Nah fields, rings, groups, etc are some of the most straight forward and easiest stuff I had in university. Then again we never had Lie Groups and algebras, or any others of the "really complicated shit" in algebra. Finite fields of non prime power seem a bit complicated too (at least compared to normal finite fields), but manageable.
@@glacide8883I have never heard of that notation before, but if you by Mn(R) mean the ring of square matrices M_(nxn) (R)? Sure! GLn(R) is a group and M is a ring, which means that we only have to take into account one of the operations (+,*) for the statement to be true. Let’s choose matrix multiplication as our operation (since that is the operation on GLn(R) and absolutely nothing else would make sense. The field we’re working over is the real numbers. Therefore the topology of the real numbers are inherited to GLn(R) through M_(nxn). Hence it is dense over the group action. Done. 😊
Calculus is not a math branch and Analysis is. And you didn't talk about Topology, Set Theory, Complex Analysis, Stochastic Processes, Logic, Category Theory, Optimization, PDE, Functional Analysis, Machine Learning (an applied topic but huge)
@@100iqgaming Well, I'm not sure I'm better at math. I really hope that my comment was not understood that way, in turn I hope that my comment can be read by someone who doesn't know much about mathematics but is interested in learning.
@El0melette nah i didnt mean that, my point is that i study further maths A-Level, if asked to split maths into groups id probably give his categories + linear algebra, idrk abt the stuff you said, the person making this video is prob about my level, i didnt mean it like you where being a dick
Set Theory, Complex Analysis, Functional Analysis (including Real Analysis) comes under Mathematical Analysis. PDE, Opt and Processes aren't wide enough to be considered and Category Theory and Logic comes under Mathematical Logic.
@@youssefchihab1613nope, not even if it was a 2d square theres 6 colours and 9 available spaces on a 3 by 3 grid each square space has 6 possible colours right? and theres 9 spaces so if im right here, on a 3 by 3 2 dimensional square with 6 possible colour options for each section, there is 6^9 possible combinations of colours.
@@greengreen110 thats dedication for doing those calculations but you actually need to divide the final number by 12. Why? Because having a fully solved cube but 1 corner twisted is’n’tpossible, or 2 corners twisted is also not possible. And these twisted corners dont only apply to the solved state, but all scrambled states of the cube. So you need to divide by 3. Then, having 1 edge flipped is also not possible so divide by 6 now. But, having 2 edges/corners swapped (it does the same thing) is also not possible so 6x2 is 12 and you need to divide by 12 as 11/12 of the states you computed could never happen without taking out pieces and the fact that you got so close though is amazing. The final answer is around 43 quintillion.
Id like to give a small note to the point of naming stuff after people. First, lets start with a quote by Hilbert: “One must be able to say at all times-instead of points, straight lines, and planes-tables, chairs, and beer mugs.” This was referring to Euclids Elements. The point (unintended pun) of this is, that if you use words that already have meaning outside of math, it is easy to make unstated assumptions. (For example that if two lines cross, they must have a point in common). In this sense i find it reasonable to name objects/theorems after the people that discovered them, since 1. it honors those who made contributions to the field and 2. it makes you restrict yourself to the definition/axiom actually given and not some non-rigorous intuition. Arguably, for Leibniz Notation this isn't a valid argument, but if you name almost anything after people, why stop there and make the naming pattern inconsistent again? After all, we humans are most often doing what we are in the habit of doing. After you then disconnected yourself from the non-rigorous definition you can then build up a new intuition based on the axioms and definitions given. For example, a mathematician would have an intuition of what it means for a (topological) space to be hausdorff or a sequence to be cauchy.
Oh hey, just as I’m studying cauchy sequences. Also i agree yes, it feels as if most things in math are too abstract for any name that references another concept to do them justice
yeah but then when you have to teach it to others you add the extra difficulty of memorizing the terms purely based on the history of mathematics. If I have to memorize Lagrange's theorem I have to memorize the theorem itself and then link the theorem with the guy's name, if I instead name Lagranges theorem as the mean value theorem I actually have a clue on what the theorem should do regardless of how well I have learned it. Also, as you can see with Lagrange's case, If one mathematician creates multiple theorems and all of the theorems he discovered are then linked to his name you even add in an element of confusion, it's typically not too big since it's not too hard to identify which theorem is being mentioned, but still an element of confusion you could easly avoid. It's just the same argument for using the traditional nomenclature rather than IUPAC's in chemistry.
@@franconullo4228 yeah my paradise somewhat crumbles in this respect. There are also certainly opportunities to combine the two, for example when you look at the list of things named after euler, many theorems will either have a second name or a description of what it is about. In this sense, you could also call MVT the Lagrange mean value theorem or something like that. (Since there are also different versions of the mean value theorem). Memorization is a topic i avoided, but it should be adressed nontheless. To me, mathematics is about understanding and not about memorization. But of course memorization of certain aspects is a necessary requirement of understanding. This could be tried to minimize by making tests open book, but of course that also has some drawbacks.
@@julianbruns7459 the best solution I can think of is trying to combine the 2 methods and get something like "Lagrange's mean value theorem" and similar for other theorems, although this naming process could make things a bit too long to name
All of it is S tier, because I haven't burrowed into everyone of them, but how much I've played around with algebra, geometry, an ency weency bit of calculus and now starting to figure out combinatorics on my own, I can confidently say that they're all S tier for me. All of them you can play and mess around with and get beautiful results.
