FOM: distributive law
Thomas Forster
T.Forster at dpmms.cam.ac.uk
Wed May 24 15:26:57 EDT 2000
My tame history of maths person here says:
From p.bursill-hall at dpmms.cam.ac.uk Wed May 24 12:35:19 2000
Return-path: <p.bursill-hall at dpmms.cam.ac.uk>
Envelope-to: T.Forster at dpmms.cam.ac.uk
Delivery-date: Wed, 24 May 2000 12:35:19 +0100
Received: from [131.111.24.17] (helo=piers.dpmms.cam.ac.uk)
by emu.dpmms.cam.ac.uk with esmtp (Exim 3.01 #5)
id 12uZRW-0001x2-00
for T.Forster at dpmms.cam.ac.uk; Wed, 24 May 2000 12:35:18 +0100
Message-Id: <4.3.1.2.20000524121631.01fe0bf0 at dpmms.cam.ac.uk>
X-Sender: piers at dpmms.cam.ac.uk
X-Mailer: QUALCOMM Windows Eudora Version 4.3.1
Date: Wed, 24 May 2000 12:35:47 +0100
To: Thomas Forster <T.Forster at dpmms.cam.ac.uk>
From: Piers Bursill-Hall <P.Bursill-Hall at dpmms.cam.ac.uk>
Subject: Re: query
In-Reply-To: <E12uLY3-0003aG-00 at jay.dpmms.cam.ac.uk>
Mime-Version: 1.0
Content-Type: text/plain; charset="us-ascii"; format=flowed
Status: RO
> How were algebraic manipulations justified (say by 16th century
> algebraists) before the formalization of `Tarski's high school
> identities'
> in the late 19th century by Boole, Peano etc. ?
>
> Specifically, I am wondering if the distributive law was explicitly
> cited.
Well, the distributive law (in geometric form, of course) is in Euclid's
Elements book II.
Algebra, in a reasonably modern sense of the term (say, something like the
abstract science of equations) emerges in the late 16th century; there is
no clear moment when one can say "this person is clearly doing algebra" ..
but I think an historically sensitive argument can be made for the latter
16th century. Bombelli and Buonasoni are the sorts of persons in whom we
see this, although one could go back so far as Cardano or so late as Vieta
(even Descartes, if you want). Before that we have things that are easily
translated into algebra, but are just generalised numerical equation
manipulation, or the rules for generalised numerical equation manipulation
(Al-Khowarizmi, in the 9th century, or Chuquet and Pacioli at the end of
the 15th century are good examples of this).
Algebraic manipulations were not, as such, justified. There was no
mathematically reasonable "foundation" (in a modern sense) to algebra until
the 19th century. Rather, what we see as algebra (abstract or generalised
study of equations) was just the extended and natural generalisation of
arithmetic manipulations; so the 'justification' of algebra was that it was
doing to things - letters - what one can do to numbers.
Rules of algebra, like algorithms, were sometimes 'proved' by demonstrating
their geometric analogues. The most pretty example of this is
Al_Khowarizmi's justification of the quadratic formula, which is an obtuse
(for the 9th century) piece of geometric reasoning .. that does show a case
of the general quadratic formula in a geometrical language.
Late in the 16th century there was a growing sense of a 'mutual assistance
society' between algebra and geometry, and the likes of Bombelli and others
sometimes used geometric arguments to justify algebraic results. Most
significant, because their texts of Euclid were corrupt, they read the
definitions of Book V (esp. 3,4,5) in a completely un-Greek way, and this
allowed them to equate (in effect) geometric proportions and ratios with
what we call fractions and equations; thus a:b::c:d ["a and b enjoy a
relationship in size that is of the same kind and amount as that enjoyed
between c and d" ] came to be conceived of as the algebraic statement a/b =
c/d, which is NOT what the greeks meant!
The lack of any more solid foundation to algebra beyond arithmetic and
occasional geometric analogies began to bother a few mathematicians in the
latter 18th century, but despite a certain amount of chatter about the
problem, it was not clear just how one might 'justify' algebra beyond
arithmetic. It wasn't a crucial problem until the constraints of
arithmetic were broken in the mid 19th century, of course.
hope this helps
piers
Piers Bursill-Hall
Department of Pure Mathematics
University of Cambridge
16 Mill Lane
Cambridge CB2 1SB
phone: +44.1223.337923
messages +44.1223.337999
fax: +44.12223.337920
More information about the FOM
mailing list