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Hello there, am an independent researcher interested in foundations of mathematics though without comprehensive mathematical background. Last year, it caught my attention that physics is formalized in richer kinds of homotopy theory in Science_Of_Logic. There is mention of universal algebra but they choose to work in a kind of synthetic homotopy theory but in my view continuing in universal algebra is more generic and could actually lead to a foundation of math. My math is bad but there could be some truth :
In my view, this should be formalized in enriched category theory so that every category turns out to be such a tower. The truth of this may be predicted using the idea that enriched categories are just free (co)completions and as such they are equivalent to eilenbergmoore categories, leading to my final step. 7.The 2-category above can be viewed as the category of all categories. The importance of this approach is that universal algebra is equivalent to inductive/recursive types and we can take advantage of this to inductively/recursively define the whole of mathematics and prove all its theorems in a big-bang kind of manner.
As a remark, I observe how bireflection seems to be equivalent to the physical notion of time, as defined in the nlab page, causal locality. This gives me the impression that math is inherently dynamic, that math actually exists objectively. Either way, what prevents a mathemetical theory from being real except causality ? I posted a related question on MathOverflow has there been any serious attempt at a circular foundation of mathematics. I at-least expect that the Hegelian pseudo-code can be formalized on such lines. I would appreciate it if some more advanced mathematician embarks on this or corrects me. Thank you
Considering that Hegel claims to have logically derived his philosophy in a somewhat causal order, is it possible that there is a pre-mathematical way of understanding math (using dialectics maybe) ? Maybe the author, Urs can help me with this ..please
I find that – once one sees it – the fact that Hegel speaks a pseudocode of adjoint modal type theory is so obvious and compelling that I wouldn’t quite know what else to say about it beyond what I already said.
(And of course Lawvere said it first, I just added the more intrinsic modal perspective and a higher progression of adjoint modalities.)
You may disregard the homotopy-theoretic aspect, if that is a stumbling block, since this is not necessary to see just the progression of dualities of opposites. (It makes for more interesting models, though, and is needed to understand the self-reflection of the "Essence", but you can leave that for later.)
First of all, thank you for your response. I am not disregarding the homotopy-theoretic aspect, am generalizing it instead. My current emphasis is that universal algebra is as general as it sounds which may have been overlooked due to traditional treatments which focused on special cases(e.g Lawvere theories). An example of their generality ; eilenbergmoore coalgebras over a linear exponential comonad is a model of multiplicative intuitionistic linear logic which subsumes both classical and intuitionistic(topos) logic, Linear Logic(forms an SMCC). While unity of unities of opposites corresponds to a tripple adjoint, aufhebung corresponds to iterated lifting(seems like aufhebung is the real progression), at the convergence of this self opposition of identity appears(Essence). That’s the tower, a 0-cell, continued lifting yields a tower of towers as a 1-cell(I think this is self-reflection) but further iteration seems degenerate so I settle on the 2-category I defined above. Guess this also corresponds to something here..(I had to rewrite this all over as it got lost)
Above, I have a typo, a tower of towers is a 1-cell.
Let me warn you than some of what you wrote in #1 above sounds like pure crackpottery. Such as this statement:
universal algebra is equivalent to inductive/recursive types and we can take advantage of this to inductively/recursively define the whole of mathematics and prove all its theorems in a big-bang kind of manner.
and this one:
bireflection seems to be equivalent to the physical notion of time, as defined in the nlab page, causal locality. This gives me the impression that math is inherently dynamic, that math actually exists objectively. Either way, what prevents a mathemetical theory from being real except causality ?
Several other of the statements above above remain unitelligible to me.
I am doubtful that the nForum is the right place for the discussion that you are after.
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