There’s a pseudo-theorem in math that is sometimes given to 1st year graduate students (at least in my case, 35 years ago), which is that
All natural numbers are interesting.
Natural numbers consist of {1, 2, 3, …} -- actually a recent hot topic of conversation on LW (“natural numbers” is sometimes defined to include 0, but everything that follows will work either way).
The “proof” used the principle of mathematical induction (one version of which is):
If P(n) is true for n=1, and the assertion “m is the smallest integer such that !P(m)” leads to a contradiction, then P(n) is true for all natural numbers.
and also uses the fact (from the Peano construction of the natural numbers?) that every non-empty subset of natural numbers has a smallest element.
PROOF:
1 is interesting.
Suppose theorem is false. Then some number m is the smallest uninteresting number. But then wouldn’t that be interesting?
Contradiction. QED.
The illustrates a pitfall of mixing (qualities that don’t really belong in a mathematical statement) with (rigorous logic), and in general, if you take a quality that is not rigorously defined, and apply a sufficiently long train of logic to it, you are liable to “prove” nonsense.
(Note: the logic just applied is equivalent to P(1) ⇒ P(2) ⇒ P(3), …which is infinite and hence long enough.)
It is my impression that certain contested (though “proven”) assertions about economics suffer from this problem, and it’s hard, for me at least, to think of a moral proposition that wouldn’t risk this sort of pitfall.
Okay but if I honestly believe that all natural numbers are interesting and thought of this proof as pretty validly matching my intuitions, what does that mean?
Unless you turn “interesting” into something rigorously defined and precisely communicated to others, what it means is that all natural numbers are {some quality that is not rigorously defined and can’t be precisely communicated to others}.
I guess I feel that even if I haven’t defined “interesting” rigorously, I still have some intuitions for what “interesting” means, large parts of which will be shared by my intended audience.
For example, I could make the empirical prediction that if someone names a number I could talk about it for a bit and then they would agree it was interesting (I mean this as a toy example; I’m not sure I could do this.)
One could then take approximations of these conversations, or even the existence of these conversations, and define interesting* to be “I can say a unique few sentences about historic results surrounding this number and related mathematical factoids.” Which then might be a strong empirical predictor of people claiming something is interesting.
So I feel like there’s something beyond a useless logical fact being expressed by my intuitions here.
I can’t tell what this is. The first link might imply that Gwern thinks I misstated the Interesting Number Paradox (I looked at the Wikipedia article before I wrote my post, but went with my memory, and there are multiple equivalent ways of saying it, but if you think I got it wrong ….? Or maybe it was offered as a handy reference.
The Berry Paradox sounds like a very different kettle of fish … with more real complexity.
More meta: Perhaps your priors for “if someone replies to my comment, they disagree with me” are too high. ;-) Maybe not for internet in general, but LW is not an average internet site.
There’s a pseudo-theorem in math that is sometimes given to 1st year graduate students (at least in my case, 35 years ago), which is that
All natural numbers are interesting.
Natural numbers consist of {1, 2, 3, …} -- actually a recent hot topic of conversation on LW (“natural numbers” is sometimes defined to include 0, but everything that follows will work either way).
The “proof” used the principle of mathematical induction (one version of which is):
If P(n) is true for n=1, and the assertion “m is the smallest integer such that !P(m)” leads to a contradiction, then P(n) is true for all natural numbers.
and also uses the fact (from the Peano construction of the natural numbers?) that every non-empty subset of natural numbers has a smallest element.
PROOF:
1 is interesting.
Suppose theorem is false. Then some number m is the smallest uninteresting number. But then wouldn’t that be interesting?
Contradiction. QED.
The illustrates a pitfall of mixing (qualities that don’t really belong in a mathematical statement) with (rigorous logic), and in general, if you take a quality that is not rigorously defined, and apply a sufficiently long train of logic to it, you are liable to “prove” nonsense.
(Note: the logic just applied is equivalent to P(1) ⇒ P(2) ⇒ P(3), …which is infinite and hence long enough.)
It is my impression that certain contested (though “proven”) assertions about economics suffer from this problem, and it’s hard, for me at least, to think of a moral proposition that wouldn’t risk this sort of pitfall.
Okay but if I honestly believe that all natural numbers are interesting and thought of this proof as pretty validly matching my intuitions, what does that mean?
Unless you turn “interesting” into something rigorously defined and precisely communicated to others, what it means is that all natural numbers are {some quality that is not rigorously defined and can’t be precisely communicated to others}.
I guess I feel that even if I haven’t defined “interesting” rigorously, I still have some intuitions for what “interesting” means, large parts of which will be shared by my intended audience.
For example, I could make the empirical prediction that if someone names a number I could talk about it for a bit and then they would agree it was interesting (I mean this as a toy example; I’m not sure I could do this.)
One could then take approximations of these conversations, or even the existence of these conversations, and define interesting* to be “I can say a unique few sentences about historic results surrounding this number and related mathematical factoids.” Which then might be a strong empirical predictor of people claiming something is interesting.
So I feel like there’s something beyond a useless logical fact being expressed by my intuitions here.
http://en.wikipedia.org/wiki/Interesting_number_paradox and http://en.wikipedia.org/wiki/Berry_paradox
I can’t tell what this is. The first link might imply that Gwern thinks I misstated the Interesting Number Paradox (I looked at the Wikipedia article before I wrote my post, but went with my memory, and there are multiple equivalent ways of saying it, but if you think I got it wrong ….? Or maybe it was offered as a handy reference.
The Berry Paradox sounds like a very different kettle of fish … with more real complexity.
I would bet on this one.
More meta: Perhaps your priors for “if someone replies to my comment, they disagree with me” are too high. ;-) Maybe not for internet in general, but LW is not an average internet site.