The power of this thread will be that if you survive it all of higher mathematics, computational theory, and Logic open up to you. Category Theory is a Rosetta Stone.
That's my true intention, a few pay attention and then later down the line in their horror they find themselves comprehending things I say despite having no interest in it.
But it is easy to think we’re in agreement, when really we’re not. Modeling our thoughts on heuristics and pictures may be convenient for quick travel down the road, but we’re liable to miss our turnoff at the first mile. The danger is in mistaking our convenient conceptualizations for what’s actually there.
Data. Hard evidence. Physical reality. It is here that science touches down and heuristics evaporate.
So let’s look again at the diagram on the cover. It is intended to evoke an idea of how science is performed. Is there hard evidence and data to back this theory up? Can we set up an experiment to find out whether science is actually performed according to such a protocol?
To do so we have to shake off the stupor evoked by the diagram and ask the question: “what does this diagram intend to communicate?”
This though is where it aligns again...
This course is an attempt to extol the virtues of a new branch of mathematics, called category theory, which was invented for powerful communication of ideas between different fields and subfields within mathematics. By powerful communication of ideas I actually mean something precise. Different branches of mathematics can be formalized into categories. These categories can then be connected together by functors. And the sense in which these functors provide powerful communication of ideas is that facts and theorems proven in one category can be transferred through a connecting functor to yield proofs of an analogous theorem in another category. A functor is like a conductor of mathematical truth.
I must be reading it wrong, but did this dude just try to tell me that "Math has found a way to speak in sentences" in an unnecessarily complex way?
I believe that the language and toolset of category theory can be useful throughout science. We build scientific understanding by developing models, and category theory is the study of basic conceptual building blocks and how they cleanly fit together to make such models. Certain structures and conceptual frameworks show up again and again in our understanding of reality.
Okay now it just sounds like sorting logic into boxes, which has been around for a while right?
For example, when we speak of a material that is both lightweight and ductile, we are intersecting two sets.
But what is the use of even mentioning this fact? The answer is that when we formalize our ideas, our understanding is almost always clarified. Our ability to communicate with others is enhanced, and the possibility for developing new insights expands. And if we are ever to get to the point that we can input our ideas into computers, we will need to be able to formalize these ideas first.
Okay now it sounds like programming.
I like the author's enthusiasm and flow in the Introduction, it keeps my attention and translates more readily than the last guy, but some part of me is still like "Dude needs an editor".
But it is easy to think we’re in agreement, when really we’re not. Modeling our thoughts on heuristics and pictures may be convenient for quick travel down the road, but we’re liable to miss our turnoff at the first mile. The danger is in mistaking our convenient conceptualizations for what’s actually there.This is very true, but typically from here is where I've leaned on perspective taking within the conceptual adjacency of similar cases, be that disorder clusters, subcultural subscriptions, etc. Psychology and Acting are always like "But what do they mean by that, and how can you become like it?" rather than "But what do you mean by that, and how could others come to understand you rather than presume your meaning?".
The writer's style gives me the impression that it is their intent to communicate more than interpret, which is interesting for me to read when I am comparatively more used to converting nonsense into common variables, rather than finding the best way to portray common variables to translate towards even those outside of convention.
I rather enjoy Heuristics in that they, like Traits, have patterns and origin points which can prove to be symptoms of larger clusters of behavior. While this can tend to make the subject look foolish and silly once it's reduced to baseline terms without as much flowery language, it does otherwise illustrate the differences that led to them portraying a similar set of events in their own unique way.
In a sense, people are so prone to patterns that I'm typically categorizing them in relation to an understood idealized canon of "Normalcy", or "Ordered Thinking", rather than categorizing the concepts they are talking about. I'm effectively an interpreter while this guy is trying to be a streamlined complex speaker, it feels similar yet backwards.
Data. Hard evidence. Physical reality. It is here that science touches down and heuristics evaporate.That is so idealistic, heuristic influence on our data is near-inescapable.
Category Theory is just another formalism that allows us to interpret knowledge in a novel way, but as with any formalism its faces the paradox you seem to be alluding to here.
The formalism itself allows us to concretely talk about certain things with as little ambiguity as we can muster, which allows for very clear and concise communication of ideas. However, by removing ambiguity we sometimes also remove some of the richness inherent to the subject.
This is a long known issue in formal language theory and one of my favorite expositions on this paradox comes from beginning of Stephen Kleene's Mathematical Logic,
Kleene said:Now we are proposing to study Logic, and indeed by mathematical methods. Here we are confronted by a paradox. For, how can we treat logic mathematically (or in a systematic way) without using logic in the treatment?
