The question at 51:09 is really good, asking if compositionality is the nature of things, or simply about the limit of our ability to understand. I think it's analogous to types in programming. Are static types the true nature of computations, or just about our ability to understand them? While sound type systems are necessarily incomplete (i.e. they fail to accommodate some perfectly valid programs), the idea of static typing has proven valuable for reliably constructing and reasoning about programs. Historically, untypable but perfectly working programs have been an important source of inspiration for improving the expressiveness of static types (e.g. various forms of polymorphism, path-sensitive reasoning, delimited continuations, etc.). If one day we come across a perfectly clear and correct program that's untypable, that needs massaging just to appease the type-checker, such program will not be a sign that we should abandon the idea of static typing, nor should we dismiss such program as invalid just because a particular static discipline fails to explain it. Each instance of incompleteness can serve as an opportunity to improve the type system. Similarly, if one day we come across an effective machine learning model that cannot be explained by the current building blocks, that won't mean that the idea of searching for compositional building blocks is worthless, and we obviously shouldn't dismiss that model's existence. That will simply mean we should revise the building blocks. After all, mathematics can't derive all truths, but it's the best tool we have for understanding truths.
Yes, your reply is directly in line with the incompleteness theorem in mathematical logic. Any logic based system we come up with to reason about a domain (a domain that is not 100% knowable) will inherently be incomplete. We will have to continue learning and improving our understanding until either the domain of knowledge is completely covered (all questions are answered) and if we find something we can’t explain, figure out how we have to revise our models to explain it.
I have just found recently that the way I think and model problems always matched the way category theory does. It's a very exciting area to explore :)
@@samueldeandrade8535 I would definitely not generalize my statement. I can spot, for example, some modern / contemporary artists wouldn't think in a compositional way (although classic artists would tend to).
We've all been there, but this time, tell yourself you are taking a mental break, so that you can return harder. I have a strong sense that this is indeed worth your time. Do 20 minutes a day, take notes. You don't need to fully understand all the mathematical terms swung around. Cheers.
Here’s another amazing series that you might be interested. Search group theory and deep learning. I also left it mid way when they started talking about steerable convolutions over spherical symmetries xd. But i like meeting fellow youtubers with same mindset. Thanks for sharing your story
Good video. At time 10:35 suddenly a new word "Model of the system" added to explain compositionality. Now to understand compositionality, you need to understand what is "Model of system". Could you please first explain what is meant by "Model of the system" ?
I think there's a far more fundamental reason why Category Theory is important to AI, beyond a simple belief that it will be useful and everyone should want it. Firstly, we should understand that information is derived from knowledge, not the other way around, because information is data with a meaning, and meaning is defined in terms of knowledge. So, if Information is created from knowledge, then what is knowledge? I think the distinction is that while information is the subject of Set Theory, knowledge is the subject of Category Theory. They're like the inverse of each other. While Set Theory deals with what is in predefined sets, Category Theory is about defining the sets in terms of the relationships between sets. When we consider our existential circumstances as embedded observers in the universe, we are not afforded any absolute or privileged frame of reference. All we get to do is to observe and model the relationships between the things we observe (all measurement is comparison), and the nature of knowledge becomes obviously aligned with Yoneda's Lemma, in which an object in a category is completely determined by its relationships to all other objects... and then our biological representation of that looks like a 100 billion neurons with a trillion dynamic connections for relationships ... and then our AI representation of that looks like either neural networks or the equivalent associations by proximity in very high dimensional vector spaces (aka, embeddings in vector stores). The formalisms of Category Theory are then required to have a way to reason about knowledge, rather than the Set Theory we use to reason about information. This should provide for a comprehensive set of knowledge representations, paving the way for explainable AI, as well as improvements to model representations and structured reasoning . This distinction is also at the centre of most arguments about whether computers can ever be considered to "know" anything. They can't as compositions of information constructs where the meaning is imposed by the viewer, but they can as compositions of knowledge constructs, where everything is known in terms of everything else.
Structuralism, a 100 year old linguistic theory, finally mating with Math and Computer Science, making cute babies. Maybe we can name the first two babies Ferdinand and Claude.
The question at 51:09 is really good, asking if compositionality is the nature of things, or simply about the limit of our ability to understand. I think it's analogous to types in programming.
