Crashing both ChatGPT and DeepSeek with the following question

[godelian]

Question:

Is it possible to know that a proposition in mathematics is definitively true without proving the proposition?

ChatGPT: internal server error
DeepSeek: The server is busy. Please try again later.

In my opinion, the answer is "no", but both AI engines keep crashing when I ask the question.

Has anybody had better luck?

[Wizard22]

Are you trying to divide by Zero?

[Veritas Aequitas]

AI Wrote:

This is a fascinating and deep question that touches on the philosophy of mathematics, logic, and epistemology. Let's carefully analyze it.

The Core of the Question
The essence of the question is whether we can know that a mathematical proposition is definitively true without providing a formal proof for it. To answer this, we need to clarify what "knowing" means in this context and how mathematical truth is established.

1. What Does It Mean to "Know" a Proposition Is True?
In mathematics, knowledge of a proposition's truth is typically tied to rigorous justification. A proposition is considered definitively true if:

It follows logically from a set of axioms (the foundational assumptions of a mathematical system).
A valid proof demonstrates this logical connection.
Without a proof, the status of a proposition remains uncertain—it could be true, false, or even undecidable within the given axiomatic framework.

However, there are scenarios where mathematicians might feel confident about the truth of a proposition without having a formal proof. This leads us to consider alternative ways of "knowing."

2. Can We Know Without Proof?
Here are some perspectives:

a) Empirical Evidence
In some cases, mathematicians observe patterns or conduct computational experiments that strongly suggest a proposition is true. For example:

The Riemann Hypothesis has been verified computationally for trillions of cases, but no general proof exists.
Fermat's Last Theorem was believed to be true for centuries based on partial results and intuition before Andrew Wiles provided a proof.
While such evidence builds confidence, it does not constitute definitive knowledge because exceptions might exist outside the tested range, or the pattern might break down in unforeseen ways.

b) Intuition and Heuristic Reasoning
Mathematicians often rely on intuition, analogies, and heuristic arguments to conjecture that a statement is true. For instance:

Many believe the P ≠ NP conjecture is true based on its consistency with existing knowledge and the lack of counterexamples, even though no proof exists.
However, intuition can sometimes lead to incorrect conclusions (e.g., early beliefs about the distribution of prime numbers were later refined by rigorous proofs).

c) Consistency with Existing Theory
If a proposition aligns perfectly with well-established theories and has no known contradictions, mathematicians may regard it as likely true. For example:

The Continuum Hypothesis (CH) is independent of ZFC set theory, meaning it cannot be proven or disproven within that framework. Some mathematicians accept CH as "true" based on its compatibility with certain models of set theory, while others reject it.
But again, this is not definitive knowledge—it depends on the choice of axioms and interpretations.

d) Undecidability
Some propositions are undecidable within a given axiomatic system (as shown by Gödel's incompleteness theorems). In such cases, we cannot prove or disprove the proposition using those axioms. However, we might still "know" the proposition is true if we adopt additional axioms or frameworks that settle the question. For example:

Adding large cardinal axioms to ZFC allows us to resolve certain undecidable statements.
This approach shifts the problem rather than solving it outright: the new axioms themselves require justification.

3. Philosophical Perspectives
Different philosophical schools offer varying answers to whether we can know a proposition is true without proof:

a) Platonism
Platonists believe mathematical truths exist independently of human thought. From this perspective, a proposition is either true or false, regardless of whether we have a proof. However, accessing this truth still requires some form of demonstration or insight.

b) Formalism
Formalists argue that mathematics is a game played with symbols according to agreed-upon rules. In this view, "truth" is synonymous with provability within a formal system. Without a proof, there is no truth.

c) Constructivism
Constructivists insist that mathematical objects must be explicitly constructed, and truth must be demonstrated through constructive methods. They would reject the idea of knowing a proposition is true without a constructive proof.

