Big

About big numbers, by Scott Aaronson, via Steve Landsburg.

Beyond ordinary notions of bigness, like exponentials and factorials, says Scott, lie more exotic numbers defined by recursive functions and Turing machines. These big numbers make my head hurt, for reasons also explained by Scott.

Whence the cowering before big numbers, then? Does it have a biological origin? In 1999, a group led by neuropsychologist Stanislas Dehaene reported evidence in Science that two separate brain systems contribute to mathematical thinking. The group trained Russian-English bilinguals to solve a set of problems, including two-digit addition, base-eight addition, cube roots, and logarithms. Some subjects were trained in Russian, others in English. When the subjects were then asked to solve problems approximately—to choose the closer of two estimates—they performed equally well in both languages. But when asked to solve problems exactly, they performed better in the language of their training. What’s more, brain-imaging evidence showed that the subjects’ parietal lobes, involved in spatial reasoning, were more active during approximation problems; while the left inferior frontal lobes, involved in verbal reasoning, were more active during exact calculation problems. Studies of patients with brain lesions paint the same picture: those with parietal lesions sometimes can’t decide whether 9 is closer to 10 or to 5, but remember the multiplication table; whereas those with left-hemispheric lesions sometimes can’t decide whether 2+2 is 3 or 4, but know that the answer is closer to 3 than to 9. Dehaene et al. conjecture that humans represent numbers in two ways. For approximate reckoning we use a ‘mental number line,’ which evolved long ago and which we likely share with other animals. But for exact computation we use numerical symbols, which evolved recently and which, being language-dependent, are unique to humans. This hypothesis neatly explains the experiment’s findings: the reason subjects performed better in the language of their training for exact computation but not for approximation problems is that the former call upon the verbally-oriented left inferior frontal lobes, and the latter upon the spatially-oriented parietal lobes.

One problem with the biggies is that they are hard to pin down. It's known that the sequence of so called Busy Beaver or BB numbers grows faster than any computable function, the first few are not so bad BB(1) = 1, BB(2) = 6, BB(3) = 21 and BB(4) = 107. It took mathematicians more than twenty years to calculate all these. Bigger BB numbers are not known, but it is known that:

Then, in 1989, Heiner Marxen and Jürgen Buntrock discovered that BB(5) is at least 47,176,870. To this day, BB(5) hasn’t been pinned down precisely, and it could turn out to be much higher still. As for BB(6), Marxen and Buntrock set another record in 1997 by proving that it’s at least 8,690,333,381,690,951.

All these biggies have their origin in the recursive properties of language. Of course it's also known that recursive thinking can lead to paradoxical conclusions.