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A universe without mathematics is beyond the scope of our imagination

Mathematics is the language of the universe.
Mathematics is the language of the universe. (Image credit: Shutterstock)

This article was originally published at The Conversation. (opens in new tab) The publication contributed the article to Space.com's Expert Voices: Op-Ed & Insights.

Peter Watson (opens in new tab), Emeritus professor, Physics, Carleton University

Almost 400 years ago, in The Assayer, Galileo wrote: "Philosophy is written in this grand book, the universe … [But the book] is written in the language of mathematics (opens in new tab)." He was much more than an astronomer, and this can almost be thought of as the first writing on the scientific method.

We do not know who first started applying mathematics to scientific study, but it is plausible that it was the Babylonians, who used it to discover the pattern underlying eclipses (opens in new tab), nearly 3,000 years ago. But it took 2,500 years and the invention of calculus and Newtonian physics (opens in new tab) to explain the patterns.

Since then, probably every single major scientific discovery has used mathematics in some form, simply because it is far more powerful than any other human language. It is not surprising that this has led many people to claim that mathematics is much more (opens in new tab): that the universe is created by a mathematician (opens in new tab).

So could we imagine a universe in which mathematics does not work?

The language of mathematics

The Sapir-Whorf hypothesis (opens in new tab) asserts that you cannot discuss a concept unless you have the language to describe it.

In any science, and physics in particular, we need to describe concepts that do not map well on to any human language. One can describe an electron, but the moment we start asking questions like "What color is it?" we start to realize the inadequacies of English.

The color of an object depends on the wavelengths of light reflected by it, so an electron has no color, or more accurately, all colors. The question itself is meaningless. But ask "How does an electron behave?" and the answer is, in principle, simple. In 1928, Paul A.M. Dirac wrote down an equation (opens in new tab) that describes the behavior of an electron almost perfectly under all circumstances. This does not mean it is simple when we look at the details.

For example, an electron behaves as a tiny magnet. The magnitude can be calculated, but the calculation is horrendously complicated (opens in new tab). Explaining an aurora, for example, requires us to understand orbital mechanics, magnetic fields and atomic physics, but at heart, these are just more mathematics.

But it is when we think of the individual that we realize that a human commitment to logical, mathematical thinking goes much deeper. The decision to overtake a slow-moving car does not involve the explicit integration of the equations of motion, but we certainly do it implicitly. A Tesla on autopilot will actually solve them explicitly.

When overtaking a car, a Tesla will explicitly calculate what a human driver processes implicitly. (Image credit: Shutterstock)

Predicting chaos

So we really should not be surprised that mathematics is not just a language for describing the external world, but in many ways the only one. But just because something can be described mathematically does not mean it can be predicted.

One of the more remarkable discoveries of the last 50 years has been the discovery of "chaotic systems (opens in new tab)." These can be apparently simple mathematical systems that cannot be solved precisely. It turns out that many systems are chaotic in this sense. Hurricane tracks in the Caribbean are superficially similar to eclipse tracks, but we cannot predict them precisely with all the power of modern computers.

However, we understand why: the equations that describe weather are intrinsically chaotic, so we can make accurate predictions in the short term, (about 24 hours), but these become increasingly unreliable over days. Similarly, quantum mechanics provides a theory where we know precisely what predictions cannot be made precisely. One can calculate the properties of an electron very accurately, but we cannot predict what an individual one will do (opens in new tab).

Hurricanes are obviously intermittent events, and we cannot predict when one will happen in advance. But the mere fact that we cannot predict an event precisely does not mean we cannot describe it when it happens. We can even handle one-off events: it is generally accepted that the universe was created in the Big Bang and we have a remarkably precise theory of that (opens in new tab).

Designing social systems

A whole host of social phenomena, from the stock market (opens in new tab) to revolutions (opens in new tab), lack good predictive mathematics, but we can describe what has happened and to some extent construct model systems.

So how about personal relationships? Love may be blind, but relationships are certainly predictable. The vast majority of us choose partners inside our social class and linguistic group, so there is absolutely no doubt that is true in the statistical sense. But it is also true in the local sense. A host of dating sites make their money by algorithms that at least make some pretense at matching you to your ideal mate.

A universe that could not be described mathematically would need to be fundamentally irrational and not merely unpredictable. Just because a theory is implausible does not mean we could not describe it mathematically.

But I do not think we live in that universe, and I suspect we cannot imagine a non-mathematical universe.

This article is republished from The Conversation (opens in new tab) under a Creative Commons license. Read the original article (opens in new tab).

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Peter Watson
Peter Watson

Peter Watson joined the Carleton University Department of Physics in 1974 and was promoted to Full Professor in 1984. He became chair of the Dept in 1990 and Dean of Science in 1997. He has taught more than 25 different courses, at all levels,  and supervised many graduate and undergraduate students. His research career has mostly been in theoretical physics, with over 50 published papers, and recently he has been working on the Sudbury Neutrino Observatory. He also worked at Ahmadu Bello University, in Nigeria, and spent 3 sabbaticals working abroad at Villigen, in Switzerland,  at the University of Edinburgh and C.E.R.N. in  Geneva and the University of Oxford. Although he retired in June 2008, he has continued to teach and do research