**Think Maths**

Is Mathematics the Grand Design for the Universe, or
Merely a Figment of the Human Imagination?

Where does mathematics come from? Is it already out
there, waiting for us to discover it, or do we make it all up as we go along?
Plato held that mathematical concepts actually exist in some weird kind of
ideal reality just off the edge of the Universe. A circle is not just an idea,
it is an ideal. We imperfect creatures may aspire to that ideal, but we can
never achieve it, if only because, pencil points are too thick. But there are
those who say that mathematics exists only in the mind of the beholder. It does
not have any existence independent of human thought, any more than language,
music or the rules of football do.

**Nature’s
Patterns**

So who is right? Well, there is much that is
attractive in the Platonist point of view. It’s tempting to see our everyday
world as a pale shadow of a more perfect, ordered, mathematically exact one.
For one thing, mathematical patterns permeate all areas of science. Moreover,
they have a universal feel to them, rather as though God thumbed His way
through some kind of mathematical wallpaper catalogue when He was trying to
work out how to decorate His Universe. Not only that: the deity’s pattern
catalogue is remarkably versatile, with the same patterns being used in many
different guises. For example, the ripples on the surface of sand dunes are
pretty much identical to the wave patterns in liquid crystals. Raindrops and
planets are both spherical. Rainbows and ripples on a pond are circular.
Honeycomb patterns are used by bees to store honey (and to pigeonhole grubs for
safekeeping),and they can also be found in the geographical distribution of
territorial fish, the frozen magma of the Giant’s Causeway, and rock piles
created by convection currents in shallow lakes. Spirals can be seen in water
running out of a bath and in the Andromeda Galaxy Frothy bubbles occur in a
washing-up bowl and the arrangement of galaxies.

With this kind of ubiquitous occurrence of the same
mathematical patterns, it is no wonder that physical scientists get carried
away and declare them to lie at the very basis of space time and matter. Eugene
Wigner expressed surprise at the “unreasonable effectiveness” mathematics as a
method for understanding the Universe. Many philosophers and scientists have
seen mathematics as the basis of the Universe. Plato wrote that “God ever geometrises”. The physicist James Jeans declared that God
was a mathematician. Paul Dirac, one of the inventors of quantum mechanics, went
further opining that he was a pure mathematician. In the past few years Edward Fredkin has argued that the Universe is made from
information, the raw material of mathematics.

This is powerful, heady stuff, and it is high
appealing to mathematicians. However, it is equally conceivable that all of
this apparently fundamental mathematics is in the eye of the beholder, or more
accurately, in the beholder’s mind. We human beings do
not experience the Universe raw, but through our senses, and we interpret the
results using our minds. So to what extent are we mentally selecting particular
kinds of experience and deeming .them to be important, rather than picking up
things that really are important in the workings of the Universe? Is
mathematics invented or discovered?

If pushed, I would say that it is a bit of both
because neither word adequately describes the process. Moreover they are not
alternatives they are not opposites, and they do not exhaust the possibilities.
They are not even particularly appropriate. We use discover for finding things
that already exist in the physical world.

However, when

It is the same with mathematics. What to the outside
world looks like invention often feels more like discovery to insiders. The distinction is made all the more tricky
because mathematical objects lead a virtual existence, nor a real one: they
reside in minds, not embodied in any kind of hardware. But unlike, say, poetry
that virtual world obeys rigid rules, and those rules are pretty much the same
in every mathematical mind.

In a way, the world of mathematical ideas is a kind of
virtual collective comparable to Jung’s famous “collective unconscious” - e
idea that all human minds have access to vast, evolutionarily ancient,
subconscious structures and processes that govern much of our behaviour. But in what sense are they “collective”? A
crucial distinction has to be made here between a single unconscious entity
into which we all dip, and a large number of distinct but very similar unconsciousnesses, one for each of us. It is the difference
between a community with a single municipal swimming pool, and one in which
every back garden has its own pool.

From the point of view of specific action, the
distinction is not terribly important: you can discuss the problems of keeping
leaves out of “the pool” with your neighbor without ever making it clear
whether you think of it as a single common pool or a typical representative of
the individual pools that everybody has. But if you want to understand what’s
going on in general, then it does make a difference. The notion of a single
unconscious mind for all of humanity is a mystical and rather silly concept
that leads in the direction of telepathy. A collection of more or less identical
individual subconsciousnesses, rendered similar by
their common social context, is considerably more prosaic but a great deal more
sensible.

