paradigms in large number construction
You have fifteen seconds. Using standard math notation, English words, or both, name a single whole number—not an infinity—on a blank index card. Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature.
So starts this article on big numbers by scott aaronson. The previous test, being conducted on school students tends to bring up solutions using many 9's. However, in the science of big numbers who wins is who is able to deal with the strongest mathematical paradigm, not who writes faster. Read the link for an account on the history of exponentials, multiple exponentials, Ackerman's function and Busy Beavers.
As is pointed out by the author, there is a large correspondence between the status of science in each era, and the progress in our capacity to name big numbers. As you progress through the article it becomes clear how new concepts such as recursion and computability have to be developed to name bigger and bigger numbers.
The article ends with a reference to a study which appeared 1999 in Science in which a group of Neuropsychologists developed an experiment to associate language with our capacity to compute and predict (ie, our capacity to make approaximated computations). The authors of the study, cited below, conclude that different parts of the brain are used when doing precise computations and approximations. Interestingly, when doing a precise computation the part of the brain that is used is the same part that deals with language. Why? Because computation depends on the capacity to manipulate symbols, be it mathematical symbols or linguistic symbols. On the other hand, they argue that approximations are done using a mental image of a "number line". It's an interesting point. However I don't understand some parts of their methodology, particularly when they claim that a larger percentage of the test individuals correctly solved computation problems stated in the language in which they learned to solve them, rather than in a different tongue. I'm not sure what they mean and why this happens. It should not be difficult to translate a mathematical problem, and trying to solve it using an unknown symbol system is something that I can't even imagine... I guess I have to read the original source :)
Here's the reference for this paper:
S. Dehaene and E. Spelke and P. Pinel and R. Stanescu and S. Tsivkin, "Sources of Mathematical Thinking: Behavioral and Brain-Imaging Evidence," Science, vol. 284, no. 5416, May 7, 1999, pp. 970- 974.
posted by miki at
5:26 AM
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