Combinatorics (specially combinatory analisys the field mentioned in the vídeo) is famous for being extremely restrict to a formulaic aproach and overall a very confusing field, where you could find that the best 3 combinatorics professors would gave 3 different answer to the same problem, every single one of them would fight for their absoluty correct resolution and the 3 would still get the problem wrong, that is my problem with combinatorics, and that is the point that I'm making.
You didnt mention linear algebra nor group theory in the algebra category, despite it being very interesting. Also where do you put set theory and topology in your categories ?
I was waiting for topology… And why is there no probability? Statistics is based on likelihood where probability is based on.. probability. These are totally different fields. Putting linear algebra and group theory in algrebra can make sense but probability is not a sub-subject of stats at all.
Love geometry!-spoken from someone who's in highschool and doesn't understand even a fraction of geometry(aside from the Pythagorean theorem of course).
I think generally, there are 6 branches of pure mathematics. -Mathematical Logic (Set Theory, Category Theory, Type Theory) -Algebra (Abstract, Linear) -Analysis (Calculus, Numerical Analysis, Real Analysis) -Number Theory -Geometry and Topology -Discrete Mathematics (Vague term but includes Graph Theory and Combinatorics) I think stuff like Statistics and Computer Science are more applied mathematics.
Isn't combinatorics also colouring the chessboard (or sth similar to chessboard), invariants, graphs theorem and many, many more? You may don't like it, but don't say it's so boring
Gonna join the "didn't mention {whatever}" crowd to suggest that foundations (eg. set/model theory and reverse mathematics), order theory, and game theory don't fit nicely into any of those 6 boxes. Also, if you go by the Mathematics Subject Classification (which amusingly lists 97 different fields which is almost definitely too many for this kind of video lol) you'll see that those guys list computer science as a subfield of mathematics. If we can grab statistics then I say we get our hands on computation too. Also also combinatorics is about counting more different kinds of stuff than just combinations and permutations. It also encompasses some less-counting-ish stuff like the pigeonhole principle, Ramsey's theorem, and (arguably) the entirety of graph theory.
But I agrre that the combinatorics showed (combinatory analisys) is definity the most sufferable part of math, and no one deserves to suffer that punish (specially because it is very interpretative, and formulaic most of the time you elaborarem a question that was "way harder" that you though and you spend most of your time calculating that just to get the wrong answer)
Bro I had a "short" 3 months long combinatorics course and I feel like I haven't scratched the surface, could easily take a full year to learn just all the basic things(college level) in combinatorics
To me, the rating you gave to statistics is unfair and really indicative to me that you did not spend anytime in statistics A lot of variety of topics, and actually a lot of **mathematical** difficulties arise from the real world data Defining estimators and deriving the rate of convergence of those is tricky, and there are also a lot of beautiful objects that are relevant to statistics you would not even know : - one for instance is functional data analysis, where your random variables are function-valued. Which of course causes a lot of head-aches because those are infinite dimensional objects, and real world data is finite dimensional. To define a "mean-function" one has to build an integral on infinite dimensional spaces, (which is analysis but here is very well motivated by statistics), also in functional data, we both observe IL^2 random elements but one could also consider to introduce them as continuous random processes (like brownian motion) - one notion is global, the otherone is pointwise in the introduction of randomness, question : when are those equivalent ? In the functional data analysis setting it makes sense for us to be both at the same time. How do we approximate such data for our finite dimensional computational world without losing information about the data ? - One could also talk about inference, with parametrical inference, likelihood, ... - There is also information theory really closely related to statistics and therefore I will include with it because frequently one shed some light to the other. Modelling what information is ( for instance Martingale theory is good to discover how we model growing knowledge in our system ) - how do we make the least number of assumption on our population, because those assumptions can be really far from the truth ( non parametric statistics ) ? When does it make sense to approximate with parametrical distributions ? Statistics is so rich it really hurts to hear the argument you made, it's like putting algebra in F tier because "yeah sure you got addition and subtraction, but in the end not that interesting though." PS : the video was good nonetheless, I just want to get this out of my chest
Calculus in A tier is crazy. 99% of algebra and geometry is derived using differential or integral operations and calculus gives us the ability to perform operations beyond elementary mathematics. Calculus deserves an S tier. Our world does not exist without calculus
I can already see how this video will get trash talked on r/mathmemes from people with different opinions. On an unrelated note, here is my favorite (combinatorics?) problem: Given an equilateral triangle, show that if 5 points are choosen randomly inside the triangle, there exists a pair of points such that their distance is less than the height of the triangle :D Hint: | | \ / Tri-force
I might be wrong, but doesn’t this problem work with only 4 points? Or a better question: does there exist a configuration of 4 points, chosen randomly, where the distance between all pairs of points is less than the height of the triangle?
