Resolution of the question as to whether math is discovered or invented

[Skepdick]

Choosing another T merely means choosing another map. Of course, you will see different things.

So when maps disagree amongst themselves which one do you trust to faithfully represent the terrain and guide you through the Platonic realm?

[godelian]

Choosing another T merely means choosing another map. Of course, you will see different things.

So when maps disagree amongst themselves which one do you trust to faithfully represent the terrain and guide you through the Platonic realm?

In first-order logic, ZFC shows more than PA. Then, you have the somewhat crippled maps, such as Robinson, Presburger, or Skolem, that show a lot less.

In fact, I don't know of maps that support a concept of numbers, are incompatible with PA, but are still consistent. It's certainly not excluded that they exist. It seems to be easier to discover alternative geometries than alternative arithmetics.

DeepSeek came up with one but it is not particularly impressive. It looks more like a contorted corner case:

DeepSeek

Robinson Arithmetic (Q) extended with the axiom ∃c (Sc = c).
Incompatible with PA because PA proves that no number is equal to its own successor.

This theory demonstrates that a consistent numerical system can exist that contradicts PA, leveraging the absence of the full induction schema in Q to allow non-standard elements like the fixed-point number c.

[Skepdick]

Choosing another T merely means choosing another map. Of course, you will see different things.

So when maps disagree amongst themselves which one do you trust to faithfully represent the terrain and guide you through the Platonic realm?

In first-order logic, ZFC shows more than PA. Then, you have the somewhat crippled maps, such as Robinson, Presburger, or Skolem, that show a lot less.

In fact, I don't know of maps that support a concept of numbers, are incompatible with PA, but are still consistent. It's certainly not excluded that they exist. It seems to be easier to discover alternative geometries than alternative arithmetics.

DeepSeek came up with one but it is not particularly impressive. It looks more like a contorted corner case:

DeepSeek

Robinson Arithmetic (Q) extended with the axiom ∃c (Sc = c).
Incompatible with PA because PA proves that no number is equal to its own successor.

This theory demonstrates that a consistent numerical system can exist that contradicts PA, leveraging the absence of the full induction schema in Q to allow non-standard elements like the fixed-point number c.

Your head's up in the sky again. Or rather - right up your ass.
You keep retreating into technical complexity while avoiding the simplest fucking point right at your feet.

One map says 0 exists.
Another map says it doesn't.

Now what?

Either mathematical truth is discovered and there is an objective answer to the question; or mathematics is invented and the answer is down to an arbitrary social convention.

Which one is it?

[godelian]

One map says 0 exists.
Another map says it doesn't.
Now what?

Look at how addition and multiplication are defined in PA:

Addition:
base: a+0=a
recurse: a+S(b)=S(a+b)

Multiplication:
base: a*0=0
recurse: a*S(b)=a+(a*b)

Without a definition for 0, you have to scrap the definition for the base case for the addition. The recursion wouldn't terminate anymore. So, you won't be able to define the addition, which is in turn necessary for the recurse case in the multiplication. So, you won't be able to define the multiplication either.

Consequently, addition and multiplication are no longer defined. Therefore, it is no longer an arithmetic.

Maybe there are workarounds and alternative solutions possible by judiciously hacking PA, but simply removing the definition for zero won't work.

[Skepdick]

Look at how addition and multiplication are defined in PA:

Addition:
base: a+0=a
recurse: a+S(b)=S(a+b)

Multiplication:
base: a*0=0
recurse: a*S(b)=a+(a*b)

Without a definition for 0, you have to scrap the definition for the base case for the addition. The recursion wouldn't terminate anymore. So, you won't be able to define the addition, which is in turn necessary for the recurse case in the multiplication. So, you won't be able to define the multiplication either.

Consequently, addition and multiplication are no longer defined. Therefore, it is no longer an arithmetic.

Maybe there are workarounds and alternative solutions possible by judiciously hacking PA, but simply removing the definition for zero won't work.

How 0 functions in any given theory is neither here nor there.

It has nothing to do with the question.

The mechanics of formal systems don't resolve the discovery vs. invention debate

Either mathematical truth is discovered and there is an objective answer to the question; or mathematics is invented and the answer is down to an arbitrary social convention.

Which one is it?

[godelian]

How 0 functions in any given theory is neither here nor there.

That remark is a bit facile. You originally asked about PA with or without zero. The answer is that PA without zero is no longer an arithmetic theory, because the definitions of addition and multiplications would blow up. It is not possible to arbitrarily remove something or change the definition of PA, because it all hangs together. Your question amounts to asking if you can just remove an arbitrary line here or there from an existing program. Chances are that you can't.

