These examples of pushbacks and pullbacks are very helpful. After your lectures, I began to see categorical patterns unexpectedly in everyday life! Thank you!
I'm really glad to hear it. Yea I think it's really useful to play with these ontology logs, to help internalise the notions of limits/colimits. Although next time I will go back to discussing some of the deeper ideas from pure category theory
Yes, maybe at some point. There is still so much nice category theory to cover. I'm currently working on my next video on the category of dynamical systems, and the category of graphs. There will I discuss easy ways to make categories of structured sets. I think the underlying techniques should be useful for understanding how lots of interesting algebraic structures live in categories.
@@RichardSouthwell dude that's awesome, both of those subjects sounds awesome Especially because graphs are used to represent categories themselves (so you got a bit of recursion there :)), and because dynamical systems are the language of nature :) I look forward to your videos man :3
I see your Venn diagram showing the intersection of Cat. Th. with Maths, Comp, Phy, Linguistics, but as yet I fail to see what it will allow me to answer that set theory will not. Apart from having a 1st in Mathematics I did a few courses from the OU back in the day and was quite into computing. I did a course on relational data bases which model things the way you describe herein. Ultimately Rel. DB design 'relations' are manifest by assigning and posting 'primary keys' (numbers usually) in corresponding 'tables' (read sets?). What are the profound results of Cat. Th. that I cannot reach built upon the logic of set theory. ie Group Theory gives me insight into why there can never be a quadratic style formula for higher polynomials than n=5. Linear mathematics gives me insight into generalised spaces dim = 0...inf that fit almost all notions of functions, including answers in quantum mechanics. As an aside, I note in your ontology log you restrict yourself somewhat ? Is it OK to include the relation: A PLANET ----- has ------> A TRAJECTORY ? .... directly ? (the diagonal relation?)
Thank you for your comment. I think you see things correctly, regarding the ontology logs. There is nothing here that could not be formulated in set theory. I mean, from one perspective, these ontology logs are just fancy ways of encoding things that can be done using sets and functions. However, I do think ontology logs are easier to appreciate when combined with category theory (the diagrams, the notion of commuting paths, the underlying types of limits/colimits that occur etc.). Have you watched my video on universal properties or limits ? I think universal properties are quite profound. That the same type of concept can encode the idea of natural numbers, products, disjoint union, exponentiation, equation solving etc, and that it can do so for general categories (so these concepts can be generalized for graphs, groups, monoids, etc. etc.). That seems quite profound to me, but I suppose it is a bit subjective what one considers to be a "profound result". Another thing I find quite profound is that many areas of mathematics can be modeled by categories. For example, there is a category with objects as vector spaces and arrows as matrices. There is also the category of groups. Perhaps, on the surface, that does not sound very profound, but when you think that you can use the same category theory ideas to study all these areas (e.g., the notion of a categorical product (which is a kind of limit) allows you to think about what it means to take the "product" of many different types of mathematical structures (sets, graphs, groups etc.) I find that quite profound). Also, these `categories of structured sets' are deeply interrelated. If you watch my next video, when its done, and see how naturally the category of graphs is defined, you will hopefully see what I mean. But coming back to your opening sentence, I do think most things in category theory could be reached with set theory alone. But I have been a professional mathematician for ten years, and for me, I have not found any other subject so insightful as category theory. But I suppose it is a matter of perspective. I do agree that linear mathematics and group theory are extremely beautiful. Also, perhaps they are more useful for physics. If you are looking for other profound ideas from category theory, I recommend you check out the curry-howard-lambek correspondence and the idea of adjoint functors.
@@RichardSouthwell Thanks for the reply. Though I did well and love abstract algebra, i could never internalise it very easily without some kind of profound motivations. I did a math degree because I knew maths was key to physics these days, or so they say. I think I'm a physicist at heart. I like to try to wrangle with interpretations of Quantum mechanics a lot. I've watched two other vids of yours on Category theory. I like its relation to 'linguistics' as I consider myself a bit of an amateur philosopher, and so much goes wrong with language. I'll keep watching and see where you take me. I enjoy the vids.