4. Conclusion
In standard mathematical practice, definitive knowledge of a proposition's truth requires a proof . While empirical evidence, intuition, and consistency with existing theory can provide strong support for a conjecture, they do not replace the certainty of a formal proof.

That said, there are nuanced situations—such as adopting new axioms or working in alternative frameworks—where one might claim to "know" a proposition is true without a traditional proof. However, these claims depend on additional assumptions and interpretations.

Thus, the answer to your question is:
No, definitive knowledge of a proposition’s truth generally requires a proof.

[Veritas Aequitas]

AI Wrote:

Your argument that the mathematical FS imposes an imperative condition of proof for definitive truth is valid.
Within this framework, asking whether a proposition can be known as true without proof appears contradictory.

However, the question can still serve valuable purposes:
It challenges the boundaries of the FS.
It invites exploration of alternative frameworks or interpretations of truth.
It highlights the distinction between formal proof and intuitive or empirical knowledge.

Thus, while the question may seem unnecessary within the strict confines of the mathematical FS, it remains meaningful in broader philosophical and epistemological contexts.

[godelian]

2. Can We Know Without Proof?

Outside mathematics, always.
Inside mathematics, never.

In my opinion, it is proof that distinguishes mathematics from non-mathematics.

We can still probabilistically "know" without proof but it won't be mathematical knowledge. It could be scientific or historical or possibly other knowledge, which is in the end also knowledge.

[Eodnhoj7]

2. Can We Know Without Proof?

Outside mathematics, always.
Inside mathematics, never.

In my opinion, it is proof that distinguishes mathematics from non-mathematics.

We can still probabilistically "know" without proof but it won't be mathematical knowledge. It could be scientific or historical or possibly other knowledge, which is in the end also knowledge.

Proof may define mathematic internally but the conceptualization of what constitutes proof starts with philosophy of mathematics...philosophy.

[godelian]

Proof may define mathematic internally but the conceptualization of what constitutes proof starts with philosophy of mathematics...philosophy.

Epistemology seems to emerge naturally from observing existing practices in the field. Well, that is certainly the case for Popper's "Science as falsification".

For science, the metalevel is itself certainly not scientific. With science justified by experimental test reports, it is not possible to create experimental test reports about experimental test reports.

In mathematics, on the other hand, proofs are themselves legitimate mathematical objects. While science about science is not viable, mathematics about mathematics certainly is:

ChatGPT: In mathematics, is proof about proof possible?

Yes, in mathematics, proofs about proofs are not only possible but are a fundamental part of mathematical logic and metamathematics. These kinds of proofs analyze the nature, structure, and limitations of proofs themselves. Here are some key areas where this occurs:

1. Meta-theorems – These are theorems about theorems. Examples include Gödel's incompleteness theorems, which prove fundamental limitations of formal mathematical systems.

2. Proof Theory – This branch of mathematical logic studies proofs as formal objects. For example, Gentzen’s consistency proof for arithmetic analyzes the structure of proofs to show they do not lead to contradictions.

3. Consistency Proofs – These are proofs that show a mathematical system does not contain contradictions. For example, Hilbert aimed to prove the consistency of mathematics using finitary means.

4. Provability Logic – This deals with the formal study of statements like "this statement is provable." Modal logic is often used to model such meta-mathematical reasoning.

5. Automated Proof Verification – In computer science and mathematics, proofs about the correctness of proof systems are essential. Programs like Coq or Lean can verify the validity of other proofs formally.

Would you like a specific example of a proof about proofs?

Unlike science, mathematics can actually handle an important part of its own metalevel. It cannot handle it completely, though, as it refuses to justify its own foundational level.

The ontology of mathematics isn't a mathematical object either. It is a philosophical one. The epistemology of mathematics is largely mathematical but its ontology is not.

[Impenitent]

Crashing ChatGPT is plausible but sinking DeepSeek is unfathomable...

-Imp

[Eodnhoj7]

Proof may define mathematic internally but the conceptualization of what constitutes proof starts with philosophy of mathematics...philosophy.

