The n-Category Café

Ancient Greek mathematicians, so I’ve heard, understood multiplication geometrically. To reason about $A \times B$ they would draw line segments of length $A$ and length $B$ and form a rectangle with those as sides. This has area $A \times B$.

Fine and dandy. But then what does an equation like $2x^2 = x + 3$ mean? $2x^2$ has units of area, $x$ has units of length and 3 is dimensionless. That doesn’t make sense! Your science teachers probably told you this. When you add or subtract quantities, they have to have the same units.

People thought about this for centuries. Eventually Newton sorted it out. Now mathematicians don’t worry about units, usually, leaving the other scientists to deal with them.

I believe, but haven’t checked, that a bunch of ancient Greeks dealt with the problem this way:

When we write $2x^2 = x + 3$, $x$ is short for the area of a rectangle whose sides are $x$ and 1. 3 is short for the area of a rectangle whose sides are 3 and 1. So now everything is an area.

This trick may seem like a pain in the butt, but it’s actually important in algebraic geometry! It’s called ‘homogenization’.

I’ll say more about this, but first let’s see how an 11th-century Arabic algebra textbook dealt with this issue. For this I’ll quote Keith Devlin’s column Algebra: it’s powerful, but it’s not what it was:

For most of history, what we today think of as a number (i.e., an object, or noun) was in fact the adjective in a multitude-species pair. People conceived numbers as real-world objects viewed from the perspective of quantity; so numbers came attached to objects (though for mathematicians they could be abstract objects). We are familiar with this today with currency and geometric angles. We have monetary amounts such as “5 dollars and 20 cents” or angles such as “15 degrees and 30 seconds”. The numerals in those expressions are adjectives that modify the nouns (those nouns being dollars, cents, degrees, and seconds, respectively).

Those multitude-species pairs were the entities that equations were made up of; they were their “numbers”. For example, the following pre-modern quadratic equation comes from an eleventh-century Arabic algebra textbook

“eighty-one shays equal eight māls and one hundred ninety-six dirhams”.

Here the English version consists of translations of the Arabic words from that textbook, apart from the terms “māls”, “dirhams”, and “shays”. The word shay referred to the unknown in the equation, māl is the term they used to denote the square of the unknown, and dirham denoted the unit term. The plural “s”s in our presentation of the equation are English additions (hence not italicized); Arabic does not designate plurals that way.

Contorted into modern notation, that pre-modern equation is this one:

$$81x = 8x^2 + 196$$

He goes on to explain what these māls, dirhams and shays are — read his column! But very very roughly speaking, their quadratic equation involves units.

Next let’s see what Descartes had to say about the question of units back in 1637, in his book La Géométrie. It may be confusing, but at least you’ll see he took the problem seriously. On page 5 he writes:

Here it must be observed that by $a^2, b^2$ and similar expressions, I ordinarily mean only simple lines, which, however, I name squares, cubes, etc., so that I may make use of the terms employed in algebra.

It should also be noted that all parts of a single line should always be expressed by the same number of dimensions, provided unity is not determined by the conditions of the problem. Thus, $a^3$ contains as many dimensions as $a b^2$ or $b^3$ […]

It is not, however, the same thing when unity is determined, because unity can always be understood, even where there are too many or too few dimensions; thus, if it be required to extract the cube root of $a^2 b^2 - b$ we must consider the quantity $a^2 b^2$ divided once by unity, and the quantity $b$ multiplied twice by unity.

Huh?

I think he’s basically saying that $a^2 b^2$ has dimensions of length$^2$ and $b$ has dimensions of length. To be allowed to subtract them, we’d better make them have the same dimensions. So he divides $a^2 b^2$ by a quantity with units of length, which he calls ‘unity’. He also multiplies $b$ by unity${}^2$, which has units of length${}^2$. Now they both have dimensions of length${}^3$, so you can subtract them.

Even better, you can now take the cube root of the result, and it will have dimensions of length!

So the trick is to make up a thing called ‘unity’ that looks a lot like 1, but has units of length, like 1 meter.

I thank Ulrik Buchholtz for pointing out this passage by Descartes!

