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

[godelian]

it's just wishful thinking with formal notation.

Constructivism amounts to saying:

If it is not on the map, then it doesn't exist.

We know, however, that every map is necessarily unreliable and often even misleading.

Much of the unknown territory (the "truth") is invisible on the map (the collection of provable truths).

This understanding started growing long before Godel published his incompleteness theorem.

So, by exploiting a very subtle flaw in the PA map -- the diagonal lemma -- Godel proved that there exist features in the actual territory that are not on the map or (unknown) map features, that do not exist in the actual territory and are just bullshit.

Godel never gave an example, though. Godel claimed that his incompleteness theorem would be "intuitionistically unobjectionable", but that is obviously not true, because he was unable to give an example of an unmapped truth about the territory or a bullshit map feature, not even to save himself from drowning.

Goodstein later gave the first example, even though it is a bit shoehorned.

Gentzen's equiconsistency theorem says that PA and PRA are maps with exactly the same (unknown) bullshit features. It is also considered an elusive example for Godel's incompleteness theorem, even though it isn't.

Constructivism is very naive:

-There is nothing in the territory that is not on the map --> not true.
- The map doesn't have bullshit features on it --> not true.

[Skepdick]

All this time you were equating classical mathematics with ZFC, and now... you are using more axioms.

ZFC is an incomplete map to an otherwise unknown territory, i.e. the abstract Platonic world of mathematical objects.

That's only true if Maths is discovered. Otherwise ZFC is just another construction/invention in an otherwise empty Platonic world.

If you don't put any axioms to generate structures - the Platonic world is completely void.

If you carefully add more axioms to the map, it may reveal more about the unknown territory. It may, however, also reveal things that are not even there. That is why adding axioms is so dangerous. ZFC itself already contains two potentially dangerous axioms: infinity and choice.

So tell me something about this Platonic world if you have NO axioms.

Con(ZFC) is an implicit axiom. If you agree to use ZFC, you accept its terms and conditions, which in the fine print mentions Con(ZFC).

That fine print is only necessary if you are a metaphysically committed to LEM.

In a setting without LEM not(Con(ZFC)) doesn't imply Inconsistent(ZFC).
In a constructive setting: Lack of proof for Con(ZFC) implies neither Con(ZFC) nor not(Con(ZFC)).

In general, every time you do the following:

T ⊢ S

Your action implies the hidden belief in Con(T). If you do not believe in Con(T), you should not use T to begin with.

The fact that the following is typically false:

T ⊢ Con(T)

Does not exempt you from believing that Con(T) is true.

If you say that something such as S follows from ZFC, you implicitly believe in Con(ZFC), even though any proof of Con(ZFC) would immediately invalidate the very belief in Con(ZFC).

None of that follows; unless you've smuggled in LEM + a bunch of other stuff.

Who says I have to believe in Con(T) to use it?
Absence of evidence of inconsistency is enough.
Ability to handle exceptions/inconsistencies without the systems exploding in triviality is also enough.

That's how real-world engineering works. Nobody invents fragile systems.

Our maps are notoriously bad and sometimes even misleading.We know that the unknown territory cannot possibly be the same as it deceptively looks like on the maps. Unfortunately, the maps are all we have.

You are fundamentally confused here. Classical logic is an ontological thesis!

You aren't constructing a map!
You are constructing the teritory!

Reverse mathematics is a philosophy about mathematical theories. It is not meant to be mathematics. Treating T as an unknown is not a mathematical act of deduction:

T ⊢ S

Exploring the world of mathematical theories to discover theories that satisfy the equation, is an act of philosophical induction.

It is not wrong, though. It is just something else.

It's a computational satisfiability search.

find axioms T such that T ⊢ S

I believe that every legitimate T is a map to the unknown abstract Platonic territory. That is not a mathematical belief but a philosophical one.

No shit. So tell us something about the abstract objects which exist in the Platonic teritory in a theory without axioms.

[Skepdick]

it's just wishful thinking with formal notation.

Constructivism amounts to saying:

If it is not on the map, then it doesn't exist.

No. It amounts to saying "If you don't put it on the teritory - it's not there". It's ontology engineering.

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

It's world-building, not cartography.

We know, however, that every map is necessarily unreliable and often even misleading.

We also know that all of our inventions (including inventions such as teritories/ontologies) are imperfect and contain unintended consequences; as well as badly specified requirements; and emergent complexity. Because we aren't omniscient - that's precisely why we reject LEM!

