Subject: Re: [HM] Mathematics and Time
From: David Wilkins (dwilkins@maths.tcd.ie)
Date: Wed Mar 08 2000 - 17:12:33 EST
Gordon Fisher wrote:
> Also William Hamilton, he of the quaternions, etc., wrote rather
>copiously on time.
This is as good an opening as I could desire to advertise one of the
latest additions to my collection of Hamilton papers that I have
recently made available online:
Theory of Conjugate Functions, or Algebraic Couples; with a
Preliminary and Elementary Essay on Algebra as the Science of
Pure Time
The URL is
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/PureTime/
The paper itself is available as Plain TeX source, and as DVI,
PostScript, and PDF. (There is no HTML version.)
It is also long (111 pages), and falls into three parts. The
first of these is `General Introductory Remarks', which is worth
reading, for those who do not want to work through the entire
paper.
The `Essay on Algebra as the Science of Pure Time' is a fairly
thorough investigation of the foundations of algebra (and
analysis), with particular emphasis on ontological questions.
(There is an interesting gap in the proof on the existence of
square roots, and more general nth roots, of positive quantities,
located in the vicinity of Lemma II and its corollary, but other
proofs are quite impressive for their rigour, such as the very
careful proof that the exponential of a sum of two complex
numbers is the product of their exponentials. I assume that
Hamilton had been reading Cauchy, and wonder to what extent he is
following Cauchy; he does not discuss his indebtedness here.)
In my opinion, this paper does not deserve the scorn directed at
it by the likes of E. T. Bell. I wonder if E. T. Bell had
actually read it?
It is interesting to note that associativity is discussed for
addition in article 11 (though not named) and there is a back
reference to this article when he comes to discuss properties of
multiplication. The essay on `Algebra as the Science of Pure
Time' dates from 1835: Hamilton was only to name and emphasize
the property of associativity round about 1844-6, after the
discovery of quaternions. According to a note in Hankins's
biography of Hamilton, Helen Pycior has suggested that it was the
discovery of the non-associativity of the octaves (Cayley
numbers) by John T. Graves and Arthur Cayley that led to the
recognition of the role of associativity.
In connection with associativity, it is worth noting that, in
Hamilton's theory of `sets' (i.e., ordered n-tuples of moments
of time, time steps, or numbers) which he developed in his
`Researches respecting Quaternions: First Series' (also recently
made available online), real numbers, couples, quaternions, and
any n-dimensional generalizations of these within the context
of Hamilton's setup, are defined by their action by left
multiplication on n-tuples of time steps; and, in consequence,
any algebra produced in this way is automatically associative.
This is probably why Hamilton did not regard the Cayley numbers
as being the generalization of his quaternions that he was to
seek in subsequent research.
I would also suggest that, in order to make sense of what
Hamilton is doing in his `Researches respecting Quaternions:
First Series', it is probably necessary to have first read the
essay on `Algebra as the Science of Pure Time'.
--------
Finally, I would like to conclude, if I may, with a progress
report on my ongoing project to transcribe, edit, and make
available online, Hamilton's mathematical papers (excluding at
this stage the two books, and material from his Nachlass).
This project is almost complete: I estimate that I have made
available around 90% of this material on the Web at
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Papers.html
(I estimate that the amount of material currently available
online represents over 1,200 A4 pages of text.) The remainder of
the papers are in preparation, and I am hoping that I might
complete the task in around a fortnight.
In particular I have made available online all mathematical
papers listed in the bibliography included in volume 3 of Robert
Perceval Graves's `Life of Sir William Rowan Hamilton' prior to
the discovery of quaternions in 1843, and also many papers
published subsequent to that date.
This year I have also thoroughly re-checked the major papers on
dynamics and on the insolvability of quintic polynomials by
radicals. I discovered a number of typographical errors in the
dynamics papers when I was working through them over the
Christmas/New Millenium period, and therefore felt it necessary
to revisit the older material which I put on the Web in 1998:
each of those papers has been re-proof-read at least five times
this year, so I hope that the texts are now sufficiently
accurate. Apologies for not having corrected the typographical
errors in these papers earlier.
The URL for Hamilton material is
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/
Greetings from overcast Dublin,
David Wilkins
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