The question is whether we can communicate our mathematics to non-humans, whose senses may be radically different from our own.
And likewise, would we be able to understand a non-human proof?
I'm a puzzled by the confidence here. I would assume pg is at least minimally familiar with some of the key philosophical themes and schools of thought around this topic.
> For intuitionistic mathematics ... truth is a mental construct of a proof in a language, and only shared by communicating that construct to other minds.
Even assuming you are fully and accurately representing the intuitionist view, you must be aware that there are competing schools of thought with strong pedigrees, like mathematical platonism, that are grounded in a more realist view of mathematical objects.
PG didn't go so far as to stake out that position here in the essay, but his thought experiment leverages a view of mathematical truth that hues closer to this (platonist) camp.
> The question is whether we can communicate our mathematics to non-humans ...
That is an interesting question, but pg did not ask that question in the essay, and its answer doesn't seem relevant to the point he was trying to make.
I think pg is educated, at least the broad outline, of the age-old controversies in this topic and I'm grateful as a reader that he spares me the details and assumes some preliminary context.
His style of pop philosophy is useful in startups because it does not require context. It is ungrounded in actual philosophy, and would struggle to find an audience outside of this group.
But it is not straightforward to me. Or, in my impression, most philosophers. I'm reluctant to agree that aliens will assign any importance or even understand concepts like geometry or calculus. So to me, it is not preliminary context but it is a critical part of the discussion.
But even a brief foray into discussions by actual philosophers will show that to not be true. What Paul Graham is doing is providing small subcultural insights tailored to an audience with a passing interest in philosophy.
I think the word in that idiom is 'hews', which means to adhere strictly to a standard, probably from the sense of the word meaning to strike or cut or beat - often used to talk about cutting a tree into shape.
I'm not sure why people find this so relevant. Regardless of the sense, there must be enough discernible structure to detect whether 0 of something is there, whether 1 of something is there, or whether there are many more than 1.
Human sense of smell might top out at differentiating maybe 4 or 5 different things, dogs can probably sense a lot more, but either way it lets us set up basic counting, and that's generally all you need for most of our math.
How can we communicate actual axioms to extra-terrestrials?
Presumably we can create a kind of Rosetta Stone with different representations of logical and algebraic expressions, and maybe they can decode that if they can figure out how to decode whatever "broadcast" representation of this we come up with. And that assumes that they recognize such a broadcast as something from intelligent life.
peano arithmetic (an axiomatiazation of counting) isn't how we count. It is how we make other things count for us.