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

[Skepdick]

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.

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. The process of turning something abstract into something ontological is called "reification" in computer science; and in ontology engineering. It's the epitome of invention.

https://en.wikipedia.org/wiki/Reificati ... r_science)
https://en.wikipedia.org/wiki/Ontology_engineering

Simultaneously the cognitive process of abstracting (generalizing from particulars) doesn't create ontological entities - it creates concepts in minds.

Abstract objects are the product/output of abstraction - itself a cognitive process. Do you think something your mind creates is "discovered"?
Perhaps you are discovering the abstractions other people have invented?

That's really trivial since they exist independently of time. Once could use different notions; what matters is what is true under which definitions.

But they don't exist until they were imagined into existence! Are imaginations discovered too?

Again, under common ontological notions, physical objects come into existence, therefore they can be invented, whereas specifications as abstract objects cannot.

So specifications are discovered; not invented?!? Don't they come into existence when they are defined/specified; and therefore by your very own criterion that makes them invented?

I think I made myself clear

Clear as mud. Yes.

From ontological perspective, all abstract things exist in a way, but therefore nothing abstract could be entirely new

So that's just Platonic realism in new dressing. Do abstract objects exist before they are imagined?

If you want to stick to the "discovered" narrative - how does one explore the search-space for abstract objects in order to discover them?
How would we distinguish between "undiscovered abstract objects" and "nonexistent abstract objects" in this search space?

One could make arbitrary definitions, and I doubt it is possible to show any single one of them as particularly relevant

But it is absolutely relevant to identify whether abstract objects exist before they are discovered or defined!

Because then you'll absolutely have to explain to me how to discover (and more importantly - IDENTIFY!) an abstract object a priori defining; or specifying it!

Where do I go to discover the number 7; a priori any definition or conception of what a number is ontologically speaking?

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

[godelian]

Now we understand the logical structure of mathematical theories at a deeper level than the previous "build everything from one axiomatic base" approach.

Historically, they started with classical Euclidean geometry, which did have an axiomatic base, expounded in Elements, and Diophantes' arithmetic, which did not. As late as Immanuel Kant's Critique of Pure Reason, they believed that arithmetic did not have an axiomatic foundation, until in the late 19th century, Peano and Dedekind finally proposed one (PA). After that, we ended up with the dominant axiomatizations, PA and ZFC.

So, it looks like we are operating in a multi-axiomatic world, but that is actually not true either.

In "On interpretations of Arithmetic and Set Theory", 2007, Kaye and Wong point out that PA and ZF-inf are actually bi-interpretable. So, PA and ZF-inf are in fact "views" on the same, single, unknown Platonic world of mathematical objects -- with "views" as in SQL views. There actually is one single axiomatic base but it cannot be uniquely expressed in language. There happen to be alternative ways of expressing essentially the same thing. This was to be expected, because mathematical objects are never truly unique. They are only unique up to isomorphism. That is apparently also the case for the core axiomatizations themselves.

PA and ZF-inf are almost surely not the only ways to express the foundations of mathematics. It would be more typical to assume that there are probably an infinite number of distinct ways to express them. In absence of expressing mathematics, there actually does seem to be a unique Platonic world of mathematical objects.

[Skepdick]

It would be more typical to assume that there are probably an infinite number of distinct ways to express them. In absence of expressing mathematics, there actually does seem to be a unique Platonic world of mathematical objects.

I really have no idea how you reconcile the "uniqueness" of this Platonic world with the foundational plurality you just acknowledged.

What is unique is the computational substrate within which you can construct; or manipulate symbols, objects, shapes, geometries or whatever else you consider as "first class citizen" in your abstract universe.

It's universal insofar as it can "execute" all of those different ways of Mathematical thinking.

But even this universality is questionable. My wife simply struggles with abstract thought.

[godelian]

It would be more typical to assume that there are probably an infinite number of distinct ways to express them. In absence of expressing mathematics, there actually does seem to be a unique Platonic world of mathematical objects.

I really have no idea how you reconcile the "uniqueness" of this Platonic world with the foundational plurality you just acknowledged.

What is unique is the computational substrate within which you can construct; or manipulate symbols, objects, shapes, geometries or whatever else you consider as "first class citizen" in your abstract universe.

Well, how does it come that numbers (PA) and sets (ZF-inf) describe exactly the same abstract Platonic world? There is a one-to-one mapping between the true statements in their models. My interpretation is that it is, in fact, the same world. You can express it by means of numbers. You can express it by means of sets. There is essentially no difference between both. So, isn't that the same unique world?

[Skepdick]

Well, how does it come that numbers (PA) and sets (ZF-inf) describe exactly the same abstract Platonic world?

