I wrote this letter as an introduction of myself at the Azimuth Project online forum.
David, John and all, Hello!
I became interested in the Azimuth Project because of John Baez's mathematical expositions. I'm truly grateful for his videos and blog posts. I'm currently interested to get a grip on the big picture in mathematics, and he makes me feel that he is too, and that it's within reach, however outrageous that may seem. I especially liked his videos of his favorite numbers because he is able to toss a line from the most concrete to the most abstract. Coincidentally, the numbers 8 and 24 are at the heart of my own explorations. And he knows so much about subjects that I think I should, too. Subjects I didn't even knew existed! Finally, he wants to save the planet. Well, I'm part of the planet, so maybe he will care about me, too.
But I was definitely encouraged by David Tanzer's wiki page. He writes: "My plan is to study math and science and then teach it to colleagues in software development. We need more scientists to solve the myriad of problems that beset the human race, and the world of programmers looks like a good recruitment base for the sciences. In the process I hope to develop myself as a scientist!" Kirby Urner at the Math Future google group expressed a very similar strategy. And my own wish is to learn more math. So I thought I might have a chance here.
For the sake of being myself, I will present my own goals here, as I imagine them, starting with the most reasonable and continuing towards the ever more unreasonable.
My starting point is that "advanced math literacy" might be a very practical endeavor for Azimuth Project. Mostly because John excels at fostering that and also at attracting people like David and me who care about that. But also because a lack of such literacy is arguably a major reason why global warming is not being taken seriously and why we have a shortage of scientists who might address it.
Personally, in learning advanced math, I am making a map of the areas in math. I have set up a page Andrius Kulikauskas at the Azimuth Project wiki where I have started writing about that: http://www.azimuthproject.org/azimuth/show/Andrius+Kulikauskas Indeed, I link to a map of some 200 nodes that I drew with yEd software. I would very much appreciate discussion to make my map as comprehensive and profound as possible. And also to consider how such a map could be helpful for math literacy and for saving the planet. So that seems a most reasonable goal here. I have no idea where I should pursue that at the wiki so for now I'm active at my own page there.
I would also love help to learn particular areas of math, many of which John writes and talks about. My big goal in life since childhood has been "to know everything and apply that knowledge usefully". Along the way I got a B.A. in Physics at the University of Chicago (1986) and a Ph.D. in Math at UCSD (1993). In 2014, I wrote an illustrated summary: http://www.selflearners.net/wiki/Truth/Book I'm now thinking through a detailed book which I plan to write next year. As part of that, I want to show that my philosophy can say useful things about Math and Physics, especially the big picture. So there is a lot for me to learn, even though I need to be very selective. That's where I appreciate help, fool that I am, and that's where I find John and Urs's writings so helpful.
My own background is in algebraic combinatorics. I thought of combinatorics as the "basement" of math from which arose the concrete objects of math. I became interested in the question, Why is it that the symmetric functions have infinitely many bases, but precisely six bases seem interesting to the human mind? (elementary, homogeneous, power, monomial, Schur, forgotten). I realized an interesting fact that we can calculate the symmetric functions of the eigenvalues of an arbitrary matrix (the determinant being the product and the trace being the sum, in particular). We thus generate collections of all manner of cycles, walks, words, Lyndon words. And if we set the off-diagonal matrix elements to zero, then the eigenvalues are on the diagonal, and so we recover the original symmetric functions. So this is a strange case where specializing (to the eigenvalues) makes a function more general. And since the symmetric functions are the basis for all of combinatorics (everything that has labels which can be permuted) I was able to show how that could be taken to arise from facts about arbitrary matrices. http://www.ms.lt/derlius/AndriusKulikauskasThesis.pdf
One area that I would like to learn the key ideas of is category theory. In my philosophy, perspectives play an important role. I imagine there must be an "algebra of perspectives". For example, if a lost child is smart, then they realize that they are the child and not the parent, and so they should not look for their parent, but should rather go where their parents think they would be. We thus have "the child's view of the parent's view of the child's view of the parent's view of the child's view". In which case, the child's view and the parent's view coincide, even though they are not in communication! So this is the kind of thing that I think I could model with category theory if I was fluent in it.
Another area to learn is Lie groups and Lie algebra because that seems central to Math but certainly for Physics. For my philosophy, I studied about 200 different ways that I had figured things out. I came up with a system of 24 ways, a "house of knowledge". Then I realized that I could come up with such a system for discovery in Math, which I did. Now I'd like to do that for Physics, for which I need to learn more Math, starting with tensors.
