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I ask here with a unique idea: Our entire mathematical corpus has at its core hidden axioms, all involving words.

Words are capable of performing immutable perfect symbology. Every truth is able to be exactly written as a statement composed of words. Every sentence of certain linguistic capacity (using logically tuned words) is a perfect capture of any situation that meets such relationships.

These axioms can be tuned, I am giving a heuristic here. To rewrite the ideas, the axioms are that words span our mathematical thought-space entirely whilst possessing immutable perfection, and sequences of words can generate mathematical thoughts.

So can wordless mathematics be performed? Is there a wordless apparatus that is capable of surviving such axioms? This aught to be true, else it suggests that words are the basis of truth. Our mathematically precise words here are comparable to computer code, where you can transplant it and verify it across different computers (people) and will independently claim a unified truth/false on the statements given.

But objects inherently possess qualities, beyond needing the description of such. And words themselves possess failures too, as seen in Gödel's Incompleteness Theorems.

Imagine if words were not to exist, how would we perform mathematics then? Would it be descriptive, using object manipulation as a metaphor (think of an abacus)? Or would it be necessary to invent words altogether as to perform mathematics.

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    – Geoffrey Thomas ♦
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  • Define "words". e.g., if we say "->" instead of "implies", is it a word?
    – Ray
    Commented Oct 25 at 16:14

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No.

See Yuri Manin, A Course in Mathematical Logic for Mathematicians (2010), page 45 :

A proof becomes a proof only after the social act of “accepting it as a proof.” This is as true for mathematics as it is for physics, linguistics, or biology. The evolution of commonly accepted criteria for an argument’s being a proof is an almost untouched theme in the history of science. In any case, the ideal for what constitutes a mathematical demonstration of a “nonobvious truth” has remained unchanged since the time of Euclid: we must arrive at such a truth from “obvious” hypotheses, or assertions that have already been proved, by means of a series of explicitly described, “obviously valid” elementary deductions.

Thus, the method of deduction is a method of mathematics par excellence.

[...] Every proof that is written must be approved and accepted by other mathematicians, sometimes by several generations of mathematicians. In the meantime, both the result and the proof itself are liable to be refined and improved.

The historical stability of the criteria for an acceptable proof does not imply that mathematics and proofs are supra-human : they are human (and social) activities.

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  • "A proof becomes a proof only after the social act of accepting it as a proof" - Or after the act of a fully formal proof being accepted by a computerized proof-verifier, such as in the formal verification of computer programs. It need not be a human who verifies a proof. There is the reply that "but it's not really a proof until a human accepts the proof-verifier's answer." However, some proofs, such as zero-knowledge proofs in cryptography or many deductions made by a compiler, never see the eyes of a human and are entirely written and read by computers, and are useful nonetheless.
    – causative ♦
    Commented Oct 24 at 18:03
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    Also the idea that a proof only becomes a proof by a social act, does not agree with how we usually use the term. If a mathematician labors in private, speaking to no one, to write a proof of a conjecture C, and then he claims, "On this day, Oct 24 2024, I have proved C!" - and on Oct 24, 2024 no one besides him has even read his proof, let alone accepted it - is he lying? By our ordinary use of language, no - provided his proof is indeed valid, it was a proof the moment his pen left the paper. Social verification comes afterwards.
    – causative ♦
    Commented Oct 24 at 18:30
  • @causative I think you're trying to anthropomorphize your cake and eat it, too. If a computer is in some way aware of what a proof is and conscious of having found one, it is a kind of person, and a person (I don't think anyone would suggest it has to be a human person) has in fact verified the proof. If a computer is not in any way conscious and aware, a computer is a circuit with a mechanical algorithm for flipping switches, and what has happened is turning on a current in a wire.
    – g s
    Commented Oct 25 at 1:35
  • I have no opinion about whether the time of the proof is assigned when a person comes along and symbolically interprets the computer state, or later when they do some social activity regarding the computer state and the proof, or at the time of the computer operation that created the computer state which is being symbolically interpreted. But I think that until there's somebody to interpret it symbolically as a claim about an abstract object, calling a hardware state a proof is just a useful anthropomorphic fiction.
    – g s
    Commented Oct 25 at 1:48
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    @mudskipper That is true, but the Manin quote actually didn't restrict itself to deductive proofs, including also "proofs" in "physics, linguistics, or biology." These other fields make use of proof in the informal sense of "strong empirical evidence," so it's also relevant that a non-deductive proof (zero-knowledge proof) could be checked by machine. I also mentioned proofs generated by compilers, many which of which are deductive (e.g. type deduction).
    – causative ♦
    Commented Oct 25 at 23:23

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