Foundations 4: Logic and Partially Ordered Sets
ฝัง
- เผยแพร่เมื่อ 2 ม.ค. 2021
- In this series we develop an understanding of the modern foundations of pure mathematics, starting from first principles. We start with intuitive ideas about set theory, and introduce notions from category theory, logic and type theory, until we are in a position to understand dependent type theory, and in particular, homotopy type theory, which promises to replace set theory as the foundation of modern mathematics. We also take an interest in computer science, and how to write computer programming languages to formalize mathematics.
In this video we introduce classical logic, using functional representations of and or, not and implies, and then we consider posets of subsets, introduce Boolean algebras, and eventually discuss Heyting Algebras as bicartesian closed categories, and discuss how our universal constructions of categorical products and coproducts give us ideas like AND and OR in intuitionistic logic. In this way we start to understand computational trinitarianism (which relates to the Curry-Howard-Lambek correspondence, and the Brouwer-Heyting-Kolmogorov interpretation).
“Hating algebra.” Got that one down in 9th grade…
I can't wait to go through the series (once I find the time), thank you so much for doing this!
Thank you so much! These videos have made some concepts so beautiful that it is hard to mentally separate them anymore! 🙂
Hi Richard! Have you checked in this regard August Stern's Matrix Logic or operator logic which is unifying all logic (fuzzy, quantum, classical and intuitionistic). As my research I connect it to algebraic topology and category theory and got to type theory! While I searched wheter someone else is doing similar I got to your videos!:)) which are great! Thanx and I hope I could give you some other hint to go and research! All the Best
Thank you Richard
Hello there Richard, Thanks for that lecture, which is extremely interesting. I've been doing things on my own that really gravitate around what you're saying. But it's hard to relate it directly to the usual math, since I've done that in a completely isolated and free-thinking manner. Basically, I'm trying to "fuse" together physics and math into a metaphysical "archetype". It involves thinking multidimensionally, and more specifically, in 4 "dimensions" (I'm putting quotes because my idea of a dimension is not quite what people usually think of - but it's hard to explain in a few words). In that context I'm trying to work with the 16 two-dimensional binary matrices, making them correspond to the 16 possible binary logic gates by changing the rules of multiplication. This in turn would be an "algebra" (it's not at all like the usual stuff - it's completely outside the scopes of linear algebra and field theory - I don't even know if I can make sense of it) for reality, thought as a projective 4-dimensional entity encompassing 4 "usual dimensions" of space and four "usual dimensions" of time, the whole thing within the context of a "hyper-cubical" co-ordinate system. I know that what I just said might sound crazy, but it's actually not that far away from the kinds of things you're talking about, so your videos allow me to make my own ideas more precise (The stuff about the product/co-product duality, and in general anything about duality).
Since I'm at it, may I ask you a question about a "mereological" model of reality I tried to formulate some years ago but never had time to develop? With your knowledge you might be able to relate it to some existing theoretical framework. The model is based on the following two definitions:
Def : an entity exists "within" some thing iff it "is contained" within that thing.
Def: an entity exists "without" or "outside of" some thing iff it "contains" that thing.
other words, an entity may be contained within something that may or may not contain it. Conversely, an entity may contain something that in turn is not contained within the former.
With that I started defining several kinds of metaphysical and physical entities, of which I termed "real objects" or "parts" of real objects (which correspond precisely to those entities that are contained within things that contain them). I then tried to determine the connections between the different types of entities through deduction from my definitions. So to make a long story short, would you know if that kind of game relates to something in existence out there in the mathematical world?
Thanks in advance for your time and feedback.
Hello. I really like physics, projective geometry and logic so what you are saying sounds very interesting. I definitely think there is something fundamental about reality and the number four. I have a friend Jos Hoebe who developed some very interesting ideas about this (related a lot to the tetrahedron), and I always enjoy contemplating original ideas about metaphysics. If you want to email me richardsouthwell254@gmail.com maybe we could further discuss your ideas. Regarding mereology, I suspect you would find topos theory very interesting. I have a video about it. In fact, everything I said in this video just consists of special cases of topos theory for the topos of sets. I would say that topos theory provided a general theory of "what a thing contains", where the "thing" may be a more complex structure than just a set. However, I'm not sure about the other direction of "what contains a thing". I think I want more examples, to see how to think about that better, in a mathematical way. The theory of subobjects of an object (from topos theory) can be dualized to give the interesting theory of partitions and epimorphisms, but I suspect that is different to what you are talking about.