They don't. You are just interpreting it that way.

You know how you can write two totally different programs (using either the same; or different programming languages) that produce the exact same output? You know how generators work in computation?

Axioms are generators. When you "drop them into" your computational apparatus they generate/compute an output.

That two different encodings can produce the same output is neither here nor there.
That shouldn't be surprising to you at all. It's a trivial instance of equifinality.

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

You can invent the same thing via many different means.

There is a one-to-one mapping between the true statements in their models. My interpretation is that it is, in fact, the same world. You can express it by means of numbers. You can express it by means of sets. There is essentially no difference between both. So, isn't that the same unique world?

Your line of reasoning is as idiotic as "You can express a fibonacci generator in 100 different ways and you can do a 1:1 mapping between the output of each implementation!"

Isn't that evidence that all implementations are accessing the exact same Unique Fibonacci World?

The magic isn't in the objects PA/ZF-inf generate.
The magic is in the function you construct to translate/map between them.

Because you haven't made this function explicit and it falls outside of your peripheral vision you think some kind of magic is going on.
If you use category theory to make the construction explicit the functor (structure preserving map) between ZF and PF-inf is just another construction.

The illusion of some Platonic realm comes from cognitive sleight of hand - talking about the equivalence without showing the actual construction.

[godelian]

Your line of reasoning is as idiotic as "You can express a fibonacci generator in 100 different ways and you can do a 1:1 mapping between the output of each implementation!"

Isn't that evidence that all implementations are accessing the exact same Unique Fibonacci World?

Yes, my interpretation is that there is an unknown unique Fibonacci world for which we can have lots of different maps, i.e. Fibonacci generators, i.e. implementations that are essentially views on something known only through these maps.

You cannot see the Fibonacci world without having access to at least one of its maps.

In general, numbers (PA) and sets (ZF-inf) are maps to the same unknown territory.

From very subtle faults in these system-level maps, we can detect that most of the unknown territory is not even visible on these maps.

This is the genius of Godel's incompleteness theorem: the understanding that our map is incomplete.

The things that are not visible on any of our maps, massively distort these maps. These inexplicable distortions are so fundamental to the things visible on the map that our maps create a fundamentally wrong impression of the actual territory.

For example, the maps give a very orderly impression, while the territory is highly chaotic. That is essentially what Noson S. Yanofsky points out in his paper "True but unprovable";

http://www.sci.brooklyn.cuny.edu/~noson ... ovable.pdf

Formalism (and constructive mathematics) insist that you should only look at the map. If it's not on the map, then it does not exist. I cannot reconcile myself with this approach.

For example, we know that inexpressible numbers exist, but because we cannot express them, the constructivists insist that they do not exist. I disagree. I prefer classical mainstream mathematics. As far as I am concerned, these inexpressible numbers do exist.

[Skepdick]

Yes, my interpretation is that there is an unknown unique Fibonacci world for which we can have lots of different maps, i.e. Fibonacci generators, i.e. implementations that are essentially views on something known only through these maps.

You cannot see the Fibonacci world without having access to at least one of its maps.

That's a pretty convoluted way to describe the unit type. https://en.wikipedia.org/wiki/Unit_type

Any computation which outputs the Fibonacci sequence.

In general, numbers (PA) and sets (ZF-inf) are maps to the same unknown territory.

But the word "same" is doing all the heavy lifting here. And you are equivocating the colloquial use of "map" with the mathematical use of "map".

PA is "the same" as ZF-inf directly translates into Mathematical language (we can construct a a 1:1 map!) as "We can construct a 1:1 map between PA and ZF-inf".

Spot the recursion?

All you are saying is "We can construct a structure preserving map (a bijection!) between two structures". They are equivalent up to isomirphism.
OK and?

That's exactly how equivalence works computationally! Continuous transformations to and fro.

To save ourselves from devolving into mystical experiences and getting sore jaws from expressing this mouthful we give such transformations a name. Homotopies. This is *exactly* what Homotopy Type Theory formalizes. Computationally! When we say two mathematical structures are "the same," we're not appealing to some Platonic realm - we're providing actual computational evidence.

The bijection is actually constructed as a program you can run on your computer.

A function f: A → B
A function g: B → A
Proofs that g(f(x)) = x and f(g(y)) = y

That's all there is to this "Platonic realm"

From very subtle faults in these system-level maps, we can detect that most of the unknown territory is not even visible on these maps.

Which maps are "not even visible"? The one we literally and explicitly construct; and then execute on computers?

This is the genius of Godel's incompleteness theorem: the understanding that our map is incomplete.

