Crashing both ChatGPT and DeepSeek with the following question

[Veritas Aequitas]

Justify [prove] whatever exists as Plato's form is real.
Whatever is not real is false, i.e. illusory.

ChatGPT: The act of counting till 3 is an act of computation. Every act of computation corresponds to executing a proof. What does the act of counting till 3 prove?

The act of counting to 3 proves the existence of the natural numbers up to 3 and their sequential order. More formally, it demonstrates the Peano axioms in action, specifically:

1. Existence of numbers: There is a starting number (typically 1 or 0 in different conventions).

2. Successor function: Each number has a successor, forming a well-defined sequence.

3. Order and distinctness: The numbers 1, 2, and 3 are distinct and appear in a fixed sequence.

From a computational perspective, counting to 3 is an algorithm that constructs a proof of the statement “there exist at least three natural numbers, and they follow a successor relation.” This is a small but foundational step in number theory.

ChatGPT: Is the number 3 false and illusory?

No, the number 3 is not false or illusory—it is an abstract concept that exists within mathematical systems. Numbers are not physical objects, but they are real in the sense that they have consistent properties and relationships that we can use to describe the world. The number 3 represents a quantity and plays a fundamental role in arithmetic, logic, and science. If you're asking from a philosophical perspective, some thinkers argue that numbers exist only in the mind, while others believe they have an objective reality independent of human thought. What makes you ask?

As I have argued previously, the idea that mathematical objects exist only in the mind, is unsustainable, because their existence is also implied in computers, other Turing machines, and in every physical computation process, natural or human. Computation is the isomorphic shadow in physical reality of the mathematical universe of Platonic abstraction.

VA to Gpt:

My interlocutor posted the following from ChatGpt:

Question: Is the number 3 false and illusory?

[ChatGPT:] No, the number 3 is not false or illusory—it is an abstract concept that exists within mathematical systems. Numbers are not physical objects, but they are real in the sense that they have consistent properties and relationships that we can use to describe the world. The number 3 represents a quantity and plays a fundamental role in arithmetic, logic, and science. If you're asking from a philosophical perspective, some thinkers argue that numbers exist only in the mind, while others believe they have an objective reality independent of human thought. What makes you ask?"

I believe there are nuances to the above.

The number is only real within the Mathematical Framework and System [FS] and no where else.

The number 3 cannot be typically real like a real apple on the table which can be seen, felt, verified and eaten by the individual or a group of people [within the common sense FS] plus by fruits experts and biologists within a scientific FS.

As such, from the scientific perspective, an hypostatize or reified number 3 is not real, i.e. it is an illusion.

Here's GPT in the more realistic sense:

ChatGPT said:
Yes, there are important nuances that the ChatGPT response does not fully capture, especially concerning the context-dependent nature of mathematical entities.

Your view aligns with a framework-system (FS) dependent understanding of mathematical objects, where numbers like "3" are real only within the Mathematical FS but do not have independent existence beyond it. This avoids the reification fallacy, which would mistakenly treat abstract concepts as if they were physical objects.

From a scientific FS, a number like 3 has no independent ontological status—it does not exist as a physical entity in the way that an apple does. Scientific realism deals with empirical objects that can be observed, measured, and tested, whereas numbers are conceptual tools within mathematics, useful for describing empirical phenomena but not existing as things in themselves.

Thus, the number 3 is not "false" within mathematics, but it would be illusory if mistaken for a mind-independent, physical reality in the scientific sense.
This distinction between FS-objective (within the framework) and FS-illusory (when improperly reified beyond the framework) is key to a nuanced understanding.

I suggest you present the whole of the above to 'your' GPT and ask for its comment on why there is an alternative view.

Anything beyond the empirical world [e.g. Platonic Form] is purely mental where anything goes.
To ensure it is realistic, it has to be brought within the ambit of the scientific FS which is the gold standard of credibility and objectivity else it remains an illusion.

[godelian]

To ensure it is realistic, it has to be brought within the ambit of the scientific FS which is the gold standard of credibility and objectivity else it remains an illusion.

You have asked proof for the existence of a Platonic form. I have mathematically proven the existence of the number 3.