[luberti]

Under common ontological notions, you obviously cannot describe a class that doesn't already exist, since they all exist as abstract objects as soon as they are classes. Also a class didn't come into existence. It exists independently of time since it is neither a physical nor a mental object.

So you've invented a new ontological category of "neither physical nor mental abstract objects" to account for pre-existing classes.

I didn't invent anything there, that's just what was taught as most essential preliminaries at my university in the most basic philosophy class there was ("Philosophische Propädeutik").

My point was that under common ontological notions, abstract objects cannot be invented but are always discovered.

Under common ontological notions abstractions are not ontological. [...]

You're full of shit, or that's a straw man (idk what you mean by "ontological" or why you believe it would even matter). You cannot talk about something when it is not a thing. Ontology is about categorizing _all_ things. Anyway, I'm not gonna argue definitions. When you refuse to use certain definitions, you simply cannot talk about certain things. Not my problem.

So I don't see a point in continuing this since it is already solved and it concerns just basic semantics.

Solved? You didn't even define/specify what the question was. What is this abstract "problem" you are solving? Where can I discover it?

No, this is not "just basic semantics".
It's just basic grammar.

What do you mean by "exists" (a verb - and therefore a temporal notion)... "independently of time".
This is grammatically incoherent.

Your basic use of language is broken

That's bullshit. Grammar is syntactical, i.e. concerns form, whereas meaning is semantical, i.e. concerns models.

Also the solved question was obviously the one stated in the title of the thread ("question as to whether math is discovered or invented").

I don't know what's the issue with you, but you seem to be arguing for the sake of arguing rather than caring about what is true under which definitions. That's just corrupted troll behavior. I can't take you any serious anymore.

[godelian]

Either mathematical truth is discovered and there is an objective answer to the question; or mathematics is invented and the answer is down to an arbitrary social convention.
Which one is it?

Until Peano and Dedekind finally managed to discover one in the late 19th century, there was no axiomatic theory for arithmetic, to the extent that a century earlier Kant had even claimed in his Critique of Pure Reason that there simply wasn't one:

Arithmetic has no axioms. ... Its propositions are all synthetic, without being derived from general concepts alone, but through the construction of magnitudes in time.

Critique of Pure Reason, A164/B205

If axiomatic theories are mere inventions, then why didn't some inventor just liberally invent an axiomatic theory for arithmetic? That should have been easy, no? If it is just an invention?

Furthermore, Peano and Dedekind proved Kant to be utterly wrong. There definitely is an axiomatic theory for the natural numbers.

Kant couldn't discover one, but that does not mean that there isn't one. It just means that Kant was not capable of discovering one, not even to save himself from drowning.

I have said this many times before. Kant's Critique of Pure Reason is a lengthy and meandering word salad in which he pretty much never commits to anything actionable or falsifiable. However, the few rare times that he actually does, his position always turns out to be plain wrong.

If you try to construct an axiomatic theory for the numbers from scratch, you are facing a harsh task because its structure is incredibly unforgiving. It will literally not add up! If you remove any essential element, the entire thing will just fall apart. You cannot just invent that structure. You have to carefully and painstakingly uncover it.

[attofishpi]

I swear you are talking to yourself!

[Skepdick]

I didn't invent anything there, that's just what was taught as most essential preliminaries at my university in the most basic philosophy class there was ("Philosophische Propädeutik").

OK, so somebody else invented it. You discovered, or more precisely - you were indoctrinated with the invention.

You're full of shit, or that's a straw man (idk what you mean by "ontological" or why you believe it would even matter).

I mean the same thing everyone means when they use the word. I also mean NON-epistemic.

You cannot talk about something when it is not a thing. Ontology is about categorizing _all_ things.

Aaaah, you see! There lies your confusion. Ontology is about what exists. Categorizing existence (synthesizing taxonomies) is epistemic.

Anyway, I'm not gonna argue definitions.

We aren't arguing definitions. We are arguing fundamental categories. And more importantly - we are arguing about how ontology relates to epistemology and vice versa.

When you refuse to use certain definitions, you simply cannot talk about certain things. Not my problem.

I guess we'll have to agree to disagree. I use language to invent definitions - I don't presuppose them.

That's bullshit. Grammar is syntactical, i.e. concerns form, whereas meaning is semantical, i.e. concerns models.

I know that. Which is why I was explaining your grammatical/syntax error.

Non-temporal existence is a category error.

So it's practically impossible for you to be saying anything meaningful using an ill-formed expression.

Also the solved question was obviously the one stated in the title of the thread ("question as to whether math is discovered or invented").

You appear to have misunderstood the assigned. You failed to define the question.

