Count on Number One
Nowadays, you can count on number one. But that wasn't always the case. To the ancient Greek philosophers, one wasn't a number. We'll talk with a mathematician about the history of the number one - after this on Earth and Sky.
Date
JB: This is Earth and Sky, on the number one. If you wanted to count a pile of beans, you'd start with one. But this wouldn't have made sense to the ancient Greek philosophers, and the mathematicians who worked under their rules. We spoke with Dr. Hendrik Lenstra, a mathematician at the University of California at Berkeley, about the history of the number one:
(Tape 0:04:21-0:04:56) The Greeks did not view one as a number. One, that was the unit by which you measure, I would say, the other numbers, but what they called numbers. I think the idea behind this -- and that was influenced by the philosophers that played an important role in Greek intellectual life -- the idea is that numbers are meant for counting, and if you have only one of something, well, then you don't feel that you need to count it. So the numbers for the Greeks, so they started at two. And that is not mathematically a very sensible distinction.
JB: Although the mathematicians started counting at two, it's likely that practical folks weren't influenced by philosophers -- so they could count with one all along. But since the eighteenth century, philosophers have lost their control over math:
(Tape 0:05:14-0:05:32) But nowadays mathematicians don't have the inclination to listen to philosophers anymore. If we would listen to philosophers, we couldn't even use negative numbers. And certainly mathematicians feel that they have the freedom of definition. And they use this freedom in order to make life as convenient as possible.
JB: Thanks to Dr. Hendrik Lenstra for speaking with us. And with thanks to the National Science Foundation, I'm Joel Block, for Deborah Byrd, for Earth and Sky.
Author: Beverly Wachtel
Thanks to the following individual for aiding in the preparation of this script:
Dr. Hendrik Lenstra
Department of Mathematics
University of California
Berkeley, CA
Hwl@math.berkeley.edu
Dr. Lenstra adds: In the `Elements' of Euclid (3rd century B.C.), the first true textbook of mathematics, `1' is not called a number, whereas `2', `3', `4', ... ARE called numbers. This makes Euclid's exposition sometimes a bit cumbersome, since sometimes he needs to distinguish cases in the enunciations of his theorems and in his proofs according as some quantity (my term!) that he encounters equals 1 or equals a number. Had he viewed `1' as a number, then his entire exposition would have gotten much `slicker' and more `economical'?Most likely Euclid observed this himself too. For a modern mathematician, this state of affairs would have been sufficient reason to bluntly change the definition and make other mathematicians accept the change on the grounds of esthetics, economy, and common sense (cf. changing to the metric systems of weights etc!). However, it is my understanding that Euclid labored under restrictions that did not permit him to do this, namely the opinions of philosophers on the matter. The latter would declare, in an a priori manner, that `1' is `the measure of all things', or `the unit by means of which everything is measured', that therefore `cannot itself measure something', and therefore `1' should not be considered a number. It sounds maybe like a play of words (as much of philosophy)?
Now this is not to say that ordinary practical people and traders would not use `1' as a counting number in Euclid's time. Most likely they did. The influence of philosophers did not extend beyond academic circles. In fact, philosophers also had something against fractions in those days (something along the lines of `the unit being indivisible', since if you break up your measuring unit into smaller ones you are effectively replacing it by a smaller measuring unit!); this was a prejudice that mathematicians got around (by looking at ratios instead of fractions), but practical people happily used fractions anyway.
If you enjoyed this program, you may be interested in the following websites:
History of Mathematics:
http://www-groups.dcs.st-and.ac.uk/~history/
Interesting Numbers:
http://www.abc.se/~m9847/constans.html
Mathematical Sciences Research Institute:
http://www.abc.se/~m9847/constans.html