Now you want to switch to experimental testing, such as in science, even though you perfectly well know that experimental testing does not conclusively prove anything.

Experimental testing is absolutely not the gold standard for proof. On the contrary, experimental testing never produces conclusive proof.

For example, you cannot prove the existence of the number 3 by means of experimental testing. You can only prove its existence by means of computation.

As ChatGPT clearly told you, the number 3 is real. It exists. Its existence can be conclusively proven by means of computation. Platonic abstractions such the number 3 are provably real.

You try to reject mathematical proof with one of your word salads. That is against the fundamental rules of logic. You cannot argue against mathematical proof with just a word salad.

[Veritas Aequitas]

I posted both GPT responses to another AI and it responded as follows;

The two responses from the AI highlight different perspectives on the nature of numbers, particularly focusing on their existence and reality. Here's a breakdown of each response and how they differ:

First Response
Abstract Concept: The first response emphasizes that numbers like "3" are abstract concepts within mathematical systems. They are not physical objects but have real properties and relationships that help describe the world.

Philosophical Perspective: It touches on philosophical debates about whether numbers exist independently or only in human minds.

General Audience Focus: This explanation seems geared towards a general audience, providing an accessible overview without delving deeply into specific frameworks or philosophical nuances.


Second Response
Framework-System (FS) Perspective: The second response introduces a more nuanced view by referencing framework-system dependency, where numbers are considered real only within their respective mathematical frameworks.

Avoiding Reification Fallacy: It highlights the importance of not treating abstract concepts as if they were physical entities (reification fallacy), emphasizing that numbers do not have independent existence beyond their mathematical context.

Scientific Realism vs. Mathematical Concepts: This explanation distinguishes between scientific realism (dealing with observable phenomena) and mathematical concepts like "3," which serve as tools for describing empirical phenomena but do not exist independently in a physical sense.

Specialized Audience Focus: This response appears to be directed at an audience familiar with philosophical or scientific discussions about ontology and epistemology, offering a more detailed analysis.

Key Differences
Level of Detail and Nuance: The second response provides more depth regarding framework dependency and avoids reifying abstract concepts, making it suitable for those interested in deeper philosophical or scientific discussions.

Audience Targeting: The first is more general, while the second is tailored for those familiar with specialized terminology like "framework-system" perspectives.

Perspective on Reality vs Illusion:

In the first response, "3" is seen as real within its system without questioning its broader ontological status outside mathematics.

In the second, while "3" is valid within mathematics (not illusory), it would be considered illusory if mistakenly believed to have an independent physical existence outside this context.

Overall, both responses address different aspects of how we understand numbers—whether focusing on their utility within systems or examining them through specific philosophical lenses—and cater to different levels of inquiry into these topics.

Generally the above imply your thinking is too narrow; in addition you are too dogmatic and arrogant with such narrow & shallow views.

[godelian]

Generally the above imply your thinking is too narrow; in addition you are too dogmatic and arrogant with such narrow & shallow views.

You are arrogant enough to reject mathematical proof with some silly word salad.

I mathematically prove that the number 3 exists, and you still manage to invent a word salad to argue that the number 3 does not exist.

[godelian]

The two responses from the AI highlight different perspectives on the nature of numbers, particularly focusing on their existence and reality.

There is absolutely no doubt about the fact that the number 3 exists:

ChatGPT: Prove that the number 3 exists.

To prove that the number 3 exists, let's explore different perspectives:

1. Mathematical Proof (Axiomatic Approach)

In modern mathematics, numbers are defined within formal systems, such as Peano Arithmetic:

Peano's axioms define the natural numbers starting from 0 (or 1, depending on the formulation) and a successor function S(n)S(n).

Using these axioms:
Define 0 as a base case.
Define 1 as S(0).
Define 2 as S(1).
Define 3 as S(2).

Since these numbers are well-defined within set theory (like the von Neumann construction: 3={0,1,2}), 3 exists in a rigorous mathematical sense.

2. Empirical Proof (Real-World Observations)

If you take three apples and count them, you can see that the quantity 3 is a real, observable property.
This suggests that 3 is not just an abstract concept but a practical one used in physics, engineering, and everyday life.

