Andrius Kulikauskas

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See: Math, Clifford algebra, Geometry

Think of angles as bivectors - the basis for conformal geometry? And vectors for projective geometry? Trivectors for symplectic geometry?

Kirby, Andrius,

I've stumbled on an interesting presentation of Clifford Algebra, by Steven Lehar. It's special appeal is that it is profusely illustrated with color diagrams, some animated, which is only fitting for a *geometric* topic. So far I have just skimmed it, but I plan to peruse it in more detail.

Lehar identifies himself as"an independent researcher <>with a novel theory <> of mind and brain, inspired by the observed properties <>of perception."

Joe Austin

Steve, I am very grateful to you for your persistence in championing your mind-brain theory. I'm very interested to see the big picture of mathematics, its "grand unification". So I look forward to studying your pages and learning advanced mathematics from you and alongside you.

In particular, I'm intrigued by your writings about characterizing different kinds of geometry, such as projective geometry: and conformal geometry

The latter made me think of the Mandelbrot set in that it does appear to be an inverse image of a world. That is, it does appear to be connected together in intricate ways which would make more sense if that universe was physically "expanding" rather than "shrinking". I imagine the p-adic integers are exactly like that. That makes me think in my own work that the point of characteristic p is that it p=0=infinity.

I have noticed about the Mandelbrot set what I think is a great simplification, that is, that it can be generated directly by plugging in each complex number z into the generating function for the Catalan numbers, which happens to count all manner of things that are processed by context-free grammars (push-down automatas), anything with finitely many "obligations" that have to be met, as with left hand parentheses that need to be balanced by right hand parentheses, or walks up tree branches that need to be balanced by walks back down, thus anything that makes use of a finite number of memory cells. Here is my letter about that to this group:

I am also interested in how you are thinking about infinity. I'm starting to realize that the concept of "infinity" is one that I think could be treated differently, perhaps for what it is, as a convenient fiction. I'm thinking that it could be just a construct that means "we have enough vertices" and that concept could be given by a number p (as in the characteristic for a finite field). That number could be variable and would, I imagine, be prime simply because otherwise we could work with a smaller p in whatever we were doing. If we want to have p be unlimited, then we would set p=0 so that we have a "field with one element" (a mysterious object that math hasn't figured out yet what it means) and where 0=p=infinity. I will write about that separately.

I am very glad that you are succesfully learning by teaching. Over the last few months I have been trying to understood the tensor product because I think it is basic to everything. I think that it is "trivial" in the sense that "linearity" is trivial. So it is the background assumption in everything, in all of geometry, but I have yet to understand this triviality. I do have the sense, though, that it is grounded in a duality of "bottom-up" (building up) and "top-down" (tearing down) views.

Steve, in my own words, I would say that what you and I and Nunez/Lakoff are doing in our different ways is a "science of math". That is, we can look at the enormous output of mathematical activity and say that there is a way to make sense of the big picture. But to do that we need to use not only math, but also other tools and approaches, both science and aesthetics and even some politics to have a relevant conversation.

Math Future is a welcoming place for us because there is an understanding, I feel, that as educators we need to develop our own "worlds" and look for how they relate. I think many of these worlds are very sympathetic to Geometric Algebra. Ted Kosan is leading lessons in computer algebra software MathPiper Kirby Urner is a modern day Buckminster Fuller who, among so many things, is interested to learn more about Clifford Algebras, as am I. Joe Austin champions a teaching approach that is rooted in physical thinking, including visualizing geometry. Bradford Hansen-Smith is a pioneer of circle folding, which I'm realizing, is very central. Yesterday was my niece Ona's birthday so I folded her a sphere (a cuboctachedron) from four circles which I found instructive (it was surprisingly taut when I put it together with paperclips). I sent her a link and a photo of me with the sphere and her name on it. Which is to say that we affect each other in large and small ways.

I want to tell you about three other groups where some day, sooner or later, you might be successful in starting conversation and collaboration towards a grand unification of math. They are the Foundations of Mathematics email list, the nLab wiki and forum, and the Azimuth Project.

The Foundations of Mathematics email list is moderated by Martin Davis, one of the solvers of Hilbert's Tenth Problem. It is dominated by Harvey Friedman, who shares his thinking-out-loud on the open problems in set theory / foundations of mathematics which he finds most interesting. The archives are open but you have to request permission to be a member and have a chance to write. You should be accepted because you have a Ph.D., as do I. But my first letter, in which I introduced myself, was rejected because by the moderator as too long and meandering. I sent the same letter to Harvey Friedman but he didn't reply. Later they approved my letter on the Mandelbrot set and the Catalan numbers. Now I'm writing a long letter to them which I hope they might accept.

The most interesting mathematician, who shares his thinking about all the math I would like to know, is John Baez. He is famous for his blog but especially for his earlier blog "This Week's Finds in Mathematical Physics", basically 318 essays that he wrote full of mathematical intuition which I keep bumping up against when I google or read Wikipedia. He is also one of the founders of the "n-category Lab", which is a loose affiliation of the initial group blog, the "n-category Cafe" There is also the "n-Lab" wiki and the related "n-Forum" for discussing changes made to the wiki.

