math/physics dump

Hi. I'm Sharan, I'm from Singapore, doing my National Service and all, and this blog is like, a repository for my mathematics and physics notes as I attempt to study the entire edifice of mathematics and physics on my own, with a few intrusions by my other interests, namely philosophy and literature.

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"O austere mathematics! I have not forgotten you since your learned teachings, sweeter than honey, distilled themselves through my heart like refreshing waves. Instinctively, since the day of my birth, I have aspired to drink from your spring more ancient than the sun and I still continue to frequent the courtyard of your solemn temple: I, the most faithful of your initiates." - Les Chants de Maldoror, Comte de Lautréamont

Hi. As the description in my blog may tell you, this is like a collection of notes and attempts to understand various topics and features in both mathematics and physics and the intersection between the two, while I’m doing my two-year National Service. 

Lots of the concepts here have been culled from various places, including but not limited to Tom M. Apostol’s Mathematical Analysis, William Lawvere’s Conceptual Mathematics, Leonard Susskind’s The Theoretical Minimum series of books, The Variational Principles of Mechanics by Cornelius Lanczos, Classical Mechanics by John R. Taylor, and many other sources. 

If you want to see my mathematics notes, #math or #mathematics or #mathblr is the way to go. 

Physics stuff goes on #physics and #physicsblr

All study-related stuff is on the master tag #studyblr, which includes both my physics and mathematics related items. There’s a lot of overlap between the two, too.

I’ll be posting more items on #personal, which is mostly me blogging about things adjacent to most of the items here. Suffice to say, I find that mathematics, physics, and my own amateur explorations in philosophy and literature mean much to me on an emotional level, and I consider them important parts of my life (so much so that some days can be summed up with “I just finished reading X” or “Y absolutely blows my mind”. But I try to be well-rounded outside of my academic studies, too - I enjoy reading and writing, mostly science fiction (when I was 15, I attended the Creative Arts Programme for writing), and listening to music - I mostly listen to rock, and have a healthy interest in the history and culture of music, and I’ve of late been trying to learn to appreciate classical music. That’s it from me - peace out.

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My most focused work on category theory, taken from Conceptual Mathematics, that goes up to the chapter on Cantor’s diagonal argument, taken with a categorical twist. 

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Look who needs mathematics help! - me, to be precise. It’s an exercise from Conceptual Mathematics that *seems* easy, but I’m not sure about my approach. For all I know it actually is easy, and I’m not seeing something really, really obvious.

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@cecilias-cool-stuff makes a great point in a comment to my dashed-off post on this exercise from Conceptual Mathematics. It’s a great point that the existence of such f such that g = rfs isn’t exactly the most acceptable of academic hypotheses. It makes sense that for given g, sgr is an endomorphism of X. 

And being an endomorphism of X, there’s some map x from a test-object T to X such that sgrx = x. Now it remains to find some a such that ga = a. Looking at a diagram like this, it seems to me that the answer is obvious, something that should be findable just by composing graphs, but I’m at a loss as to how to use the fixed point property. I feel that there’s some sort of error in how I’m drawing up my diagram, thinking about the problem - I have the feeling that if I can just get this part the problem would be solved.

Feels like there’s some issue here - which of these maps actually compose? Could I say that a = sx? Hmph. 

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Look who needs mathematics help! - me, to be precise. It's an exercise from Conceptual Mathematics that *seems* easy, but I'm not sure about my approach. For all I know it actually is easy, and I'm not seeing something really, really obvious.

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I was free, so I got around to proving a short exercise, a corollary to Brouwer's fixed point theorem, from the book Conceptual Mathematics.

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26 May, 2022. 

This happens to be the first of many theorems of Conceptual Mathematics by William Lawvere and Stephen Schanuel which I still have issues with. It certainly means that I need greater study in this area. Here’s the category theoretical presentation of Cantor’s diagonalization argument. While I can follow each step by step, especially if I’ve had a cup of coffee, I admit I’m not very sure what’s the significance of an set that may or may not have an endomap with a fixed point. It obviously plays an important role in how Lawvere conceptualizes the proof! I mean. I really have to look this over again and again until I can totally grok it, as Michael Valentine Smith would say. The proof itself is apparently Lawvere’s brainchild, apparently.  

(See: TIL Gödel's incompleteness theorem, Russell's paradox, Cantor's theorem, Turing's halting problem, and Tarski's undefiniability of truth are all mere instances of one theorem in category theory: Lawvere's fixed point theorem)

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Equalizers and graphs (not graphs as in graph theory, but the Cartesian diagrams), in Conceptual Mathematics by William Lawvere and Stephen Schanuel. 26 May 2022.

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25 May 2022. Some notes on the product of mathematical objects in categories that have products.

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19 May 2022 - here I am taking notes on figures and cofigures, and incidences between figures, as presented in William Lawvere and Stephen Schanuel’s Conceptual Mathematics.

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5 May 2022. Here was me, hard at work, attempting to reconstruct from memory the categorical proof as given in Bill Lawvere’s Conceptual Mathematics of the Brouwer fixed-point theorems. Most was done straight from memory, though I did consult my workings in my earlier pages in my journals, to make sure that I got everything right without reference to the textbook.

[ishasideas]

How? Just how?