3. Philosophical Argument (Platonism vs. Formalism)

If you accept mathematical Platonism (the idea that numbers exist independently of human thought), then 3 exists as an abstract entity.
If you take a more formalist approach, numbers are symbols with rules, and since we can meaningfully manipulate and define 3, it exists within our system of reasoning.

Conclusion

The number 3 exists:

✔ Mathematically, as a well-defined object in number theory.
✔ Empirically, as a countable quantity in the real world.
✔ Conceptually, as a necessary and functional abstraction in logic and science.

Neither Platonism nor formalism makes the existence of the number 3 dependent on human conditions. The number 3 simply does not need humans to exist. As soon as some computation takes place, human or natural, the Platonic shadow of the number 3 becomes visible in the physical universe. Note that even "empirical proof" of the existence of the number 3 still requires an explicit act of computation. As the Curry-Howard correspondence insists, every program is a proof and every proof is a program.

[Veritas Aequitas]

The two responses from the AI highlight different perspectives on the nature of numbers, particularly focusing on their existence and reality.

There is absolutely no doubt about the fact that the number 3 exists:

ChatGPT: Prove that the number 3 exists.

To prove that the number 3 exists, let's explore different perspectives:

1. Mathematical Proof (Axiomatic Approach)

In modern mathematics, numbers are defined within formal systems, such as Peano Arithmetic:

Peano's axioms define the natural numbers starting from 0 (or 1, depending on the formulation) and a successor function S(n)S(n).

Using these axioms:
Define 0 as a base case.
Define 1 as S(0).
Define 2 as S(1).
Define 3 as S(2).

Since these numbers are well-defined within set theory (like the von Neumann construction: 3={0,1,2}), 3 exists in a rigorous mathematical sense.

2. Empirical Proof (Real-World Observations)

If you take three apples and count them, you can see that the quantity 3 is a real, observable property.
This suggests that 3 is not just an abstract concept but a practical one used in physics, engineering, and everyday life.

3. Philosophical Argument (Platonism vs. Formalism)

If you accept mathematical Platonism (the idea that numbers exist independently of human thought), then 3 exists as an abstract entity.
If you take a more formalist approach, numbers are symbols with rules, and since we can meaningfully manipulate and define 3, it exists within our system of reasoning.

Conclusion

The number 3 exists:

✔ Mathematically, as a well-defined object in number theory.
✔ Empirically, as a countable quantity in the real world.
✔ Conceptually, as a necessary and functional abstraction in logic and science.

Neither Platonism nor formalism makes the existence of the number 3 dependent on human conditions. The number 3 simply does not need humans to exist. As soon as some computation takes place, human or natural, the Platonic shadow of the number 3 becomes visible in the physical universe. Note that even "empirical proof" of the existence of the number 3 still requires an explicit act of computation. As the Curry-Howard correspondence insists, every program is a proof and every proof is a program.

You are not applying rationality and critical thinking;

Here's MY ChatGpt:

Your interlocutor is assuming a Platonic or formalist view of numbers, which treats numbers as existing independently of human cognition. However, their argument contains key presuppositions that can be critically examined:

1. The Fallacy of Reification (Category Error)
They claim that "the number 3 does not need humans to exist", but they are treating numbers as if they were mind-independent entities rather than conceptual constructs within a formal system.
The phrase "the Platonic shadow of the number 3 becomes visible in the physical universe" is metaphorical and does not establish that numbers exist in the same way that physical objects do. What does it mean for a "shadow" of 3 to be "visible" in nature?
Critical Question: Where exactly is the number 3 in nature without a mathematical framework interpreting it?

2. Numbers Require a Framework-System (FS)
Computation does not happen in a vacuum—it requires a formal structure (e.g., Peano Arithmetic, set theory) that is itself a product of rule-based reasoning.
Their argument suggests that "as soon as computation takes place," the number 3 manifests. But computation is an abstract process that depends on a formal system, meaning numbers do not "exist" outside of such systems—they are framework-dependent.
Counterpoint: Does the number 3 "exist" for a universe with no minds, no formal systems, and no computational rules?