@@gruk3683 Well just choose 4 points really close to each other. I think you meant to ask if there is a configuration of 4 points such that all distances between pairs is GREATER that the height of the triangle. And this can't happen since with 3 points yes but with 4 no. The only possible configuration with 3 points where this works is if the 3 points are close to the corners and adding a fourth one would just decrease some distance between a pair of points. This would have bern possible if we were working in an equilateral tetrahedron and placing the 4 points close to the 4 corbers( I think). If you want another challenge: Prove the same problem again but instead of height of triangle its about half the side of the triangle! ( i mean to say: Given 5 random points inside an equallat. triangle, you can always find a pair of points whose distance is smaller than half the side lenght of the triangle) :D
It definitly doesn't seem to fit in the combinatorics definition presented (combinatory analisys) and the problem you presented fit more in the geometry category (specificaly finding a "geometric place" of "all possible solutions") but as well the categories presesnted are very broad and can fit everything in (I could say it is discrete analisys and fit the problem with the algebra category or something of this sort)
Statistics has a lot of philosophical problems. How can you say that the probabilities you calculated are objective or subjective? How can we be sure in the truthfulness of our models and our results? A lot of the replication crises in science is a result of, in large part, the overeliance on p-values for determining significance, which is a frequenting concept (the idea that parameters are fixed and data is random), but this might not be how we ought to understand probabilities for this problem. It's very hard to trust statistical inference with the same kind of confidence as other mathematical derivations. Other parts of math typically don't have glaring foundational problems in their interpretations.
1) Calculus Foundations Contradictory: Newtonian Fluxional Calculus dx/dt = lim(Δx/Δt) as Δt->0 This expresses the derivative using the limiting ratio of finite differences Δx/Δt as Δt shrinks towards 0. However, the limit concept contains logical contradictions when extended to the infinitesimal scale. Non-Contradictory: Leibnizian Infinitesimal Calculus dx = ɛ, where ɛ is an infinitesimal dx/dt = ɛ/dt Leibniz treated the differentials dx, dt as infinite "inassignable" infinitesimal increments ɛ, rather than limits of finite ratios - thus avoiding the paradoxes of vanishing quantities. 2) Foundations of Mathematics Contradictory Paradoxes: - Russell's Paradox, Burali-Forti Paradox - Banach-Tarski "Pea Paradox" - Other Set-Theoretic Pathologies Non-Contradictory Possibilities: Algebraic Homotopy ∞-Toposes a ≃ b ⇐⇒ ∃n, Path[a,b] in ∞Grpd(n) U: ∞Töpoi → ∞Grpds (univalent universes) Reconceiving mathematical foundations as homotopy toposes structured by identifications in ∞-groupoids could resolve contradictions in an intrinsically coherent theory of "motive-like" objects/relations. 3) Foundational Paradoxes in Arithmetic Contradictory: - Russell's Paradox about sets/classes - Berry's Paradox about definability - Other set-theoretic pathologies These paradoxes revealed fundamental inconsistencies in early naive attempts to formalize arithmetic foundations. Non-Contradictory Possibility: Homotopy Type Theory / Univalent Foundations a ≃ b ⇐⇒ α : a =A b (Equivalence as paths in ∞-groupoids) Arithmetic ≃ ∞-Topos(A) (Numbers as objects in higher toposes) Representing arithmetic objects categorically as identifications in higher homotopy types and toposes avoids the self-referential paradoxes. 4) The Foundations of Arithmetic Contradictory: Peano's Axioms contain implicit circularity, while naive set theory axiomatizations lead to paradoxes like Russell's Paradox about the set of all sets that don't contain themselves. Non-Contradictory Possibility: Homotopy Type Theory / Univalent Foundations N ≃ W∞-Grpd (Natural numbers as objects in ∞-groupoids) S(n) ≃ n = n+1 (Successor is path identification) Let Z ≃ Grpd[N, Π1(S1)] (Integers from N and winding paths) Defining arithmetic objects categorically using homotopy theory and mapping into higher toposes avoids the self-referential paradoxes.
include all the topics of math upto grad school those are all the math most of us will ever need to worry about or if you're an ai u could try to rank all 500ish math fields
Alright, I should clarify a couple things. When I said that "combinatorics can be learnt in 2 weeks" and "it only had 2 operations", I meant the basics. Obviously you can't learn the whole field inside out in 2 weeks and I understand there exists other aspects of the field I didn't mention. Sorry if its place in F tier offended any of you, one of the fields had to be there.
Calculus easy S tier Geometry A tier They are fun and tricky Algebra not so fun anymore, pretty mashine-like calculations C tier Number theory, sometimes boring, but it can be fun B tier Combinatorics are also mashine-like so C tier Statistics, although very usefull, it is shit to calculate. 0% fun (you get it?) It is trash F tier or worse if I can
combinatorics being mechanical is a crazy take. there are so many beautiful and creative proofs out there of combinatorial identities and connections to other fields like number theory. math is not about calculations and formulas, and this video does an awful job at communicating that
@@kevinstreeter6943 If we consider that we know basics in every topic here presented, I rated those in case of getting deeper into them, not in case, that you cannot run further without knowing basics. Going with your thinking path, we have to give GEOMETRY an S tier, because that is how all of the math was created. Ancient philosophers had known only geometry. But we don't want to get so deep, so I'm rating every topic separatelly. (You ain't reading that, right?)
Not really. Numerical Analysis is a little bit of everything because the theory is basically just “how do we solve things numerically and how do we approximate solutions?” That has applications in Calculus, Linear Algebra, Number Theory, etc.
