The Language of Mathematics

Philosophy is written in this grand book ― I mean the Universe ― which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.

"The Assayer", Galileo Galilei, 1623

Logarithm

During a regular group discussion on the structures and etymology revealed by English spelling, I was lured in to the origins of the word <logarithm>.

It sparked off various thoughts, which I've now kindled into a slightly more coherent if thinly sinuous flow of dim light: I checked out a few things after our session. My special interest was the presence of the base <log(e)> and why it was there. Perhaps over-ambitiously, I see in it a story of human wonder at the world and our need to find order within it.

John Napier

Logarithms were invented by the Scottish mathematician John Napier in the decades leading up to publication in 1614 in his mid sixties. They play a central role in modern maths and science, but Napier had a more limited, utilitarian aim: to simplify increasingly complex arithmetical calculations in science and finance.

He laboured for years producing the first of the kind of dense table of obscure-looking numbers that would leave most modern on-lookers cold and many first-time students perplexed. Ironically, it was the concept that endured: Napier's own tables were soon rendered redundant by a more usable variation of his approach. Built on Napier's insights, this variation endured as a calculation aid for three centuries, until just a few decades ago. In a more abstract setting, the logarithm is one of the key agents in the move towards modern mathematics.

Etymology

Logarithms are also used nowadays in some physical measures: eg. decibels for sound and pH for acidity. Rather more commonly, though less overtly, they're at the heart of musical scales: a musical interval (eg. octave, fifth, harmonic semi-tone) is deeply logarithmic in character.

Napier himself coined the name <logarithm>. One can see how the base <arithm> meaning "number" might echo his aim of simplifying calculations in <arithmetic> (the final <etic> here looks intriguing). The initial base <log(e)> is less obvious, especially given M's circumspection over the otherwise plausible rendering "to reckon" as one denotation of its Greek root <logos>

My Oxford Etymology lists <logos> as "account, ratio, reason, argument, discourse, saying, (rarely) word". Etymonline's laconic and apposite rendition of <logarithm> is "ratio-number". The link between ratios and reason seemed to be worth exploring.

Rational Numbers

The linguistic ray I was following led me to think about what mathematicians call rational and irrational numbers. For a long time, this seemed an odd choice of terminology to me, especially the notion that a number could be irrational. Why might some numbers seem more or less reasonable than others? The idea, I think, has roots in Ancient Greek maths, as expounded by Euclid, so I was jarred by M's observation that <rational> has a Latin root, even though if pressed I might have ventured Latin nominative <ratio> and genitive <rationis>.

Euclid studied in Athens and taught in Alexandria. He was contemporary with Ptolemy I, at around 300 BC. He wrote in Greek, most famously (and fittingly for our Band) authoring his Elements, which is an astounding mathematical text of lasting elegance, subtlety and rigour. The Elements mostly deal with geometry, but he also writes about ratios, from two quite different perspectives.

Euclid embraced a method of proof in his work that still holds sway today and is a tribute to Greek intellect of the time. Ratios of whole numbers form a system that we would now equate with rational numbers, which are numbers which are expressible as whole-number fractions. For example, a ratio 1:2 equates to the number 1/2 and 103:4 to 103/4 (equivalently written 25.75 in modern decimal notation). They lead to relatively simple proofs for Euclid.

Pythagoras (of triangle fame and a couple of centuries before Euclid) found special significance in the ratios between two whole numbers. For example, he's credited with realising that whole number ratios were intimately linked with musical intervals.

Pythagoreans were both mystical and mathematical in their view of the world, seeing it as ordered by number, namely whole numbers and the ratios between them. According to some accounts, when one of his successors discovered that the relative lengths of the diagonal of a square compared to the length of one of its edges couldn't be expressed as a whole number ratio, it caused such a crisis that the discoverer was killed to hush it up. They had seen reason in number only to discover unreason: they considered such relationships in geometrical measure to be irrational.

This was, I think, the origin of our modern recognition of rational and irrational numbers. Irrational numbers still cause us problems: they're still harder to reason about and perform calculations on. (For example, while a computer can (in principle) manipulate rationals exactly, there's no reliable method to accurately represent irrational values.)

Not only was this inescapable irrationality an affront to the Pythagorean view of the ordered universe, they posed a problem for Euclid and his fellows in developing rigorous proofs involving magnitudes in geometry such as the diagonal of a square or the area of a circle. It's hard for me to put across the amazement I feel when I read my translation of Euclid, that he and his contemporaries first of all recognised this need and then found a way to address it. In listing the properties of ratios, he typically presents both the simple proof for whole number ratios and its sophisticated counterpart for the more subtle irrational case.

I'd always assumed that, being a mathematical concept expounded by Euclid, the modern word <ratio> had its ultimate roots in ancient Greek, but our discussion shook this long-held presumption. My modern translation tells me that Euclid's term for ratio was λογος or <logos>, which by his time he applied to pairs of any magnitude. Elsewhere, I read that the ancient Greeks dubbed irrational numbers ἄλογος or <alogos>.

Ratios ― an Example

And this brings us back to Napier's incorporation of a base meaning "ratio" in his coinage of the word <logarithm>. In the spirit of Napier, we can write down a sequence of numbers, starting at 1 and with each next number in constant ratio with the number before it. To achieve, this we can pick a fractional value such as 1.1 and proceed from 1 by repeated multiplication. The first term will be 1; the second 1 x 1.1 = 1.1; and the third 1.1 x 1.1 = 1.21; etc.

Hence (rounding to 2 decimal places):

    1 -> 1.1 -> 1.21 -> 1.33 -> 1.46 -> 1.61 -> 1.77 -> ..etc..

Each term here is 1.1 times its predecessor (or in the ratio 11 : 10).

The arrows help us count the steps from 1 to a given number. Each step corresponds to a single multiplication of 1.1 and this is true no matter which number you start from. And equal steps will therefore correspond to equal multiplications (and equal intervals to equal ratios between start and finish).

It takes 2 steps to get from 1 to 1.21, so to multiply any number by 1.21 we can just move forward 2 steps from it to get the answer. For example, 1.46 is 4 steps (4 arrows) away from 1, so 1.21 x 1.46 must be 2 + 4 = 6 steps from the number 1, ie. equal to 1.77 in our sequence: Napier's method has reduced the multiplication 1.21 x 1.46 to the simple act of adding 2 + 4. The arrow count is the logarithm (with base 1.1) of any number we see in our sequence.

Of course, this particular sequence based on 1.1 is too thinly spread to be of practical use. Instead, a base much closer to 1, such as 1.001 or 1.0001 or even 1.0000001, leads to smaller steps and greater numeric coverage, albeit at the cost of much more preparatory calculation. In fact, Napier chose his base to be smaller than 1, rather than larger, but the principle still applied that his logarithmic method reduced multiplication to an equivalent addition sum. His (and later) coverage was complete enough to work with logarithms of any number of interest with reasonable accuracy.

This is a bit of a winding path, but one I wanted to have a go at sharing.

Posted by Neil