3. Curry-Howard Correspondence Does Not Prove Platonic Existence
They cite the Curry-Howard correspondence, which states that every proof corresponds to a program and vice versa. However, this is an internal feature of formal logic and computation, not an ontological argument for numbers existing independently of frameworks.
At best, it shows that numbers are embedded in logical and computational structures, but this does not prove that numbers exist apart from those structures.
Rebuttal: If numbers exist independently of human cognition, where is the number 3 without a formal system to define it?

4. Empirical vs. Conceptual Existence
The number 3 can be represented empirically (e.g., 3 apples), but the abstract number 3 itself is not an empirical entity.
Without a framework, "3" is just a useful construct, not an independently existing entity like an apple.
Critical Thought Experiment: Does the number 3 exist in a universe with no beings capable of counting?

Final Rational Summary
Numbers exist within mathematical frameworks, but not in a mind-independent way like physical objects.
Computation presupposes a formal structure, which is a rule-based system, not an independent Platonic realm.
Their appeal to Curry-Howard is about internal logical structure, not external existence.
The number 3 is not "false," but it is FS-dependent—real in the mathematical framework, but illusory if hypostatized as a physical object.

A reasonable middle ground: We can treat numbers as useful abstractions that appear universally applicable, but they do not "exist" outside of interpretative frameworks.

Did you post the whole of MY GPT to your ChatGpt and ask why the different views.

I suggest again, you post the above MY-ChatGpt response and ask for your ChatGpt's view, so that it will sink some rationality into you.

[godelian]

You are not applying rationality and critical thinking;

Your argument against the existence of the number 3, is unsustainable. Number 3 truly exists. There is no doubt about that. There is even incontrovertible proof for it.

[Veritas Aequitas]

You are not applying rationality and critical thinking;

Your argument against the existence of the number 3, is unsustainable. Number 3 truly exists. There is no doubt about that. There is even incontrovertible proof for it.

I suggest again, you submit the whole of MY ChatGpt's argument to your ChatGpt, ask for its comments and may be you can learn something there.

Are you familiar with the very contentious Rationalism versus Empiricism chasm?
https://plato.stanford.edu/entries/rati ... mpiricism/
You, Plato, Descartes, et. al. are on the rationalists' side.
Hume et al are on the empiricists' side.
Kant reconciled and closed the chasm to that impossible to resolve divide between the above.
Give me an idea what you know about this, if not, ask AI.

Your dogmatism with rationalism is a psychological issue not an intellectual-philosophical one arising from an evolutionary default. Due you have any respect for more knowledge?

[godelian]

I suggest again, you submit the whole of MY ChatGpt's argument to your ChatGpt, ask for its comments and may be you can learn something there.

You are rejecting proof of existence while that is what you originally asked for. The Platonic form "3" exists and I have provided mathematical proof for that.

[Veritas Aequitas]

I suggest again, you submit the whole of MY ChatGpt's argument to your ChatGpt, ask for its comments and may be you can learn something there.

You are rejecting proof of existence while that is what you originally asked for. The Platonic form "3" exists and I have provided mathematical proof for that.

I have asked for justification [proof] that the number 3 exists as real as the keyboard you are typing on is real, i.e. the keyboard can be verified, justified and proven to be real, and it is not unreal nor an illusion.
The 'keyboard' is just an example, it can be anything that can be recognized as real, even gravity, wind, water (H20) and atoms [H, 0, Fe].

I don't want a mathematical proof which is ultimately illusory, I want a proof of that it is really real as above.

[godelian]

I don't want a mathematical proof which is ultimately illusory, I want a proof of that it is really real as above.

There is no other incontrovertibly conclusive proof possible than mathematical proof. It is your false, pagan belief in the existence of physicalist proof that is ultimately illusory.

[Veritas Aequitas]

I don't want a mathematical proof which is ultimately illusory, I want a proof of that it is really real as above.

There is no other incontrovertibly conclusive proof possible than mathematical proof. It is your false, pagan belief in the existence of physicalist proof that is ultimately illusory.

Strawman.

To ensure your are not confused I had used the term 'justification' with a bracketed 'prove'.