See! That perpetual sleight of hand you pull on yourself, the way you pull your strings! You just equivocated "map" from its Mathematical to cartographic meaning!

The things that are not visible on any of our maps, massively distort these maps.

These inexplicable distortions are so fundamental to the things visible on the map that our maps create a fundamentally wrong impression of the actual territory.

Everything is visible on those maps. That's how you know there is a bijection between ZF-inf and PA!

The map is constructed and made explicit. That is what a proof IS under Curry-Howard!

This mystical woo woo shit doesn't work in a complete system where everything is explicit.

For example, the maps give a very orderly impression, while the territory is highly chaotic. That is essentially what Noson S. Yanofsky points out in his paper "True but unprovable";

http://www.sci.brooklyn.cuny.edu/~noson ... ovable.pdf

Non-sequitur.

Formalism (and constructive mathematics) insist that you should only look at the map. If it's not on the map, then it does not exist. I cannot reconcile myself with this approach.

Equivocation. The "map" is precisely what you should look at when you say X is "the same" as Y.

In constructive mathematics, there's no gap between the mathematical object and its construction via maps. When we say two structures are equivalent, we're not pointing to some pre-existing Platonic equivalence - we're constructing the equivalence through explicit structure-preserving mappings.

We assert the existence of such bijection by constructing it!

For example, we know that inexpressible numbers exist, but because we cannot express them, the constructivists insist that they do not exist. I disagree. I prefer classical mainstream mathematics. As far as I am concerned, these inexpressible numbers do exist.

Are we going to go over this nonsense again? You are failing to distinguish between two very very important categories:

A. knowable AND not(expressible)
B. not(knowable) AND not(expressible)

Which is it that you claim exists: A; or B?

If A: How do you know? I don't need you to express the number (lets call it X), but I do need you to tell me how you established which disjunct holds in the expression: ∃X ∨ ¬∃X
If B: You know that unknowable numbers exist? Contradiction.

So go ahead and prove the existence of X without expressing X.

In Mathematical practice lacking knowledge of HOW to establish existence IS equivalent to not knowing that it exists.
There is no distinction to be drawn between know-THAT and know-HOW any more than you can know THAT you can ride a bicycle without knowing HOW to ride one.

[Skepdick]

For example, we know that inexpressible numbers exist, but because we cannot express them, the constructivists insist that they do not exist. I disagree. I prefer classical mainstream mathematics. As far as I am concerned, these inexpressible numbers do exist.

See, this is ALL I hear when you tell me about an "inexpressible" number...

X ∈ R, and X is unbound to a value. OK.. so it's a value of type Maybe<Real>.

The logical implications of which is such that any mathematical operation (except identity) returns the Maybe monad/option type.

X > 0 ? maybe.
x < 0? Maybe.
X = 0? Mabe.
X > π? Maybe.
sin(X)? Maybe.

x = x? Yes.

Such an object is trivially expressible in ANY programming language which safely handles NULL-values!

Even if I don’t know what X is, I can still represent it as an object with unknown contents in code, which makes it expressible in the programming sense. Even it that means it's inexpressible in the Mathematical sense.

Guess whose language is more expressive?

Code: Select all

enum InexpressibleNumber {
    Unknown
}

There's your operational semantics. Every operation is well-defined and returns Option<T>.

Your mystical inexpressible number is a data constructor in a sum type.

you can even type-check it if you want to.

Trivial procedure!
Is any operation (except identity) defined on the object?

Code: Select all

impl InexpressibleNumber {
    // Identity - type checks ✓
    fn equals(&self, other: &Self) -> bool {
        matches!((self, other), (Unknown, Unknown))
    }
    
    // All other operations - type checker enforces Option<T> ✓
    fn greater_than(&self, _other: &Self) -> Option<bool> { None }
    fn add(&self, _value: f64) -> Option<f64> { None }
    fn sin(&self) -> Option<f64> { None }
    // ... etc
}

[godelian]

Are we going to go over this nonsense again? You are failing to distinguish between two very very important categories:

A. knowable AND not(expressible)
B. not(knowable) AND not(expressible)

Which is it that you claim exists: A; or B?

If A: How do you know? I don't need you to express the number (lets call it X), but I do need you to tell me how you established which disjunct holds in the expression: ∃X ∨ ¬∃X
If B: You know that unknowable numbers exist? Contradiction.

So go ahead and prove the existence of X without expressing X.

In Mathematical practice lacking knowledge of HOW to establish existence IS equivalent to not knowing that it exists.
There is no distinction to be drawn between know-THAT and know-HOW any more than you can know THAT you can ride a bicycle without knowing HOW to ride one.

