The smallest non-interesting number

There’s an argument I’ve heard which I think is fallacious.

The basic idea is that numbers have various interesting properties. Two is the only even prime; three is the only prime triangular number; four is the smallest square; five is the smallest pyramid number; six is the only number that is both the sum and the product of three consecutive positive numbers; and so on.

And the debate is whether or not every number has an interesting property.

Now the argument I’ve heard is that every number must have an interesting property. Because suppose you’re going through the numbers and you reach a number with no interesting properties - let’s say 11,630 for the sake of argument. The argument is that 11,630 has an interesting property - it’s the smallest number with no interesting properties.

Okay, I can accept that. But here’s where the argument falters in my opinion. It then continues the same line of reasoning. You supposedly can continue to work through the numbers and if you find another number with no interesting properties - let’s say 12,407 this time - then once again it’s made interesting by its lack of interesting properties.

And that’s what I dispute. The interesting property of 12,407 is not that it’s the smallest number with no interesting properties. Because 11,630 is still the smallest number with no interesting properties. So 12,407 can’t claim that distinction so it is in fact as uninteresting as it first appeared.

Well I think seven is pretty dull.

(‘Interesting’ is an amorphous human construct. They’re just numbers.)

Well unless you are stating that there is more than one uninteresting number than 12,407 would be the largest uninteresting number, which of course makes it interesting, so it isn’t. Which is why there is no uninteresting number, or any other thing which you can name, because doing so makes it interesting. Like silence, saying the word destroys it.

ETA: You are going to have to work hard to start the most absurd thread in GD today.

Like so many other apparent mathematical paradoces, this one goes away when you express it rigorously. How does one define “interesting”? One way would be “can be expressed in a way that’s shorter than its direct representation”. For instance, the number 65536 is interesting, because it can be expressed as 2^16, using only 4 characters instead of 5 (of course, this will vary depending on the precise alphabet and rules of representation used to represent numbers, but that’s the general idea).

Except, by that standard, 1 is uninteresting. There’s no way of expressing 1 that’s shorter than just 1. Certainly, the expression “the smallest uninteresting number” is not a shorter expression for it, so (despite being accurate) that description does not suffice to make 1 interesting.

How are the other single digit numbers represented with fewer characters?

ETA: I’ll grant 0 could be represented by no characters.

The argument is a proof by contradiction. You start by assuming that not all numbers are interesting, which leads to a contradiction. Therefore, your assumption must be wrong.

When I’ve seen this argument, it has been presented as humor, not as a serious piece of mathematics, because it rests on the idea of “interesting” being a meaningful, well-defined term.

Big friggin’ whoop. All this really means is that two is the only prime divisible by 2. But so what? Three is the only prime divisible by 3; seventeen is the only prime divisible by 17; and, in general, p is the only prime divisible by p.

Well I think it’s an interesting question (which may work against you), prime-arily because a recent XKCDcomic led me to the List of Numbers Wiki page.

On that page is a list of types of numbers. Among the trivial are the types of things you’ll find in various textbooks (e.g., prime, perfect, named). If one takes the OP’s discussion to mean a mathematically significant type of number–and even allows such things as scientifically recognized named numbers (e.g., Planck’s constant), and doesn’t recognize ad hoc arbitrary properties (e.g., it’s the only 7-digit sequence in the Xth position of Pi), then what is the first number to be ‘boring’. (I don’t know whether limiting it to integers is necessary.)

If one does start with that kind of question, than does it imply the question of whether or not to include the property of ‘boring’ among those that disqualify a number? At what point does it?

I heard somewhere that if a number is a bore it is equal to an integer times Planck’s constant divided by 2

That’s what reminded me of this topic.

Personally, I don’t see much point in getting into a debate of how to define “interesting”. Nobody protested the existence of interesting properties when the issue was first raised. Arguing it now is backtracking.

And it really isn’t necessary to what I think is the central point. You could make the same argument about the unexpected hanging paradox. I don’t think anyone would claim that the terms “judge”, “prisoner”, “execution”, “surprise”, and “one day next week” lack sufficient rigor.

But the same problem arises. The logic of the hanging paradox is also based on repeatedly applying the same solution without proving that the solution can be validly repeated.

Heh, I almost mentioned that your post reminded me of the Unexpected Hanging.

It’s a joke. It’s not worth this level of analysis. Go outside and get some fresh air.

Bingo.

For instance, 43 is the smallest non-negative integer absent from David Wells’ Book of Curious and Interesting Numbers. That might make it interesting, in a more general sense. But the second smallest non-negative integer (51) that Wells fails to find interesting is going to be considerably less interesting by virtue of its omission, and when the integers he skips over get downright frequent (54, 57, 58, 62, 67, 68, 74, 75…) the argument that they’re each really interesting by virtue of each one taking a quick turn as first uninteresting number rings rather hollow.

One is the loneliest number.

This is essentially the starting point for Kolmogorov-Chaitin complexity.

There is no debate. Numbers are an invention of the human intelligence, they are not real. You can easily define a new property that any number possesses and so any number can be “interesting”, whatever that means.

Agreed, however it is more poetic to quote Kronecker: “God invented the integers; all else is the work of man.”

I think the point of the paradox is not to prove anything about any numbers, but rather to point out that a description which appears to be meaningful (“the smallest number that does not have any interesting properties”) is, in fact, nonsensical because it can have no referent. The paradox is about semantics, not numbers.

I’d go with the KISS principal. If it makes it a pain in the ass to bump the first non-interesting item to being interesting for that reason, well don’t do that. The smallest non-interesting number is the smallest non-interesting number, regardless of how much fame might surround it for that distinction.

There are no gods. That line is meaningless, at the least.

When I saw this, I immediately thought, “One plus one is two. Six plus three is nine. Nine plus two is eleven.”

Cool number. :wink: