It made me giggle, and so I had to share.

Anyone else get theirs to make a neat number? Any 1.23s or...?

Technically, your average rank is approximately one-eighth of one percent greater than pi.

---

Unofficial Improbable Island out-of-character chat is at irc.foonetic.net, channel #iisland - come on by!

Shhh! It only counts to two decimal places, so it counts.

Pi is my GPA for last semester.
Unless you count the as-yet-unreported A I made in history...

All numbers are neat! (there are some links at the bottom of the page to other pages with interesting things about numbers, also fun.)

You may have heard the anecdote that G.H. Hardy related about Ramanujan:

As a future Math teacher I highly approve of this message. You are fabulous Beeker.

There is a proof that all positive integers are interesting:

Assume that there exist positive integers that are NOT interesting. Then there must be a lowest uninteresting number. However, such a number would be interesting because it has that unique property of being the Lowest Uninteresting Number. Since this is a contradiction, the conclusion is that all positive numbers are interesting.

Hey, I didn't say the proof would be interesting.

Identify that unique property as "Lowest Otherwise Uninteresting Number", and the paradox is eliminated, isn't it? Unless there's another paradox that arises. I can't think of one off the top of my head.

---

Unofficial Improbable Island out-of-character chat is at irc.foonetic.net, channel #iisland - come on by!

But then the next uninteresting number would be the lowest otherwise uninteresting number, and the next one, and the next one...

No, the first lowest otherwise uninteresting number is the only lowest otherwise uninteresting number. The next is the second lowest otherwise uninteresting number, which I don't think reaches the proper threshold for being interesting; at least, not uniquely so.

---

Unofficial Improbable Island out-of-character chat is at irc.foonetic.net, channel #iisland - come on by!

This isn't really a paradox, just a proof that every positive integer is interesting. If you're still determined to resolve this 'paradox', just note that 'interesting' is hardly well defined.

If you want a real paradox, then what's the smallest positive integer that can't be defined in less than twenty words?

There's only finitely many words in the English language, and so only finitely many sentences with less than twenty words in them. (Most of these don't even make sense, let alone define a number.)
Hence these sentences can only define finitely many positive integers.
Since there are infinitely many positive integers, there must exist some (in fact infinitely many) that aren't so definable.
The positive integers are well ordered so there must be a minimum such integer.
Hence this looks very much like a well defined number, with the minor difficulty that you've just defined it in a sentence of less than twenty words. What's gone wrong here?

By the way, this doesn't work in German, (I believe, someone correct me if I'm wrong) as in that language, you concatenate the number words to make one big word. So every number can be defined in one word. Instead you have the impossibility of ever writing a formally complete dictionary, and the rather odd idea that out of all the words in the German language, 100% of them never have been and never will be used, or even refer to an idea that's flitted across anyones head.

We implicitly assumed that one sentence can only define one, or at least finitely many integers. Which is of course wrong. A single word "integers" defines infinite number of integers, for example.
Was it the right answer?

Concerning the main topic...
Restate my assumptions: One, Mathematics is the language of nature. Two, Everything around us can be represented and understood through numbers. Three: If you graph the numbers of any system, patterns emerge. Therefore, there are patterns everywhere in nature.
- Pi

If I'm understanding this correctly, the paradox could be more succinctly described as follows:
"The smallest positive integer undefinable in under ten words."
That's a nice one!

Does it? Is defining a numerical set the same as defining each (or any) particular member of the set? I think we're getting bogged down in pure semantics, here...

---

Unofficial Improbable Island out-of-character chat is at irc.foonetic.net, channel #iisland - come on by!

You have a point here. This paradox needs more subtle approach.

And I think I found one. The sentence "lowest integer impossible to define in ten words or less" does not define any integer at all. It is grammatically correct, but does not refer to an actual object. Examples of similar sentences would be: "highest integer", "an integer between two and three", "positive integer lower than zero", or "the pigeon's teeth".

That means that, indeed - there is only finite number of integers defined in sentences of ten words or less.

That's a perfect way of stating what I was trying to get at, I thought I ought to go into a bit of detail about why it was a paradox.

Onedoesnot: I don't have the answer I'm afraid. I first heard this one years ago, and I still don't have what I think of as a proper answer.

Your answer: I don't think that really works, most sentences don't define a number at all. The large majority of sentences anywhere on this forum for example. We can just tighten it up to say the sentence has to uniquely define the integer if we must, but I would say that that is implicit in the meaning of the word 'define'. The word 'Integers' doesn't define any number at all. It defines a whole set of numbers. That's how it appears to me, but please feel free to disagree.

I think that this is very similar, formally at least, to the Russel paradox in set theory, and so we have to be a lot more careful about what it means to define a number (so sort of what Onedoesnot said). It's the self reference which is causing the problem. What rules do you use to decide when a sentence properly defines a number? Haven't a clue.

In set theory, they're very careful about what constitutes a set. You can't have the set of all sets (although you can have the class of all sets), and the whole thing gets damned complicate really fast. But that's going into really strange places, that probably aren't really appropriate here.

Yeah, I was going to point out earlier that mathematically, you can (and should) define your concepts without the use of structured English sentences at all, but rather with mathematical notation and language. Or if necessary, a combination of the two. Which leads me to believe that the biggest sticking point about this particular paradox is that it mixes language with mathematics and therefore has no logical resolution, because mathematics and language operate in separate logical systems. The amount of overlap between the two systems (e.g., the concept of singularity and plurality) confuses the matter and makes a non sequitur appear to be a paradox.

...OR DOES IT? I have to get back to my paper now. Glhlarhhg.

---

Unofficial Improbable Island out-of-character chat is at irc.foonetic.net, channel #iisland - come on by!

Hrmn...

I was just about to disagree with you ("Say what? Language and mathematics are inextricably intertwined!"), then I realized that I couldn't 'cause I didn't know what you meant by "logical system".

Besides, I do not wish to detract from the point of this thread which is "Hey, I got a rank of a number that I thought was fun, maybe other people have fun ranks, too!" Which is a most excellent sentiment. Yes! More cool ranks!