Are static types the true nature of computations, or just about our ability to understand them? While sound type systems are necessarily incomplete (i.e. they fail to accommodate some perfectly valid programs), the idea of static typing has proven valuable for reliably constructing and reasoning about programs. Historically, untypable but perfectly working programs have been an important source of inspiration for improving the expressiveness of static types (e.g. various forms of polymorphism, path-sensitive reasoning, delimited continuations, etc.). If one day we come across a perfectly clear and correct program that's untypable, that needs massaging just to appease the type-checker, such program will not be a sign that we should abandon the idea of static typing, nor should we dismiss such program as invalid just because a particular static discipline fails to explain it. Each instance of incompleteness can serve as an opportunity to improve the type system.
Similarly, if one day we come across an effective machine learning model that cannot be explained by the current building blocks, that won't mean that the idea of searching for compositional building blocks is worthless, and we obviously shouldn't dismiss that model's existence. That will simply mean we should revise the building blocks. After all, mathematics can't derive all truths, but it's the best tool we have for understanding truths.
Yes, your reply is directly in line with the incompleteness theorem in mathematical logic. Any logic based system we come up with to reason about a domain (a domain that is not 100% knowable) will inherently be incomplete. We will have to continue learning and improving our understanding until either the domain of knowledge is completely covered (all questions are answered) and if we find something we can’t explain, figure out how we have to revise our models to explain it.
Thanks for publishing these lectures!
I have just found recently that the way I think and model problems always matched the way category theory does. It's a very exciting area to explore :)
Well, it is not the "way you think". It is the way WE think, in general.
@@samueldeandrade8535 I would definitely not generalize my statement. I can spot, for example, some modern / contemporary artists wouldn't think in a compositional way (although classic artists would tend to).
@@0Tsutsumi0 ok. But no. And now I understand a little better what you tried to say.
Thank you for publishing these!
Absolutely amazing playlist!
thank you very much for this amazing content
There I go, starting another amazing lecture series and leaving it when it gets hard 😩
please man do not give up!!
We've all been there, but this time, tell yourself you are taking a mental break, so that you can return harder. I have a strong sense that this is indeed worth your time. Do 20 minutes a day, take notes. You don't need to fully understand all the mathematical terms swung around. Cheers.
Whitesnake
Here’s another amazing series that you might be interested. Search group theory and deep learning. I also left it mid way when they started talking about steerable convolutions over spherical symmetries xd. But i like meeting fellow youtubers with same mindset. Thanks for sharing your story
We need more content from you bring more theory
Good video. At time 10:35 suddenly a new word "Model of the system" added to explain compositionality. Now to understand compositionality, you need to understand what is "Model of system". Could you please first explain what is meant by "Model of the system" ?
What is the Zulip server URL?
Great analysis
Thanks for the great intro! Could you share the Zulip link?
Great introduction tutorial!
I think there's a far more fundamental reason why Category Theory is important to AI, beyond a simple belief that it will be useful and everyone should want it.
Firstly, we should understand that information is derived from knowledge, not the other way around, because information is data with a meaning, and meaning is defined in terms of knowledge.
So, if Information is created from knowledge, then what is knowledge?
I think the distinction is that while information is the subject of Set Theory, knowledge is the subject of Category Theory. They're like the inverse of each other.
While Set Theory deals with what is in predefined sets, Category Theory is about defining the sets in terms of the relationships between sets.
When we consider our existential circumstances as embedded observers in the universe, we are not afforded any absolute or privileged frame of reference. All we get to do is to observe and model the relationships between the things we observe (all measurement is comparison), and the nature of knowledge becomes obviously aligned with Yoneda's Lemma, in which an object in a category is completely determined by its relationships to all other objects... and then our biological representation of that looks like a 100 billion neurons with a trillion dynamic connections for relationships ... and then our AI representation of that looks like either neural networks or the equivalent associations by proximity in very high dimensional vector spaces (aka, embeddings in vector stores).
The formalisms of Category Theory are then required to have a way to reason about knowledge, rather than the Set Theory we use to reason about information.
This should provide for a comprehensive set of knowledge representations, paving the way for explainable AI, as well as improvements to model representations and structured reasoning
.
This distinction is also at the centre of most arguments about whether computers can ever be considered to "know" anything. They can't as compositions of information constructs where the meaning is imposed by the viewer, but they can as compositions of knowledge constructs, where everything is known in terms of everything else.
Structuralism, a 100 year old linguistic theory, finally mating with Math and Computer Science, making cute babies.
Maybe we can name the first two babies Ferdinand and Claude.
Is there any Google Colabs for these videos
Yes that’s what i need
👀👀
Are guys ready for a corny ass joke: functors are funtors :D