To get on my soapbox for a second – the fact that mathematics works to describe our physical universe tells us: (1) that the physical universe is logically consistent (for the most part); and (2) that there are symmetries in the physical universe (as stated in a previous post), which we know exist up to a certain level of accuracy.

When nature approaches a logical inconsistency (such as 1/x), there must be a way to get around that inconsistency; e.g., by shooting out a photon of light. If there was an observable logical inconsistency in nature, it would lead to paradoxes that we probably could not wrap our heads around, and almost certainly wouldn’t be able to use mathematics to describe. That the universe is logically consistent even in the infinite spaces we don’t yet understand is a matter of philosophy at this time.

The fact that there are symmetries to the physical universe up to a certain level of accuracy is evident by the fact that we can write equations at all that say “this equals that.” Without symmetry, we wouldn’t have nature’s building blocks, and the simplest of structures would require too much energy to create. But we wouldn’t have known without studying the physical world that these symmetries existed, and if these symmetries didn’t exist then none of this beautiful algebra would be useful. That the universe is symmetric below the level of accuracy we can measure is, again, a matter of philosophy at this time.

On a side note – I don’t share your dis-satisfaction of the cross-product (and the fact that it is non-associative as pointed out by the previous post). These are simply limitations of our views of vectors. As Clifford and Geometric Algebra bring to light, but which we still haven’t been able to fully internalize because our formal education gets in the way, is that we have to stop thinking about vectors and scalars by themselves. When you think of vectors and scalars together, the cross-product is much more beautiful then your writings suggest. Plus, I always get a little weary when someone claims that a mathematical structure that clearly has tremendous applicability to physics does not have deep meaning behind it.

I’m going to be writing a book in the near future on a mathematical framework that shows many of these algebraic ideas in a framework that I think is much more intuitive and easy to understand. So easy, in fact, that I plan on teaching it to my three young kids when they get to middle school. This is based on 15 years of my personal self-reflections (and rotations) on this topic. I was a physics and math undergraduate and went on to get an M.S. in mathematics – even though I work as an engineer for my day job, I have continued to advance this work in my spare time. I did all this in a vacuum, only to become aware more recently of Clifford and Geometric Algebras as I started doing research to support a Ph.D. thesis on this topic that I will be pursuing starting next year. I think these previous algebras are good starting points, but they are missing some basic principles that make them more complex and less applicable than they should be. If you are interested in talking about this more let me know – I have been struggling to find anybody I can share this with and based on work you’ve done I feel like you have a deep appreciation for the beauty and connected-ness of this type of mathematics.

]]>(AxB)xC does not, in general, equal Ax(BxC). The lack of associativity can be seen from the connection of the wedge product (which is associative) and the cross product:

AxB = *(A^B) and so you get into trouble with triple cross products

(AxB)xC = *( *[A^B] ^ C)

Ax(BxC) = *( A ^ *[B^C])

and since the right-hand-sides are not, in general, equal the cross product is not an associative product.

That seems rather unnatural to me.

Yours, John Smith

]]>What I may guarrantee you though is that whenever I talk to someone who studies cognitive/neuroscience I’ll come up with your blog and ideas, I’m already a GA advocate, and am gonna stimulate its use in this field too.

Hopefully in a near future I’ll be in condition to fully understanding your work, as many others I think this language will help me with. Please, keep developing and publishing it! Me from the future certainly thanks you.

]]>The Perceptual Origins of Mathematics

(work in progress)

I had been bothered for a long time about i, the square root of -1. The concept of “complex” numbers was shrouded in the deepest mystery. Then I ran into BetterExplained and his Visual Intuitive Guide to Imaginary Numbers I was blown away that such an abstruse abstraction could become intuitively obvious in the right spatial context was quite a revelation to me.

But in my Google searches I kept getting hits “Clifford Algebra”, “Clifford Algebra”, “Clifford Algebra”… and I kept thinking I’m too old to be learning a whole new algebra! But one Sunday lying in bed smoking pot and surfing the internet I thought “What the hell!” and clicked on a Clifford Algebra link. I have never been the same since!

I have also made a connection between mind and harmonic resonance

Harmonic Resonance In the Brain

and provided an

Intuitive Explanation of Phase Conjugation

which I see as a model of resonances acting like physical gears, pushing and pulling on each other like genuine physical objects. The theory of computation was revolutionized by concepts like the transistor, the op-amp, amplification, that involve an input, output, and “gate” signal that controls the flow from input to output.

I see phase conjugation as a similar concept except, instead of single one-bit inputs, outputs, and controls, there is an input IMAGE, output IMAGE, and control IMAGE (in 2 or 3 dimensions!) where the control image modulates the input image to produce the output image.

Do you see where I am going with this stuff? Have you seen my

Constructive Aspect of Visual Perception

to understand how spatial waves can interact as a computational process? If you begin to see the huge potential of this direction of investigation please help me! I can’t do it all on my own! There is a huge HUGE payoff when we ultimately discover the (analog analogical spatial) computational principle of the human brain as a direct counterpoise to the world of digital logic. There is so much here to be discovered once you understand the potential significance, I’m an old man, I won’t be around for too many more years. Don’t let these ideas die with me! Help me think this through if you can see the potential that I see. I think it is the funnest, most intriguing problem that science has ever faced. The theory of our own minds.

]]>1. Yes, it’s a great video, but it’s [unfortunately] not me! It’s from 3Blue1Brown.com

2. You can make neat computer vision algorithm with tauquernions.

-m

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