A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics by David Stipp
My rating: 5 of 5 stars

The God Equation

Turning the infinite into the finite (and back again) is no mean feat. It is what Christians claim God did with Jesus. It is also just one of the things that the equation produced by the Swiss mathematician, Leonhard Euler, in the late 18th century does in mathematics. Or to put it another way, Euler’s equation shows how things that we know are real are actually manifestations of something inconceivably alien, not simply infinite but also entirely other. I think it would only diminish the significance of Euler’s triumph to even state it in mathematical terms to non-mathematicians. So here’s a descriptive rather than analytical summary of Stipp’s highly accessible book.

The ancient Greeks learned how to deal with the idea of infinity by treating it as unreal and illusory, as something which didn’t really exist outside our heads. Since we couldn’t see it, touch it or feel it, infinity wasn’t, as today’s jargon has it, ‘a thing.’ Euler didn’t prove that infinity was a real thing (that was Georg Cantor a century later); but he did show that the infinite went into the construction of real things. Thus whatever the infinite was, it wasn’t only in our heads. Infinity may be irrational, that is beyond our thought of concrete objects, but it is still real as shown in Euler’s equation.

The fact of something real called infinity is relatively easy to deal with in the scheme of things, however. An infinite amount or number or degree of something is always in relation to something we already know about - people, distances, speeds, or indeed numbers - just many, many more or much, much larger; or alternatively much much smaller with many, many more fine divisions. The infinity we talk about, therefore, is merely a collection of these things that we are familiar with. Infinity is the sum, as it were, of these real things.

But things get considerably stranger when we leave the domain of the senses. Even quantum physics may seem to make relative sense. Euler’s equation introduces an entirely novel ‘substance’ into our thinking about what constitutes reality. This substance is something no one has ever seen. It is the dark matter and dark energy of mathematics. The substance is composed of what mathematicians call transcendental numbers.

Transcendental numbers, like π, can only be expressed as an unpredictably infinite set of decimals (technically speaking they never converge). It is their unpredictability that makes them so strange - in terms of how many there might be, where they might be found, and their full identities. Transcendentals are certainly numbers but they can’t be expressed as numbers other than as themselves - not as fractions, formulas, or combinations of other numbers.

In simple terms, transcendental numbers don’t follow any of the normal mathematical laws, even those of arithmetic. They stand in splendid isolation. We know they exist but we don’t know how many there are or what they’re made of. Yet Euler’s equation shows that they too are a constituent of reality. In fact they are so important that they are the rough equivalent of the Higgs boson in particle physics - in a sense promoting the existence of the numbers we measure for things we can feel and see in everyday life - from circles to the mysteries of alternating current.

Some object to this treatment of numbers as if they themselves are real things. They call this view ‘Platonist’ and criticise it as mystically religious. But even if one adopts the view that numbers are all in one’s head tout court, the implication is that mathematics is how our minds consistently (and effectively) work in dealing with the world.

Whether the world is ‘really’ mathematical or only appears that way because of how our minds work would therefore seem a meaningless distinction. All of science, indeed all of thought, is as much a discovery of the reality of ourselves as it is of the universe. And what we discover are patterns that have a remarkable aesthetic attraction. That is to say, we find beauty.

Finally, there are those most mysterious of all numbers, the so-called imaginary numbers. Everyone agrees that this is an inapt description but we’re stuck with the term. A better term than ‘imaginary’ would probably be ‘impossible.’ This is the number i, which is the square root of negative 1. On the face of it, such an entity indeed appears impossible. There is no number, neither a negative nor a positive, which can be the square root of a negative number since whatever number is used will always yield a positive value.

And yet not only does i exist in mathematics, it is of widespread relevance in mechanical, electrical, electronic, and computer engineering, to name just a few of its applications. Infinity may be irrational; transcendentals may be irrational and unseeable; but the imaginary number i is not just irrational, and unseeable, it is entirely incomprehensible. And yet it too is shown by Euler to be a fundamental constituent of the world.

Benjamin Peirce, The 19th century American mathematician (and father of the first great American philosopher, Charles Sanders Peirce) summed up the import of the Euler equation rather succinctly. “It is absolutely paradoxical,” Stipp quotes from Peirce’s published lectures, “We cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth.” I have never encountered a better or more inspiring claim to divine revelation.

Postscript: for a further literary interpretation of the Euler formula, see: https://www.goodreads.com/review/show...

Further postscript: https://www.wired.co.uk/article/googl...

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