FOM: distributive law

[Michael Detlefsen]

John Baldwin asks:

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.

I don't find the question entirely clear. It could mean 'Before the late
19th century, did people write algebraic derivations in line-by-line
fashion explicitly state the justification (including, explicitly a law of
distribution) for each of the lines?'. Or it could mean 'Before the late
19th century, did people explicitly identify a law equivalent to our law of
distribution and use it as a rule of algebraic manipulation?'. Or it could
mean 'Before the late 19th century, were mathematicians explicitly aware of
distributivity as a law governing a variety of different mathematical
systems?' The answer to both the first two is 'yes', though there is
obvious room for disagreement as to what should count as line-by-line
procedure. Various algebra books in the 17th century explicitly stated a
rule or law of distribution (though it was not called that). Thus, for
example, I have the algebra text of John Ward before me now (A COMPENDIUM
OF ALGEBRA, 1695). Ch. III, paragraph 3 states a general rule of
multiplication of compound quantities. It states: 'Multiplication of
Compound Quantities is the same with that of Single Quantities; for the
Product of each Member of the Multiplier, into all Members of the
Multiplicand, (respect had to signs) is the Product'. He then gives, as his
first example, the multiplication of a compound quantity 3a+5d by a simple
quantity b. The product is 3ab+5db. To get the entire rule, one obviously
needs to add the rule for multiplying simple quantities (i.e. single termed
expressions rather than polynomials). Ward states this in three explicit
rules in ch. II, paragraph 3. The three rules are these: a rule dealing
with simple terms having no coefficients and like signs (join the
quantities together and prefix a plus sign); a rule dealing with terms
having coefficients and like signs (multiply the coefficients, concatenate
(Ward's term 'joyn') the signs for the quantities); and a rule dealing with
terms having unlike signs (handle coefficients in the manner of rules 1 and
2 and prefix a minus sign). There are similar rules stated in other algebra
texts from the 17th century.

In addition, the answer to the third reading of the question may also be
'yes'. Various 17th century algebraists had an understanding of algebra as
a generalized arithmetic ... a system of rules that could be applied to
quantities generally, and not just the recognized numbers of the day. In
particular, they proposed that the rules be seen as applying to quantities
other than the recognized numbers of the day--most specifically, to the
'imaginary' quantities mentioned by Cardano and Descartes. To give even a
creditable sketch of that, however, is much too complex a thing to attempt
here. (The interested reader can consult Pycior's 1997 study on the
subject. It is entitled SYMBOLS, IMPOSSIBLE NUMBERS, AND GEOMETRIC
ENTANGLEMENTS.)

>From a warm and fragrant South Bend,
Mic Detlefsen

Hope this helps