Skips to the recap part to know if his opinion is worthy enough to be heard. Calculus not in S tier and Combinatorics in F tier. WTF!!! Invalid Opinion
You group together calculus and numerical analysis? WOW! That is a brave move. You consider combinatorics to be a special field of mathematics? Alright….. CRIKEY! Anyway, your motivations for which mathematical fields that are important or not are extremely vague throughout the video. It’s an absolute shame to say that calculus (or any of the other fields) doesn’t play a role as big as algebra. If you would have put number theory on S and all of the other fields that you bring up on A, you might have had my sympathies. But as you put it now, it’s nothing but very awkward and extremely weird. With best regards // an algebraist
Hello, here is a fun exercice for an algebraist like you. Lets consider G a finished, abelian group. When can we calculate the product of every element of G? Calculate it
@@sydneythesurfboards5903 Sorry, Im not English so maybe the names of objects are different. By a "finished group", I mean a group with a certain number of elements. Not infinite.
@@glacide8883 I don’t understand the problem setting actually… What exactly do you want me to calculate? To calculate the sum or product of all elements of a group doesn’t really make sense I think… Are the elements of the group known? Do you have a specific type of group in mind? What do you mean by product? What kind of operation is defined on the group?
your take on algebra is too simplistic, to the point of being wrong Algebra as something toddlers learn and algebra as a field in university are pretty different, because as you already said, every single field in math is just letters for numbers, that's a stupid definition. Also you totally threw away my field 😭😭 which is logic/foundations, but i guess if you really want to you could put it in algebra.
"algebra makes sense"
group theory and abstract algebra: allow me to introduce myself
Well they still makes sense
@@thundercraft0496 well, it depends what @nerd5865 means by "abstract algebra".
How does that not make sense?
Its beautiful too
Relatively speaking it’s just as difficult if not more so than a lot of the other more complicated fields he referred to.
Nah fields, rings, groups, etc are some of the most straight forward and easiest stuff I had in university. Then again we never had Lie Groups and algebras, or any others of the "really complicated shit" in algebra. Finite fields of non prime power seem a bit complicated too (at least compared to normal finite fields), but manageable.
I'd call this "highschool math tierlist"
Annoyed combinatorics fan here! You seem to have no idea what combinatorics actually is. Cheers.
To be fair the creator doesn't seem to know anything about these topics besides surface level things you learn in highschool
@@jadonjones4590 When he started saying that statistical stuff came from outside math... He's very clearly not aware of mathematical statistics.
I hate combinatorics, too much logic and i hate it
@@jadonjones4590 Can you prove that GLn(R) is dense in Mn(R)?
@@glacide8883I have never heard of that notation before, but if you by Mn(R) mean the ring of square matrices M_(nxn) (R)? Sure! GLn(R) is a group and M is a ring, which means that we only have to take into account one of the operations (+,*) for the statement to be true. Let’s choose matrix multiplication as our operation (since that is the operation on GLn(R) and absolutely nothing else would make sense. The field we’re working over is the real numbers. Therefore the topology of the real numbers are inherited to GLn(R) through M_(nxn). Hence it is dense over the group action. Done. 😊
The fact that linear Algebra isn’t part of this tier list is outrageous 🗿
guy probably just started college
@weirdo911aw that other guy probably started highschool💩
Subfield of algebra
@@Qq-lp5xg"Subfield" is strictly a field-theoretic term in abstract algebra.
@@Gordy-io8sblittle man, he is clearly not referring to a field definition in algebra.
Calculus not in S Tier, opinion invalid.
You mean geometry.
It's all nonsense across the board number theory in B is just insane
It is though
Calculus is the worst thing, no proof full calcul
Calculus that you learn in college are not real math course.
Calculus is not a math branch and Analysis is. And you didn't talk about Topology, Set Theory, Complex Analysis, Stochastic Processes, Logic, Category Theory, Optimization, PDE, Functional Analysis, Machine Learning (an applied topic but huge)
congratulations you are better at maths than the person who made this video, the calculus section alone should make it clear that bossman is like 18
@@100iqgaming Well, I'm not sure I'm better at math. I really hope that my comment was not understood that way, in turn I hope that my comment can be read by someone who doesn't know much about mathematics but is interested in learning.
@El0melette nah i didnt mean that, my point is that i study further maths A-Level, if asked to split maths into groups id probably give his categories + linear algebra, idrk abt the stuff you said, the person making this video is prob about my level, i didnt mean it like you where being a dick
Set Theory, Complex Analysis, Functional Analysis (including Real Analysis) comes under Mathematical Analysis. PDE, Opt and Processes aren't wide enough to be considered and Category Theory and Logic comes under Mathematical Logic.
How many permutation does a rubiks cube have?
Well I guess I’ll never know because combinatorics is in F tier.
isn't it just 81! ?
@@youssefchihab1613 How is it possibly 81! Thats MANY magnitudes larger than the actual amount.
@@youssefchihab1613nope, not even if it was a 2d square
theres 6 colours and 9 available spaces on a 3 by 3 grid
each square space has 6 possible colours right? and theres 9 spaces
so if im right here, on a 3 by 3 2 dimensional square with 6 possible colour options for each section, there is 6^9 possible combinations of colours.