Mainstream classical mathematics views the matter differently:

ChatGPT: Do unknowable numbers exist?

Yes, in a meaningful sense, unknowable numbers do exist—depending on how we define "unknowable." There are different layers to this idea, ranging from mathematical logic to computability theory. Here's a breakdown:

1. Uncomputable Numbers
These are real numbers whose digits cannot be generated by any algorithm.
Example: Chaitin's constant (Ω), which is defined in terms of the halting problem and is algorithmically random. No algorithm can produce its digits.
Why they're unknowable: There is no finite process that allows you to determine their exact value or even their digits beyond a certain point.

2. Non-Definable Numbers (in a language)
There are uncountably many real numbers, but only countably many possible definitions in any formal language.
Hence, most real numbers are undefinable—they can’t be described, named, or singled out in any systematic way.
Why they're unknowable: You can't even describe or refer to them, much less compute them.

3. Epistemic Unknowability
Even if a number is theoretically describable or computable, it might be unknowable in practice due to resource constraints (like needing infinite time or memory). Why they're unknowable: Humans or machines can never access the full information due to physical limits.

Summary
Type Example Why Unknowable?
Uncomputable Chaitin's Ω No algorithm can compute it
Undefinable Most real numbers Cannot even be described in language
Practically unknowable Very complex numbers Too complex or resource-intensive to understand

Philosophical Implication
This shows that "number" as a concept goes far beyond what can be grasped or calculated. In fact, the overwhelming majority of real numbers are unknowable in any concrete sense.

Of course, I am not interested in "Practically unknowable" numbers. As far as I am concerned, it is only about theoretically unknowable ones. In classical mathematics, there are uncomputable and undefinable numbers. We know that these uncomputable or even undefinable mathematical objects exist because we can see that there must be large collections of them. So, we can prove the existence of these collections but we cannot pinpoint one single element in them. So, you can actually prove that a particular set exists while you may not be able to compute or define a single element in it. For constructive mathematics, this is unacceptable. I am, however, okay with this.

[Skepdick]

Mainstream classical mathematics views the matter differently:

ChatGPT: Do unknowable numbers exist?
...

That's hardly the view of "Mainstream classical mathematics".

That's the view of a stochastic parrot.

Yes, in a meaningful sense, unknowable numbers do exist—depending on how we define "unknowable."

Gee wiz! The genius! Doesn't how we define "unknowable" depend on how we define "define"?

Don't you think you should distinguish between unknown and unknowable?

Of course, I am not interested in "Practically unknowable" numbers. As far as I am concerned, it is only about theoretically unknowable ones.

Distinction without a difference "theorizing" is a verb; and therefore a practice.

So while practicing theorizing how do you come to know an unknowable number?

In classical mathematics, there are uncomputable and undefinable numbers. We know that these uncomputable or even undefinable mathematical objects exist because we can see that there must be large collections of them.

The equivocation continues... This is so fucking tiresome.

[Skepdick]

...

Lets pick up on this point (and connect everything back to the question in the OP); shall we?

T ⊢God exists

Are you discovering foundations for the existence of ℝ; or asserting the existence of ℝ as a foundational?

Which mode of reasoning are your arguments being made in?

Mode A: ∃ℝ ⊢ T (eg. ∃ℝ is an axiom with implications. )
Mode B: T ⊢ ∃ℝ (e.g ∃ℝis an axiomatizable theorem)

[godelian]

That's hardly the view of "Mainstream classical mathematics".
That's the view of a stochastic parrot.

I am sure that constructive mathematics rejects the notion of inaccessible cardinal:

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

The existence of a strongly inaccessible cardinal is equivalent to the existence of a Grothendieck universe. If κ is a strongly inaccessible cardinal then the von Neumann stage V κ is a Grothendieck universe. Conversely, if U is a Grothendieck universe then there is a strongly inaccessible cardinal κ such that V κ = U. As expected from their correspondence with strongly inaccessible cardinals, Grothendieck universes are very well-closed under set-theoretic operations.

Platonically, this is perfectly fine. Constructivistically, it is not:

Existence of a proper class of inaccessibles

There are many important axioms in set theory which assert the existence of a proper class of cardinals which satisfy a predicate of interest. In the case of inaccessibility, the corresponding axiom is the assertion that for every cardinal μ, there is an inaccessible cardinal κ which is strictly larger, μ < κ. Thus, this axiom guarantees the existence of an infinite tower of inaccessible cardinals (and may occasionally be referred to as the inaccessible cardinal axiom). As is the case for the existence of any inaccessible cardinal, the inaccessible cardinal axiom is unprovable from the axioms of ZFC. Assuming ZFC, the inaccessible cardinal axiom is equivalent to the universe axiom of Grothendieck and Verdier: every set is contained in a Grothendieck universe.

