# Philosophy of Math Meme

2005-11-13T17:20:00Z

I filled this meme in quite a while ago. If you answer it, feel free to mail in your answers.

1. Do you agree that it is impossible to define a total function from the reals to the reals which is not continuous?

Yes.

2. Do you agree that the intermediate value theorem does not hold the way that it is normally stated?

Yes. Here I am taking “does not hold” to mean “one cannot prove” rather than “is not true”.

3. Do you agree that there are only three infinite cardinalities?

No. I understand that we can prove ℕ, ℕ ⇒ ℕ, (ℕ ⇒ ℕ)⇒ ℕ, … all have different cardinalities, even if the representatives of these types are all members of the countable set of some formal language.

4. Do you agree that the continuum hypothesis is a meaningful statement that has a definite truth value, even if we do not know what it is?

Mu. I don’t know the answer to this. The statement would be, “does ℕ ⇒ ℕ have the same cardinality as the least uncountable ordinal”. I’m not sure if least uncountable ordinal is meaningful. If it is meaningful, then it has a definite truth value.

5. Do you agree that the axiom which states the existence of an inaccessible cardinal is a meaningful statement that has a definite truth value, even if we do not know what it is?

Yes. I taking inaccessible cardinal to mean inaccessible ordinal. I’m not certain about my answer here, but I am (more or less) in favour of large ordinals that give us lots of induction to prove functions terminate. Bigger ordinals are better, even if they are impractical, so long as they are actually well founded.

6. Do you agree that for any mathematical question it is easy to build a machine with two lights, yes and no, where the light marked yes will be on if it is true and the light marked no will be on if it is false?

No. For any given mathematical statement is may not be easy to build a machine such that if the mathematical statement is true then the yes light lights up. For any given mathematical statement if the statement is true it is easy to build a machine such that the yes light lights up.

7. Do you agree that for any two statements the first implies the second or the second implies the first?

No.

8. Do you agree that a constructive proof of a theorem gives more insight than a classical proof?

Yes. In general theorems are what give insights, and proofs give no insights. But sometimes proofs give insights, and even more rarely constructive proofs give more insight than classical proofs.

9. Do you agree that mathematics can be done using different kinds of reasoning, and that depending on the situation different kinds of reasoning are appropriate?

No. I cannot think of any situation where constructive mathematics is not the most appropriate to be using; however I am open to the possibility that there are some.

10. Do you agree that all mathematical truths are true, but that some mathematical truths are more true than other mathematical truths?

No. All truths are equally true.

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