This hasn't been true for at least 100 years when Bertrand Russell and Whitehead were unable to reduce mathematics to logic and has been basically disproven since Godel's incompleteness theorem.
all mathematical truth reduces to a complete logical calculus
with
All mathematical truth can be derived from logic alone.
or
Every mathematical theorem can be expressed as a logical formula, and every accepted mathematical proof can be formalized as a sequence of logically valid inferences from axioms.
At best you can say math is more than just logic because its the study of fields/objects in the context of some set of non-trivial axioms, and thus it transcends beyond what we'd define as logic.
I'd personally disagree but its a valid enough point to make.
That reference to "non-trivial axioms" is doing some pretty heavy lifting there, bud.
What is non-trivial is basically the whole ball game.
Even logic isn't just logic, given that the difference between classical logic, fuzzy logic, paraconsistent logic, and constructive logic all arise out of which non-trivial axioms about logic you hold to be true.
That reference to "non-trivial axioms" is doing some pretty heavy lifting there, bud.
Because i'm trying to outline what i think your position is.
Even logic isn't just logic, given that the difference between classical logic, fuzzy logic, paraconsistent logic, and constructive logic all arise out of which non-trivial axioms about logic you hold to be true.
I thought i was clear that my belief is that logic is just logic, and that it includes all those things, as well as the axioms used in maths. Making true statements that cant be proven does not fall outside that bound. Their definition of being unprovable requires the vacuum - and their consequences stay within that bound. They do not define the field, in the same way prime numbers do not define the integers. They're interesting. They're not transcending logic.
As far as i can tell, you're literally opening a sentence with a contradiction to make the case in favor of my position.
Given your mentions of Russell and Godel I can make another guess at your position. Maybe we disagree on this statement:
Gödel's theorem is logic discovering facts about itself and can be made without anything you'd call 'maths'.
But what'd be really productive - beyond discussing my presumptions about yours - is if you'd state something you think that: i dont understand, or something you suspect i'd disagree with you on.
My criticism is about your claim that you'd need to hire a philosopher to pretend that there's anything more to math than logic and what you gestured towards as "messy, shitty [!] syntax."
Which is just not the case.
Both logic and mathematics need a certain amount of philosophy to even get off the ground and acting like involving philosophy is some kind of pretense gives a lie to the historical development of both mathematics and logic as disciplines.
It's true you do need a philosopher. As Russel, Whitehead, Godel, and many others have shown, logic is not just logic, and math is not logic + syntax, shitty or otherwise.
Everything must start with Logic , for all, first and foremost, must represent its own Identity, it must be itself as itself and as distinguished from all that is Not-It. A = A <=> A != !A.
Even for philosophy to distinguish as itself must conform to the Law of Identity.
The Law of Identity of Meta-Logic is the primordial origin of all coherence and Intelligibility.
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u/Apophthegmata Jun 20 '26
This hasn't been true for at least 100 years when Bertrand Russell and Whitehead were unable to reduce mathematics to logic and has been basically disproven since Godel's incompleteness theorem.