In a sense, that wiki is the closest thing there is to collaborative work on a "grand unification" of Mathematics. The approach that they are taking is called "n-category theory". The best source which I have found for that is the paper by John Baez and Aaron Lauda, "A Pre-History of n-Categorical Physics" See especially page 104 on Baez-Dolan (1995) which discusses this paper: Such papers may seem impossible but I am realizing, as I think you as well, that if I have the fruitful attitude: "What is the simplest issue of the deepest consequence?" then I have the machete with which to cut through the thickest weeds. In my case, it means that nobody knows what the -1 simplex is nor the "field with one element". In your case, if you can find the right issue, which relates to what they fail to do in their world, then you will get the chance to say what you want to say about that and everything else. So I'm curious not only what your own deepest insights are (you seem to be able to write about that) but also what particular math solution you provide might get others interested in our hope for collaboration on a grand unificiation.

The point of "n-category theory" is that it can formulate mathematical intuitions in homology, homotopy and other advanced fields which are completely ignored by the classical "set theory" foundations of mathematics. N-category theory is related to "homotopy type theory" and there will be a conference in Munich on "Foundations of Mathematical Structuralism" which I think I'll submit an abstract to, due June 30, 2016. N-category theory has many layers of abstraction that serve to identify and describe what it means for mathematical equivalences, transformations, objects to be "natural". It's just all extremely abstract and "unnatural" to learn. There are some exchanges by John Baez and Harvey Friedman where they would like to have a basis for fruitful discussion but they can't find it and seem to have better things to do. Harvey Friedman's position is that set theory is what works and that it doesn't matter which approach is more "natural" but if it can address mathematicians' problems that the classical foundations can't, then please speak up. This is why I'm focusing on the field with one element and the negative-one-dimensional simplex and I think that's proved very fruitful for me but we'll see what they say. I should mention that I've learned that the flip-side of the abstractness of "categorification" is the concreteness of "decategorification" as in algebraic combinatorics, my own specialty in math. I learned that from Jeff Hicks's e-book "Categorification": I'm surprised he's just a grad student. Anyways, in my own work, it means that a variable q which we use for tracking some feature of enumerated objects (such as the way a path Pascal's triangle swings left or right), can be set to equal 1 (in which case we are simply "counting") or it can be "categorified" in some way to describe the actual objects, which might be, for example, strings of generators of the symmetric group. Which is to say, there is a flip-side that is a concreteness to the abstractness. You might find that in your own work as well and that might help you communicate it both concretely and abstractly.

I tried to introduce myself at the nLab wiki and then I created pages on the "big picture", "beauty", "discovery" but they were deleted. You can see the discussion at the nForum: But I think if you write about the subjects that you know well and link to your articles then you might be well received. It's worth trying.

I noticed that John Baez is active in the Azimuth Project which he started for mathematicians and scientists to work together now in response to climate change. I created a page for myself at the wiki: And introduced myself at the forum and participated in a couple of threads: I proposed to work together on helping others, and each other, to learn advanced mathematics, as you are doing, and so is the wiki administrator David Tanzer. I proposed to work on a graph of all of the areas in math. I didn't get any response but I wasn't kicked out yet, either. I should note that John Baez is also known as the author of the "crackpot index": which is understandable given his active participation in the online world.

Steve, I look forward to your letters!


Andrius Kulikauskas +370 607 27 665

Dear Andrius,

Thank you very kindly for alerting me to the existence of the mathfuture group! And thank you also for promoting my visual introduction to Clifford Algebra / Geometric Algebra.

You and the group may also be interested in this paper of mine on the Double Conformal Mapping, an extension to David Hestenes' Conformal Geometry extension to Geometric Algebra, which relates directly to my theory that the origins of mathematics lie in the laws of perception.

I also have a book in progress (not yet complete) titled The Perceptual Origins of Mathematics.

As with my Visual Introduction to Clifford Algebra, I prefer to explain math in pictures rather than equations, wherever possible, to clarify the connection to perception.

Indeed the extraordinary Grand Unification of math accomplished by Clifford Algebra stems from the discovery that all of algebra is a branch of geometry, and that most mathematical operations can be represented as spatial operations on spatial structures. This makes my writing immediately accessible to the non-professional mathematician.

I intend one day to write a book that explains all the most interesting aspects of math in simple intuitive terms that most anyone can understand.

Thanks again for making contact with me!

  Steve Lehar

On Wed, Jun 15, 2016 at 10:33 PM, Andrius Kulikauskas < <>> wrote: Dear Joseph Austin,

Thank you for alerting us to Steven Lehar's very helpful page on Clifford Algebra and also his thoughtful independent research which I look forward to looking over. I take the opportunity to let him know about the MathFuture google group and share at least the beginning of my letter on "implicit math" which might interest him and you and Kirby Urner as well. I am writing the letter to the Foundations of Mathematics group and so many possibilities are opening up that I simply have to go through the most basic of them.



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