Assuming first-order logic:
T ⊢ S
S can be unknown. T cannot.
DeepSeek.
Undecidability: In first-order logic, solving for T given S is generally undecidable (per Gödel/Turing).
Resolution of the question as to whether math is discovered or invented
Re: Resolution of the question as to whether math is discovered or invented
That's another deflection.
Assuming nothing.
The classes A and B are unconstrained and maximally general; thus contain theories of any and all orders.
Do you need it spelled out in detail?
Any order: First-order logic, second-order logic, higher-order logics, infinitary logics
Any system: Classical logic, intuitionistic logic, paraconsistent logic, modal logic
Any framework: Set theory, type theory, category theory, topos theory
Any formalism: Axiomatic systems, natural deduction, sequent calculus
So on and so on and so on.
A ⊢ ∃0
B ⊢ ¬∃0
You are being presented with an decision problem!
Does the Platonic realm resolve it; or not?
Is the Platonic realm an exception to general undecidability or not?
If it is not an exception then it is necessarily vacuous. Because that's what exceptions are - information!
Re: Resolution of the question as to whether math is discovered or invented
The symbol "0" is just a unicode character. The number zero has a particular impact on the theory in which it is mentioned. If you look at PA:
∀x¬(S(x)=0)
∀x(x=0∨∃y(x=S(y)))
==> it has no successor
==> So, it must be a theory where S(x) is defined. Otherwise, zero is pointless.
∀x(x+0=x)
==> It is the neutral element for the addition
==> Has addition even be defined in the theory?
∀x(x×0=0)
==> It absorbs the multiplication
==> Has multiplication even be defined in the theory?
[φ(0)∧∀x(φ(x)→φ(S(x)))]→∀xφ(x)
==> It is the base case for an induction schema
==> Does the theory support induction?
Does theory A or B have all these axioms in which zero is mentioned? Otherwise, is it even the same zero? The theory could use the unicode character 0 for some completely incompatible purpose, or it could use another character for a notion that is actually compatible with zero.
Furthermore, zero may not even be mentioned in the axioms of the theory, but the theory may be able to define it indirectly, such as ZF-inf does. Without ever mentioning zero in its axioms, ZF-inf is perfectly bi-interpretable with PA.
Hence, the expression "A ⊢ ∃0, B ⊢ ¬∃0" means absolutely nothing. It is just bullshit.
Re: Resolution of the question as to whether math is discovered or invented
Why are you evading?godelian wrote: ↑Wed Jun 11, 2025 11:42 amThe symbol "0" is just a unicode character. The number zero has a particular impact on the theory in which it is mentioned. If you look at PA:
∀x¬(S(x)=0)
∀x(x=0∨∃y(x=S(y)))
==> it has no successor
==> So, it must be a theory where S(x) is defined. Otherwise, zero is pointless.
∀x(x+0=x)
==> It is the neutral element for the addition
==> Has addition even be defined in the theory?
∀x(x×0=0)
==> It absorbs the multiplication
==> Has multiplication even be defined in the theory?
[φ(0)∧∀x(φ(x)→φ(S(x)))]→∀xφ(x)
==> It is the base case for an induction schema
==> Does the theory support induction?
Does theory A or B have all these axioms in which zero is mentioned? Otherwise, is it even the same zero? The theory could use the unicode character 0 for some completely incompatible purpose, or it could use another character for a notion that is actually compatible with zero.
Furthermore, zero may not even be mentioned in the axioms of the theory, but the theory may be able to define it indirectly, such as ZF-inf does. Without ever mentioning zero in its axioms, ZF-inf is perfectly bi-interpretable with PA.
Hence, the expression "A ⊢ ∃0, B ⊢ ¬∃0" means absolutely nothing. It is just bullshit.
Tell me that you don't know whether 0 exists or not.
Tell me that you don't know whether 0 denotes any object in Platonic reality.
Tell me that PA is your religion; and that other religions exist.
You know that I know that you know that I know that you know that I know that you know that I know.... that Platonism is vacuous.
Why are you so committed to blatant fucking intellectual dishonesty?!?
What we DO when we DO Mathematics is invention. Followed by the discovery of the semantic properties/invariants of our inventions. Stop lying about it already.
Last edited by Skepdick on Wed Jun 11, 2025 11:58 am, edited 1 time in total.
Re: Resolution of the question as to whether math is discovered or invented
The notion of 0 in PA also exists in ZF-inf as {}, even though it is never mentioned in its axioms. Concerning whether it exists in an arbitrary theory A or B, I don't know. It depends on the details of A or B.
Re: Resolution of the question as to whether math is discovered or invented
Immaterial. The notation accounts for this.
Either A ⊢ ∃0 axiomatically (e.g explicitly stated); or A ⊢ ∃0 as a theorem (e.g implicit given the axioms).
Either way: A ⊢ ∃0
No it doesn't depend on the details of A or B. Not on Platonism.
Either 0; or {}; or ∅; or whatever other symbol/representation denotes an object in Platonic reality; or it doesn't.
Stop fucking lying.
Re: Resolution of the question as to whether math is discovered or invented
0 is defined by its role in the structure of PA. There is no decidable procedure to figure out if there is any symbol that could potentially, even implicitly, have the same role in an arbitrary structure A.
You won't figure out the ontology of the thing represented on a map, merely by staring at the map. The ontology of mathematical objects is simply not a deductive claim in mathematics. The philosophy of mathematics is philosophy and not mathematics. With your A and B theories, you are trying to solve a problem by means of mathematics that mathematics itself cannot solve.
You either believe in Platonism or you don't. You cannot prove or disprove it.ChatGPT: According to mathematical Platonism, is 0 an object in Platonic reality?