The solution of this paradox is simple, though it will take some time before we can appreciate fully how it works. We simply put we are studying into one compartment, and the logic that we are using to study it in another. Instead of "compartments", we can speak of "languages". Where are studying logic, the logic we are studying will pertain to one language, which we call the object language, because this is the object of our study. Our study of this language and its logic, including our use of logic in carrying out the study, we regard as taking place in another language, which we call the observers language.
In our study of the formal system known as category theory we must be aware of the distinction between that formal system and how we talk about it. This distinction is necessary because the formal system itself is fine-tuned via definition to be unambiguous, yet it is necessary that we use our everyday speech, which is ambiguous, in our study of it especially if we are to talk about the subject together.
This is true once we go beyond category theory and start applying it to certain subjects whether it be mathematics, computer science, physics, or psychology. The very process of distilling a set of elements from a subject into categories and mapping the relationship between those categories relies on ambiguity inherent in ourselves, and to
This course is an attempt to extol the virtues of a new branch of mathematics, called category theory, which was invented for powerful communication of ideas between different fields and subfields within mathematics. By powerful communication of ideas I actually mean something precise. Different branches of mathematics can be formalized into categories. These categories can then be connected together by functors. And the sense in which these functors provide powerful communication of ideas is that facts and theorems proven in one category can be transferred through a connecting functor to yield proofs of an analogous theorem in another category. A functor is like a conductor of mathematical truth.I must be reading it wrong, but did this dude just try to tell me that "Math has found a way to speak in sentences" in an unnecessarily complex way?
Not in sentences but in Diagrams.
Surely you are familiar with basic Algebra, you've at least experienced it at one point. If we treat algebra as an object of study itself we find that an Algebra is a set of objects and the rules of manipulating those symbols while maintaining some equivalence. Usually the objects in question our numbers, but over the 19th and 20th century those objects became increasingly abstract until literally branches of mathematics itself were the objects and the operations between them were mappings (or ways to symbolize certain relationships). It turned out if you form the branches into categories and mappings, which is a kind of diagram, once branch would be found to be structurally equivalent to another branch and as such proofs of theorems could be translated between them. In short, category theory is the Algebra of Mathematics itself.
The real power is that we can generalize further, and instead of making making Mathematics our object we can choose another. Hence, effective applications of category theory have been used in Physics, Computer Science, and even Music.
The Topos of Muisc: Geometric Logic of Concepts, Theory, and Performance
It seems Davids goal via his Ologs formalism is to generalize category theory further by making it possible to apply it to traditionally less formal subject matters, that is he hopes to show that those subjects are valid objects to be treated algebraically.
Category Theory is just another formalism that allows us to interpret knowledge in a novel way, but as with any formalism its faces the paradox you seem to be alluding to here.
Until we can ask another lifeform with fundamentally different senses what they are perceiving, we cannot get away from it. As is we can only subdivide between different kinds of people, rather than anything outside of people.
We are more likely to achieve this with AI than Aliens or Undersea Mysteries.
The formalism itself allows us to concretely talk about certain things with as little ambiguity as we can muster, which allows for very clear and concise communication of ideas. However, by removing ambiguity we sometimes also remove some of the richness inherent to the subject.
So... sentences?
This is a long known issue in formal language theory and one of my favorite expositions on this paradox comes from beginning of Stephen Kleene's Mathematical Logic,
Kleene said:Now we are proposing to study Logic, and indeed by mathematical methods. Here we are confronted by a paradox. For, how can we treat logic mathematically (or in a systematic way) without using logic in the treatment?
The solution of this paradox is simple, though it will take some time before we can appreciate fully how it works. We simply put we are studying into one compartment, and the logic that we are using to study it in another. Instead of "compartments", we can speak of "languages". Where are studying logic, the logic we are studying will pertain to one language, which we call the object language, because this is the object of our study. Our study of this language and its logic, including our use of logic in carrying out the study, we regard as taking place in another language, which we call the observers language.
The structure of this language... I feel like it could be done in a lot less words: "We are using math as a language, weird right?".
In our study of the formal system known as category theory we must be aware of the distinction between that formal system and how we talk about it. This distinction is necessary because the formal system itself is fine-tuned via definition to be unambiguous, yet it is necessary that we use our everyday speech, which is ambiguous, in our study of it especially if we are to talk about the subject together.
This is true once we go beyond category theory and start applying it to certain subjects whether it be mathematics, computer science, physics, or psychology. The very process of distilling a set of elements from a subject into categories and mapping the relationship between those categories relies on ambiguity inherent in ourselves, and to
The enthusiasm here makes me feel like I must be missing something fundamental towards the idea, as this looks to me like trying to use one medium to accomplish another, like trying to use paint to look like a charcoal art piece.
A part of me starts to become like "Why not use charcoal?", but right now I am on the denser end of the Dunning Kruger model for this subject. It likely looks simple to me because I do not understand it.
