I am an independent researcher with some radical ideas about harmonic resonance and spatial standing waves as the computational principle behind perception and cognition.

1. Your blog helped me understand, finally, how Grassmann and Clifford algebras are related. Thank you! 🙂

2. Vinícius says:

Just couldn’t find out how/why as a cognitive researcher did you come up to Geometric Algebra! As an engineering student, it took me too long to discover this formulation, thus it would be really interesting to hear about how a researcher in a seemingly distant area ended up writing something about it. Amazing to see how different backgrounds can lead to the same ideas. The articles are great, thanks!

• slehar says:

Thanks for the appreciation! Yes it does seem like a remote connection. But I have always sought out remote connections because that is where the most interesting insights are to be found. Descartes: geometry and algebra; Darwin: inheritance, random mutation, and survival = evolution; Newton: earthly & celestial mechanics; These were the most intriguing puzzles of their day. I always knew that we must have our own era specific blindness, and how much fun it would be to pierce the veil and discover what have we been seeing wrong all along. When I started studying mind and brain, I was convinced from the outset that if our brain produces our experience, then there must be artifacts in that experience that reveal the underlying principles and mechanism. I came to believe that all of mathematics was an artifact of how our mind represents reality.

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.

• Vinícius says:

I’m completely astonished with this whole thing! You might be sure that I see all this potential you mentioned, but now I cannot commit myself with studying and developing such a theory – I’m still trying to learn GA (while also smoking some pot hahah) and apply it to my own studies.

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.

3. corvus says:

Such an excellent, excellent post. Many thanks for taking the time to put it together and share 🙂