I know that you will argue against the existence of a proper class of inaccessibles because it is absolutely impossible to express even one witness of this class. The language of ZFC cannot handle it. It contains transfinite numbers that are too big for that.

So, you reject this research. I accept it.

[Phil8659]

the abstract is written by a liar, a liar by being illiterate.
Every system of grammar starts with a convention of names, and each convention of names is based on the definition of a thing, aka, a unit, aka, 1.
The only difference between the members of our Grammar Matrix, is how we recursively use that standard of measure.
And, as every grammar starts with a given, the relative difference, aka, verb, and the only thing you learn is how to assert correlatives to it, then where, in all of this is invention? Nature applies correlatives in a seemly random manner, but grammar teaches us how to apply correlatives, which are not things at all, and incapable of being any thing at all. So, apparently you are as illiterate as the person you quoted.
Grammar Systems are, by design only methods of standardizing our mental behavior. Have you ever invented, by any grammar system, a unique relative difference. The verb is what we manage in grammar, and we do it by applying limits or boundaries, or again, correlatives to it.
Are you one of those fools who actually believe that a chef can make a meal by waving his knife in the air an infinite number of times?

[Skepdick]

That's hardly the view of "Mainstream classical mathematics".
That's the view of a stochastic parrot.

I am sure that constructive mathematics rejects the notion of inaccessible cardinal:

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

The existence of a strongly inaccessible cardinal is equivalent to the existence of a Grothendieck universe. If κ is a strongly inaccessible cardinal then the von Neumann stage V κ is a Grothendieck universe. Conversely, if U is a Grothendieck universe then there is a strongly inaccessible cardinal κ such that V κ = U. As expected from their correspondence with strongly inaccessible cardinals, Grothendieck universes are very well-closed under set-theoretic operations.

Platonically, this is perfectly fine. Constructivistically, it is not:

Existence of a proper class of inaccessibles

There are many important axioms in set theory which assert the existence of a proper class of cardinals which satisfy a predicate of interest. In the case of inaccessibility, the corresponding axiom is the assertion that for every cardinal μ, there is an inaccessible cardinal κ which is strictly larger, μ < κ. Thus, this axiom guarantees the existence of an infinite tower of inaccessible cardinals (and may occasionally be referred to as the inaccessible cardinal axiom). As is the case for the existence of any inaccessible cardinal, the inaccessible cardinal axiom is unprovable from the axioms of ZFC. Assuming ZFC, the inaccessible cardinal axiom is equivalent to the universe axiom of Grothendieck and Verdier: every set is contained in a Grothendieck universe.

I know that you will argue against the existence of a proper class of inaccessibles because it is absolutely impossible to express even one witness of this class. The language of ZFC cannot handle it. It contains transfinite numbers that are too big for that.

So, you reject this research. I accept it.

How did you just go on this tangent?

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

Obviously - the more axioms you add - the more stuff you can prove.
But that's neither here nor there - if you want to prove more, you have to assume more. And more. And more....

Such as adding the large cardinal axiom. Their existence implies Con(ZFC); which you were babbling on as unprovable given Godel's work.

But if you are a Platonist why can't you just assume everything? If you assume LEM + Platonic existence, then every mathematical question has a definite answer. Even all instances of the halting problem.

If Platonists can just assume definite answers exist for everything, why do we need mathematicians at all? Just consult the Platonic realm! Surely it's not strongly inaccessible for any answers?

Just add an axiom for strong accessibility to the Platonic realm and get the answer to P vs NP. It's not mathematics anymore - it's just wishful thinking with formal notation.

It sure sounds like reverse mathematics; not mathematics.

You want P = NP? NO problem! Just add the axiom of Polynomial Solvability: Every problem in NP has a polynomial-time algorithm!
To put it in a language you clearly understand: The polynomial for solving the general case exists, but is inexpressible. Perhaps even incomprehensible to a human mind.

[godelian]

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.

Obviously - the more axioms you add - the more stuff you can prove.
But that's neither here nor there - if you want to prove more, you have to assume more. And more. And more....

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.

Such as adding the large cardinal axiom. Their existence implies Con(ZFC); which you were babbling on as unprovable given Godel's work.

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).

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).

If Platonists can just assume definite answers exist for everything, why do we need mathematicians at all? Just consult the Platonic realm! Surely it's not strongly inaccessible for any answers?

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.

It sure sounds like reverse mathematics; not mathematics.

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.

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.