жыл бұрын

Idk proof theory (and math proofs) really dovetails with epistemology so for a philosophy student it's probably the best start in fact until I did math proofs nothing was intuitive.

@bitesizedmath9686 Жыл бұрын

@@faraminesmailiseraji7777 I really like Goldblatt's book! It's a comfy introduction. However, it doesn't cover the more interesting results IMO. If you want more category theory, Emily Riehl's "Category Theory in Context" is fundamental and contains content which is applicable to everything in category theory. If you want to continue in topos theory, a natural extension of Goldblatt's book is Mac Lane and Moerdijk's "Sheaves in Geometry and Logic" but this book is slightly more advanced. If you want to dive into set theory, I am told by set theorist friends that Thomas Jech's book is very good, at least for the first 10 or so chapters. I learned that logic content from Peter Hinman's "Fundamentals of Mathematical Logic" but apparently that book is much harder to read.

@faraminesmailiseraji7777 Жыл бұрын

@@bitesizedmath9686 Now it seems like i have a proper study plan! thank you sir.

@gimmerain4days 7 ай бұрын

had to scroll way too far to see this. My thoughts exactly, love the guy but he completely missed the mark here.

@FlaminTubbyToast Жыл бұрын

It saddens me that most people’s conception of math is computation heavy instead of the creative humanity it is.

@richardgray8593 Жыл бұрын

Naught plus naught is naught. Naught plus one is one. Naught plus two is two ... -- Jethro Bodine

@ghenno74 29 күн бұрын

Your hair style reminds me that of Sir Izaac Newton

@acamarocutcher8845 Ай бұрын

I apologize for posting a reference to math foundations. I believe some of the contents could be wrong, I apologize for the misleading knowledge that could be there. For future inquires about its contents and what is right and wrong or what is acceptable you can talk with a minister in the Truth of God under Pastor Gino Jennings. Thank you.

@glormorbylationglormorbylation Жыл бұрын

Grad student in set theory here 👋 You have no idea what he's asking. He wants to know about Foundations as a proper field of study. None of the things you mention are relevant. He wants to learn things about first order logic, formal deductions, completeness, proper set theory, forcing, and philosophy adjacent to these things. Don't open up an abstract algebra book, find section 1.1, and call it Foundations.

@mingmiao364 Жыл бұрын

Well, in that case (meaning understanding the word “foundations” as axiomatic theory, mathematical logic), it’s a known fact that 95% of professional mathematicians don’t care about foundations.

@martinepstein9826 Жыл бұрын

@@mingmiao364 Just because they don't use it in their day-to-day work doesn't mean they don't care. Joel David Hamkins got to be the most highly rated user on mathoverflow mainly by writing about foundations.

@whitb6111 Жыл бұрын

Great comment. Such an interesting field you get to study!

@declanfarber Жыл бұрын

Starting to understand the foundations of math from Bertrand Russell… hoo boy. I’d suggest you start from a more “originalist” place, I.e. understanding where the major ideas came from. Please not Euclid, that’s so constraining and parochial. Try to find a book called “The Rainbow of Mathematics” by Ivor Grattan-Guinness, which is a marvelously accessible (to the intelligent layman) discussion of what led UP to some of the stuff that Russell was thinking about. There are other much more elaborate treatises on the history of the ideas of mathematics, but they also tend to be more dry.

@shawnruby7011 Жыл бұрын

That's a cool book recommendation but foundations of math isn't a historical process. It's about identifying the underlying form that derives the disciplines you're interested in. More Descartes finding a way to mix algebra and geometry and less cavemen having a "base" 2 or 3 system.

@declanfarber Жыл бұрын

@@shawnruby7011 The foundations of math IS a historical process, how math has been understood has evolved over time. IMO understanding how math got to be the way it is, is more important than trying to make it fit the latest philosophical straitjacket. Corollary: You can put a sweater on your cat, but the cat won’t like it.

@shawnruby7011 Жыл бұрын

@@declanfarber I mean you're using a computer while you say that. Logic was developed after analysis to be the foundation of math and we've gotten a lot out of it because of that. Foundations of math is generally an analytic subject and analytic philosophy is generally ahistorical. Math is by definition ahistorical, formalist or not.

@martinepstein9826 Жыл бұрын

It's the other way around. If you want to understand modern type-theoretic foundations then reading Russell and Whitehead is *too* originalist. The theory has been greatly streamlined and improved since then.

@akumar7366 Жыл бұрын

Surely the most important foundation of maths is the Indian discovery of Zero , which all mathematics is based upon , proof of which is in the British Museum London.

@bdinh3130 Жыл бұрын

The foundations of mathematics is something to build on. Sick insight. Kind of like how people lay concrete down for buildings. I have no idea what that would be called though.

