Subject: Re: [HM] Mathematics and Time
From: David Wilkins (dwilkins@maths.tcd.ie)
Date: Thu Mar 09 2000 - 17:28:13 EST
Al Barron wrote:
>
> Hamilton may have been influenced by the writings of Kant directly
> or through their cultural influence, but it's problematic that a
> criteria for either noncommutativity or commutativity follow from
> an existential predisposition toward the nature of algebra. What
> would have been his argument ?
>
> If he cited algebra as the science of pure time, did he have insight
> into its relation to space as well ? This seems most likely.
>
The following is an off the cuff response. I could respond
at greater length, or with more precision, but it is getting
quite late at night.
First point: Hamilton was keenly interested in the writings of
Kant, mastering German to the extent that he could read Kant in
the original. The `Life' by R. P. Graves includes a letter from
Hamilton to one of his correspondents (either Lord Adare, or
Aubrey de Vere, I forget which) giving his overview of Kant's
ideas. But, if I remember correctly, he claimed that he had
already developed his ideas concerning the connection between
algebra and time before he started reading Kant.
Second point: Hamilton's ideas on the nature of algebra and its
relation to time and space evolved over several decades.
Third point: Hamilton explained that, to him, algebra was the
science of `pure time', a mental science, divorced from `outward
chronology' and from physics. He also described it as the
science of `order in progression', and was particularly
associated with the notion of continuity. I suggest that if one
were to replace `pure time' by `the real line' or `the
continuum', this would be pretty much in the spirit of what
Hamilton was about.
Fourth point: in a letter (to De Morgan, if I remember
correctly), Hamilton observed that if he thought of a line
segment then he thought of that segment as being drawn from
beginning to end over time.
Fifth point: Hamilton argued strongly against the idea that the
`variables' in algebra were devoid of reference, and that algebra
was a purely syntactical exercise defined in terms of certain
`laws' such as commutativity. He referred to this as a
`philological' approach, viewing algebra purely as a language.
Instead he emphasized models: real numbers as ratios of time steps,
complex numbers as ordered pairs of real numbers, quaternions as,
well, ordered quaternions. This meant that he could DEFINE in
as `natural' fashion as possible, appropriate binary operations
of addition, multiplication etc. on these objects, and THEN
INVESTIGATE, the properties such as commutativity, associativity,
distibutivity that these operations have, rather than having these
laws built into his system right from the beginning.
Sixth point: since he had a concrete models in mind, which he
investigated with great thoroughness, he could be assured of the
consistency of the algebraic systems he introduced (to the
standards of the time, not those of Hilbert and Goedel, of
course).
Seventh point: I mentioned above that Hamilton's ideas on the
nature of algebra evolved over several decades. He investigated
the foundation of his theory of quaternions from both algebraic
and geometric perspectives. In the geometric approach,
quaternions were viewed as quotients of directed line segments in
space. One could specify certain natural properties that such
quotients should have: for example, they should remain unchanged
under translation of the line segments, and under rotation of
both line segments about an axis perpendicular to both. One
can then seek the most natural definitions of addition,
multiplication etc. of these quotients. One then discovers that
the resulting algebra must then be isomorphic to the algebra of
quaternions. Hamilton first summarized this approach in a
communication
`Illustrations from Geometry of the Theory of Algebraic
Quaternions'
(Proceedings of the Royal Irish Academy, vol. 3, Appendix, pp.
xxxi-xxxvi.) [This paper is not yet available here online; it
can be found in volume III of the RIA edition of the
`mathematical papers'.] A thorough development of this approach
is to be found in the paper
`On symbolical geometry'
published installments in the `Cambridge and Dublin Mathematical
Journal' between 1846 and 1849. This paper is available here
online, at
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/SymGeom/
Eighth point: as a result, there is an interesting parallel
between real numbers, or `scalars' (Hamilton's word), and
quaternions: scalars are viewed as quotients of steps in time,
whereas quaternions are viewed analogously as quotients of steps
in space.
Ninth point: In the introduction to `On symbolical geometry',
Hamilton himself acknowledged that his views as to the nature
of algebra were converging to some extent on those of Peacock,
and the title of his paper was chosen to reflect this.
One could say a lot more (e.g., about commutativity), but I hope
that the above is enough for now.
David Wilkins,
Trinity College, Dublin, Ireland.
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