In a recent work of mine I described something very similar to the amodal / modal divide. I just called it “world-sheet” and “texture”. The world-sheet is the depth-map that results from network of subjective measurements of distance that we entertain in our mind. Texture is the modal-specific qualia patterns (e.g. in visual experience these would be mongrels) that enrich the information content and experiential quality of the models we hold in the world-sheet.

(If you are interested, here is the article in which I talked about this: https://qualiacomputing.com/2016/12/12/the-hyperbolic-geometry-of-dmt-experiences/)

]]>There are plenty of other areas of math out there, say topology, number theory, probability, category theory, to name a few broad fields. You seem to say that mathematics is about discovering the structure of our mental representations. Does that mean category theory is not ‘true’ math? If so, then what you mean by math is not what anyone else means by math.

And furthermore we can write down plenty of perfectly self-consistent geometrical theory that doesn’t necessarily correspond to either the physical world or to perception. You present some theory about some particular projections that you think match our visual perception, but you could easily write other projections (say, replace the sphere with an ellipsoid). And at most one of them would truly represent perception. Would you say the other one wasn’t ‘true’?

It makes much more sense to think of all the theories as ‘true’ math, and say that the question you’re interested in is a scientific question about what math describes the structure of your thoughts, rather than being in any sense about which math is ‘true’.

]]>But then I suspect you won’t agree with that one either. The deep mystery is what makes it interesting! You are missing the best part of the story!

]]>But in the end, either algebra is just a set of symbols on a page – they have their structures independent of the minds that conceive them, and independent of the world. The theories can be used to describe the world because the world has the same structure. (I similarly don’t think there’s any paradox in mapping infinite domains into finite ones – as long as the math doesn’t generate any contradictions (and it provably doesn’t), there’s no reason this should be impossible. Whether it’s physical is a separate question). We don’t see periodicity because our brains have periodicity but because rotations just are periodic.

There is a connection to minds, but I think you have the wrong one – I think our minds have representations that are described efficiently by GA because our minds are adapted to a world that is described efficiently by GA. Note that physics isn’t made of math, either, on this view; it’s just isomorphic to math. Isomorphism is the key really — there’s nothing mysterious going on.

]]>My hypothesis is that we have a mental coordinate system similar to that of a globe.

On the globe longitude angular separations are greater near equator and smaler near the poles.

The moons diameter is in fact constant 0.5 degree, put persieved greater near the horison and smaller near zenit. Maybe this could be explained even better with your ideas of our visual perception. ]]>

A Mathematician’s Lament

https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

We are on the same page re Traditional Notation: http://philiprhoades.org/music/gnote2_1.pdf

What language did you use for your NN/BP exercise? Would you share it?

BTW, I have signed up with Better Explained and left a note for Kalid re NNs. ]]>