Mathematics is one of our most powerful and perplexing inventions.
From one perspective, it's just a system for making various sequences of marks on pieces of paper (or computer keyboards, etc.). A mathematical system tells you which sequences of marks are "allowed" or not; and then the doing of mathematics consists of figuring out which sequences of marks are allowed in a certain system. Sort of like a language where the basic grammar principles are known, but are so tricky that it's a hard puzzle to figure out which sentences are grammatical or not.
The purpose of this "mathematical marks game" is that some people find it beautiful and entertaining ... and that people know how to correlate some of the marks with actions and perceptions in the world, thus allowing mathematics to be used in physics, biology, sociology and so forth.
From another perspective, mathematics describes realities beyond the one we live in. For instance, there are various theories of huge infinite sets. The "existence" or otherwise of these sets can never be validated by science, because science ultimately has to do with finite sets of finite-precision data. But, mathematical theories involving these sets may nevertheless be very useful to science.
And the communication of information about these infinite sets via language is an interesting thing -- because language, like science, has to do with finite sets of data (finite texts composed of characters drawn from a finite alphabet).
But if we remember that language doesn't encapsulate knowledge -- it rather serves to channel shared understanding -- then this isn't so mysterious. If we human minds have shared understanding of these infinite sets, then language can serve to coordinate and channel this shared understanding. This is what it feels like is happening when mathematicians discuss abstract mathematics.
Can Digital Computer Programs Understand Mathematics?
Humans' apparent ability to intuit infinite sets makes things interesting for the AI theorist -- because, what would it mean to say that a mind implemented as a finite, digital computer program could enter into a shared understanding of an infinite set?
Some AI theorists (for instance, Selmer Bringsjord) argue that digital AGI programs are only capable of understanding infinite sets indirectly, as certain finite arrangements of symbols -- whereas we humans can apprehend them directly
This is possible, but I'm skeptical.
Rather, I think what's happening here is well-understood in Peircean terms as a confusion between Firsts, Seconds and Thirds.
Infinite sets have their own unique Firstness ... but in their Thirdness (not their Firstness or Secondness) they are reducible to symbol manipulations, to sequences of characters.
I doubt that, when we humans intuit infinite sets, our brains are doing something fundamentally different from when we intuit the number "5", which mathematics models as a finite set.
It seems quite feasible that advanced digital computer programs will, like humans, be able to experience a Secondness, in which their own Firstness collides with the Firstness of infinite sets.
One wonders if some future discipline might weave aspects of current mathematics into other aspects of experience. If a future science of consciousness brings subjectivity and objectivity together in some novel way -- will it come along with some allied novel discipline binding the formalism and experience of infinity?