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Mathematics usually stays in the shadows, never getting the kind of exposure enjoyed by religion or physics

Last Updated 23 December 2022, 12:37 IST

In 1977, as a new bachelor’s student at the Institute of Science in Mumbai, I came the closest I’ve ever been to having a religious experience. My algebra professor, the late great Prof. M S Huzurbazar, related to us the German mathematician Kronecker’s famous quote, “God made the whole numbers, everything else is the work of man”. “Except, we don’t need God,” Huzurbazar declared. “We can make the whole numbers ourselves as well – that too, out of pure emptiness!”

To demonstrate, he equated emptiness to zero, and then for any number, used set theory to define its successor. It was like lighting the fuse to a chain reaction: one emerged from zero, two from one, three from two – suddenly, an explosion of numbers seemed to erupt through the classroom. I felt the walls around me dissolve, the ceiling above me part, to reveal waves of numbers streaming through time and space. The whole experience felt cosmic, as if I was at the dawn of creation – it’s where the title of my new book, The Big Bang of Numbers comes from.

Now, many people would associate creation with a supreme being. Brahma blows out the universe in a single breath, the biblical God fashions the cosmos over the first six days of Genesis, the ancient Egyptian deity Atum summons up the world out of a sea of chaos. The more scientific-minded amongst us might associate creation with physics – the universe emerges from a singularity of infinite density in the Big Bang. Mathematics, which is so abstract and so independent of physical reality, seems like an unlikely candidate.

And yet, Prof Huzurbazar’s construction came closer to the ideal of creatio ex nihilo, or “creation out of nothing,” than either religion or physics can. In religion, you have to assume the existence of a supreme being, while in physics, you need a singularity. In mathematics, all Prof Huzurbazar used was a version of nothing called the empty set.

Fine, but that only takes care of the numbers. Is mathematics able to create anything else? That’s the question I began to consider decades later. Could one continue the construction and build the universe using only maths?

There were good reasons to pursue such a thought experiment. Mathematics usually stays in the shadows, never getting the kind of exposure enjoyed by religion or physics. Why not have it step out and show its mettle in the arena of public discourse, in an easy-to-understand, non-technical way? Even if the construction ultimately didn’t quite succeed, pursuing it would reveal how intricately maths is woven into our universe’s fabric.

And perhaps with the addition of only a few more extra ingredients, we could actually succeed! Let’s note here that God and physics need such extras as well – for instance, both merrily use numbers, without creating them.

So, what would our next project be after creating the numbers? If we’re starting with complete nothingness, then we’d need to take care of something rarely talked about: empty space. Notice that space – even a vacuum – is different from pure “nothing”. God certainly takes such space for granted, creating heaven and earth without the slightest mention of any prior preparations to set up the empty stage. As mathematicians, we can’t be as blithe as God. We’re aware of the need for a matrix of locations to harbour all these creations, so it behoves us to fill this gap.

Over two millennia ago, the Greek mathematician Euclid postulated that between any two points, there exists a straight line, and moreover, that this line can be extended indefinitely in either direction.

This gives us a recipe for creating space. Start with two points, and (assuming Euclid’s postulate holds) you can get an infinite line; add another point, and it turns out you can draw a series of lines that comprise a plane; add one more point and you can generate a multitude of parallel planes that stack together to form 3-D space.

Interestingly, the Russian artist Wassily Kandinsky used a similar point-line-plane progression to map the surface of his blank canvas.

The above construction is quite easy and intuitive (for details, see my book). The thing to note is that with just a couple of extra ingredients – a few points, plus Euclid’s postulates – mathematics can build up empty space. Actually, it can do much more – it can also create several alternatives to this space!

For one, the power of abstraction lets us extract the gist of this idea and repeat it again and again. Points, stacked together, form a one-dimensional line; lines, stacked together, form a 2-D plane; planes, stacked together, form 3-D space. What if we stack together copies of 3-D space? The answer becomes obvious – we should get four-dimensional space! Mathematics allows us to access this idea intellectually, even though, stuck as we are in 3-D, we can’t visualize it.

In fact, maths makes us wonder whether there might indeed be higher dimensions to reality. In case this idea sounds too far-fetched, note that superstring theory requires space to have at least nine dimensions, and there have been other physics theories which posit that our universe may be a lower-dimensional facet of a higher-dimensional reality (just like an edge or face is a lower-dimensional facet of a block).

