FOM: distributive law

Thomas Forster T.Forster at
Wed May 24 15:26:57 EDT 2000

My tame history of maths person here says:

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	To: Thomas Forster <T.Forster at>
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	>         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 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

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