@@Hythloday71 I am glad you enjoy the vids. I think quantum mechanics is very interesting, but I have not been able to get my head around it very well. At one point I knew how to do the basic maths of it (Schrödinger's equation for the hydrogen atom etc.) but that was a long time ago, and I have not yet studied quantum field theory etc. I have been interested in knowing what the best way to interpret quantum mechanics is (Copenhagen vs many-worlds interpretations etc.) but I have not yet put in the study time to be able to get a deep understanding of the underlying issues. I am curious about which interpretation you find most palatable.
@@RichardSouthwell After reading all the 'lay stuff' possible at the time and getting the same old sell on the mystique of QM, I found Leonard Susskinds detailing the linear mathematics of QM in his Quantum Entanglements TH-cam series the BEST. I think Murray Gell-Mann's consistent histories would be a favourite along with generally 'non local hidden variable' theories. But I have my own limited interpretation which suggests the notion of 'variable' is wrong. Bell's inequality famously discounts local variable theorems, as evidenced in Alan Aspect's experimental confirmation of the violation of Bell's 'SET THEORETIC' inequality. The problem is, set theory deals with properties as if they are wrote on the objects, little flags on perhaps, but this is clearly absurd. Properties are dynamic, I have proven to my own satisfaction that the probabilities involved with single electron spin measurements can have a physical / geometrical interpretation. I believe, whilst not rigorously or formally doing so, I have essentially re-invented Sir Roger Penrose's 'Spinors'. I checked his book 'Road to Reality' he speaks of a clear geometric structure underlying the QM in regards of spin and its analysis through the 'spinor'.
33:23 Goddam, that voice and knocking on the background scared the hell out of me. In my headsets sounded like it was next to me. It took me a while to notice it was someone in the video.
These examples of pushbacks and pullbacks are very helpful. After your lectures, I began to see categorical patterns unexpectedly in everyday life! Thank you!
I'm really glad to hear it. Yea I think it's really useful to play with these ontology logs, to help internalise the notions of limits/colimits. Although next time I will go back to discussing some of the deeper ideas from pure category theory
same! By far the most intuitive examples I’ve come across while learning category theory basics.
Would we get categories of all mathematical proofs?
yep
Are you ever thinking of doing a series on abstract algebra?
Yes, maybe at some point. There is still so much nice category theory to cover. I'm currently working on my next video on the category of dynamical systems, and the category of graphs. There will I discuss easy ways to make categories of structured sets. I think the underlying techniques should be useful for understanding how lots of interesting algebraic structures live in categories.
@@RichardSouthwell dude that's awesome, both of those subjects sounds awesome
Especially because graphs are used to represent categories themselves (so you got a bit of recursion there :)), and because dynamical systems are the language of nature :)
I look forward to your videos man :3
I see your Venn diagram showing the intersection of Cat. Th. with Maths, Comp, Phy, Linguistics, but as yet I fail to see what it will allow me to answer that set theory will not. Apart from having a 1st in Mathematics I did a few courses from the OU back in the day and was quite into computing. I did a course on relational data bases which model things the way you describe herein. Ultimately Rel. DB design 'relations' are manifest by assigning and posting 'primary keys' (numbers usually) in corresponding 'tables' (read sets?).
What are the profound results of Cat. Th. that I cannot reach built upon the logic of set theory. ie Group Theory gives me insight into why there can never be a quadratic style formula for higher polynomials than n=5. Linear mathematics gives me insight into generalised spaces dim = 0...inf that fit almost all notions of functions, including answers in quantum mechanics.
As an aside, I note in your ontology log you restrict yourself somewhat ? Is it OK to include the relation:
A PLANET ----- has ------> A TRAJECTORY ? .... directly ? (the diagonal relation?)
