**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” -the 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)*