In Math, I read George Polya's book "How to Solve It". He has a "pattern of two loci" which is at work in trying to draw an equilateral triangle given one side AB. You draw circles centered at A and B and see where they intersect. I realized that when we solve this problem, what is happening in our mind is that we're working with a lattice of conditions given by the plane (no conditions), circle A (one condition), circle B (another condition), and the two intersections (both conditions). Thus the crux of the discovery has nothing to do with triangles but is given by a math structure. That structure is implicit in our mind. So this is a way to see what kind of math is natural, what kind we actually leverage in our minds as we do math. I studied Paul Zeitz book and found 24 such structures/methods. I wrote them up here: http://www.ms.lt/sodas/Mintys/MatematikosRūmai and I should work on that further.
I also want to do that for Physics. I'm studying and sorting physics experiments listed at Wikipedia. I also need to learn a lot of related Math to appreciate how things are figured out in theoretical physics.
Another of Math/Physics that I want to learn key ideas from include Entropy because I want to sharpen my concepts of grace and justice which I picture entropically as the distinction between an open system (fed by an infinitely loving sun) and a closed system (where everything tends to fall apart). Real life is quite ambiguous and I'd like to model that ambiguity and its implications.
I'm intrigued to learn some Geometric Representation Theory because I want to know how different areas in Math are linked. And I realized from my map that geometry must be quite fundamental. But who can tell me, What is geometry? I'm guessing that it's the way that a lower dimensional space is embedded in a higher dimensional space. And tensors must be crucial in that they are the "trivial" answer to that, but especially as they model the duality between the top-down and bottom-up views. So I need to learn tensors well. I'm not there yet.
I also hope at some point to be able to share useful ideas in Math that originate from my philosophy. For example, I first started thinking about Network Theory when I visited a hermit here in Lithuania in 1998. He told me of his vision that every school child's education consist of writing three books: one sequentially (like blogs today), one hierarchically (a thesaurus) and one as a network (an encyclopedia). I'm always looking for such systemizations so that I intrigued me. I decided to collect examples of such visualizations. To my surprise, I never found any example that would involve just one of these structures. Instead, they always came in pairs:
I wrote a paper about that "Organizing Thoughts into Sequences, Hierarchies and Networks": http://www.ms.lt/derlius/organizingthoughts.html So I think that could be relevant for your Network theory. One way to test this is to go through yEd's example graphs at: http://www.yworks.com/products/yfiles/gallery
Also, I went through all the paradoxes listed at Wikipedia and was able to organize a taxonomy based on the visualizations above. I write about them in my book, pages 167 to 175, "Chronicle of How God Grows Warm".
In Recursive Function theory, I believe the Arithmetic hierarchy is very important. In particular, I'd like to learn the Yates Index theorem which says that the triple jump tells you everything about the recursive functions. In my philosophy, a key concept is the "divisions of everything", an example of which is the division of everything into four perspectives (whether, what, how, why), which comes up whenever we deal with issues of knowledge. So I think the triple jump is an example of how the "why" relates to "whether", which is to say, consciousness.
These "divisions of everything" are a key structure in my philosophy which I write about in my book, pages 94 to 125, "2) I Learn from Them". Well, it turns out that there are eight of them, with the eighth collapsing into the nullth. And there are three operations on them, +1, +2 and +3. Well, I was quite intrigued to learn about Bott periodicity, Clifford Algebra periodicity and clock shifts. So that's the kind of coincidence that makes me want to learn more math. And perhaps some day it will turn out that my philosophical insights can inform the math that models it.
I want to know everything so I spend my best energies trying to imagine God's point of view. For example, I'm trying to describe God's dance which is the kernel from which all arises. Basically, God asks, Is God necessary? And pulls aside, making the most disadvantageous conditions, but then appears anyways, as in a proof by contradiction. So there is God who understands and also God is comes to understand. And how do they know they are the same God? Because they both understand the same God. So this is a trinity, basically the Father, Son and Spirit. But this is from the Father's point of view, God as I, who loves himself. There is another 4+4 = 8-fold structure (like the prayer "Our Father") which says the same from the Son's point of view about God as You, who loves each other. And there is another 4+6 = 10-fold structure (like the 10 commandments) that says the same from the Spirit's point of view about God as He who loves all. And then these three unities (of God, of individual, of individuals) are united by people through a three-cycle (the division of everything into three perspectives): taking a stand, following through, reflecting. So that is a total of 3 + 8 +10 + 3 = 24 perspectives. So we have the same favorite numbers.
Mathematically, I think of God as the state of contradiction in which all things are true. So then I imagine how that contradiction divides itself, for example, into two parts to give a proof by contradiction (if God exists, then God exists (as in the spiritual world); if God does not exist, then yet God exists (as in the physical world). And into three parts, as in Godel's incompleteness theorem (a system that is complete, consistent and sufficiently creative will be contradictory), and so on. Until by the division of everything into seven perspectives you get a logical system that is not contradictory. But adding the eighth perspective (all are good & all are bad) means that the system is empty and we collapse back to the division of everything into zero perspectives, namely God.
