Re: [tied] Re: About methodology

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Sent: Sunday, August 27, 2000 3:38 PM

Subject: [tied] Re: About methodology

Nemo wrote:

To begin with, everything in this world is imperfect, including mathematics. Even the concept of proof in mathematics isn't quite certain. What is a valid proof for a formalist, may be rejected by an intuitionist or constructivist and so on. (The very existence of those –ists is a little suspicious, isn't it?).

 

100% certainty is possible perhaps in logic, but is logic a science? Science is, after all, the quest for new information; purely logical reasoning is just a chain of tautologies. Input = output, nothing new. The price for absolute certainty is emptiness.

Notwithstanding all the "isms", mathematicians normally agree whether something has been proved or not. Of course mathematics is not about tangible and visible things but about abstract objects, rules and relations. Some "ists" don't accept certain rules or axioms that others consider self-evident, but these differences are philosophical rather than practical. If you lay out your axiomatic system in detail, any proof can be evaluated with regard to it with 100% certainty. Of course, as demonstarted by Gödel, mathematics is incomplete -- it can be formally proved that there are theorems that can't be formally proved; but the fact that mathematics can analyse the limitations of its own formalism is impressive rather than embarrassing.

 

Tautologies are "empty" in the sense that they state the identity of immaterial abstractions, but it's not quite true that they carry no new information. Any theorem is derived via a chain of tautological statements, but the internal structure of the chain may be very complex and encode a lot of meaningful information. Namely, it contains the explanation WHY the theorem holds. A theorem can't prove itself, even if mathematicians say that it "follows automatically". It takes a human mind to crack any non-trivial mathematical problem, so a proof is also a formalised summary of the mathematician's mental work. They say that Gauss was never content with just one proof of a difficult theorem and often published a number of elegant alternatives. An innocent-looking equation like Euler's formula

e^(Pi*i) +1 = O

may encapsulate numerous important generalisations and symbolically connect several departments of mathematics, because it took a lot of intellectual effort to derive it.

Nemo: The perversity of science is that you may be always wrong. Your pet theories may be more or less correct, partly correct or whatever; but it is also possible that you are totally and massively wrong. This is perhaps the only absolute certainty in science.

Very true. The only comfort is that it happens to the best of scientists.

Glen: Comparative linguistics defies yet to be properly measured.

Nemo: This is what concerns me. Why so little effort is made to use new tools in comparative linguistics in order to get more objective and verifiable results? Which tools? I don't know. Perhaps more statistics, probability, deterministic chaos, fuzzy sets...

It is being done, though like much interdisciplinary research it's gaining momentum rather slowly, partly because universities are so stupidly bureaucratised and compartmentalised. Computational techniques developed by geneticists and evolutionary biologists are increasingly used in lexicostatic and cladistic analyses of language groups -- with interesting results, though it's all little better than preliminary at present. As regards quantitative work, we're still far behind genetics or archaeology, but ask again in a few years' time.

 

Donald A. Ringe analyses the probabilistic limitations of the comparative method in the following oft-cited articles:

1992, "On calculating the factor of chance in language comparison", Transactions of the American Philosophical Society 82, 1-110.

 

1995, "'Nostratic' and the factor of chance", Diachronica 12(1), 55-74.

I also know of a series of interesting but partly unpublished papers on modelling language evolution written jointly by April McMahon, Marisa Lohr and Robert McMahon. I've only read one of them, "Family trees and favourite daughters", published in the proceedings of the 1998 Cambridge conference on Nostratic (Nostratic: Examining a Linguistic Macrofamily, edited by Colin Renfrew and Daniel Nettle), pp. 269-285.

 

Piotr