Tupper’s self-referential formula describes a function whose graph (for inputs within a certain range) is the formula of the function itself.

Links: Wolfram, Wikipedia.

Hofstadter should probably be informed.

And here’s the accompanying GEB dialogue.

I recently finished reading William Goldbloom Bloch’s The Unimaginable Mathematics of Borges’ Library of Babel. As was the case with the Atlas of Remote Islands, most of my reading time was spent beaming at the very unlikeliness of the book’s existence. Borges’ work could be described as “literary nerd-sniping,” so the notion of a mathematician devoting a book to an analysis of one of his stories makes perfect sense — it’s just so rare to see those spheres overlapping.

The mathematics the book employs isn’t terribly difficult, since it’s written for the interested layman. Conversely, if you did your undergrad degree in math (or something else sufficiently mathly) you’ll probably find yourself skimming occasionally, but there’s still quite a lot in there to enjoy. The book is very good at communicating the pleasure of doing mathematics for its own sake, and that aspect really struck a familiar chord with me. Who doesn’t enjoy discovering that The Library would contain enough books to fill 10 ^ 1,834,013 universes?

Finally, any writer who in the preface defines his intended audience as “Umberto Eco” has won a fan for life.

I’ve seen references to four- and five-dimensional chess (which look like they might still be borderline-human-playable games), but a search for “n-dimensional chess” shows that not a lot of progress has been made for the general case. It seems like it wouldn’t be too hard to develop rules… rooks could move an arbitrary distance on paths parallel to the basis vectors, bishops could move diagonally on any plane, etc. Certainly the number of pieces would have to scale up, and it might be tricky to prove that checkmate is possible for all n, but someone who wasn’t prepping for finals could probably develop a consistent set of “playable” rules without too much trouble.

The real trick, of course, would be a consistent set of rules for chess in Hilbert space, which I think I should leave to the real mathematicians.

Mr. Smith and his wife invited four other couples to a party. When everyone arrived, some of the people in the room shook hands with some of the others. Of course, nobody shook hands with their spouse and nobody shook hands with the same person twice.

After that, Mr. Smith asked everyone how many times they shook someone’s hand. He received different answers from everybody.

How many times did Mrs. Smith shake someone’s hand?

Written by Peter Ross, Napier U. ]]> http://www.blogonauts.com/eats-the-world/puzzle-time/feed 3 http://www.blogonauts.com/eats-the-world/godel-in-single-syllables http://www.blogonauts.com/eats-the-world/godel-in-single-syllables#comments Wed, 27 Oct 2010 21:14:14 +0000 Harry http://www.blogonauts.com/eats-the-world/?p=1906 The logician George Boolos once published an article called Gödel’s Second Incompleteness Theorem Explained in Words of One Syllable. In full, the article reads:

First of all, when I say “proved,” what I will mean is “proved with the aid of the whole of math.” Now then: two plus two is four, as you well know. And, of course, it can be proved that two plus two is four (proved, that is, with the aid of the whole of math, as I said, though in the case of two plus two, of course we do not need the whole of math to prove that it is four). And, as may not be quite so clear, it can be proved that it can be proved that two plus two is four, as well. And it can be proved that it can be proved that it can be proved that two plus two is four. And so on. In fact, if a claim can be proved, then it can be proved that the claim can be proved. And that too can be proved.
Now, two plus two is not five. And it can be proved that two plus two is not five. And it can be proved that it can be proved that two plus two is not five, and so on.

Thus: it can be proved that two plus two is not five. Can it be proved as well that two plus two is five? It would be a real blow to math, to say the least, if it could. If it could be proved that two plus two is five, then it could be proved that five is not five, and then there would be no claim that could not be proved, and math would be a lot of bunk.

So, we now want to ask, can it be proved that it can’t be proved that two plus two is five? Here’s the shock: no, it can’t. Or, to hedge a bit: if it can be proved that it can’t be proved that two plus two is five, then it can be proved as well that two plus two is five, and math is a lot of bunk. In fact, if math is not a lot of bunk, then no claim of the form “claim X can’t be proved” can be proved.

So, if math is not a lot of bunk, then, though it can’t be proved that two plus two is five, it can’t be proved that it can’t be proved that two plus two is five.

By the way, in case you’d like to know: yes, it can be proved that if it can be proved that it can’t be proved that two plus two is five, then it can be proved that two plus two is five.

It would appear that Benoît Mandelbrot passed away yesterday. =\

A man who is always asking, “Is what I do worth while?” and “Am I the right person to do it?” will always be ineffective himself and a discouragement to others. He must shut his eyes a little and think a little more of his subject and himself than they deserve.

The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. … It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind — we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it.

¹ The usual formulation of Hardy’s rule is that, “if a mathematician’s going to do any significant work, it’ll be done before they’re thirty.” This is true so long as we ignore the later work of Archimedes, Cauchy, Descartes, Euler, Fermat, Frege, Gauss, Hilbert, Newton, Peano, Poincare, Russell, von Neumann, Weierstrass, and most recently Andrew Wiles. I would guess that Hardy’s opinion on the matter was influenced by his relationship with the mathematical prodigy Ramanujan, who died at 33.

One especially interesting result was the notion that people intuitively distribute numbers on a logarithmic scale rather than a linear one. As it turns out, children do this, too.

It is Pica’s belief that understanding quantities in terms of estimating ratios is a universal human intuition, due to the fact that ratios are much more important for survival in the wild. Historically, faced with a group of adversaries, we needed to know instantly whether there were more of them than us. When we saw two trees, we needed to know instantly which had more fruit hanging from it. In neither case was it necessary to enumerate every enemy or every fruit individually. The crucial thing was to be able to make quick estimates of the relevant amounts and compare them; in other words to make approximations and judge their ratios.