Ramble, Part 7: Uncertainties

[stoutfellow]

Some years ago, I read a book entitled Ancient History: Evidence and Models, by M. I. Finley. The main lesson I took away from that reading was just how little we know, with any certainty, about the classical world. So much material was simply lost, and so much of what survived is unreliable. (The case of Thucydides, who freely confesses to making up many of the speeches he presents, is notorious, but he was far from alone.) In reference to classical mathematics, one might point to the fact that Archimedes' letter to Eratosthenes, mentioned in the last Ramble, was not only lost but forgotten (if it was ever widely known) for more than two thousand years, and there are many other important works that we know to be lost.

The next mathematician I want to take up is Diophantus of Alexandria, and the problem I just mentioned is rather severe here. He stands outside the Euclidean tradition, interested neither in geometry nor, even, in the definition-theorem-proof mode of presentation. He seems like an outlier, almost, but he can't have been. Some of his work definitely seems to be derived from the Pythagorean tradition, though without the "all is number" triumphalism of that school. More strikingly, there are similarities of style and technique between Diophantus and the mathematicians of ancient Mesopotamia. It seems clear that there was at least one flourishing mathematical tradition, outside what we see as the main line, that stretched across millennia from Babylon to Alexandria - but we know next to nothing about the means of transmission.

Thus, I want to raise a cautionary flag. When I speak of mathematics in the classical era, for the most part I mean as it had impact on latter-day mathematics - not, primarily, what it was like in itself. There were mathematical schools other than the one that runs through Euclid, but the Euclidean tradition is the primary one which survived and nourished post-classical mathematics.

Not everything is simple, though. In the early late seventeenth century, Diophantus' Arithmetica was translated from the Arabic (in which it had survived) into Latin from Greek, and a copy found its way into the hands of a fellow by the name of Pierre de Fermat...

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Comments

[pompe.livejournal.com]

I realize this is a difficult question, but how much of what we consider math do you think is a legacy of classic culture - that is, do you have any speculations of how math could have looked?

(As an example, in cartography a lot of concepts are directly classical but not necessarily given.)

[stoutfellow.livejournal.com]

Mm, yes, that's a hard one. Part of the problem is that the mathematics of Egypt, of Mesopotamia, of Greece, and of the Hindu and Muslim civilizations interpenetrate so heavily that they can't be considered independent - and modern Western math is heir to all of them.

The only lead I can see rests with Chinese mathematics, up through the early Ming dynasty. (What follows is from memory; it's been a few years since I've looked at this stuff, and my resources for it are at the office.) The Chinese may have gotten the idea of positional notation from India, but their version of that is quite different. With that exception, Chinese math developed almost independently of the rest of Eurasia, and developed in very interesting ways. For one thing, the notion of "proof" took a different form; Chinese mathematical arguments were as likely to draw on religious or philosophical ideas as on what Westerners would consider acceptable reasoning. More importantly, Chinese mathematics was heavily algorithmic from the beginning, and their understanding of the construction of algorithms was quite sophisticated - they had something of a "meta-algorithmics", if you will. They used computational devices more extensively than the civilizations to their west did; in addition to the abacus, they had devices which allowed computations involving fractions and negative numbers, and also permitted work with variables.

Unfortunately, legal changes sometime in the Song dynasty hampered the further development of Chinese mathematics. Specifically, mathematics was dropped from the civil service examinations; as a result, math became primarily the preserve of independently wealthy dilettantes, and, the supply of those being limited, progress slowed considerably. (There were still a number of interesting developments, such as the discovery of the arithmetic triangle a century or more before Pascal came up with it.) With the Ming dynasty and the entry of Jesuit missionaries - many of whom were themselves at least amateur mathematicians - the indigenous tradition withered, and Chinese math was absorbed into the Eurasian consensus.

So, speculation: an uninterrupted Chinese-style mathematics could, perhaps, have developed into something like a hybrid of Western-style math and computer science (rather like what Wolfram has been touting lately, maybe). It's hard for me to be any more specific than that; I'm not sufficiently familiar with the details. That their path was a viable and productive one seems clear, and if you want a non-classical mathematics, that's probably the first place to look for ideas.

[pompe.livejournal.com]

Thanks. I have Georges Ifrah's history somewhere, but it is more about numbers than math.

[filkferengi]

You are *such* a tease! ;)

[stoutfellow.livejournal.com]

:smirk: