what do mathematicians mean by "universal"?

[@CompagnoArti]

So what you're saying is that a tensor is a thing that transforms like a tensor

[@amelie-sophiegluck7576]

Nah he is saying that a tensor is an Element of the tensorproduct of a vector space, and that tensorproduct is fully described by a universal property lol

[@eiseks3410]

This topic is very interesting. I think we would all appreciate another (longer) video about this topic.

[@antoniusnies-komponistpian2172]

So the universal property for math is basically what behaviourism for psychology? Like "I stop caring about what it is, I just care what it does".

[@_PFD_]

Interesting! Universal properties do come up a lot in category theory. Let's note that universal properties usually give unicity (up to isomorphism), but not existence. So you need to come up with a construction. For example: - natural numbers (with zero) = Peano system & construction by minimal inductive set, - real numbers = real number system & construction by Dedekind cuts or Cauchy sequences, - product topology = product universal property in Top category & construction by initial topology with respect to the projection maps, - disjoint union topology = coproduct universal property in Top category & construction by final topology with respect to the canonical injections, - etc

[@aradarbel4579]

"A general tenet of modern category theory is that cleaner, more conceptual arguments shall be preferred against element-wise proofs that are evil in spirit, if not in shape" ~Fosco Loregian In my opinion universality is one of the most beautiful concepts of modern math. It can be found *everywhere*; geometry, topology, algebra, logic, etc. The idea behind it is so versatile yet so fundamental to the way we do and think about mathematics, I'm excited to see you exposing more people to it and to abstract thinking :)

[@emanuellandeholm5657]

I'd love to learn more about category theory. As a computer programmer.

[@mraarone]

Same, how does category theory handle the lambda function category?

[@aradarbel4579]

@@emanuellandeholm5657 I feel obligated to point out that category theory tends to be over-hyped in programming circles (and especially functional programming). I'm not saying it's useless, just saying you shouldn't treat it like magic, and it's not as relevant for programmers as it is for mathematicians, unless you do something highly theoretical like programming language theory or categorical semantics, at which point I'd argue you're closer to doing math than CS anyway. Usually when computer scientists talk about CT they mean something like Milewski's stuff, though this is very different from what mathematicians mean when they talk about CT. If your goal is to write better code, CT will probably be nearly useless. However if your goal is to become more mathematically-minded, and just expand your views on abstraction in general, then you could really enjoy it. I've heard good things about Tom Leinster's short book, "Basic Category Theory", as a wonderful first introduction.

[@emanuellandeholm5657]

@@aradarbel4579 Thanks, and I think I know where you're coming from. I'm not one of those Haskell nerds, and I'm not specifically looking to improve my coding game. Just looking to learn more general things in math and maybe develop another special interest or so.

[@emanuellandeholm5657]

@@aradarbel4579 Just to further trigger the F#/Haskell nerds, I'd like to announce that the perfect functional language, Scheme, was invented way back in the 70s. And it doesn't have "monads"! :D

[@curtiswfranks]

The determinant is the first time in which I somewhat consciously encountered this concept, but I did not really know why we cared, so it felt like a round-about way of explaining the determinant. I do not think that we even explicitly discussed how to calculate it after that point, which was annoying for homework purposes. However, I later understood the importance of that style of introduction and do believe it to be the proper way of doing it, although I would provide at least one algorithm for actually implementing the object.

[@soyoltoi]

Especially for the determinant, where the characterization is the simplest way to show how all the different ways of describing the determinant are actually the same.

[@umbraemilitos]

The Clifford / Geometric algebra definition, of a determinant as a special case of outermorphisms, made the most sense to me.

[@PRIYANSH_SUTHAR]

The derivative proof was so elegant. It flowed so naturally even with mathematical induction.

[@iabervon]

A neat thing to do next is to show that the function for the signed area of a parallelogram is an alternating bilinear function from R²xR² to R and the area of a square formed by the basis vectors is 1, all of which is pretty obvious, so it must be the determinant, which wouldn't have been obvious previously.

