Sorry to have been uncharacteristically absent all week; I’ve been busy in a good way, though I hope and expect to get back to more regular blogging before long. In the meantime, to keep you busy, let me give you a pointer to a marvelous essay I’ve long been a fan of, and just happened to get reminded of today: Scott Aaronson’s take on the old riddle of who can name the biggest number. Have fun with this, and I’ll see you soon.
I once ran a competition for elementary school kids (through my website) to write down the biggest number they could in a 2″x4″ box.
Can you believe – there was a tie for first place?
Details at http://bit.ly/e3LV8m
Here’s a big number I just learned about:
“The TREE sequence begins TREE(1) = 1, TREE(2) = 3, then suddenly TREE(3) explodes to a value so enormously large that many other ‘large’ combinatorial constants, such as Friedman’s n(4), are extremely small by comparison. A lower bound for n(4), and hence an extremely weak lower bound for TREE(3), is A(A(…A(1)…)), where the number of A’s is A(187196), and A() is a version of Ackermann’s function: A(x) = 2↑↑…↑x with x-1 ↑s (Knuth up-arrows)”
http://en.wikipedia.org/wiki/Kruskal's_tree_theorem
I write 9 followed by as many ! as I can print in 15 seconds.
Ok, that settles it. Now you *must* blog about the connection between fast-growing functions and provability. :)
From the article:
“Such paradigms are historical rarities. We find a flurry in antiquity, another flurry in the twentieth century, and nothing much in between.”
I didn’t know that the concept of giving notation sequential operations didn’t come about until the 20th century.
Does anyone know if there is some simple notation for expressing something like multiplying all numbers up to a certain number? E.g. 1*2*3*4*5. This could get pretty amazingly big!!!!. So maybe a notation like !!(10) might work.
If there is such a notation which 20th century mathematician came up with this.
Also does anyone know the 20th century mathematician that came up with the first concise/precise definition of “e” to the power of x?
If the backward scientists and mathematicians of the 17th, 18th and 19th century who were mired in Euclidean Geometry had access to this kind of knowledge, they could have done some great things.
@Will A: Exactly. In the nineteenth century Gallois was just a mispelt cigarette.
Maybe the first big mathematical “discovery” of the 21st century will be a way to define pi using only ASCII characters so that it can be posted on a blog. E.g.:
pi = 4 * WA(k, 0, inf, ((-1)^k)/(2*k+1)))
WA stands for Will A and is basically a replacement for sigma. I call it WA because of my huge ego and because I’m too lazy to come up with a way to express my ideas using normal classical mathematical convention.
3 advantages:
1) It makes it more difficult to read/write and would help to discourage students who become math majors so they won’t have to write as many papers and want to skate through college.
2) Once this notation becomes common place and all formulas are rewritten using this notation, future mathematicians might think that all important math notations (and hopefully ideas) were a result of the 21st century.
Ok end of rant
Who can name the biggest infinity?