I don't want a mathematical proof which is ultimately illusory, I want a proof [justification] of that it is really real as above.

If a person was raped and murdered, the court will surely required proofs beyond reasonable doubts to convince the murderer to capital punishments.
It is the same that the prosecutor will provide the 'proof' i.e. all relevant evidences and justifications.
Such proofs will be justified via 'physical' proofs, e.g. DNA.

Any mathematical proof will be secondary.

[godelian]

If a person was raped and murdered, the court will surely required proofs beyond reasonable doubts to convince the murderer to capital punishments.
It is the same that the prosecutor will provide the 'proof' i.e. all relevant evidences and justifications.
Such proofs will be justified via 'physical' proofs, e.g. DNA.
Any mathematical proof will be secondary.

There is no mathematical proof about the physical universe. Hence, it won't even be secondary. It will simply be nonexistent. You are again making a category error here. As Walter Pauli famously stated:

Das ist nicht nur nicht richtig; es ist nicht einmal falsch.
(Not only is this not right; it is not even wrong.)

What you are arguing, is not even wrong.

[Veritas Aequitas]

If a person was raped and murdered, the court will surely required proofs beyond reasonable doubts to convince the murderer to capital punishments.
It is the same that the prosecutor will provide the 'proof' i.e. all relevant evidences and justifications.
Such proofs will be justified via 'physical' proofs, e.g. DNA.
Any mathematical proof will be secondary.

There is no mathematical proof about the physical universe. Hence, it won't even be secondary. It will simply be nonexistent. You are again making a category error here. As Walter Pauli famously stated:

Das ist nicht nur nicht richtig; es ist nicht einmal falsch.
(Not only is this not right; it is not even wrong.)

What you are arguing, is not even wrong.

Strawman.

Yes, there are no real mathematical proofs in the real physical universe.
There are only mental mathematical proofs about the real physical universe.
1 + 1 = 2 is merely mental and not a real object.
What is real is 1 real apple + 1 real apple = 2 real apples.
1+1=2 is secondary to reality.

1. What is critical is 'what is real' real enough for humanity to survive, flourish and have peace.

Mathematical proofs albeit useful are only valid within its qualified domain, framework and system; they are secondary to 1 above.

As sounded by ChatGpt and in general on the basis of rationality and critical thinking, mathematical elements and proofs are merely mental tools to facilitate 1.

If you insist mathematical objects are real as in 1, you are chasing illusions.

My point:
There are a wide range of meanings to 'proof'.
https://www.dictionary.com/browse/proof
There are 17 meanings in the above link.

The term 'proof' cannot be confined to only Mathematics as you insist.

[Veritas Aequitas]

Note: Here are 17 meanings for the term 'proof'

https://www.dictionary.com/browse/proof

1. evidence sufficient to establish a thing as true, or to produce belief in its truth.
Synonyms: support, corroboration, demonstration, confirmation

anything serving as such evidence:
What proof do you have?

the act of testing or making trial of anything; test; trial:
to put a thing to the proof.
Synonyms: assay, examination

the establishment of the truth of anything; demonstration.

Law. (in judicial proceedings) evidence having probative weight.

the effect of evidence in convincing the mind.

an arithmetical operation serving to check the correctness of a calculation.

Mathematics, Logic. a sequence of steps, statements, or demonstrations that leads to a valid conclusion.

a test to determine the quality, durability, etc., of materials used in manufacture.

Distilling.
the arbitrary standard strength, as of an alcoholic liquor.
strength with reference to this standard: “100 proof ” signifies a proof spirit, usually 50% alcohol.

Photography. a trial print from a negative.

Printing.
a trial impression, as of composed type, taken to correct errors and make alterations.
one of a number of early and superior impressions taken before the printing of the ordinary issue:
to pull a proof.

(in printmaking) an impression taken from a plate or the like to show the quality or condition of work during the process of execution; a print pulled for examination while working on a plate, block, stone, etc.

Numismatics. one of a limited number of coins of a new issue struck from polished dies on a blank having a polished or matte surface.

the state of having been tested and approved.

proved strength, as of armor.

17. Scots Law. the trial of a case by a judge alone, without a jury.