Much of the unknown territory (the "truth") is invisible on the map (the collection of provable truths).

This understanding started growing long before Godel published his incompleteness theorem.

That's generally what happens when your constructions outgrows you and becomes too complex to fully understand.

Godel merely shattered the illusion that you can ever keep up.

Constructivism is very naive:

Naive? You mean the guys who directly state what it IS we are doing (when we DO mathematics) - we are inventing notations/computations/formal systems/creating mathematical objects are naive?

And the clowns who think they are discovering are the realists here?

I guess there's part-truth to it. You are discovering other people's inventions...

1000 points for irony.

The people honestly admitting they're building stuff are called "naive," while the people who think they're cosmic archaeologists discovering eternal truths are considered "realistic."

[godelian]

That's only true if Maths is discovered. Otherwise ZFC is just another construction/invention in an otherwise empty Platonic world.

Platonism is a commitment to the belief that math is discovered.

If you don't put any axioms to generate structures - the Platonic world is completely void.
So tell me something about this Platonic world if you have NO axioms.
No shit. So tell us something about the abstract objects which exist in the Platonic teritory in a theory without axioms.

No axioms means that you have an empty map. Such empty map says nothing about the territory.

Who says I have to believe in Con(T) to use it?

If you prove S from T, then you believe that the map does not contain bullshit features. So, in that case, you believe in Con(T). Otherwise, you would not even want to prove S from T.

Absence of evidence of inconsistency is enough.

How do you even know?

You are fundamentally confused here. Classical logic is an ontological thesis!

It is Platonism that is the ontological thesis.

You aren't constructing a map!

Yes, you are, because a theory is a map.

You are constructing the teritory!

No, this is not how model theory works. We construct a map, i.e. a theory. The territory are all the models that interpret the map. It happens to be a quirk of first-order logic that if a theory has an unbounded standard interpretation, that it also has an infinite number of non-standard interpretations. The "interpretation" or "model" is the territory:

https://en.wikipedia.org/wiki/L%C3%B6we ... em_theorem

Löwenheim–Skolem theorem

It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ, and that no first-order theory with an infinite model can have a unique model up to isomorphism. As a consequence, first-order theories are unable to control the cardinality of their infinite models.

So, for example, for Peano Arithmetic (PA) as a map, the territory (the "model") initially looks like just the natural numbers. That is incorrect:

https://en.wikipedia.org/wiki/Non-stand ... arithmetic

In mathematical logic, a non-standard model of arithmetic is a model of first-order Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934).

Most numbers in the territory are non-standard.

[godelian]

And the clowns who think they are discovering are the realists here? The people honestly admitting they're building stuff are called "naive," while the people who think they're cosmic archaeologists discovering eternal truths are considered "realistic."

It is even literally classified like that:

Gemini

Mathematical realism, a philosophical stance, proposes that mathematical truths exist independently of human minds, similar to how physical objects exist in the world. It suggests that mathematicians discover these truths rather than creating them through human thought. This view is often associated with Platonism, where mathematical concepts are considered timeless and unchanging entities in an abstract realm.

Platonism is indeed considered to be a form of "mathematical realism".

[Skepdick]

Platonism is a commitment to the belief that math is discovered.

OK so discover some math! What do you need axioms for?

No axioms means that you have an empty map. Such empty map says nothing about the territory.

That's weird. You can't say "numbers exist" In plain English without PA axioms?

If you prove S from T, then you believe that the map does not contain bullshit features. So, in that case, you believe in Con(T). Otherwise, you would not even want to prove S from T.

Wow! The mental gymnastics. Every map contains bullshit features. The map is never the territory.

How do you even know?

Because you have no proof for Con(ZFC) when using ZFC!

It is Platonism that is the ontological thesis.

Really? Tell me something about your ontology when I remove identity, LEM and non-contradiction.

Yes, because a theory is a map.

A theory is a theory! Law of identity.
"A theory is a map" is an equational statement!

No, this is not how model theory works. We construct a map, i.e. a theory. The territory are all the models that interpret the map.

Really? What's your model/map of a Platonic universe without any axioms?

It happens to be a quirk of first-order logic that if a theory has an unbounded standard interpretation, that it also has an infinite number of non-standard interpretations. The "interpretation" or "model" is the territory:

What? That's completely backwards. The theory is the teritory. The models are the interpretations/maps!

