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Comment by twelfthnight | Hacker News Reader

> Why do we care that there are only five Platonic Solids? The true answer is because there is an answer, and it would be intolerable not to know it

I see a few holes with this argument:

1) Who is this "we"? I'm sure not all non-"non-mathematicians" agree with this sentiment.

2) If a mathematician is still looking for an answer to a question, they don't yet know if an answer exists or not (see: Godel, halting problem). Perhaps this should be rephrased "because there might be an answer and it would be unbearable not to know it if it did exist."

3) The argument is circular (calling not knowing bad doesn't answer why knowing is good).

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1) Since he seems to be talking about mathematicians, that's actually likely - but not necessarily relevant.

2) Mountaineers climb mountains. Musicians make music. Artists make art. Developers make code. To varying degrees, everyone suffers from the same problem. And people who are exceptional in their fields are far more likely to find not pursuing their passion intolerable.

3) Considering how useful pure maths turn out to be - often in completely unexpected ways - I'm quite happy to leave pure maths types to do what they do.

Even if the hit rate were one gamechanger a century, that's still an exceptionally good return. (The actual return seems to be much higher than that.)

With regards to 2), there's a difference between trying to find an answer to a single yes/no question, and trying to find a general method for solving an infinite class of such questions.

If a single yes/no question is clearly stated, then it does have an answer. This just a tautology - that's what it means to be clearly stated. For example, does there exist integers x, y, and z such that x^3 + y^3 + z^3 = 33? No one knows, but at least the question is clearly stated. The answer is definitely yes or no.

More generally, x^3 + y^3 + z^3 = 33 is an example of a Diophantine equation (an equation between multivariate polynomials with integer coefficients and integer variables). Suppose we are not so interested in this particular equation, but instead want to find an algorithm that can always tell if a Diophantine equation has a solution. This kind of question might not have a solution, since such an algorithm might not exists. And in fact, it doesn't (Hilbert's tenth problem).

> If a single yes/no question is clearly stated, then it does have an answer.

Without having to reach for fancy counterexamples, how about the Empedoclean "Is the answer to this question 'no'?"?

Simple answer on the metalevel: It's not clearly stated, because it's self referential. It's not a question by the GP's standards.

Well, to be sure, if the definition of 'clearly stated' is made to exclude counterexamples to the assertion, then there are no counterexamples. :-) I don't think many people would include 'not self referential' in the definition of 'clearly stated'; for example, "is this question more than eight words long?" is pretty clear.

To put my words in context, I was thinking about mathematical questions, such as "does there exist a curve that goes through every point in a square?". Everyone has some kind of intuitive understanding of what's a "curve", but mathematicians need a precise definition in order to make this question clear enough for mathematical study. (There is such a curve: the Hilbert curve.)

It's not at all easy to recognize when a question is stated "clearly enough", and your questions provide an excellent example of this. The question "is this question more than eight words long?" refers to question itself (as a string), and in general, this kind of self-reference is perfectly harmless. The question "Is the answer to this question 'no'?" refers to its own answer, and this kind of self-reference is potentially problematic.

Also, it occurs to me that the self reference can be avoided by an epistemologically insignificant dodge: "Are you about to say 'no'?" (This dodge is, I admit, a little dodgy; it admits things like "I wasn't planning to" as a perfectly sound answer. However, I'm taking it as read that we understand all questions as yes / no questions.)

Implicitly, the "no" references the answer to the question itself, hence it's still selfreferential.

In other words, the fallacy is to assume the following response was fixed to be an answer to the question. If I answered "no" it would be obvious that i wasn't planning to say "no" prior ("about") to the question. There is just no context in which the question would make sense the way you suggest. It's non-sense.

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