@@crunchysnails well that's for one side
you didn't consider that for each permutation of colors the other side of the cube can be rotated
@@greengreen110 thats dedication for doing those calculations but you actually need to divide the final number by 12. Why? Because having a fully solved cube but 1 corner twisted is’n’tpossible, or 2 corners twisted is also not possible. And these twisted corners dont only apply to the solved state, but all scrambled states of the cube. So you need to divide by 3. Then, having 1 edge flipped is also not possible so divide by 6 now. But, having 2 edges/corners swapped (it does the same thing) is also not possible so 6x2 is 12 and you need to divide by 12 as 11/12 of the states you computed could never happen without taking out pieces and the fact that you got so close though is amazing.
The final answer is around 43 quintillion.
Id like to give a small note to the point of naming stuff after people.
First, lets start with a quote by Hilbert: “One must be able to say at all times-instead of points, straight lines, and planes-tables, chairs, and beer mugs.”
This was referring to Euclids Elements. The point (unintended pun) of this is, that if you use words that already have meaning outside of math, it is easy to make unstated assumptions. (For example that if two lines cross, they must have a point in common). In this sense i find it reasonable to name objects/theorems after the people that discovered them, since 1. it honors those who made contributions to the field and 2. it makes you restrict yourself to the definition/axiom actually given and not some non-rigorous intuition. Arguably, for Leibniz Notation this isn't a valid argument, but if you name almost anything after people, why stop there and make the naming pattern inconsistent again? After all, we humans are most often doing what we are in the habit of doing.
After you then disconnected yourself from the non-rigorous definition you can then build up a new intuition based on the axioms and definitions given.
For example, a mathematician would have an intuition of what it means for a (topological) space to be hausdorff or a sequence to be cauchy.
Oh hey, just as I’m studying cauchy sequences. Also i agree yes, it feels as if most things in math are too abstract for any name that references another concept to do them justice
yeah but then when you have to teach it to others you add the extra difficulty of memorizing the terms purely based on the history of mathematics. If I have to memorize Lagrange's theorem I have to memorize the theorem itself and then link the theorem with the guy's name, if I instead name Lagranges theorem as the mean value theorem I actually have a clue on what the theorem should do regardless of how well I have learned it. Also, as you can see with Lagrange's case, If one mathematician creates multiple theorems and all of the theorems he discovered are then linked to his name you even add in an element of confusion, it's typically not too big since it's not too hard to identify which theorem is being mentioned, but still an element of confusion you could easly avoid. It's just the same argument for using the traditional nomenclature rather than IUPAC's in chemistry.
@@franconullo4228 yeah my paradise somewhat crumbles in this respect. There are also certainly opportunities to combine the two, for example when you look at the list of things named after euler, many theorems will either have a second name or a description of what it is about.
In this sense, you could also call MVT the Lagrange mean value theorem or something like that. (Since there are also different versions of the mean value theorem).
Memorization is a topic i avoided, but it should be adressed nontheless. To me, mathematics is about understanding and not about memorization. But of course memorization of certain aspects is a necessary requirement of understanding.
This could be tried to minimize by making tests open book, but of course that also has some drawbacks.
@@julianbruns7459 the best solution I can think of is trying to combine the 2 methods and get something like "Lagrange's mean value theorem" and similar for other theorems, although this naming process could make things a bit too long to name
All of it is S tier, because I haven't burrowed into everyone of them, but how much I've played around with algebra, geometry, an ency weency bit of calculus and now starting to figure out combinatorics on my own, I can confidently say that they're all S tier for me. All of them you can play and mess around with and get beautiful results.
Exactly, math is S tier
Best comment ever
@@massipiero2974 ❤️
limit as x approaches fr = infinity @@massipiero2974
I agree but statistics is really ass bro 😭
Saying combinatorics can be learned in two weeks really shows you don't really know much about it
Never let this guy rate again 🗣🗣🗣🔥🔥🔥🔥
Never let This guy comment again 🗣🗣🗣🔥🔥🔥🔥
@@PareidolicPineappl Never let this guy reply again 🗣🗣🗣🔥🔥🔥🔥
Bro what the fucking Combinatorics should atleast be in A. I'd give it an S but atleast A bro give it the goddamn respect man
Combinatorics (specially combinatory analisys the field mentioned in the vídeo) is famous for being extremely restrict to a formulaic aproach and overall a very confusing field, where you could find that the best 3 combinatorics professors would gave 3 different answer to the same problem, every single one of them would fight for their absoluty correct resolution and the 3 would still get the problem wrong, that is my problem with combinatorics, and that is the point that I'm making.
You didnt mention linear algebra nor group theory in the algebra category, despite it being very interesting.
Also where do you put set theory and topology in your categories ?
bro the person making this video probably hasnt done that yet, judging by the way they are talking they probably arent at uni yet
I was waiting for topology…
And why is there no probability? Statistics is based on likelihood where probability is based on.. probability. These are totally different fields. Putting linear algebra and group theory in algrebra can make sense but probability is not a sub-subject of stats at all.
@@wallahitsnotmine4255 fuck stats, it would be in F----- tier
Topology and Logic go fully unnoticed.
This guy has no idea what hes saying
"how many fields of mathematics are there"
me: "a field is a triple (F,+,*) such that..."
combinatorics not in A is crazy
Stats and probability require all those tiers and require a fairly deep intuition of all of them as well
And the most useful of all of them
Love geometry!-spoken from someone who's in highschool and doesn't understand even a fraction of geometry(aside from the Pythagorean theorem of course).
blud has got no clue of what he's talking about 😭
I think generally, there are 6 branches of pure mathematics.