It implies that if a countable first-order theory has an infinite model

Is a first-order theory without any axioms "countable"?
Does it have infinite models?

Most numbers in the territory are non-standard.

Wait! So numbers exist?

[Skepdick]

And the clowns who think they are discovering are the realists here? The people honestly admitting they're building stuff are called "naive," while the people who think they're cosmic archaeologists discovering eternal truths are considered "realistic."

It is even literally classified like that:

Gemini

Mathematical realism, a philosophical stance, proposes that mathematical truths exist independently of human minds, similar to how physical objects exist in the world. It suggests that mathematicians discover these truths rather than creating them through human thought. This view is often associated with Platonism, where mathematical concepts are considered timeless and unchanging entities in an abstract realm.

Platonism is indeed considered to be a form of "mathematical realism".

So communicate a mathematical truth already. Tell me something about Mathematical objects in a Platonic realm without any axioms!

Platonists wouldn't be the first cranks to call themselves "realists".

The only thing constructivists are "realists" about are the nature of Mathematical practice. It's invented, not discovered!

This is what Mathematicians DO:

This is but some of the things Mathematicians actually DO:

Invent notation systems
Construct formal proofs
Build computational procedures
Create axiom systems
Design algorithms
Engineer logical frameworks

You know what Mathematicians don't DO? Discover mystical truths in a mysterious realm.

[godelian]

OK so discover some math! What do you need axioms for?

Without axioms, no map.

That's weird. You can't say "numbers exist" In plain English without PA axioms?

Yes, but you cannot prove it. For that, you need the PA axioms. Saying that "numbers exist" does not prove that they do. The map consists of what you can prove from its axioms. The map shows you the provable elements of the territory. However, (1) the map does not show the entire territory (2) there is no guarantee that the map does not contain bullshit features that do not correspond at all with the territory.

Wow! The mental gymnastics. Every map contains bullshit features. The map is never the territory.

Yes, that is why the theory is never the model. In fact, we even know that the map does not show just one territory. There are also non-standard territories.

Because you have no proof for Con(ZFC) when using ZFC!

Yes, you have absolutely no proof that the ZFC map does not contain bullshit features.
You just assume it.

It is Platonism that is the ontological thesis.

Really? Tell me something about your ontology when I remove identity, LEM and non-contradiction.

Platonism does not seem to be compatible with constructive mathematics.

ChatGPT: Is mathematical Platonism compatible with constructive mathematics?

Mathematical Platonism and constructive mathematics are generally seen as philosophically incompatible, though in practice they can overlap in some ways. Here’s a breakdown of the core issues and nuances:

Philosophical Incompatibility
1. Mathematical Platonism
Ontology: Believes in the independent existence of mathematical objects (e.g., numbers, sets) in a non-physical realm.
Epistemology: Humans discover truths about these pre-existing objects.
Truth: A statement is true if it corresponds to the facts about this mathematical realm, regardless of whether we can prove it.

2. Constructive Mathematics (e.g., Intuitionism, Type Theory)
Ontology: Denies or is agnostic about a platonic realm of mathematical entities.
Epistemology: Mathematics is a mental or constructive activity—objects exist only when they can be explicitly constructed.
Truth: A statement is true only if we can construct a proof of it.

Thus, a Platonist may accept the Law of Excluded Middle (LEM) and non-constructive existence proofs, whereas a constructivist does not.
Can They Coexist?

In Practice (Maybe):

Some mathematicians pragmatically use both frameworks depending on context.
Some results in constructive mathematics can be embedded into classical mathematics and vice versa (e.g., via realizability, double-negation translations, or Kripke models).

Philosophically (Tension):
A Platonist believes in truths independent of construction, which a constructivist rejects.
Constructive mathematics is often motivated by anti-Platonist philosophies (e.g., Brouwer’s intuitionism was a direct challenge to Platonism).

Examples of Conflict
Existence proofs:
Platonist: "There exists an x such that P(x)" can be true without knowing x.
Constructivist: Only acceptable if we can actually construct such an x.

Continuum Hypothesis:
A Platonist may believe this has a definite truth value, even if undecidable.
A constructivist might argue it lacks a meaning without constructive interpretation.

Possible Reconciliations?