Yes, according to mathematical Platonism, 0 is considered an object in Platonic reality.
Here's why:
Mathematical Platonism is the view that mathematical entities (like numbers, sets, functions, etc.) exist independently of human minds in a non-physical, abstract realm. According to this view:
Numbers are not inventions or mere symbols, but discovered entities that exist timelessly and necessarily.
Just as the number 1 or π is a real object in this Platonic realm, so is the number 0.
The status of 0:
Despite being a relatively late historical development (not widely used in ancient mathematics until Indian and Islamic mathematicians formalized it), Platonists would argue that 0 always existed in the Platonic realm, even before humans conceptualized it.
0 represents a cardinal number (the size of the empty set), and in most modern mathematical frameworks (e.g. set theory), it is defined in terms of other well-formed abstract concepts.
So, from a Platonist point of view, 0 is as real and objective as any other mathematical object.
Summary:
Under mathematical Platonism, 0 is an abstract object that exists in the Platonic realm, independent of space, time, and human thought.
Re: Resolution of the question as to whether math is discovered or invented
ChatGPT: Can you prove or disprove mathematical Platonism?
Mathematical Platonism is a philosophical position, not a mathematical theorem, so it cannot be proved or disproved in the same way that a mathematical statement can. However, we can analyze arguments for and against it and evaluate its plausibility within different philosophical frameworks.
Can It Be Proved or Disproved?
No. Here's why:
Mathematical Platonism makes metaphysical claims, not empirically testable or deductive ones.
Just like you can’t "prove" that moral facts exist independently of humans, you can't prove that numbers do either.
It’s subject to the same types of debate as realism vs anti-realism in other domains (e.g., ethics, aesthetics).
Re: Resolution of the question as to whether math is discovered or invented
That is PRECISELY what makes it epistemically vacuous!godelian wrote: ↑Wed Jun 11, 2025 12:42 pmChatGPT: Can you prove or disprove mathematical Platonism?
Mathematical Platonism is a philosophical position, not a mathematical theorem, so it cannot be proved or disproved in the same way that a mathematical statement can. However, we can analyze arguments for and against it and evaluate its plausibility within different philosophical frameworks.
Can It Be Proved or Disproved?
No. Here's why:
Mathematical Platonism makes metaphysical claims, not empirically testable or deductive ones.
Just like you can’t "prove" that moral facts exist independently of humans, you can't prove that numbers do either.
It’s subject to the same types of debate as realism vs anti-realism in other domains (e.g., ethics, aesthetics).
It is just another instantiation of an undecidable problem!
It is an informal encoding of an undecidable proposition!
A computation without a value assignment!
A meaningless computation.
Skepdick wrote: ↑Wed Jun 11, 2025 11:51 am Tell me that you don't know whether 0 exists or not.
Tell me that you don't know whether 0 denotes any object in Platonic reality.
Tell me that PA is your religion; and that other religions exist.
You know that I know that you know that I know that you know that I know that you know that I know.... that Platonism is vacuous.
Why are you so committed to blatant fucking intellectual dishonesty?!?
Re: Resolution of the question as to whether math is discovered or invented
NO SHIT sherlock! That's precisely what we have proven constructively.
ALL meta-circular evaluators are vacuous.
They are just fixed point computations.
Re: Resolution of the question as to whether math is discovered or invented
In the Cartesian multiplication (inductive/deductive) concerning (physical universe, Platonic world of ideas/abstractions) we have four possibilities:
(1) inductive - physical universe ==> science and observational studies
(2) inductive - Platonic world ==> philosophy
(3) deductive - Platonic world ==> mathematics
(4) deductive - physical universe ==> not possible because there is no ToE (Theory of Everthing) available for that
The epistemic domain (inductive - Platonic world), i.e. philosophy, exists because deduction is not always possible in the Platonic world. Mutatis mutandis, that is also the reason why science exists. If the epistemic domain (deductive - physical universe) were possible, we would not even use science.
Just like there is no ToE available for the physical universe, there is not one for the Platonic realm either. Philosophy is not vacuous. We use it because we do not have anything better available. I am fine with it because I accept the limitations described above.
Re: Resolution of the question as to whether math is discovered or invented
All you have done is listed methods of inference.godelian wrote: ↑Wed Jun 11, 2025 1:32 pmIn the Cartesian multiplication (inductive/deductive) concerning (physical universe, Platonic world of ideas/abstractions) we have four possibilities:
(1) inductive - physical universe ==> science and observational studies
(2) inductive - Platonic world ==> philosophy
(3) deductive - Platonic world ==> mathematics
(4) deductive - physical universe ==> not possible because there is no ToE (Theory of Everthing) available for that
The epistemic domain (inductive - Platonic world), i.e. philosophy, exists because deduction is not always possible in the Platonic world. Mutatis mutandis, that is also the reason why science exists. If the epistemic domain (deductive - physical universe) were possible, we would not even use science.
Just like there is no ToE available for the physical universe, there is not one for the Platonic realm either. Philosophy is not vacuous. We use it because we do not have anything better available. I am fine with it because I accept the limitations described above.
You've said nothing about what it is you input into them.
You've constructed elaborate intellectual machinery, but there's nobody there to operate it.
A mechanical contraption without a machinist.
Re: Resolution of the question as to whether math is discovered or invented
I have just pointed out what the fundamental difference is between mathematics, science, and philosophy. I don't think that it is necessary to give examples. Everybody is supposed to know examples already. There are millions of machinists out there.
Re: Resolution of the question as to whether math is discovered or invented
Where is the machinist in the Platonic realm?