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Stream: theory: philosophy

Topic: All visual art on canvases is the same?!?

Keith Elliott Peterson (Jan 25 2024 at 03:39):

I just realized a strange consequence of mathematics is that all art on a rectangular canvas (with no holes), whether it is digital or not, is isomorphic to each other. How strange.

John Baez (Jan 25 2024 at 04:52):

Everything is isomorphic to everything else if you put enough invertible morphisms into your category.

Keith Elliott Peterson (Jan 25 2024 at 06:18):

John Baez said:

Everything is isomorphic to everything else if you put enough invertible morphisms into your category.

I mean sure. $\mathscr{C}\coprod\mathscr{C}^\text{op}/\text{Obj}(\mathscr{C})$ will contain all invertibles for any category $\mathscr{C}$.

However, the invertibility of pictures is more of a fact that a picture is kind of like a matrix, and so zero morphisms exist. That is to say, such a category of pictures, $\text{Pict}$, naturally has invertible morphisms, which is somewhat surprising. Or maybe I'm just easily impressed.

Morgan Rogers (he/him) (Jan 25 2024 at 07:42):

Zero morphisms are typically not isomorphisms.

Damiano Mazza (Jan 25 2024 at 07:54):

Keith Elliott Peterson said:

I just realized a strange consequence of mathematics is that all art on a rectangular canvas (with no holes), whether it is digital or not, is isomorphic to each other. How strange.

Then mathematics is wrong :big_smile:

Keith Elliott Peterson (Jan 25 2024 at 10:16):

Morgan Rogers (he/him) said:

Zero morphisms are typically not isomorphisms.

I stand corrected.

Jean-Baptiste Vienney (Jan 25 2024 at 16:42):

Many theorems can be stated as a bijection, an isomorphism or an equivalence of categories. Isomorphic things are not exactly the same, they are just with respect to a restricted set of algebraic manipulations. If isomorphic things were exactly the same, these theorems would be useless.

Jean-Baptiste Vienney (Jan 25 2024 at 16:44):

Secondly, computers don’t understand isomorphisms, not even equality, only chains of characters. $1+2$ and $2+1$ are not even the same for a computer, they are different things in the memory.

Jean-Baptiste Vienney (Jan 25 2024 at 17:14):

Finally, they are like us, at the most basic level we consider two different drawings of a sentence isomorphic even if they are not written exactly in the same way, if they are sufficiently well-written for us to be able to identify the characters and decide that the two sentences consist of exactly the same string of characters.

Jean-Baptiste Vienney (Jan 25 2024 at 17:16):

In art, we can't even consider any isomorphism, even if you copy perfectly some artwork, it would be considered as a fake and not an isomorphic artwork.

Jean-Baptiste Vienney (Jan 25 2024 at 17:17):

Artists are very different from us, they consider all slightly different things as unique and distinct things.

Jason Erbele (Feb 20 2024 at 01:17):

Jean-Baptiste Vienney said:

In art, we can't even consider any isomorphism, even if you copy perfectly some artwork, it would be considered as a fake and not an isomorphic artwork.

I think the art situation is somewhat subtler than either extreme of "all visual art on canvases is the same" and "no visual art on canvases is the same". Even assuming there are only noninvertible morphisms from an original piece of artwork to its copies (even when the copies are perfect clones of the original), what of the morphisms between the copies? Art Book A and Art Book B both contain depictions of famous painting X. There is some sense of "sameness" between the two depictions of X.

Or consider two unused postage stamps of the same design, and even from the same location on the plate—if stamps are still printed that way—so they even have the same (lack of) defects. Are artists so different from us that they would claim the two stamps are distinguishable as works of art? (Obviously the post office would consider any two stamps to be isomorphic, even with different designs, so long as the denominations are the same.)