Epistemology seems to emerge naturally from observing existing practices in the field. Well, that is certainly the case for Popper's "Science as falsification".

For science, the metalevel is itself certainly not scientific. With science justified by experimental test reports, it is not possible to create experimental test reports about experimental test reports.

In mathematics, on the other hand, proofs are themselves legitimate mathematical objects. While science about science is not viable, mathematics about mathematics certainly is:

ChatGPT: In mathematics, is proof about proof possible?

Yes, in mathematics, proofs about proofs are not only possible but are a fundamental part of mathematical logic and metamathematics. These kinds of proofs analyze the nature, structure, and limitations of proofs themselves. Here are some key areas where this occurs:

1. Meta-theorems – These are theorems about theorems. Examples include Gödel's incompleteness theorems, which prove fundamental limitations of formal mathematical systems.

2. Proof Theory – This branch of mathematical logic studies proofs as formal objects. For example, Gentzen’s consistency proof for arithmetic analyzes the structure of proofs to show they do not lead to contradictions.

3. Consistency Proofs – These are proofs that show a mathematical system does not contain contradictions. For example, Hilbert aimed to prove the consistency of mathematics using finitary means.

4. Provability Logic – This deals with the formal study of statements like "this statement is provable." Modal logic is often used to model such meta-mathematical reasoning.

5. Automated Proof Verification – In computer science and mathematics, proofs about the correctness of proof systems are essential. Programs like Coq or Lean can verify the validity of other proofs formally.

Would you like a specific example of a proof about proofs?

Unlike science, mathematics can actually handle an important part of its own metalevel. It cannot handle it completely, though, as it refuses to justify its own foundational level.

The ontology of mathematics isn't a mathematical object either. It is a philosophical one. The epistemology of mathematics is largely mathematical but its ontology is not.

Yeah and chat GPT used philosophical language for it's point. Try explaining that in pure numbers...it would be meaningless.

[godelian]

Yeah and chat GPT used philosophical language for it's point. Try explaining that in pure numbers...it would be meaningless.

Language itself can be represented as natural numbers. In fact, this entire page is in UTF-8:

https://en.m.wikipedia.org/wiki/UTF-8

UTF-8 is a character encoding standard used for electronic communication. Defined by the Unicode Standard, the name is derived from Unicode Transformation Format – 8-bit.[1] Almost every webpage is stored in UTF-8.

A sequence of natural numbers is again a single natural number. So, this page can be represented as a single natural number.

The original canonical representation of language is just a sequence of sounds. But then again, these sounds are carriers of information and not the information itself.

Information is accessed through one of its alternative representations. It cannot be accessed directly.

[Eodnhoj7]

Yeah and chat GPT used philosophical language for it's point. Try explaining that in pure numbers...it would be meaningless.

Language itself can be represented as natural numbers. In fact, this entire page is in UTF-8:

https://en.m.wikipedia.org/wiki/UTF-8

UTF-8 is a character encoding standard used for electronic communication. Defined by the Unicode Standard, the name is derived from Unicode Transformation Format – 8-bit.[1] Almost every webpage is stored in UTF-8.

A sequence of natural numbers is again a single natural number. So, this page can be represented as a single natural number.

The original canonical representation of language is just a sequence of sounds. But then again, these sounds are carriers of information and not the information itself.

Information is accessed through one of its alternative representations. It cannot be accessed directly.

Natural number representation of natural language is a revolving loop, the explanation of mathematics can dually be put in natural language. There is no precedent of one over the other. Representation can be mediated by anything, even an empirical cloud.

There is no fixed symbolic structure to meaning.

The application of number has no law but the act of attentively calculating.