-Mathematical Logic (Set Theory, Category Theory, Type Theory)
-Algebra (Abstract, Linear)
-Analysis (Calculus, Numerical Analysis, Real Analysis)
-Number Theory
-Geometry and Topology
-Discrete Mathematics (Vague term but includes Graph Theory and Combinatorics)
I think stuff like Statistics and Computer Science are more applied mathematics.
Topology is definitely not the same category as geometry
@Archway-9 I think topology is just geometry without measure but yeah
This man needs 3Blue1Brown in his system
also he has an immediate anchor bias with algebra 😢
Nah 3blue1brown is a dog just like these other math TH-camrs. They would all get gapped by actual mathematicians.
Isn't combinatorics also colouring the chessboard (or sth similar to chessboard), invariants, graphs theorem and many, many more? You may don't like it, but don't say it's so boring
when you're in middle school and wanna make a math video:
Just for future reference it's pronounced com-BIN-atorics. Not Com-BINE-atorics.
NAHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH STATS IN D TIER IN INSANITY
Gonna join the "didn't mention {whatever}" crowd to suggest that foundations (eg. set/model theory and reverse mathematics), order theory, and game theory don't fit nicely into any of those 6 boxes.
Also, if you go by the Mathematics Subject Classification (which amusingly lists 97 different fields which is almost definitely too many for this kind of video lol) you'll see that those guys list computer science as a subfield of mathematics. If we can grab statistics then I say we get our hands on computation too.
Also also combinatorics is about counting more different kinds of stuff than just combinations and permutations. It also encompasses some less-counting-ish stuff like the pigeonhole principle, Ramsey's theorem, and (arguably) the entirety of graph theory.
But I agrre that the combinatorics showed (combinatory analisys) is definity the most sufferable part of math, and no one deserves to suffer that punish (specially because it is very interpretative, and formulaic most of the time you elaborarem a question that was "way harder" that you though and you spend most of your time calculating that just to get the wrong answer)
Bro I had a "short" 3 months long combinatorics course and I feel like I haven't scratched the surface, could easily take a full year to learn just all the basic things(college level) in combinatorics
If you cannot pronounce combinatorics, then you probably shouldn’t be able to rank it either
Nothing new in probability theory/statistics? You have obviously never taken any higher level classes in these.
To me, the rating you gave to statistics is unfair and really indicative to me that you did not spend anytime in statistics
A lot of variety of topics, and actually a lot of **mathematical** difficulties arise from the real world data
Defining estimators and deriving the rate of convergence of those is tricky, and there are also a lot of beautiful objects that are relevant to statistics you would not even know :
- one for instance is functional data analysis, where your random variables are function-valued. Which of course causes a lot of head-aches because those are infinite dimensional objects, and real world data is finite dimensional. To define a "mean-function" one has to build an integral on infinite dimensional spaces, (which is analysis but here is very well motivated by statistics), also in functional data, we both observe IL^2 random elements but one could also consider to introduce them as continuous random processes (like brownian motion) - one notion is global, the otherone is pointwise in the introduction of randomness, question : when are those equivalent ? In the functional data analysis setting it makes sense for us to be both at the same time. How do we approximate such data for our finite dimensional computational world without losing information about the data ?
- One could also talk about inference, with parametrical inference, likelihood, ...
- There is also information theory really closely related to statistics and therefore I will include with it because frequently one shed some light to the other. Modelling what information is ( for instance Martingale theory is good to discover how we model growing knowledge in our system )
- how do we make the least number of assumption on our population, because those assumptions can be really far from the truth ( non parametric statistics ) ? When does it make sense to approximate with parametrical distributions ?
Statistics is so rich it really hurts to hear the argument you made, it's like putting algebra in F tier because "yeah sure you got addition and subtraction, but in the end not that interesting though."
PS : the video was good nonetheless, I just want to get this out of my chest
Calculus in A tier is crazy. 99% of algebra and geometry is derived using differential or integral operations and calculus gives us the ability to perform operations beyond elementary mathematics. Calculus deserves an S tier. Our world does not exist without calculus
4:20 OH GOSH ITS THE HYPERBOLIC AND SPHERICAL GEOMETRYS RUN
I can already see how this video will get trash talked on r/mathmemes from people with different opinions. On an unrelated note, here is my favorite (combinatorics?) problem: Given an equilateral triangle, show that if 5 points are choosen randomly inside the triangle, there exists a pair of points such that their distance is less than the height of the triangle :D
Hint:
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Tri-force
I might be wrong, but doesn’t this problem work with only 4 points? Or a better question: does there exist a configuration of 4 points, chosen randomly, where the distance between all pairs of points is less than the height of the triangle?