While philosophical reconciliation is difficult:
One could argue for a pluralist view: different foundations serve different purposes.
Some researchers (e.g. in category theory or topos theory) work at a foundational level compatible with both constructive and classical reasoning.

Summary
Aspect Platonism Constructivism
Ontology Abstract objects exist independently Objects exist only through construction
Truth Truth independent of knowability Truth = proof
Law of Excluded Middle Accepted Rejected
Existence Proofs Non-constructive proofs allowed Only constructive allowed
Conclusion: Philosophically incompatible, but pragmatically interrelated in certain contexts of mathematical practice.

This is pretty much what I also accept on the matter. I will pragmatically accept some constructivist concerns, but fundamentally, I consider it incompatible with my Platonist commitment.

What? That's completely backwards. The theory is the teritory. The models are the interpretations/maps!

The theory is the main map. The models are more refined maps that interpret the main map. The Platonic reality is the full truth which is unknown. So, yes, I agree, the models are not the Platonic reality either.

[Skepdick]

Without axioms, no map.

So what? This has no bearing on the teritory.

Yes, but you cannot prove it.

So what? You can't prove your axioms are true either.

For that, you need the PA axioms. Saying that "numbers exist" does not prove that they do.

Q.E.D

The map consists of what you can prove from its axioms.

This mode of reasoning is absolutely idiotic; you see.

You can't prove your axioms from your axioms; unless you include identity as an axiom.
But when you include identity as an axiom (which may or may not be true)

X -> X

Tadaaaa! Proof.

The map shows you the provable elements of the territory.

How? How do you prove "0 is a natural number" corresponds to the teritory?

However, (1) the map does not show the entire territory

Does it show ANY of the teritory? How does the axiom "0 is a natural number" relate to the teritory?

(2) there is no guarantee that the map does not contain bullshit features that do not correspond at all with the territory.

OK. So "0 is a natural number" could be a bullshit feature?

Yes, that is why the theory is never the model. In fact, we even know that the map does not show just one territory. There are also non-standard territories.

non-standard territories? That's fucking hilarious.

What's the cardinality of the set of all territories?

From the sample-space of ALL territories what is your choice-function for the "standard" one?

Platonism does not seem to be compatible with constructive mathematics.

Is Platonism even compatible with itself?

This is pretty much what I also accept on the matter. I will pragmatically accept some constructivist concerns, but fundamentally, I consider it incompatible with my Platonist commitment.

What commitment? How do you establish the relationship between the map and the teritory? How do you establish that "0 is a natural number" maps to Platonic reality?

The theory is the main map.

Of what?

The models are more refined maps that interpret the main map. The Platonic reality is the full truth which is unknown. So, yes, I agree, the models are not the Platonic reality either.

You need to take a step back here... Lets not get muddled up with delusions of grandeur about Full Truths.

Is "0 is a natural number" true? Does such an object exist in the Teritory?

If yes - how do you know?

[godelian]

How? How do you prove "0 is a natural number" corresponds to the teritory?

You can't.

non-standard territories? That's fucking hilarious.
What's the cardinality of the set of all territories?
From the sample-space of ALL territories what is your choice-function for the "standard" one?

Non-standard models of arithmetic is a feature of PA. The standard model is the "intended" interpretation.

ChatGPT: standard model of arithmetic

The standard model of arithmetic refers to the conventional, intended model of the natural numbers (ℕ) under the usual operations and relations.

The standard model is unique up to isomorphism: all standard models of arithmetic are structurally the same. There exist nonstandard models of arithmetic due to limitations of first-order logic (as shown by the Löwenheim–Skolem theorem and compactness theorem). These contain “nonstandard” elements beyond the natural numbers.

This is the core of classical model theory.

[Skepdick]

How? How do you prove "0 is a natural number" corresponds to the teritory?

You can't.

So are you even saying anything about the teritory then?!? How do you know?

non-standard territories? That's fucking hilarious.
What's the cardinality of the set of all territories?
From the sample-space of ALL territories what is your choice-function for the "standard" one?

Non-standard models of arithmetic is a feature of PA. The standard model is the "intended" interpretation.

*cough*bullshit*cough* don't be heading for the clouds of abstraction. Please plant your feet on the ground first.

What is "it" that you are modeling with the axiom "0 is a natural number"?

This is the core of classical model theory.

You are absolutely conceptually confused. This is what happens when you equivocate...