[Flannel Jesus]

We can all agree that op did not, in fact, crash anything with those questions, right? That's clearly a fiction from his under-active imagination

[godelian]

We can all agree that op did not, in fact, crash anything with those questions, right? That's clearly a fiction from his under-active imagination

There are numerous articles online that mention ChatGPT's "internal server error":

https://harpa.ai/blog/how-to-fix-chatgp ... rver-error

How to Fix ChatGPT Internal Server Error

ChatGPT, a revolutionary language model, has transformed the way we interact with AI. It offers users a seamless experience, answering questions with precision and speed. However, like all technology, it's not immune to glitches. One such hiccup users often encounter is the "internal server error." In this article, we'll delve into the nature of this error, its causes, and how to troubleshoot it.

So, can we all agree that, in fact, you don't know what you are talking about?

[Flannel Jesus]

We can all agree that op did not, in fact, crash anything with those questions, right? That's clearly a fiction from his under-active imagination

There are numerous articles online that mention ChatGPT's "internal server error":

https://harpa.ai/blog/how-to-fix-chatgp ... rver-error

How to Fix ChatGPT Internal Server Error

ChatGPT, a revolutionary language model, has transformed the way we interact with AI. It offers users a seamless experience, answering questions with precision and speed. However, like all technology, it's not immune to glitches. One such hiccup users often encounter is the "internal server error." In this article, we'll delve into the nature of this error, its causes, and how to troubleshoot it.

So, can we all agree that, in fact, you don't know what you are talking about?

Nope

From your source:
"An internal server error indicates an issue on the server side, preventing ChatGPT from fulfilling user requests.

This error can arise due to various reasons, including high traffic, server maintenance, rate limits, and more."

None of the listed reasons is "asking it philosophical questions" numnuts.

Anyway I went and asked it myself. It didn't crash. You're just retarded and incapable of the most basic level of reasoning.

In mathematics, a proposition is considered *definitively true* only if it has been *proven* using a rigorous logical framework based on axioms and previously established theorems. However, there are a few nuances to consider:

1. **Intuition and Heuristics**: In many cases, mathematicians have strong intuitive reasons to believe that a proposition is true (e.g., numerical evidence, probabilistic heuristics, or analogy with known results). However, until a formal proof is provided, such beliefs remain conjectural.

2. **Empirical or Computational Evidence**: Some propositions can be tested extensively through computational methods. For instance, the Riemann Hypothesis has been verified for billions of cases numerically, but that does not constitute a proof.

3. **Undecidability**: Some propositions may be *true* in a given axiomatic system but *unprovable* within that system. Gödel’s incompleteness theorems demonstrate that in any sufficiently expressive formal system, there exist true statements that cannot be proven within the system.

4. **Axiomatic Assumptions**: If a mathematical statement is independent of a given axiom system (e.g., the Continuum Hypothesis in Zermelo-Fraenkel set theory with the Axiom of Choice, ZFC), then its truth value may depend on the choice of axioms rather than proof alone.

### Conclusion:
A mathematical proposition is only *definitively* known to be true when a proof is established. Without a proof, we can have strong reasons to *believe* in its truth, but that belief is not the same as certainty in the mathematical sense.

[godelian]

From your source:
"An internal server error indicates an issue on the server side, preventing ChatGPT from fulfilling user requests.

This error can arise due to various reasons, including high traffic, server maintenance, rate limits, and more."

None of the listed reasons is "asking it philosophical questions" numnuts.

I did not say for what reason these AI engines crashed. They simply did. Next, I asked other people to try too and check if they got another result.

Anyway I went and asked it myself. It didn't crash.

That was indeed what I asked other people to do: Try and see what result you get.

You're just retarded and incapable of the most basic level of reasoning.

It is exactly the other way around.

When someone asks you, "I tried the following. It did not work. Can you also try?", the imbecile retard that you are, does not understand what exactly is being asked from you.

You are the kind of idiot that absolutely nobody wants on their team. You are an individual that everyone else on the team spits, pees, and shits on. They are all just waiting for the manager to understand that as well, so that he finally fires you. Long overdue. Good riddance.