This course is an attempt to extol the virtues of a new branch of mathematics, called category theory, which was invented for powerful communication of ideas between different fields and subfields within mathematics. By powerful communication of ideas I actually mean something precise. Different branches of mathematics can be formalized into categories. These categories can then be connected together by functors. And the sense in which these functors provide powerful communication of ideas is that facts and theorems proven in one category can be transferred through a connecting functor to yield proofs of an analogous theorem in another category. A functor is like a conductor of mathematical truth.I must be reading it wrong, but did this dude just try to tell me that "Math has found a way to speak in sentences" in an unnecessarily complex way?
Not in sentences but in Diagrams.
What's the difference? Whether it's medical terminology or math statements, they are all sentences.
Category Theory by an asbolute surface level glance at it looks like a means of communication through different symbols and structure. When looking more at the "what" rather than the "how", they begin to look like eachother, albeit with a different foundational codex.
Surely you are familiar with basic Algebra, you've at least experienced it at one point.
If we treat algebra as an object of study itself we find that an Algebra is a set of objects and the rules of manipulating those symbols while maintaining some equivalence. Usually the objects in question our numbers, but over the 19th and 20th century those objects became increasingly abstract until literally branches of mathematics itself were the objects and the operations between them were mappings (or ways to symbolize certain relationships). It turned out if you form the branches into categories and mappings, which is a kind of diagram, once branch would be found to be structurally equivalent to another branch and as such proofs of theorems could be translated between them. In short, category theory is the Algebra of Mathematics itself.
Yes, but if we were to say "Four plus four equals eight", or even "X times 4.2 equals 15.54, solve for X", I am accomplishing the same thing as a math equation within a less efficient codex framework.
From where I am now, the author seems proud to say that they learned how to speak a semi-conventional language through math, and are surprised at it's room to translate into other fields. As someone with no real knowledge over the subject, I'm stuck wondering what the point of it is beyond how to translate it towards machines and shorthand. It gives me mild flashbacks to learning things like how Arrays and Variables work in programming.
The real power is that we can generalize further, and instead of making making Mathematics our object we can choose another. Hence, effective applications of category theory have been used in Physics, Computer Science, and even Music.
So... yes, it's a celebration over streamlining?
I am not diminishing the importance of it, less is more when it comes to getting a message across, but that's essentially it right?
The Topos of Muisc: Geometric Logic of Concepts, Theory, and Performance
It seems Davids goal via his Ologs formalism is to generalize category theory further by making it possible to apply it to traditionally less formal subject matters, that is he hopes to show that those subjects are valid objects to be treated algebraically.
A lot of this makes it sound more linguistic than mathematical, which is interesting on it's own when they could have potentially chosen existing conventions to try to get the same points across.
As posted elsewhere, Milewski has a YouTube playlist for category theory, which might be a good introductory format versus dry text.
The formalism itself allows us to concretely talk about certain things with as little ambiguity as we can muster, which allows for very clear and concise communication of ideas. However, by removing ambiguity we sometimes also remove some of the richness inherent to the subject.
So... sentences?
Categories as things in themselves are highly abstracted symbolic representations of sentences, and usually by the the point in which you've reached the the declaration of a category you're several symbolic instantiations above actual sentences.
The above is at least true for Mathematics, Physics, and Computer Science.
It does depend on what you mean by sentence. For instance, I can program a function that describes the identity of an object. That program can be described in multiple programming languages, all of which have different, more or less, syntactical styling that can be considered a symbolic schema. Are program lines or programs themselves sentences? I can also describe that program using what we colloquially consider sentences. The symbolic schema that is a program can be stated via category theory, specifically via an identity morphism.
English
A function called identity that takes in an object x of any type and returns a new value x of the same type using a type construction function.
Python
def identity(x):
return type(x)(x)
Category Theory
x---Id--->x
What can be considered a sentence here? They all certainly express the same idea. Though, it is far easier to claim that my constructions are consistent when using a formal language than something like plain English.
This is a long known issue in formal language theory and one of my favorite expositions on this paradox comes from beginning of Stephen Kleene's Mathematical Logic,
Kleene said:Now we are proposing to study Logic, and indeed by mathematical methods. Here we are confronted by a paradox. For, how can we treat logic mathematically (or in a systematic way) without using logic in the treatment?
The solution of this paradox is simple, though it will take some time before we can appreciate fully how it works. We simply put we are studying into one compartment, and the logic that we are using to study it in another. Instead of "compartments", we can speak of "languages". Where are studying logic, the logic we are studying will pertain to one language, which we call the object language, because this is the object of our study. Our study of this language and its logic, including our use of logic in carrying out the study, we regard as taking place in another language, which we call the observers language.
The structure of this language... I feel like it could be done in a lot less words: "We are using math as a language, weird right?".
