Actually I think it's not at all clear how to give any (modern) proof of the Pythagorean theorem that doesn't feel at least a little bit circular in spirit. The issue boils down to the question: where does the Euclidean distance formula come from, if it doesn't come from the Pythagorean theorem? I spent awhile thinking about this off and on and I still don't feel totally satisfied, but I think the most honest answer is that it must and can only come from our phenomenological experience of physical space (e.g. the fact that we can freely rotate our bodies).

If you don't allow the use of constants such as $\pi$ and $e$ then I think you could attempt to argue that no such closed formula can have the correct asymptotics as given by Stirling's formula.

There isn't an interesting preorder structure, you just get an equivalence relation and lose information about automorphisms.

I don't understand this claim about fiber functors, which I think contradicts some versions of the Tannaka-Krein theorem depending on what you mean by "fiber functor." Is it that you only require fiber functors to be monoidal but not symmetric monoidal?

Your definition of polynomial function makes implicit use of polarization so I am not convinced it does the right thing in positive characteristic.

The first question seems delicate but see the Weyl dimension formula. The answer to the second question should be that $G$ is virtually abelian, e.g. it could be a semidirect product $T^n \rtimes H$ of a torus by a finite group, but if the Lie algebra $\mathfrak{g}$ has a nontrivial simple factor there should be irreducibles of arbitrarily large dimension.

That phrasing is a little ambiguous; it seems clearer to say that pullback along $\phi$ defines a functor from $T$-algebras to $S$-algebras, etc. Even in familiar examples there may be more than one candidate for $\phi$, e.g. there are two homomorphisms from the monoid monad to the ring monad, corresponding to addition and multiplication.

(And our phenomenal experience of physical space also does not privilege an orientation of any particular plane sitting inside phenomenal $\mathbb{R}^3$, e.g. whatever surfaces Euclid worked geometry problems on.)

@Joel: this is a longer discussion than there's room for here, but my contention is that this is philosophically backwards. From a historical and phenomenological point of view the Euclidean plane is conceptually prior to ZFC; indeed we could not possibly have known that $\mathbb{R}$ and therefore $\mathbb{R}^2$ were objects it would be useful and meaningful for us to construct in ZFC without extensive prior experience with Euclidean geometry that was not grounded by set theory. What grounds Euclidean geometry is our phenomenal experience of physical space.

There is also a Clifford algebra-style construction where we can think of complex numbers as formal ratios $\frac{v}{w}$ of vectors in $\mathbb{E}^2$, which again does not require breaking the symmetry between $i$ and $-i$, and is directly analogous to how vectors themselves are formal differences of points. The connection between these perspectives is that given two nonzero vectors $v, w \in \mathbb{E}^2$ there is a unique homothety sending one to the other. I learned this from a paper by David Hestenes who attributes it in some form to Grassmann.

@Joel: you can construct the complex numbers starting from the Euclidean plane $\mathbb{E}^2$ (equipped with its normed vector space structure): it is the subring of the ring of linear transformations $\mathbb{E}^2 \to \mathbb{E}^2$ consisting of homotheties, namely maps that multiply distances by a fixed constant. This does not require breaking the symmetry between $i$ and $-i$, which can be identified precisely with the symmetry between the counterclockwise and clockwise orientations of $\mathbb{E}^2$.

The nLab (ncatlab.org/nlab/show/ribbon+category) says that the free ribbon category on one object is the category of "framed oriented tangles." I don't know exactly what that means but presumably that "oriented" is where the ribbon element is coming from.