[@Wielorybkek]

Really cool video! I would love to see some more universal properties.

[@schweinmachtbree1013]

Uhh... this is not what mathematicians mean by "universal" - this is just what mathematicians mean by "unique", or "characterising property". Examples of the difference between universal properties and unique properties are greatest common divisors vs "unique common divisors", or least upper bounds vs "unique upper bounds" (unique upper bounds are not very useful, but order-theoretically they are the maximal upper bounds)

[@doria_bolognese]

I was just looking videos of universal properties yesterday and then you made one on it today!

[@Nirmy001]

A video suggestion with a graph theory/combinatorics problem: work through finding a formula to count how many edges need to be added to a cycle with n vertices to make it to a complete graph with n vertices. A complete graph with n vertices is a graph where each vertex has an edge to every other vertex. A cycle is a graph with n vertices connected in a way such that each vertex is connected to 2 other vertices and if you start at a vertex you should be able to traverse the whole graph and return to the starting vertex without needing to repeat vertices.

[@meowmeowfood]

I guess my question, for the polynomial case anyway, would be: How do we know the derivative is the unique linear map satisfying the desired propertiesand and not just one of many that all happen to look the same once you restrict to the polynomials?

[@franzlyonheart4362]

9:15, LOL at "n=n" case. That would simply be the case n≥4.

[@TheEternalVortex42]

It's not clear to me that it's "better" to use these definitions. It's certainly more elegant in the sense of being simpler, but it also loses the entire motivation of these definitions in the first place, which is a loss of information. The integers are important because they map to discrete (possibly abstract) objects in the real world, whereas an "infinite cyclic group" does not. Each time you go up one level of abstraction there is something you lose, in that when things are equal up to isomorphism, the actual map producing that isomorphism is important as well. You could consider the analogy of NP-hard problems in computer science. Anything in NP is reducible to an NP-hard problem. So if we can solve one NP-hard problem efficiently we can solve all of them. But the algorithm for the reduction is important too--it's not like we would say these problems are all *the same*! I guess I'm not a big fan of category theory...

[@aradarbel4579]

The advantage of universal properties is to decouple the structure of mathematical entities from their meaning. When defining free groups, we have a certain explicit method for constructing them, but we also have an intuitive idea behind why we even want them in the first place, how we want them to behave, or relate to other groups, and that's not necessarily encoded in the explicit construction (indeed, the universal property can be applied in other situations, which might have nothing to do with groups at all). But I totally agree that the universal property loses something. And that category theorists can unintentionally go too far with the whole "up to isomorphism" idea. In real life, even if we define something by a universal property, that doesn't mean there exists an object that satisfies the property in the first place. Or sometimes there are many different (isomorphic) objects that satisfy it. So very often we still need to explicitly "choose" the specific object we have in mind. This choice is how the abstraction relates to down-to-earth calculations and other applications. That doesn't mean the universal property is not useful for other things, it helps us distill the essence of our definitions and recognize wider patterns throughout math!

[@djspacewhale]

constructions satisfying a universal property are unique up to *unique* isomorphism, i.e. there is a "canonical" isomorphism showing the two constructions agree. a paradigm in functional programming uses this, showing that a universal (more correctly, monadic) construction agrees with some element-wise construction along a unique isomorphism and then using one or the other in different cases while transferring along that iso

[@alxjones]