And even more importantly - you failed to define the methodology for discerning the right from the wrong answers.

I don't know what's the issue with you, but you seem to be arguing for the sake of arguing rather than caring about what is true under which definitions. That's just corrupted troll behavior. I can't take you any serious anymore.

You get 100000000000000000 points for irony.

I care very very very deeply as to which answer to the OP questions is true under which definitions.
And I care very very deeply about the methodology used to obtain the answers.

Which is precisely and exactly why I keep asking you to define the question using well-formed/well-defined semantics!

Can you please make the epistemic criteria for discerning one from the other well-defined.
Can you please construct a classifier - specify the criteria for identity.

Thanks.

If; or when you eventually invent the binary classifier - the question will answer itself. Until then the answer is "We don't know".

Or maybe we don't have to go that far? ALL methods for discovering/obtaining truth (whatever that is!) are of human inventions!
Mathematics is no exception.

[Skepdick]

How 0 functions in any given theory is neither here nor there.

That remark is a bit facile. You originally asked about PA with or without zero.

I didn't. Why are you lying?

My question was context-free of any theory beyond your metaphysical commitment to Platonism.

Suppose some class of theories (lets call them A) asserts the existence of 0; an another class of theories (lets call them B) asserts the non-existence of 0. To avoid bickering over notation lets just say:

A ⊢ ∃0
B ⊢ ¬∃0

Both A and B may; or may not assert other things too. That is uninteresting.
That PA (or any particular theory) happens to be of type A; and not type B is immaterial to the question.

The question being: Which class of theories corresponds to the Platonic realm?

There's no wiggle room here. A Platonist can't say "it depends on the theory" or "we need more context." They're committed to there being an objective fact of the matter that determines which class of theories corresponds to reality.

This is a direct reification of excluded middle - either ∃0 or ¬∃0 (if you are more logically inclined); or the axiom of choice (if you are Mathematically inclined). Given the set {A ⊢ ∃0, B ⊢ ¬∃0} choose one.

The answer is that PA without zero is no longer an arithmetic theory , because the definitions of addition and multiplications would blow up. It is not possible to arbitrarily remove something or change the definition of PA, because it all hangs together. Your question amounts to asking if you can just remove an arbitrary line here or there from an existing program. Chances are that you can't.

Who cares? I am not interested in a theory of arithmetic. I am interested in true statements about the Platonic realm.

Axiomatization has no bearing on arithmetic anyway. We were doing arithmetic long before we had axioms for it.

[godelian]

My question was context-free of any theory beyond your metaphysical commitment to Platonism.

There is nothing to say about "any theory". Context-free does not exist in mathematics. There is always a context. Otherwise, it is not mathematics.

Axiomatization has no bearing on arithmetic anyway. We were doing arithmetic long before we had axioms for it.

Yes, but there was nothing provable about arithmetic without theory. Same for geometry. You don't need a theory for geometry, but you won't prove anything about it, if you go ahead without a theory.

https://en.wikipedia.org/wiki/Informal_mathematics

Informal mathematics, also called naïve mathematics, has historically been the predominant form of mathematics at most times and in most cultures, and is the subject of modern ethno-cultural studies of mathematics.

Informal mathematics means any informal mathematical practices, as used in everyday life, or by aboriginal or ancient peoples, without historical or geographical limitation. Modern mathematics, exceptionally from that point of view, emphasizes formal and strict proofs of all statements from given axioms. This can usefully be called therefore formal mathematics. Informal practices are usually understood intuitively and justified with examples—there are no axioms.

Aboriginal mathematics is fine. It's what most people still use today in daily life.

[Skepdick]

My question was context-free of any theory beyond your metaphysical commitment to Platonism.

There is nothing to say about "any theory". Context-free does not exist in mathematics. There is always a context. Otherwise, it is not mathematics.

OK. Lets work with that.

In context of the Platonic ream which class of theories is true?

A or B?

I am not going to entertain the rest of the evasion.

[godelian]

My question was context-free of any theory beyond your metaphysical commitment to Platonism.

There is nothing to say about "any theory". Context-free does not exist in mathematics. There is always a context. Otherwise, it is not mathematics.

OK. Lets work with that.

In context of the Platonic ream which class of theories is true?

A or B?

I am not going to entertain the rest of the evasion.

You have to say what A and B are. Otherwise, there is nothing to say about A or B.

[Skepdick]

You have to say what A and B are. Otherwise, there is nothing to say about A or B.

Suppose some class of theories (lets call them A) asserts the existence of 0; an another class of theories (lets call them B) asserts the non-existence of 0. To avoid bickering over notation lets just say:

A ⊢ ∃0
B ⊢ ¬∃0