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Is all math/logic most fundamentally geometry?



Don't think so. Geometry requires space, which has certain features which constrain its properties (sorry for a tautology). If you avoid such constraints, you can still have math, but it doesn't make sense to call it geometry.

Looks a bit surprising math definition include concept of space. Geometry looks underappreciated, yes, but to replace the whole math...


To me, the answer here is “kinda but no”. Math in its most basic level is about studying logical connections. Sometimes being able to symbolize something in notation allows inspection of logical objects otherwise unobservable - like higher dimensional objects. But there’s the whole area of mathematical logic. I think I can say Godels incompleteness/inconsistency theorem was only about axiomatic systems with no necessary connection to geometry. Mathematics loves to study logical reductions of things and geometry can certainly be left out through reductions.

There’s something of a geometry/algebra separation in mathematics, too. The last few centuries (?) have tended toward algebraic research at the exclusion of geometry. There’s even reason to believe the two types of reasoning are separated in human brains in so far as people tend to be good at one and less good at the other.


Ah, but you can't encode math except for in some necessarily geometric form.


Are you referring to written notation? Calling that geometry is a bit of a stretch. There's also nothing geometric about maths encoded in computer code, or many types of mathematical thoughts, so I think you are just incorrect.


> Are you referring to written notation? Calling that geometry is a bit of a stretch.

Can you write without shape?

> There's also nothing geometric about maths encoded in computer code

Look at a computer chip under a microscope: nothing but geometry.

> or many types of mathematical thoughts

In re: math itself, perhaps there is such a thing as a mathematics of the formless (I doubt it but cannot rule it out) but to communicate it you are again reduced to some symbolic form.

> so I think you are just incorrect.

I've been thinking about this for a long time, and I'm still not 101% convinced, but I think it's true: you can't have information without form.

Check out "The Markable Mark" and "My Stroke of Insight". The act of distinction is the foundation of the whole of symbolic thought, and it is intrinsically a geometric act.

http://www.markability.net

> ... what is to be found in these pages is a reworking of material from the book Laws of Form.

> Think of these pages, if you like, as a study in origination; where I am thinking of 'origin' not in the historical sense but as something more like the timeless grounding of one idea on or in another.

Distinction is a physiological thing the brain does. It can e.g. be "turned off" by physical damage to the brain:

https://www.ted.com/talks/jill_bolte_taylor_my_stroke_of_ins...

https://en.wikipedia.org/wiki/My_Stroke_of_Insight

> Dr. Jill Bolte Taylor ... tells of her experience in 1996 of having a stroke in her left hemisphere and how the human brain creates our perception of reality and includes tips about how Dr. Taylor rebuilt her own brain from the inside out.

So whether you come at it from the mystical realm of pure thought or the gooey realm of living brains all math is geometric. (As far as I can tell with my gooey brain.)

Cheers!


> and it is intrinsically a geometric act.

Why? Can't you have distinction without geometry? It's not only position which can be distinct, you can have other properties.

Two digits in different position on paper can be both different - 0 and 1 - and the same - 5 and 5. You can encode them not by shape, but, say, by kind of particle?

And in general, our physical world has space - but how would you prove a world without space as we understand it can't have math?


> Why? Can't you have distinction without geometry?

Maybe but I don't see how.

> It's not only position which can be distinct, you can have other properties.

Properties like what? Color, sound, temperature, etc., all of these are geometric, no? Can you think of a concrete physical property that doesn't reduce to some kind of geometry?

> You can encode them not by shape, but, say, by kind of particle?

Sure, but then that particle must have some distinction from every other particle, either intrinsic or extrinsic (in relation to other particles), no?

Any sort of real-world distinction-making device has to have form, so that eliminates real non-geometric distinctions.

It may be possible to imagine a formless symbol but I've tried and I can't do it.

The experience of Dr. Taylor indicates to me that the brain constructs the subjective experience of symbolic distinction. (Watching her talk from an epistemological POV is really fascinating!)

So that only leaves some kind of mystic realm of formless, uh, "things". My experience has convinced me that "the formless" is both real and non-symbolic, however by the very nature of the thing I can't symbolize this knowledge.

    In the Beginning was the Void
    And the Void was without Form
If you can come up with a counter-example I would stand amazed. Cheers!


> Can you think of a concrete physical property that doesn't reduce to some kind of geometry?

How would you reduce charge to geometry? Or spin?

Can we differentiate by space the electrons in an atom of helium?

But we sort of digress. The question was if a concept of space is required to a concept of math, and specifically, if we can have distinction without space. Surely we can at least think of distinction without space, even if we'd fail to present that in our physical world?




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