I have to disagree with the idea that universal properties are less motivated definitions. To take your example with the integers, the value in them is that they represent counting forwards and backwards without stopping. Like how N represents starting somewhere and then you can keep adding 1, Z is where you start somewhere and you can keep adding 1 or subtracting 1. This is exactly what it means to be an infinite cyclic group. It's a group generated by 1 element of infinite order (if you keep adding it, you never get back where you started). You can think of it like the object you want to count, like an apple. No matter how many times you add an apple to your pile, it will never revert to an amount you had before. Because it's a group, every element has an inverse, so your "apple" has a inverse too, and so you can count backwards in apples too. If you start from the original apple and count backwards, you get to no apples, and then negative apples (however you like to interpret that, it has a reflection in the group properties). So you just started with the idea of an infinite cyclic group, and the way such a group is generated naturally mirrors counting forwards and backwards. In a sense, it's the most well-motivated definition of the integers. Similarly, the usual introduction of the determinant is just as a magical formula that you apply to matrices and it happens to be non-zero exactly when that matrix is invertible. Plus, the connection between the determinant and the exterior algebra is hidden. When you start from the universal property, you start with the desire for something alternating and multilinear, which has a natural connection to oriented "chunks of space" defined by vectors, which gets you to k-forms and how they're useful for integration. Then, we see how the determinant is just the top form (n-form in n-dimensional space) that evaluates to 1 on the basis, and heavily motivates its role in the change of coordinates for integration.

[@franzlyonheart4362]

​@@alxjones bingo!

[@TedHopp]

I think you had in mind NP-complete problems, not NP-hard. (A problem is NP-complete if it is NP-hard and also in NP.) Finding an efficient solution to any NP-hard problem means that all problems in NP can be solved efficiently, but it does not mean that all NP-hard problems have efficient solutions.

[@CarTLA]

I would enjoy a category theory series!

[@Alan-zf2tt]

Regarding views on "Catergory Theory" A great observation for sure - but is the observation a human characteristic rather than object centric characteristic? In other words Category Theory is really just a result of tidy-desk syndrome? By that I mean it neatly bundles up things into pleasant groupings that may not be supportive of how things work in those categories in general.

[@mraarone]

Why does the universal function take column vectors to prove row vector determinants?

[@goodplacetostop2973]

16:48

[@ricc3541]

You're still thereeee, i remember you from like 2019/2020

[@goodplacetostop2973]

@@ricc3541 Yep, that’s still me 😂

[@kokainum]

In last line you wrote det of transposed matrix than the one made of vector columns at the start of a proof. I think that's the mistake but it reminds of one property that is missing and that is that these properties imply equality of fs after "transposing" the vectors, but I think it's hard to prove it without defining general formula for determinant and learn a bit about permutations and their parity.

[@paperstars9078]

Does anyone have some good examples on when/how to use universal properties to proves stuff? It just takes me way too long to understand proofs that somewhere mention "by the universal property". I get it that it's supposed to be better but I never got warm with using it to proof stuff. (currently doing abstract algebra)

[@АндрейДенькевич]

In video second propety of derivative is represented by 2-convolution of binom '1 1" row_0= 1 row_1= 1 1 row_2= 1 2 1 row_3= 1 3 3 1 row_4= 1 4 6 4 1 row_5= 1 5 10 10 5 1 ... Each row is face-vector of simplex. If to replace binom "1 1" by "1 2" then we can represent derivative by cube: row_0= 1 row_1= 1 2 row_2= 1 4 4 row_3= 1 6 12 8 row_4= 1 8 24 32 16 row_5= 1 10 40 80 80 32 ... Each row is face-vector of cube. D(f*g)=1*D(f)*g + 2*f*D(g). Second derivative is DD(f*g)=1*(DD(f)*g+2*D(f)*D(g))+ 2*(D(f)*D(g)+2*f*DD(g))= 1*DD(f)*g+4*(D(f)*D(g)+4*f*DD(g)) wich is row_2 of above Pascal Matrix. If to replace binom "1 1" by 3-nom "1 1 1" ( sequence "1 1 1 1 1 . . ." , of course ,is a face-vector of sphere) and 2-convolution by 3-convolutoin then we can represent derivative by simplex3 wich appears only in odd dimensions: row_0= 1 =1 row_1= 1 1 1 =3 row_2= 1 2 3 2 1 =9 row_3= 1 3 6 7 6 3 1 =27 row_4= 1 4 10 16 19 16 10 4 1 =81 row_5= 1 5 15 30 45 51 45 30 15 5 1 =243 ... Each row is face-vector of simplex3. D(f*g*h)=1*f*D(g*h) + 1*g*D(f*h)+ 1*h*D(f*g) . Of course, D(m*n)=a*D(m)*n + b*m*D(n). a and b depends on wich nested shape we choose (see above). Unlike previose, this derivative operator is nested(dimensional). because of quantity-quality inversion happens in even dimensions. Instead of Simplex3 to be curved into shape, dimension itself is curved into operator wich became nested(dimensional).