Now let’s hear how Newton dealt with this same issue 33 years after Descartes. Again this is from Keith Devlin:

The key to today’s algebra was the creation of an abstract number system, specified by axioms, a process that was not completed until the early twentieth century. A valuable first step towards that solution was made by several seventeenth century mathematicians who defined numbers-as-objects from the multitude-species pairs that had served for so long.

Newton, for example, wrote down the proposal below in one of his Notebooks (1670). The species he was working with for calculus were line segments, which he called Quanta. To develop calculus, he looked at line segments from the perspective of their length; those were his multitude-species “numbers”, which he wanted to break free of. He wrote:

“Number is the mode or affection of one quantum compared to another which is considered as One, whereby its Ratio or Proportion to that One is expressed. [Thus b/a is the number expressing the ratio of b to a.]”

The brackets are his. He wrote “b/a” in vertical-stack form, not inline as here. His key claim was: “b/a is the number.” He had defined a ratio (hence a dimensionless object) to be his number concept.

Defining numbers to be ratios of Quanta was equivalent to adopting a unit Quantum. He formulated other variants in other notebooks.

In short, Newton said numbers are ratios of lengths, so they are dimensionless. The lengths, he called Quanta. So Newton’s classical mechanics relies on his calculus, but his calculus relies on Quanta.

So classical mechanics is really Quantum mechanics.

Finally, what do algebraic geometers think about this today? They will take a polynomial equation like

$$y = 3x^2 + 4 x$$

and use Descartes’ trick to make the dimensions match up. Suppose $x$ and $y$ have units of length. Then they let $z$ stand for 1 meter, and rewrite the equation as

$$y z = 3x^2 + 4x z$$

Notice: if we set $z = 1$ we’re back to our original equation! But now every term has the same dimensions, so it’s legit: your physics teacher wouldn’t complain about it.

This trick is called ‘homogenization’.

To be honest, algebraic geometers don’t really say they are doing this trick to get the dimensions to work right. They usually give another reason: while the original equation defines a curve in the ordinary plane, the new equation defines a curve in something called the ‘projective plane’. The projective plane has more points than the ordinary plane, and it works better in a lot of ways.

But my friend James Dolan thought about this a bunch, and decided you could understand a lot of what algebraic geometers are doing, like their apparent obsession with ‘line bundles’, by thinking about dimensional analysis. Here’s something we wrote about that:

• Doctrines of algebraic geometry.

It quickly leads into some interesting category theory. Here’s a taste:

An object $x$ in a symmetric monoidal category $K$ is called a line object if it has an inverse object with respect to tensor product, and if the canonical “switching” morphism $x \otimes x \to x \otimes x$ is the identity morphism. A section of a line object is a morphism from the unit object. We will be mainly but not exclusively interested in the case where $K$ is enriched over the category of vector spaces.

Example. In a symmetric monoidal category of vector bundles, the line objects are the line bundles.

Another name for the study of categories of line objects is “dimensional analysis”. In dimensional analysis, a physical theory is described by specifying an abelian group of “dimensions” (these are the line objects) together with a commutative algebra of “quantities” (these are the sections of the line objects) which is graded by the dimension group. We’ll call a physical theory described in this way a dimensional algebra, but the fundamental fact about a dimensional algebra is that it’s equivalent to a dimensional category, which is a symmetric monoidal category where all objects are line objects.

So, there really is some point to worrying about dimensional analysis, even in so-called pure mathematics… and it’s an old issue that concerned Descartes, Newton, and probably many others!

Further reading

When I posted about this on Mastodon, I got some suggestions for further reading. Martin Escardo said:

There is a long discussion about the subject of this thread in Bourbaki’s Elements of Mathematics, I think in the historical section of one of the two volumes on Topology, probably the second one, but I can’t check now. It took a long time until mathematicians started to consider dimensionless numbers without the fears that e.g. Descartes had!

Over on the Category Theory Community Server, JR recommended this paper:

• Mark A. Atherton, Ronald A. Bates, and Henry P. Wynn, Dimensional analysis using toric ideals: primitive invariants.

Kevin Carlson wrote:

This also reminds me of the discussion in Mayberry’s The Foundations of Mathematics in the Theory of Sets of the Greek attitude toward number, which he argues is almost indistinguishable from our concept of a finite set. (A number of sheep, for instance.)