I think the goal of knowing everything, the practical application, is to organize a culture of independent thinkers (cranks?), what Jesus called the kingdom of heaven. From 1998 to 2010, I led Minciu Sodas, an online laboratory for independent thinkers, from which I tried to make a living, but I ultimately went bankrupt. Our most amazing activity was the Pyramid of Peace in Kenya in 2008 in which we organized 100 peacemakers on the ground in Kenya and 100 online assistants to avert genocide there.
I think it speaks so well of you that you care to address global warming. That is both loving and rational. But then I will conclude with my own perspective. If a drunk driver is steering us over a cliff, then I personally don't think the problem is getting a better car. Similarly, I don't think that technology is key in this case. I think the big problem is that we, people, aren't brothers and sisters to each other. If we know how to care about each other, help each other, love and share, then we will be able to tackle any challenge. As things stand, the problem with global warming is that unseasonable weather, warm or cold, will stress the poor, multiply the refugee crisis, create nationalistic insular backlashes, elect Putin-type leaders and sycophantic parties in the US and other countries, make terrorist states attractive and spark regional nuclear wars.
Rationally, I think that it makes sense to ask whether there is a God or not, real or imaginary. If there is a God, then she may care about global warming and we should link up with her. If she's just imaginary, well, she is still a concept who can unify many of us, whatever that concept may be. If she's completely irrelevant, well, then, what about extraterrestrial aliens? Our galaxy is only 100,000 light years across and so if life like us exists there, then surely it did 1 billion years ago in which case it had time to colonize the entire galaxy. Which is to say, it is 1 billion years more advanced and either it seeded us or finds us special and interesting. It's not going to just not care about us. If they don't exist and we're all alone, well why worry?
So I think we can be quite reasonable working on math literacy, but especially enjoying pursuit of the big picture in math, which may say something about the big picture in life. So thus I have introduced myself and concluded reasonably enough, yes?
I want to write about my current interests in math. I'm wondering if some or all of it might be welcome here. You may suggest other venues.
I am currently interested in understanding the whole of mathematics, and especially, the most advanced mathematics. I'm interested in self-educating myself and learn along with others. I hope I might be welcome to share ideas that come up as I learn. I know I can learn a lot from responses I might get.
In particular, I am trying to map out all of mathematics and understand what are the key ideas. So I have drawn a map... In my map I have organized the areas in the Mathematics Subject Classification: https://en.wikipedia.org/wiki/Mathematics_Subject_Classification (I still have to add the areas in Applied Mathematics). I have tried to indicate which areas build on which other areas.
This map helps me see which concepts in math are most central and which are most basic. So, to my surprise, it seems that geometry is very basic. For example, the subject "differential geometry" = "calculus" + "geometry". But what is geometry? This is the kind of question that I would like to bring to Math Future. More generally, I appreciate any help to make the map as informative as possible.
I will write more broadly about what I'm currently up to and how that relates to math. All of my life I have been interested to know everything and apply that knowledge usefully. In 2014, I wrote an illustrated summary: http://www.selflearners.net/wiki/Truth/Book Now I'm working on a detailed handbook for investigators of big questions. This year I plan to think it all through and next year I will write it up. As part of that I want to show how it can apply to mathematics.
In 2011, I wrote a letter to Math Future where I outlined a system of 24 ways of figuring things out in mathematics: https://groups.google.com/forum/#!msg/mathfuture/50pk00XZCLQ/HnQjrun8ej8J I want to work on that further. Also, I want to do something similar for physics. I'm going through lists of physics experiments at Wikipedia such as: https://en.wikipedia.org/wiki/Category:Physics_experiments I also want to learn quantum field theory and general relativity. I have a Ph.D. in Math and a B.A. in Physics but I still have to learn a lot. So I'm realizing that I would like to take the opportunity to learn the big picture in math as much as I can. Key areas of interest for me, or simply starting points, are:
But I'd also like to know the basics of all the areas on the map, including number theory, algebraic geometry, algebraic topology, complex analysis, functional analysis, etc.
I am glad to be making use of several great resources.
There are some other math investigations that I would bring up. I've made a list at the n-Lab: https://ncatlab.org/nlab/show/big+picture For example, I think it would be very fruitful to analyze what makes math beautiful: https://ncatlab.org/nlab/show/beauty But I'm not sure how much of a future I have there. Here's a discussion about my pages so far there: https://nforum.ncatlab.org/discussion/7066/discovery/
Still, the response there was better than I got at the Foundations of Mathematics forum. I wrote a letter introducing myself and my interests but it was rejected for being "autobiographical" and "meandering".
I understand that I'm a stranger with a strange point of view. But I find it strange that people live without a "big picture" in mathematics. And topics like beauty, discovery, insight are never to be pursued. Because they are not (yet) explicit and so they are not math. So where to talk about math? That is my question.
I think you understand me.