While the post covers quite a bit of ground, it feels (to me) like it conflates knowledge representation, language, biological systems (i.e., the messiness of implementation), computability, and realism.
Regarding numbers in particular, there are a practically uncountably infinite number of mathematical truths that apply equally to numbers or to other abstract (non-numerical) mathematical ideas.
I would rather see depth in one area or another, rather than a conflation of ideas providing food for thought. Otherwise there are far too many variables to consider for a worthwhile analysis.
While the post covers quite a bit of ground, it feels (to me) like it conflates knowledge representation, language, biological systems (i.e., the messiness of implementation), computability, and realism.
I understand that many of those well-developed fields which exist on their own terms, have standard methods, standard questions and standard approaches to moving towards answers.
The article jump between these fields to ask and grope for an answer to a simple question that in many ways can't be asked or answered in these fields.
One thing to consider is that present day computers can follow the mechanical production of mathematical propositions close to completely. But computers have a lot of trouble producing or following arguments like this, in "natural language", which have a definite logic to them but whose operation is not based on only explicit, codified rules.
Edit: To me, this sort of speculation is what philosophy actually should be doing. The questions that are "ill-defined but compelling" are the questions that have lead significant intellectual progress. How Zeno's paradox lead (or at least related) to the invention of calculus, how Einstein's thought experiments lead to relativity, etc.
Because of the limitation of language there's only a countable number of mathematical truths that can be proven or written down. So for all practical purposes there's countably many.
There are countably many proofs, but a proof does not entail a single truth. Many truths are entailed by a single proof, in fact an uncountable number of truths can be entailed by a single proof.
That does not mean that all truths can be written down or enumerated, but strictly speaking it is not sufficient to conclude that due to the fact that the set of all proofs are countable, that the set of all truths entailed by the proofs must also be countable.
None of this should be taken to violate Godel's incompleteness theorems.
Finally, it's worth mentioning that not all formal systems are limited to finite proofs. There are formal systems where theorems as well as proofs can be countably infinite in length and where there are uncountably many proofs. These systems, known as infinitary logic, are often reduceable to second order logic and hence are incomplete.
No, because proofs have to consist of a finite number of words. Thus there are only countably many proofs of anything. In particular, there are only countably many reals between 0 and 1 which can be expressed in a finite number of words.
Are there a finite number of words? Language seems to grow and adapt to new concepts as needed, perhaps there is an infinity of linguistic descriptions available to us.
It's not. It's a single proof about a set, a set that's assumed to be uncountable in standard ZF set theory.
The "axiom system" that (supposedly) contain a countable number of axioms. But these too are constructs of set theory. We still create proofs one by one of theories about axiom systems with infinite axiom - so we have a countable/enumerable set of such theories.
The proof systems to we can see or touch involve this enumerable properties. Perhaps you could change that with an analogue computer that a person could input "any" "quantity" into. But that's outside math as things stand.
Do you mean the proof that 0.25 >= 0 and the proof that 1/e >= 0 count as the same one, because there's a more general proof that a set of values including those is >= 0? But then where do you draw the line? When can you consider 2 proofs different enough to count as different ones?
I think you have a slightly stricter definition of "a proof" than me. I would consider a proof that all the numbers in (0,1) are positive to also be a proof that the number 0.5 is positive, as well as the number 1/e, and Champernowne's constant.
Since the original question was about uncountably many mathematical truths I would say we have one proof that proves uncountably many mathematical truths.
It is an abstract proof of a generator for concrete proofs of specific assignments to variables. The potential is uncountable, but only a countable subset will ever be invoked.
I don't know what it really means for a proof to be invoked, and I also don't really like the idea of separating proofs into concrete and abstract proofs. Either it proves something or it doesn't.
Regarding numbers in particular, there are a practically uncountably infinite number of mathematical truths that apply equally to numbers or to other abstract (non-numerical) mathematical ideas.
I would rather see depth in one area or another, rather than a conflation of ideas providing food for thought. Otherwise there are far too many variables to consider for a worthwhile analysis.