[@АндрейДенькевич]

Moreover, we can notice that digital root2(digital sum2) of cube and simplex3 is same(=sum of 3^i for i:=0 to infinity). But digital root is sort of dimension ,for example take simplex: 2^i (1, 10, 100,1000...) is vertices, 2^i+2^j is edges (11, 101,110,1100,1010,...),2^i+2^j+2^k is triangles (111,1011,1110,11010,...). So we can conclude that n-convolution of any n-nom, wich has digital root=m, has same digital root2=sum of m^i for i:=0 to infinity. For exmple. 3-convolution of 3-nom with digital root=5 '1 1 3' (or '2 1 2'or '1 0 4' or '1 2 2'...): row_0= 1 =1 row_1= 1 1 3 =5 row_2= 1 2 7 6 9 =25 row_3= 1 3 12 19 36 27 27 =125 row_4= 1 4 18 40 91 120 162 108 81 =625 row_5= 1 5 25 70 185 331 555 630 675 405 243 =3125 or row_0= 1 =1 row_1= 2 1 2 =5 row_2= 4 4 9 4 4 =25 row_3= 8 12 30 25 30 12 8 =125 row_4= 16 32 88 104 145 104 88 32 16 =625 row_5= 32 80 240 360 570 561 570 360 240 80 32 =3125

[@TaladrisKpop]

It does not match with the notion of "universal" I am aware of, like "universal properties" (of tensor products,...). What is "universal" in cyclic groups? They are a very basic example of groups

[@robshaw2639]

I think another good example is the free group being universal over groups

[@Trumpet12333]

Just a beautiful way to relax ones mind. 😌🔥

[@jamesfortune243]

Mind expanding.

[@TomFarrell-p9z]

Very interesting! So, is it better to teach mathematics using universal properties, or through some approach that appeals more to human intuition and--roughly--the history of mathematical development? I'd contend the latter. This is, if I understand correctly, the mistake made by trying to teach the "new math" in the early 1960's. In a different field, Cal Tech made this same mistake when they allowed Feynman to teach undergraduate physics. His course was great as a review for graduate physics students, but left many of the undergraduates lost.

[@allanjmcpherson]

I would agree. One of the great challenges of teaching mathematics is that once you understand a concept it can be difficult to remember what it was like not understanding it. The more deeply you understand it, the more tempting it can be to try to jump straight to that level of understanding because you see the elegance and beauty of it. But that's not how we learn.

[@sieni221]

First time I encountered universal properties was Topology and it has alot of them.

[@xizar0rg]

Doesn't the uniqueness of the linear operator need to be shown in the first worked example?

[@TaladrisKpop]

If every linear map with the given properties is equal to the derivative, then there is only one such map.

[@xizar0rg]

@@TaladrisKpop You can't use a statement to prove itself. The work done was to show that there exists a linear map; the statement asserts uniqueness. To show that you'd need to assume a second such mapping and then show they have to be the same thing.

[@TaladrisKpop]

@@xizar0rgNo one used the statement that one wants to prove here. In the video, Micheal Penn 1) assumes with no proof the well-known fact that differentiation is a linear operator that satisfies the Leibniz Rule (fair game, it is a basic fact of Calculus); 2) then considers an arbitrary linear operator that satisfies the Leibniz Rule and shows it is equal to the differentiation operator. This is sufficient for uniqueness.

[@jakobthomsen1595]

elegant!

[@MDNQ-ud1ty]

What does any of this have to do with the fact that math is being used by psychopaths to destroy humanity?

[@stmmniko7836]

nice video

[@gp-ht7ug]

Nice video

[@Kreypossukr]

« Infinite cyclic group » ??? Wtf