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### History of Properties

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Date: 12/11/2001 at 01:32:20
From: Noah
Subject: Properties

I really wonder who invented the properties. For example the
distributive property and associative property.
```

```
Date: 12/11/2001 at 15:26:53
From: Doctor Peterson
Subject: Re: Properties

Hi, Noah.

These properties have always been there, and have probably been used
since ancient times; for example, any method of multiplying digit by
digit uses the distributive property. But it makes sense to ask who
first recognized these properties as something worth talking about.

One question to ask, then, is who came up with those names for the
properties. You can find such questions answered at Jeff Miller's
site, which is listed in our FAQ under Math History:

Earliest Uses of Some Words of Mathematics
http://jeff560.tripod.com/mathword.html

Looking up "distributive" and "associative," I find this:

ASSOCIATIVE "seems to be due to W. R. Hamilton" (Cajori 1919,
page 273). Hamilton used the term as follows:

However, in virtue of the same definitions, it will be found that
another important property of the old multiplication is preserved,
or extended to the new, namely, that which may be called the
associative character of the operation....

The citation above is from "On a New Species of Imaginary
Quantities Connected with the Theory of Quaternions," Royal Irish
Academy, Proceedings, Nov. 13, 1843, vol. 2, 424-434.

COMMUTATIVE and DISTRIBUTIVE were used (in French) by Francois
Joseph Servois (1768-1847) in a memoir published in Annales de
Gergonne (volume V, no. IV, October 1, 1814). He introduced the
terms as follows (pp. 98-99):

but about "functions" or "operators" in general]

So these terms arose in the early 1800's, when mathematicians were
starting to analyze more abstract kinds of objects than numbers (such
as quaternions and functions, in these examples), and needed to talk
about which of the properties of numbers also worked for these
objects. I'm sure that the developers of algebra in its early form
must have been aware of these properties to some extent, but they
didn't speak about them in the same terms.

Looking for more historical details, I went to

The MacTutor History of Mathematics archive
http://www-history.mcs.st-and.ac.uk/

which is also listed in our FAQ. Searching for the word
"distributive," I found this about George Boole:

In 1854 he published _An investigation into the Laws of Thought,
on Which are founded the Mathematical Theories of Logic and
Probabilities_. Boole approached logic in a new way reducing it
to a simple algebra, incorporating logic into mathematics. He
pointed out the analogy between algebraic symbols and those that
represent logical forms. It began the algebra of logic called
Boolean algebra which now finds application in computer
construction, switching circuits etc.

Boole also worked on differential equations, the influential
_Treatise on Differential Equations_ appeared in 1859, the
calculus of finite differences, _Treatise on the Calculus of
Finite Differences_ (1860), and general methods in probability.
He published around 50 papers and was one of the first to
investigate the basic properties of numbers, such as the
distributive property, that underlie the subject of algebra.

He, too, was studying new objects that follow rules similar to those
of numbers, particularly logic (treating "true" and "false" as
numbers).

A little earlier, there is George Peacock:

In 1830 he published _Treatise on Algebra_ which attempted to give
algebra a logical treatment comparable to Euclid's _Elements_. He
has two types of algebra, arithmetical algebra and symbolic
algebra. In the book he describes symbolic algebra as

the science which treats the combinations of arbitrary signs
and symbols by means defined through arbitrary laws.

He also said

We may assume any laws for the combination and incorporation
of such symbols, so long as our assumptions are independent,
and therefore not inconsistent with each other.

Peacock extended the rules of arithmetic using what he called the
principle of the permanence of equivalent forms to give his
symbolic algebra, so he was not as bold in practice as the
abstract ideas for symbolic algebra which he gives in theory. He
investigated the basic properties of numbers, such as the
distributive property, that underlie the subject of algebra.

So, again, these properties were not carefully investigated until
mathematics had reached this point in its development, where
clearly in order to make algebra rigorous and allow for study of
alternative kinds of "algebra."

I think this gives a pretty good answer to your question: the
properties we teach were first examined closely between 1810 and 1860
by a number of mathematicians who were investigating the logical basis
of algebra and working on ways to extend it. Before then, they were
used, but perhaps not discussed much.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 10/18/2002 at 18:09:19
From: Mollie King
Subject: Commutative Property

I need to know the inventors of the commutative property and
how they came to use it.

Thanks,
Mollie K.
```

```
Date: 10/18/2002 at 23:38:19
From: Doctor Peterson
Subject: Re: Commutative Property

Hi, Mollie.

As you'll see from the above, it's tricky to define what you mean by
"inventing" such a property. It might be compared to "inventing" the
law of gravity: it's always been there, and has even always been
used, yet it was not recognized as a rule worth thinking about (and
as something that moves not only rocks but the whole universe) until
the right person came along. The same is true here: people knew that
2+3 = 3+2 since ancient times, but eventually people realized that
this was a general property that could be ascribed to operations other
than addition and multiplication, and then it became something worth
naming. But it was not a single person who made this discovery.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum

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Associated Topics:
High School History/Biography
High School Number Theory

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