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I'm not sure what you mean here, actually

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18 May 2022. Here we come to the meat of Conceptual Mathematics, by William Lawvere and Stephen Schanuel. In quick succession, we have:

- initial and final objects in categories

- products in categories

- binary operations in a category

- Maps in a category parameterized by a particular object

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It’s an interesting note from William Lawvere’s Conceptual Mathematics on categories and diagrams. Much has been said about the importance of diagrams in category theory - Rocco Gangle in Diagrammatic Immanence, has related the diagrammatic character in category theory with the philosophies of immanence found in Baruch Spinoza, Charles Sanders Peirce, and Gilles Deleuze. This is interesting as it essentially gives a formalization of “diagram” (if we take graphs to be our diagram, graphs as in graph theory, not plots on a 2D plane), and gives a way to interchange diagrams and categories. Very much food for (philosophical and mathematical) thought.

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17 May 2022. William Lawvere, Conceptual Mathematics. A constant theme in Conceptual Mathematics is the usage of simpler structures to investigate larger, more complicated structures, in the same category etc. using a set to investigate the category of sets, using irreflexive directed graphs to investigate irreflexive directed graphs, and so on, using maps from and to “test objects” to do so. 

Here, for example, the set of maps from the set of natural numbers with the addition endomap to another set Y with an endomap on it can be used to investigate the structure of the object Y in the category of endomaps. It’s interesting as it 1) reframes messy diagrams into a function on the natural numbers, which makes investigation simpler and 2) provides semi-global characteristics i.e not just of individual elements, by showing the existence of n-length loops if a particular map exists. In general, all the maps x from a “test figure” object A to an object X provide all the A-figures in X, whether it comes to sets, irreflexive directed graphs, or endomaps.

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17 May 2022. Here I am wrangling about what exactly it means to interpret an object, or a particular thing, in terms of a particular category. Also some bookkeeping - how are categories specified? Based off the textbook Conceptual Mathematics

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Here is a picture of me trying to think with pen and paper, based on the tantalizing crumbs left behind by Lawvere in Conceptual Mathematics, on exactly what functors are. Conceptual Mathematics is nominally a book for those with no background in mathematics, but since it treats topics that are very advanced and very new, there can sometimes be oddities in the way the material is presented. Aside from a few notes scattered in the front, which is what this is based on, most of the presentation of key topics like functors and presheaves are concentrated in the last few chapters, where difficulty zooms up hyperbolically and frankly I’m very lost and I need to re-read those segments. Here I attempt to classify and poke around with the basic functors presented in the chapter on endomorphisms, trying to figure out what sort of structure functors have. Very interesting bits of thinking, and also probably very pretentious, but of little help when left- and right-adjoint functors jumped out at me later. Ugh. Also, some notes on retractions and injections

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5 May 2022. Here was me, hard at work, attempting to reconstruct from memory the categorical proof as given in Bill Lawvere’s Conceptual Mathematics of the Brouwer fixed-point theorems. Most was done straight from memory, though I did consult my workings in my earlier pages in my journals, to make sure that I got everything right without reference to the textbook.

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Logicism as a philosophy of mathematics, logicism as a serious program of metamathematics, I hear, because of the incompleteness theorems of mathematics of Gödel and others, and because of ZFC overturning Russellian type theory. (Here I am referencing the Stanford Encyclopedia of Philosophy's article on "Logicism and Neo-Logicism".) Regardless, I still hold some interest in the logical work of Bertrand Russell and Bertrand Russell, especially as I am interested in how it connects up to their respective philosophical systems - Russell's neutral monism and Whitehead's metaphysics of process (as given in "Process and Reality", and other books, and so ably explicated in Isabelle Stengers' “Thinking With Whitehead”, originally published “Penser avec Whitehead” in French). Even despite that, the whole system of the ramified theory of types sounds like an excellent tool to make my thinking in mathematics more discerning.

To deal with the issue of self-reference, R&W build up a system in which each proposition has a given order, and can only take lower order functions as arguments. This is the basic theory of types. (I wonder what this has to do with the first-order, second-order, ..., nth order logic studied by logicians and mathematicians.) Of course, this does not completely get rid of self-reference. And there are issues with statements like the law of the excluded middle, which we want to apply to all statements, but that would be impossible in this system.

There are still complications with variables, so now in the ramified theory of types the order of the main function is determined by the highest order of any variable in it, whether its own or of an inner function. There is now a creation of a new hierarchy, too, an infinite number of sublevels. In the ramified theory types, we can make only some statements about some, not all the functions that can take an object of any order as an argument. R&W fix it with the axiom of reducibility (which I hear has had some controversy, among logicians and mathematicians and philosophers), by collapsing all the levels into one level. So the statement "all hereditary properties n has" now becomes a conjunction of the statements "n has a hereditary property x" and "n has a hereditary property y" and so on. While I am sure this is all well and good, I admit this makes me feel a little uncomfortable - is it really possible to list all properties, hereditary or not, of a particular number? I worry it might run into a problem with Gödel-ish arguments. Hmpf.

The end here is R&W's "no-class" theory of classes, where classes here are simulated by prepositions. I now suppose there no longer can be collections of sets anymore (eg the theorems of analysis that talk about countable and uncountable covers of sets, etc.) and segues somewhat into Russell's theory of descriptions.