Thank you for your comment. I think you see things correctly, regarding the ontology logs. There is nothing here that could not be formulated in set theory. I mean, from one perspective, these ontology logs are just fancy ways of encoding things that can be done using sets and functions. However, I do think ontology logs are easier to appreciate when combined with category theory (the diagrams, the notion of commuting paths, the underlying types of limits/colimits that occur etc.). Have you watched my video on universal properties or limits ? I think universal properties are quite profound. That the same type of concept can encode the idea of natural numbers, products, disjoint union, exponentiation, equation solving etc, and that it can do so for general categories (so these concepts can be generalized for graphs, groups, monoids, etc. etc.). That seems quite profound to me, but I suppose it is a bit subjective what one considers to be a "profound result". Another thing I find quite profound is that many areas of mathematics can be modeled by categories. For example, there is a category with objects as vector spaces and arrows as matrices. There is also the category of groups. Perhaps, on the surface, that does not sound very profound, but when you think that you can use the same category theory ideas to study all these areas (e.g., the notion of a categorical product (which is a kind of limit) allows you to think about what it means to take the "product" of many different types of mathematical structures (sets, graphs, groups etc.) I find that quite profound). Also, these `categories of structured sets' are deeply interrelated. If you watch my next video, when its done, and see how naturally the category of graphs is defined, you will hopefully see what I mean. But coming back to your opening sentence, I do think most things in category theory could be reached with set theory alone. But I have been a professional mathematician for ten years, and for me, I have not found any other subject so insightful as category theory. But I suppose it is a matter of perspective. I do agree that linear mathematics and group theory are extremely beautiful. Also, perhaps they are more useful for physics. If you are looking for other profound ideas from category theory, I recommend you check out the curry-howard-lambek correspondence and the idea of adjoint functors.
@@RichardSouthwell Thanks for the reply. Though I did well and love abstract algebra, i could never internalise it very easily without some kind of profound motivations. I did a math degree because I knew maths was key to physics these days, or so they say. I think I'm a physicist at heart. I like to try to wrangle with interpretations of Quantum mechanics a lot. I've watched two other vids of yours on Category theory. I like its relation to 'linguistics' as I consider myself a bit of an amateur philosopher, and so much goes wrong with language. I'll keep watching and see where you take me. I enjoy the vids.
@@Hythloday71 I am glad you enjoy the vids. I think quantum mechanics is very interesting, but I have not been able to get my head around it very well. At one point I knew how to do the basic maths of it (Schrödinger's equation for the hydrogen atom etc.) but that was a long time ago, and I have not yet studied quantum field theory etc. I have been interested in knowing what the best way to interpret quantum mechanics is (Copenhagen vs many-worlds interpretations etc.) but I have not yet put in the study time to be able to get a deep understanding of the underlying issues. I am curious about which interpretation you find most palatable.
@@RichardSouthwell After reading all the 'lay stuff' possible at the time and getting the same old sell on the mystique of QM, I found Leonard Susskinds detailing the linear mathematics of QM in his Quantum Entanglements TH-cam series the BEST. I think Murray Gell-Mann's consistent histories would be a favourite along with generally 'non local hidden variable' theories. But I have my own limited interpretation which suggests the notion of 'variable' is wrong. Bell's inequality famously discounts local variable theorems, as evidenced in Alan Aspect's experimental confirmation of the violation of Bell's 'SET THEORETIC' inequality. The problem is, set theory deals with properties as if they are wrote on the objects, little flags on perhaps, but this is clearly absurd. Properties are dynamic, I have proven to my own satisfaction that the probabilities involved with single electron spin measurements can have a physical / geometrical interpretation. I believe, whilst not rigorously or formally doing so, I have essentially re-invented Sir Roger Penrose's 'Spinors'. I checked his book 'Road to Reality' he speaks of a clear geometric structure underlying the QM in regards of spin and its analysis through the 'spinor'.
hi sir, can you please tell me what is the difference between product and pullback? product is an object, so is pullback a process to get this object?
Pullback is an object. In this context a pullback of a pair of arrows f and g is a subset of the product of the source of f with the source of g
So how does limits and colimits refer to these ologs? 0_0
33:23 Goddam, that voice and knocking on the background scared the hell out of me. In my headsets sounded like it was next to me. It took me a while to notice it was someone in the video.
Pushouts example with cats and dogs is something like map reduce algorithm
so the question is : can we represent ALL knowledge with ontology logs and category theory ?
That is indeed the question. What is possible and how to do it ? What do you think ?
Alice and Bob color is full join from SQL:)
X = A \ B etc... mmm open and closed sets?
P😮😮of
Can you explain in cathegory theory what a woman is?