The same point lies at the heart of how I think we
should view mathematics. Because we have a single word for the virtual
collective it is tempting to think of it as a single thing – like Jung’s
mystical telepathic unconscious - into which all mathematicians dip. This is a
difficult concept to capture. Where is that thing? What is it made of? How does
it grow?

Instead, it is better to think of mathematics as being
distributed throughout the minds of the world’s mathematicians. Each has his or
her own mathematics inside his or her head. Moreover, those individual systems
are extremely similar to each other, much more so than Jungian subconciousnesses. Not in the sense that each head contains
the whole of mathematics. Mine contains dynamical systems, yours contains
analysis, and hers algebra, say. But all three are logically consistent with
each other because of how mathematicians are trained, and how they communicate
their ideas. If what is in my head is not consistent with what is in yours,
then one of us has got it wrong and we will argue until it becomes clear to us
both who it is.

**Baking Bread**

Most areas of human activity are structured in this
way. So the difficult questions of existence and discovery versus invention are
not confined to mathematics. Take medicine, for example. What is medicine?
Where does it live? Is it invented or discovered? Now replace medicine by
plumbing, ballet, football, language or cycling, and it is clear just how
widespread the structure is, and why the question doesn’t make a great deal of
sense in any area of human activity. What goes on is neither invention nor
discovery but a complex context-dependent mix of both.

When it comes to mathematics, sometimes it really does
feel like discovery.When you are carrying out
mathematical research in a previously defined area it feels like discovery
because there is no choice about what the answer is. But when you are trying to
formalise an elusive idea or find a new method, it
feels more like invention: you are floundering around, trying all sorts of
harebrained ideas, and you simply do not know where it will all lead. The more
established an area of mathematics becomes, the more strongly it feels as if
there is some kind of fixed logical landscape, which you merely explore. Once
you’ve made a few assumptions (axioms), then everything that follows from them
is predetermined. But this account misses out the most crucial features: significance,
simplicity elegance, how compelling the argument is, all
things that give the landscape its character.

But if mathematics resides in mathematicians’ heads,
why is it so “unreasonably effective”? The easy answer is that most mathematics
starts in the real world. For instance, after observing on innumerable
occasions that two sheep plus two more sheep make four sheep, ditto cows,
wolves, warts and witches, it is a small step to introduce the idea that 2 + 2
= 4 in a universal, abstract sense. Since the abstraction came out of reality,
it’s no surprise if it applies to reality.

However, that is too simple-minded a view. Mathematics
has an internal structure of logical deduction that allows it to grow in
unexpected ways. New ideas can be generated internally too, whenever anyone
tries to fill obvious holes in the logical landscape. For example, having
worked out how to solve quadratic equations, which arose from problems about
baking bread, or whatever, it is obvious that you ought to try to solve cubic
and quintic equations too. Before you can say “Evariste Galois” you’re doing Galois theory,
which shows that you can’t solve quintics, but is
almost totally useless for anything practical. Then someone generalises
Galois theory so that it applies to differential
equations, and. suddenly you find applications again, but to dynamics, not to
bakery.

**Herd of
Elephants**

Yes, there is a flow of problems and concepts from the
real world into mathematics, and a back-flow of solutions from mathematics to reality.
My point is that the back-flow may not answer the problem that you set out to
solve. Instead it may answer something just as real, just as important, but
physically unrelated. Why should this be? Well, mathematics is the art of
drawing necessary conclusions, independently of interpretations. Two plus two
has to be four whether you are discussing sheep, cows or witches. In other
words, the same abstract structure can have several interpretations. So you can
get the ideas from one interpretation, and transfer the result to others.
Mathematics is so powerful because it is an abstraction.

This is all very well, but why do the abstractions of
mathematics match reality? Indeed, do they really match, or is it all an
Illusion? Enter cultural relativism - the idea that has lately become so
fashionable in academic arts departments, which sees maths
and science as social constructs no less and no more valid than any other
social construct. Does this lead to the idea that science can be anything
scientists want it to be?

True, science is a social construct. Scientists who
claim that it is not are making the same mistake as those who think that we all
dip into the same collective subconscious. But there is something special about
science: it is a construct that has at every step been tested against external
reality. If the worlds scientists all got together and said all elephants are
weightless and rise into the air if they are not held down by ropes it would
still be foolish to stand under a cliff when a herd of elephants was leaping
off the edge. In science, there has to be a reality check. Because it is done
by beings who see reality through imperfect and biased
senses, the reality check cannot be perfect, but science still has to survive
some very stringent scrutiny

So what’s the reality check in Maths?
Well, the deeper we delve into the “fundamental” nature of the Universe, the
more mathematical it seems to get. The ghostly world of the quantum cannot be
expressed without mathematics: if you try to describe it in everyday language,
it makes no sense. Mind you, not all fields are so obviously mathematical in
their structure. The biological world, in particular, seems not to obey the
rigid rules that we find in physics. The “Harvard law of animal behaviour” - in carefully controlled laboratory conditions,
animals do what they damned well please – is more appropriate than

take in.