@@gruk3683
Well just choose 4 points really close to each other. I think you meant to ask if there is a configuration of 4 points such that all distances between pairs is GREATER that the height of the triangle. And this can't happen since with 3 points yes but with 4 no. The only possible configuration with 3 points where this works is if the 3 points are close to the corners and adding a fourth one would just decrease some distance between a pair of points. This would have bern possible if we were working in an equilateral tetrahedron and placing the 4 points close to the 4 corbers( I think). If you want another challenge: Prove the same problem again but instead of height of triangle its about half the side of the triangle! ( i mean to say: Given 5 random points inside an equallat. triangle, you can always find a pair of points whose distance is smaller than half the side lenght of the triangle) :D
It definitly doesn't seem to fit in the combinatorics definition presented (combinatory analisys) and the problem you presented fit more in the geometry category (specificaly finding a "geometric place" of "all possible solutions") but as well the categories presesnted are very broad and can fit everything in (I could say it is discrete analisys and fit the problem with the algebra category or something of this sort)
sets and category are kinda out there you should have included them very fundamental to maths
They teach combinatronics(Ik I spelled that wrong) and stats as a part of the algebra 2 curriculum in our school (Michigan US)
The tierlist is invalid without Linear Algebra listed.
I would group combinatorics and stats, then add something like real analysis or logic?
stat not in s tier when its the only form of math that can be properly understood in the real world is appauling
the fact that analysis and set theory are not even here is outrageous
But why combinatorics in F tier?
Stats BELONGS in S Tier dammit and I don't care what anybody else says
Fr,
But it really depends on how he was taught it. I don’t blame him for disliking it if it was taught formulaic
Statistics has a lot of philosophical problems. How can you say that the probabilities you calculated are objective or subjective? How can we be sure in the truthfulness of our models and our results? A lot of the replication crises in science is a result of, in large part, the overeliance on p-values for determining significance, which is a frequenting concept (the idea that parameters are fixed and data is random), but this might not be how we ought to understand probabilities for this problem. It's very hard to trust statistical inference with the same kind of confidence as other mathematical derivations. Other parts of math typically don't have glaring foundational problems in their interpretations.
Calculus is best. But the real fun starts in higher dimensions.
I feel proud that I know all of these things
this is all high school level stuff. this person in the video barely mentioned any math that is actually used by mathematicians today
You shpuld do an extensed version with the more advanced fields. Great video
How does this video only have 280 views?? It's a great watch!
It is not, Logics was not on the list, and it's a pretty fucking big and important field, more than statistics and Combinatorics
If you don't know anything about math then yes its a great watch
What if i dont like geometry at all, but i love algebra and calculus like the most in the world😅
At least put geometry in A tier, early geometry is much more fun than other things fr
1) Calculus Foundations
Contradictory:
Newtonian Fluxional Calculus
dx/dt = lim(Δx/Δt) as Δt->0
This expresses the derivative using the limiting ratio of finite differences Δx/Δt as Δt shrinks towards 0. However, the limit concept contains logical contradictions when extended to the infinitesimal scale.
Non-Contradictory:
Leibnizian Infinitesimal Calculus
dx = ɛ, where ɛ is an infinitesimal
dx/dt = ɛ/dt
Leibniz treated the differentials dx, dt as infinite "inassignable" infinitesimal increments ɛ, rather than limits of finite ratios - thus avoiding the paradoxes of vanishing quantities.
2) Foundations of Mathematics
Contradictory Paradoxes:
- Russell's Paradox, Burali-Forti Paradox
- Banach-Tarski "Pea Paradox"
- Other Set-Theoretic Pathologies
Non-Contradictory Possibilities:
Algebraic Homotopy ∞-Toposes
a ≃ b ⇐⇒ ∃n, Path[a,b] in ∞Grpd(n)
U: ∞Töpoi → ∞Grpds (univalent universes)
Reconceiving mathematical foundations as homotopy toposes structured by identifications in ∞-groupoids could resolve contradictions in an intrinsically coherent theory of "motive-like" objects/relations.
3) Foundational Paradoxes in Arithmetic
Contradictory:
- Russell's Paradox about sets/classes
- Berry's Paradox about definability
- Other set-theoretic pathologies
These paradoxes revealed fundamental inconsistencies in early naive attempts to formalize arithmetic foundations.
Non-Contradictory Possibility:
Homotopy Type Theory / Univalent Foundations
a ≃ b ⇐⇒ α : a =A b (Equivalence as paths in ∞-groupoids)
Arithmetic ≃ ∞-Topos(A) (Numbers as objects in higher toposes)
Representing arithmetic objects categorically as identifications in higher homotopy types and toposes avoids the self-referential paradoxes.
4) The Foundations of Arithmetic
Contradictory:
Peano's Axioms contain implicit circularity, while naive set theory axiomatizations lead to paradoxes like Russell's Paradox about the set of all sets that don't contain themselves.
Non-Contradictory Possibility:
Homotopy Type Theory / Univalent Foundations
N ≃ W∞-Grpd (Natural numbers as objects in ∞-groupoids)
S(n) ≃ n = n+1 (Successor is path identification)
Let Z ≃ Grpd[N, Π1(S1)] (Integers from N and winding paths)
Defining arithmetic objects categorically using homotopy theory and mapping into higher toposes avoids the self-referential paradoxes.
Are you a type theorist? Or does your specialty mostly deals with types?
you forgot linear alg, topology,analysis, abstract alg, set theory, logic .......
Look there's always that bitchy question in any field of math that u just can't solve and all of them deserve S tier
Love this, subscribed!
Number theory IS algebra!
include all the topics of math upto grad school those are all the math most of us will ever need to worry about
or if you're an ai u could try to rank all 500ish math fields
knot theory best math field
Alright, I should clarify a couple things. When I said that "combinatorics can be learnt in 2 weeks" and "it only had 2 operations", I meant the basics. Obviously you can't learn the whole field inside out in 2 weeks and I understand there exists other aspects of the field I didn't mention. Sorry if its place in F tier offended any of you, one of the fields had to be there.
i dont really think that's the issue here. i think you bit off way more than you can chew.
