> Can you give me an example of an instantiation of this "destination" that is not moving symbols around?

Visual proofs are a good example of this: https://mathoverflow.net/questions/8846/proofs-without-words The main objection against them is that diagrams can be subtly manipulated to lie or that they are overly specific in what they prove. Which leads me to...

A clearer but unfortunately unrealistic example of this would be if we could have direct mind-to-mind communication in the future. I think the bulk of mathematicians would argue we would still have mathematics in that world and any symbol written down would be only an encoding of the mathematical ideas rather than the mathematics themselves, which are transmitted directly as abstract thought.

Put another way, symbolic proofs are a crutch for mathematics born out of the frailties of the human body. If humans were capable of perfect, repeatable actions or otherwise directly transmit information, then symbols would not be needed to do mathematics. We could perform computations "directly" as it were. We use symbols because that is not possible, but that is a contingent concern, not a necessary one.

But again the most thorny question of what mathematics is about comes from the reality of the natural numbers and Godel's incompleteness theorem. Most mathematicians would agree we study the natural numbers as they exist in our reality, and yet those can't be fully captured by symbols, so what exactly are we studying then?

Now this whole line of argumentation feels a bit unnatural to me (I don't believe all of what I've been saying in these replies). As I mentioned at the beginning of all this that I'm actually quite a bit of a formalist myself despite all this and I'm really hiding behind a realist costume while I dance around here. In the absence of direct mind-to-mind communication, formalism is a great practical benchmark for rigor in mathematics (can you see how to turn this prose description into a symbol manipulation game? No? Well then the description is probably not precise enough). And that's good enough for me to put on a formalist hat most of the time when I do mathematics (there are certain intuitions in mathematical logic that are harder to grasp with a formalist hat on so that's when I take it off). I think it's as good a backup to have as "shut up and calculate" is in physics.

But as a philosophical understanding of what mathematics "really is all about," I still find it wanting, especially if I believe that there is only one "true" set of natural numbers in our universe (which sometimes I don't, but if you believe that the natural numbers are physically realized by real objects, then it's a compelling conclusion), because if you believe that, then Godel's Incompleteness Theorems are a philosophical death blow for formalism.

But... speaking for myself personally and leaving aside the persona I've adopted for the purposes of this discussion, I don't usually care what mathematics "really is all about." I do have a formalist counterargument for Godel's Incompleteneess theorems as well, but I've already written too much already and will only reply to that if anybody is interested.

EDIT: This is an even better example of visual proofs: https://moodle.tau.ac.il/2018/pluginfile.php/403616/mod_reso... again the fact that they are not 100% trustworthy is more a problem of physical deficiencies in humans and materials than anything else.

EDIT EDIT: Another example of non-symbolic computation would be using physical processes for computation. Again as a practical matter, this is an inversion of results, since we do not have a perfect understanding of physical principles, but assuming we were perfect physicists and there are indeed entirely featureless pure particles (which it seems like there are), then we could do computation via physics directly instead of symbols.

EDIT EDIT EDIT: Although this can be rephrased in a formalist framework of symbols and manipulation rules, a great practical example of this is diagram chasing in category theory. Although, unlike the case of pure particles, you can argue that this is just a symbol game in 2-d, it certainly doesn't feel like one in the heat of the moment and comes closest to approximating what rigorous, but non-symbolic proofs would "feel" like.