This is the old philosophical problem of “emergence”,
but in a new guise. Emergent phenomena are things that seem to transcend their
ingredients, like consciousness arising in a material brain. Philosophers have
a habit of discussing emergence as if it breaks the chain of causality but what
really happens is the chain of causality becomes so intricate that the human
mind cannot grasp it. Your behaviour is caused by
mathematical rules applied to your constituent atoms, in the context of
everything that is happening around you, but you can’t do the calculations to
check that because they’re too messy and too lengthy.

You could argue that this makes the whole question
academic it doesn’t matter whether this kind of mathematical basis exists for
biology because even if it does exist, it’s of no practical use. However, there
is an attractive alternative. Even very complex mathematical systems tend to
generate recognizable patterns on higher levels of description. For example,
the underlying quantum theory of a crystal involves just as many atoms as a
human being, at least if it’s a human sized crystal, and therefore runs into
the same intractable problem of emergence. But crystals exhibit clear
mathematical patterns of their own, such as a regular geometric form, and while
nobody can deduce this in full logical rigour from
the quantum mechanics of their atoms, there is a chain of reasoning that makes
it plausible that the laws of quantum mechanics do indeed lead to the
regularities of crystal structure. Roughly speaking, it goes like this: quantum
mechanics causes the atoms to arrange themselves in a minimum-energy
configuration; the overall symmetry of the laws of nature in space and time
causes such configurations to be highly symmetrical; in this case, the
consequence is that they form regular atomic lattices.

**Lottery
Illusion**

From this point of view, mathematical patterns that
arise in high-level descriptions of living organisms are evidence that biology
too, is mathematical at heart. For example, the number of petals in a flower
tends to be one of the Fibonacci numbers - 3, 5, 8, 13, 21, 34, 55 and so on, where each is the sum of the previous two.
This strange numerology can be traced to the dynamical behaviour
of the cells at the tip of a growing shoot. The “primordia”
- tiny lumps of cells from which the interesting features of plants develop -
become arranged in patterns like interpenetrating spirals, and the mathematics
of such patterns leads inevitably to Fibonacci numbers.

But do patterns like these really tell us that
mathematics is inherent in nature? Our minds certainly have a tendency to seek
out mathematical patterns, whether or not they are actually significant. This
tendency has led to

It’s worth asking how our minds developed this
tendency for pattern seeking. Human minds evolved in the real world; and they
learnt to detect patterns to help us survive events outside ourselves. If none
of the patterns detected by these minds bore any genuine relation to the real
world outside, they wouldn’t have helped their owners survive, and would
eventually have died out. So our figments must correspond, to some extent, to
real patterns. In the same way, mathematics is our way of understanding certain
features of nature. It is a construct of the human mind, but we are part of
nature, made from the same kind of material existing in the same kinds of space
and time as the rest of the Universe. So the figments in our heads are not
arbitrary inventions. There are definitely some mathematical things in the
Universe, the most obvious being the mind of a mathematician. Mathematical
minds cannot evolve in an unmathematical universe.
Only a geometer God can create beings able to come up with geometry

But that is not to say that only one kind of
mathematics is possible: the mathematics of the Universe. That seems too
parochial a view. Would aliens necessarily come up with the same kind of
mathematics as us? I don’t mean in fine detail. For example the six-clawed cat
creatures of Apellobetnees Gamma would no doubt use
base-24 notation, but they would still agree that twenty-five is a perfect
square, even If they write it as 11. However, I’m thinking more of the kind of
mathematics that might be developed by the plasma vortex wizards of Cygnus V,
for whom everything is in constant flux. I bet they’d understand plasma
dynamics a lot better than we do, though I suspect we wouldn’t have any idea
how they did it But I doubt that they would have anything like Pythagoras’
theorem. There are few right angles in plasmas. In fact, I doubt they’d have
the concept “triangle”. By the time they had drawn the third vertex of a right
triangle, the other two would be long gone, wafted away on the plasma winds.

*Ian Stewart*

*The New
Scientist (November 30, 1996)*