@@RichardSouthwell Ok, that's great. I'll communicate with you through email. Here I'll just say two short thoughts that your reply reminded me of as I was reading it. First, 2+2 = 2x2. Sounds trivial but it's not, because I developed a train of thought in which addition and multiplication are actually dual to each other. And if that's the case, the number 4 becomes important. Additionally, 2^4 = 4^2 (note that it is also 2 tetrated to the 3rd power in fully symmetric fashion). That's also important, because it's the only time we have symmetry at the level of "exponential objects". And that ties up to fundamental logic. Another important fact is that the long radius of the 4d hypercube/tesseract is exactly its edge length (it's very significant if one considers distance in 4D in terms of the taxicab metric instead of Euclid, which is maybe more appropriate in 4D, and it hints towards an explanation of irrational numbers of the linear continuum in terms of something rational in 4 and only 4 dimensions).
Another thought that your reply spontaneously evoked to me was the following fact important to physical reality and its projective nature. If you consider a 6x6x6 "rubik's cube" and ask yourself how many cubes altogether make it up and how many faces it has, the answer is 216 each time, because a cube has precisely 6 faces. In other words, you can label each smaller cube making up that big cube on its outer surface. That to me is fundamental, because it betrays the fundamentally projective nature of 3D space. I will come back to it later when we communicate.
Thanks again for your time and talk to you later now.
Check in this relation unorinted surfaces like Moebius trip or Klein buttle and its embedding into higher dimensional logic spaces or logic mebranes like Matrix logic and topologic! With that you could modell consciousness and higher states of consciousenss and describe Gödell paradoxes like you mention! All the best
Thanks for your introduction to the fascinating depths of mathematical thought!
Just a question: These indicator functions here must stand for predicates of the first-order logic, don't they? Then, is there a way to represent the universal and existential quantifiers in categories?
Hi Richard! I have a question regarding the classifying arrow for the empty set/initial object (around 59:00). How do we "get to" false when the domains are empty sets?
If we have two sets A = {1,2,3} and B = {3,4}. We saw earlier that A × B is the set of (6) pairs {(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)}. Now we see that A × B equals A ∩ B, the intersection, which is just {3}. Is this difference just the result of the attributes of the category we are in (Set vs Sub({1,2,3,4}))?
How would I be able to see the difference when viewing it all categorically, which means I am not allowed to peek inside the set A × B. Similarly for coproducts, what would be the size of A + B? Is it 5 for a tagged union or 4 for a set union A ∪ B?
Good question. Yes, the categorical product works differently in Set and Sub({1,2,3,4}) just as you said. You can check that the objects you describe, together with appropriate "projection" arrows (which in Sub({1,2,3,4}) just signify containment), do indeed satisfy the definition of a categorical product in each of these two different categories. A similar thing happens for coproducts/sums/unions just as you said. Does this answer your question ? One key difference between Set and most other categories is that in Set, if we have arrows f and g, from A to B, and they are such that f x = g x, for each arrow x from the terminal object to A, then f = g. In other words, in Set, arrows that agree on points are equal (where a point of A is an arrow into A from the terminal object). Now the categorical product definition says is x is a point of A and y is a point of B then is a point of the product of A and B. That is why the product in Set yields a set made of pairs.
@@RichardSouthwell Thanks. I was first thinking that Set was just the same as Sub(Everything), so the same rules would apply, but that is obviously not the case as Set is not partially ordered; it contains many more arrows (every conceivable function) between to Objects.
Furthermore, I would say that Sub(X) is just a subcategory of Set, so I was thinking that all the rules of Set would apply to Sub(X), but that is also not the case I recon, as pruning away all the arrows and objects we don't need would probably invalidate some of these rules.
@@rikkertkoppes Indeed, Sub(X) can just be considered to be the subcategory of Set, on objects corresponding to subsets of X, and with arrows as the distinct injections/monomorphisms between such subsets. Perhaps informally Sub(everything) looks like the subcategory of Set with all objects, and just injection functions. Although paradoxes can arise by imagining 'everything' is itself a set.
@@RichardSouthwell It dawned on me last night and the solution is actually much more simple and elegant than I anticipated.
Both the set of pairs, call it P for now, and A ∩ B (= {3}) can be considered as candidates for the product in Set. However, P is the "better" one as we can construct a unique arrow from A ∩ B into P, specifically the map 3 ↦ (3,3) that makes it all commute in the "starfleet diagram" of a product. Note that we can also conceive of an arrow from P into A ∩ B, but we cannot make that commute.
So in Set, P is the "better" limit and is labeled A × B. In Sub(X), P does not even exist, so in that category, A ∩ B is the best limit.