My man really said algebra is easy to understand.
You've done calculus wrong
Calculus easy S tier
Geometry A tier
They are fun and tricky
Algebra not so fun anymore, pretty mashine-like calculations C tier
Number theory, sometimes boring, but it can be fun B tier
Combinatorics are also mashine-like so C tier
Statistics, although very usefull, it is shit to calculate. 0% fun (you get it?) It is trash F tier or worse if I can
Algebra has simple calculations? Have you taken college level algebra? Also what about analysis?
combinatorics being mechanical is a crazy take. there are so many beautiful and creative proofs out there of combinatorial identities and connections to other fields like number theory. math is not about calculations and formulas, and this video does an awful job at communicating that
Algebra has to be S. It is the language of all other math fields.
@@kevinstreeter6943 If we consider that we know basics in every topic here presented, I rated those in case of getting deeper into them, not in case, that you cannot run further without knowing basics. Going with your thinking path, we have to give GEOMETRY an S tier, because that is how all of the math was created. Ancient philosophers had known only geometry. But we don't want to get so deep, so I'm rating every topic separatelly. (You ain't reading that, right?)
3:42 bro, euler has discovered too many things in math. its both amazing and annoying
Crazily underrated
Measure theory is still the best, statistics is just an application of it
numerical analysis is mostly linear algebra, no?
Not really. Numerical Analysis is a little bit of everything because the theory is basically just “how do we solve things numerically and how do we approximate solutions?” That has applications in Calculus, Linear Algebra, Number Theory, etc.
Wdym hyperbolic geometry is beautiful 😭
(Speaking as a physics undergrad)
Arithmetic S++ tier
Peoples who pratics googology for fun : 🗿
Discrete mathematics wasn’t included 😔
That is close to number theory and combinatorics.
arithmetic?
Mathsinanutshell is very sigma
sup tom
Bro said bcc 💀💀
Algebra has to be S. It is the language of all the other fields. All the other field would not exists without it.
This is why I get better grades in Tech lmao
calc is D tier complex analysis is S tier
Statistics are not pure maths, is applied mathematics, but probability is pure math and has many applications in pure maths, like in combinatorics
If Statistics is applied mathematics, Physics is also applied mathematics. Statistics is a separate discipline.
dafuq
try contest combinatorics....
you ll know ;_;
number theory isn't S tier. I do not recognise the opinion.
Linear Algebra 💀
I hate combinatorics if I'm being honest
Combinatorics should be lower.
Should have included Category Theory, easy S
What the fuck is Salgebra and Bstatistics
Skips to the recap part to know if his opinion is worthy enough to be heard.
Calculus not in S tier and Combinatorics in F tier.
WTF!!! Invalid Opinion
Numerical analysis is more like computer science tho not maths
Computer science is literally math. Like, computer science is mathematical logic, representation theory, finite algebras, numerical analysis, etc.
That's fair.
Combinatorics in two weeks😂😂😂. What school are you attending.
Algebra is S²
Where is game theory?
algebra is mid, linear algebra however...
Bros still in highschool
I hate statistics the most
Salgebra
Bstatistics
Linear algebra!
Where is linear algebra, group theory something!
Just say that you are in middle school, this list is missing some key elements and also has some very controversial opinions.
Here before viral
It seems that you havent understood what each field is about in any sense man
You group together calculus and numerical analysis? WOW! That is a brave move. You consider combinatorics to be a special field of mathematics? Alright….. CRIKEY!
Anyway, your motivations for which mathematical fields that are important or not are extremely vague throughout the video. It’s an absolute shame to say that calculus (or any of the other fields) doesn’t play a role as big as algebra.
If you would have put number theory on S and all of the other fields that you bring up on A, you might have had my sympathies. But as you put it now, it’s nothing but very awkward and extremely weird.
With best regards
// an algebraist
Hello, here is a fun exercice for an algebraist like you.
Lets consider G a finished, abelian group. When can we calculate the product of every element of G? Calculate it
I would gladly! But unfortunately I don’t know what a finished group is.
@@sydneythesurfboards5903 Sorry, Im not English so maybe the names of objects are different. By a "finished group", I mean a group with a certain number of elements. Not infinite.
@@glacide8883 In English that would be known as a finite group. 🙂
@@glacide8883 I don’t understand the problem setting actually… What exactly do you want me to calculate?
To calculate the sum or product of all elements of a group doesn’t really make sense I think… Are the elements of the group known? Do you have a specific type of group in mind? What do you mean by product? What kind of operation is defined on the group?
Geometry and calculus not in S tier , opinion invalidated.
your take on algebra is too simplistic, to the point of being wrong
Algebra as something toddlers learn and algebra as a field in university are pretty different, because as you already said, every single field in math is just letters for numbers, that's a stupid definition. Also you totally threw away my field 😭😭 which is logic/foundations, but i guess if you really want to you could put it in algebra.
Was this meant for highschoolers?
geometry is definitely f tier....
and how TF is combo f tier, it has literally everything
Geometry is S wdym
Blud said combyenatorics