There’s more – mathematics also allows us to construct space which is curved! Remember that postulate by Euclid we used? Well, if one replaces “straight line” by “circular arc”, then instead of lines joining up to form a plane, the same construction will give you circles merging together to form the surface of a sphere. Not only that, but substituting other types of lines for “straight line” gives you various other surfaces, including the ruffled “hyperbolic” ones found so often in nature (e.g. in corals).

As before, mathematicians have been able to extend such ideas to higher dimensions, thereby constructing (theoretically, at least) “curved” versions of 3-D space. Lest you dismiss this as something that’s too abstract or academic to be of practical interest, note that Einstein, in his General Theory of Relativity, propounded that our own spacetime is curved – a fact that has been experimentally verified.

Let me pause at this juncture to raise a fundamental question. Mathematics is often understood as something humans create to solve problems and describe the universe – for instance, Newton’s invention of calculus was inspired by a desire to analyse motion. Consider, however, curved geometries like the hyperbolic variety, which mathematicians discovered in the 1800s, after centuries of abstract thinking. It was not as if they were trying to model corals or other sea creatures, nor were they anticipating that Einstein would use their discoveries decades hence. How, then, could the esoteric, cerebral theories of mathematicians have found such profound applications in nature?

In fact, there are several similar examples – ellipses, first formulated by the ancient Greeks, turned out to be the right paths for planetary motion; logarithmic spirals showed up in nautilus shells and galaxy formation; group theory, developed to abstractly describe symmetry, proved to be the perfect setting for quantum mechanics; the abstract theory of knots popped up unexpectedly in DNA modelling. What explains this “unreasonable effectiveness” maths has in describing the universe, as Nobel laureate Eugene Wigner put it?

There is one obvious possibility: that mathematics is not manmade but, as Plato believed, exists immutably, and independently of us. That maths is the intelligence behind the universe, the “Vishnu”, if you will, that keeps everything running and orderly. Rather than us formulating mathematics to try and describe the patterns we see around us, it is mathematics, imbedded like DNA, that’s responsible for creating the patterns in the first place. That is why corals and other sea creatures could adopt hyperbolic geometry a half billion years before we humans had any inkling of it.

While one can’t make an airtight case for it, accepting this reversal of outlook makes several natural phenomena easier to interpret. Think of mathematics providing an array of shapes, equations and behavioural laws from which every component of the universe can be built. This does not necessarily contradict belief in a supreme being. God might do the implementation, but relies on mathematics to draw up the blueprints.

So, getting back to our mathematical construction, what is the next thing we should help God (or physics) with? Let’s say we’ve already created a catalogue of standard geometrical shapes like triangles, squares, circles, etc. These will suffice for many purposes, but several other applications – like the boundaries of clouds or the branching of air passages – will need more complicated interfaces called fractals. Such patterns, where similar designs occur at different scales, often show up spontaneously in nature – for instance, on the surfaces of shells.

Despite their complexity, we can generate such images using very simple rules, such as: For each black triangle, colour its centre quarter white. Suppose you start iterating, using the previous output as the new input to which the rule is reapplied. Notice how quickly you get a shape where the boundary between black and white gets very complicated. Also, how the pattern on the shell is similar to the one you get now, but just with a good dose of randomness mixed in!

The above type of input/output rules, where the output of the previous step serves as the input of the next one, gives us insight into many evolutionary processes. For instance, think of a coastline evolving in a series of such snapshots under the influence of tide. The erosive forces have similar effects both at small and large scales, which is why most coastlines end up resembling fractals.

There are many more mathematical steps we can perform towards making the universe more of a reality. Eventually, though, we’re faced with the most difficult question of all: how to create life? Surely that requires a divine spark, far beyond the capabilities of mathematics?

Scientific theories usually say that the first living matter was formed from the right kinds of molecules randomly interacting billions of times. What mathematics contributes, through the example of input-output rules and fractals, is the insight that remarkable changes in complexity can occur, from simple rules to complex outcomes, from inanimate molecules to living cells.

Perhaps, as Prof Huzurbazar said, we don’t need God after all. Perhaps mathematics can indeed bring everything to life out of emptiness.

*(Manil Suri is a distinguished mathematics professor at the University of Maryland, Baltimore County, and the author, most recently, of The Big Bang of Numbers: How to Build the Universe Using Only Maths)*

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(Published 21 December 2022, 17:56 IST)

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