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Comment by drsopp | Hacker News Reader

Is there any attempt to organize mathematics kind of like the OSI model? On the bottom we might start with the physical layer that can be paper+pencil marks on it or a soundwave, the next could be a shape, next could be what the shape stands for and so on. Further up we could find things like isomorphisms.

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Yes, typically the base layer is set theory (with ZFC being the "standard" formulation).

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

I understand what you mean, but in my context I would say that you need layers below this. You for example paper, ink, glyphs, and other things before you can start defining ZFC.

Discreteness, finitenes, causality.

You don't need paper or ink to start working with math. You don't need symbols. You can just do it in your head. That's the beauty of math. The physical instantiation of it doesn't matter.

True. But the physical instantiation is necessary. That is what I am exploring: What is necessary to start working with math, and can it be described like the OSI model?

There are other foundations for mathematics. Dependent Type Theory has a lot of support from people studying Proof Theory. It is based on typed lambda calculus. It is currently being researched and implemented in mathematical provers, like Coq and Lean.

If you want to see an organization of mathematical ideas, I'd recommend digging into https://us.metamath.org/. Great intro to formal systems, though many more layers of definitions than the OSI model.

Though, there may be canonical structures, any universal structure is illusive if not non-existent.

Or start with the morphisms, then look for the idempotents, so further up the structures fall out naturally?

(this program may have an advantage in that it motivates passing from the continuous to the discrete?)

In a sense computer and electrical engineers/scientists did largely map out the base layers, over ~150 years from the 19th century through the mid 20th. I think equipment (broadly speaking) is foundational for mathematics. The "stack", as far as I interpret it, is something like

1. Being

2. Communication

3. Equipment -- a device you can put marks on and read off of

4. Discipline -- ability to reliably and skillfully manipulate the device

5. Submission

The stage is set at this point for some "elementary mathematics" -- think back to elementary school.

6. Symbolism -- the equipment is not just equipment. Mathematical relevance springs here.

7. Geometry -- from vision we see area and edges, objects of perception, they are interpreted as mathematically relevant and hence symbolized.

8. Algebra -- manipulating our equipment with discipline, an equivalence is perceived between different sequences of operations.

9. Proposition -- conviction the relevant facts of geometry and algebra can be formulated clearly in declarations of the sort "if ... then ..., and ... (... and etc)".

Higher level mathematics

10. Refinement 11. Proof

12. Application 13. Theory

14. Computer science, engineering, and design

...

The computer and computer science is, in a sense, the result of the somewhat failed attempt to put mathematics on a solid logical grounds, in a way that resembles a stack like OSI. Gödel's incompleteness theorems and Turing undecidability results showed that it's not that easy. Briefly, the proof for the halting problem was the Turing machine.

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