The novelty is not that Mathematics is a language, that is an idea as old as the Greeks in so far as a direct reference exists. The novelty lies in the distinction between the language Mathematics and the metalanguage used to talk about Mathematics.
In our study of the formal system known as category theory we must be aware of the distinction between that formal system and how we talk about it. This distinction is necessary because the formal system itself is fine-tuned via definition to be unambiguous, yet it is necessary that we use our everyday speech, which is ambiguous, in our study of it especially if we are to talk about the subject together.
This is true once we go beyond category theory and start applying it to certain subjects whether it be mathematics, computer science, physics, or psychology. The very process of distilling a set of elements from a subject into categories and mapping the relationship between those categories relies on ambiguity inherent in ourselves, and to
The enthusiasm here makes me feel like I must be missing something fundamental towards the idea, as this looks to me like trying to use one medium to accomplish another, like trying to use paint to look like a charcoal art piece.
A part of me starts to become like "Why not use charcoal?", but right now I am on the denser end of the Dunning Kruger model for this subject. It likely looks simple to me because I do not understand it.
What's the difference? Whether it's medical terminology or math statements, they are all sentences.
Category Theory by an asbolute surface level glance at it looks like a means of communication through different symbols and structure. When looking more at the "what" rather than the "how", they begin to look like eachother, albeit with a different foundational codex.
Yes, but if we were to say "Four plus four equals eight", or even "X times 4.2 equals 15.54, solve for X", I am accomplishing the same thing as a math equation within a less efficient codex framework.
From where I am now, the author seems proud to say that they learned how to speak a semi-conventional language through math, and are surprised at it's room to translate into other fields. As someone with no real knowledge over the subject, I'm stuck wondering what the point of it is beyond how to translate it towards machines and shorthand. It gives me mild flashbacks to learning things like how Arrays and Variables work in programming.
So... yes, it's a celebration over streamlining?
I am not diminishing the importance of it, less is more when it comes to getting a message across, but that's essentially it right?
The use of and interaction with Category Theory reveals new information about the kinds of objects you apply to. Using your analogy, its more like creating the same image using charcoal and paint and noticing how differently you perceive the two, though in this case a more extreme analogy may be in order where its the difference in information that painting David and sculpting David provides.
I'll provide an example from mathematics. All of mathematics, every single result, can be constructed using different underlying schema or theories. Set Theory is the most popular of these theories and it is all about what constitutes membership of elements in a set via shared properties. Category Theory is fundamentally about compositional structures. Both can be used to describe and explore Algebra as example, but I will find it a lot easier to explore and come up with new novel applications of algebra using Category Theory than I will set theory. I may have never come across certain ideas unless I used category theory over set theory, as such there's something revelatory about it (and the same goes for set theory or whatever other foundational system you desire). Interestingly, if I come across a object using category theory i could still construct that object using set theory, I just may have never discovered it if set theory was my only conceptual language of mathematics.
Effectively, different foundational theories emphasize certain things over others which in turn causes different revelations about the same kinds of objects.
A lot of this makes it sound more linguistic than mathematical, which is interesting on it's own when they could have potentially chosen existing conventions to try to get the same points across.
I view Mathematics as formal language theory.
AliceInWonderland said:Are program lines or programs themselves sentences?
Essentially?
The novelty is not that Mathematics is a language, that is an idea as old as the Greeks in so far as a direct reference exists. The novelty lies in the distinction between the language Mathematics and the metalanguage used to talk about Mathematics.
So streamlining towards a different goal through a different codex of linguistics?
It's all shorthand, but this form of it is by design meant to achieve a faster more efficient outcome over a different set of priorities. While writing it out in plain english can accomplish the same things, it is often messier than linguistics dedicated to it, like medical terminology.
While to me, putting together multiple root phrases into a larger word-cluster seems silly and inefficient in most cases, for someone in the field of medicine those absurdly long names save them the time of having to describe entire sentences.
I may have never come across certain ideas unless I used category theory over set theory
Like what?
Seriously getting Comp Sci flashbacks over some of this, now that you mention it as one of it's applications. To borrow your Painting vs Sculpting example, this seems like comparing what a paintbrush versus an art algorythm could do over how it gets to the final product moreso than the final product itself, a study over function.
Topos Institute channel on youtube is all Category Theory content and features a lot of David Spivak as well, this video provides a solid upshot of CT.
...how the fuck did he just make Symmetry seem cooler? Turning every angle into a color array to demonstrate all the ways it could still look the same is a super cool idea, even if visually demonstrating it is less efficient than just saying 6^2.
How it becomes translated into thought processes through existing functions and heuristics from cross reference as comparison is weirdly fascinating (that and it being a video is making this way easier for me). This video now has me feeling like Linguistics are inherently Categorical, rather than Category Theory being Linguistic.