@Alexj_movieguy Жыл бұрын

Book of proof by who?

@TheMathSorcerer Жыл бұрын

Hammack

@Alexj_movieguy Жыл бұрын

@@TheMathSorcerer thank you!

@alineharam Жыл бұрын

I have a degree in philosophy and I’m watching this channel. Plato?

@TheMathSorcerer Жыл бұрын

Awesome!

@cunningham.s_law Жыл бұрын

infinite sets are not intuitive

@isabellanievesthegaminggir6558 Жыл бұрын

How does this history works

@krunoslavregvar477 Жыл бұрын

Easy task. Especially if you are positive towards KZfaq videos, there are videos from professor Norman (B?) Wildberger (NSW University, Sydney), and those are very, very good!

@elizabethharper9081 Жыл бұрын

You are joking right? This guy is arguing axiom of infinity even though he doesn't know its formulation.

@krunoslavregvar477 Жыл бұрын

@@elizabethharper9081 Nope. Well, I wrote this despite the fact that N. Wildberger is not in the mainstream of mathematics. But he served as regular professor of mathematics at NSW University (Sidney), so I am pretty sure that he knows all important axioms, but because he is indeed some kind of intuitionistic mathematician, for him infinity does not exist. However, put it in the perspective, his KZfaq channel provides a lot. Of course, if you aren't fan of intuitionistic mathematics, then you must, besides him, look for some other resources, which aren't in line with his view of mathematics. I am in neither of those two raws, but slightly towards "mainstream" mathematics, as obviously you are. Good luck!

@nicolascalandruccio Жыл бұрын

I suppose the question in the email is a query to get from you details of what are the foundations of Mathematics from both set theory and categories. I'd like you explain more of it too and show maybe other new theories. Perhaps you can talk about philosophical ideas such as in set theory one creates numbers from nothing as well as in big bang theory things come from nothing.

@rusi6219 Жыл бұрын

Review Khwarazmi's Algebra it's translated to English and available online

@superegofrued Жыл бұрын

Foundation from Arya bhatta

@krwada Жыл бұрын

In my opinion, the fundamentals of mathematics comes from something as simple as counting. This is truly amazing to me. What is so amazing is something as simple as counting, and sequences is what started all of this. And ... to this day, there appears to be STILL, unexplored regions of mathematics that is founded on something as fundamental as counting and sequences. After this, I would say geometry is probably the way that we humans attempt to describe the world around us. And, it goes on!

@douglasstrother6584 Жыл бұрын

"Gödel, Escher, Bach" ~ Douglas Hofstadter

@randomcandy1000 Жыл бұрын

I think it's interesting that some people, even with a PhD in math, reject the consistency of real numbers and only believe in integers and discrete math.

@shawnirwin6633 Жыл бұрын

Playing the devil's advocate . . . . . . . if you have a number, you cannot always find a bigger number . . . . . if the number you have is so large that you die before you or your computer can read it, then you will never find a larger number. This may seem ridiculus, but despite that, it is still a valid assertation.

@declanfarber Жыл бұрын

That’s not the way it works.

@kokwahtan8577 Жыл бұрын

Maths is not hard or difficult , but its suffering to learn well. As such, Science, Finance, & Economics, Engineering are suffering too.

@tomkerruish2982 Жыл бұрын

IMO Russell was a royal prick for what he did to Frege, and Gödel was karmic payback. Separately from that, one's position on the Axiom of Choice is almost certainly informed by one's own philosophy.

@whitb6111 Жыл бұрын

Lol, what he did to Frege? He revealed a serious error in his work. Was he supposed to not saying anything? It’s called intellectual honesty and it should be the basis of all science and mathematics.

@souravnatta2751 Жыл бұрын

Hello ! Professor, did you know that mathematics formula , founded by ancient Indians. Ancient Arabs they learned mathematics, geometry , number systems , Physics, chemistry, etc.....they European peoples adopted their theories. When 16th century came , then European peoples to come to discover secrets of Physics & mathematics. European peoples think they discovered everything, actually it's not actual real history. Lots of history of India erased in the time of colonial period by the British attackers. It's my big slap to European & Arabian peoples who wanted discriminate our Indian ancient science history from history books. Sir , it's my own request for you , when you make videos on mathematics, please make videos on " how ancient Indian Mathematicians who discover mathematics formulas ". Then European peoples would be understand. Who we (Indians) are. 😂

@Number6_ Жыл бұрын

You don't have to move on! This is where many fall down, by not appreating the basics. Set theory has it flaws and it's use as a foundation of theoretical (abstract) math is also flawed with it. It only works with the exclusive or in logic. An application of an inclusive logical or and the whole of set theory in math falls apart.