Map ( in the cartographical sense) is absolutely synonymous with colloquial use of "model". The map is the model of the teritory!

Whereas in a Mathematical sense of "model" a model is SOMETHING which satisfies the axioms and thus - makes them true.

What model satisfies the axiom "0 is a natural number" ?

[godelian]

What model satisfies the axiom "0 is a natural number" ?

The set-theoretical interpretation of zero.

[Skepdick]

What model satisfies the axiom "0 is a natural number" ?

The set-theoretical interpretation of zero.

But that's another Mathematical map! That's another model! It's not the teritory.

What does the empty set correspond to in Platonic reality?

You are going the wrong way. You are continuously equivocating "map" and "model" between their colloquial and mathematical meaning; and you use whichever meaning suits you in the moment.

For somebody objecting to constructivism all you seem to have up your sleeve is Mathematical constructions! Maps! Maps! Maps!
The person attacking constructivism can only defend their position using... constructions!

Did you notice how you ignored the question: What is the cardinality of the set of all territories?

[godelian]

What does the empty set correspond to in Platonic reality?

As soon as you express it in language, you commit to a map already. By representing the territory, it is no longer the territory but a map.

What is the cardinality of the set of all territories?

DeepSeek: how many nonstandard models of arithmetic?

The number of **nonstandard models of Peano Arithmetic (PA)** is **uncountably infinite**, and more specifically, it forms a **proper class** (too many to be a set). Here's a breakdown:

1. **Existence:** Gödel's Completeness Theorem combined with the **Compactness Theorem** proves that PA has nonstandard models. This is because PA (or even weaker systems like Robinson Arithmetic) cannot uniquely characterize the natural numbers using first-order logic.

2. **Cardinality Spectrum:** For every infinite cardinal number **κ** (e.g., ℵ₀, ℵ₁, ℵ₂, ..., ℵ_ω, etc.), there exists a nonstandard model of PA with exactly that cardinality.
* **Countably Infinite Models (κ = ℵ₀):** There are countably infinite nonstandard models. These are the smallest nonstandard models.
* **Uncountably Infinite Models (κ > ℵ₀):** For every uncountable cardinal κ, there are nonstandard models of PA with cardinality κ.

3. **How Many?**
* **Uncountably Many Models:** Since there are infinitely many infinite cardinals (ℵ₀, ℵ₁, ℵ₂, ...), and at least one model per cardinal, there are at least uncountably many nonstandard models.
* **A Proper Class:** More strongly, the collection of *all* nonstandard models of PA is a **proper class**. This means:
* There is no set that contains all nonstandard models of PA.
* For any cardinal number κ, no matter how large, there are nonstandard models of PA whose size is larger than κ.
* This follows from the **Löwenheim–Skolem Theorem** (upward direction). Given any model M of PA (standard or nonstandard) and any cardinal κ larger than the cardinality of M, there exists an elementary extension of M of cardinality κ. Since nonstandard models exist at all, this process generates nonstandard models of arbitrarily large cardinality.

4. **Comparison to the Standard Model:**
* There is exactly **one** standard model of PA (up to isomorphism): the usual natural numbers (ℕ, +, ·, S, 0).
* In stark contrast, there are **infinitely many** nonstandard models, spanning all possible infinite cardinalities, forming an uncountable collection that is too large to be a set (a proper class).

**In summary:** While there is only one standard model of Peano Arithmetic (up to isomorphism), there are **uncountably many nonstandard models**, and in fact, the collection of all nonstandard models is a **proper class**.

Of course, none of the above can be constructed. I am perfectly okay with that. I suspect that you are not.

[Skepdick]

What does the empty set correspond to in Platonic reality?

As soon as you express it in language, you commit to a map already. By representing the territory, it is no longer the territory but a map.

But I am genuinely committed to the ontological non-existence of the empty set!

Representing nothing is incoherent.

DeepSeek: how many nonstandard models of arithmetic?

That is not the question I asked you. Why do you keep deflecting?

I asked you about the cardinality of the set of all territories; not about the cardinality of the set of all maps.

The empty set exists is one model/map.
The empty set does not exist is another model/map.
Do these MODELS/MAPS correspond to the same territory; or different territories?

How many territories are there for models/maps to correspond to?

Of course, none of the above can be constructed. I am perfectly okay with that. I suspect that you are not.

Are you OK with it? Any map can be constructed. That's what ALL maps are. Constructions.