Think Maths
Is mathematics the grand design for the Universe, or merely a figment
of the human imagination, asks Ian Stewart
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.
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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" of 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 highly 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. Columbus discovered America-it was already
there, but neither he nor anyone else where he came from knew it was-and
David Livingstone discovered the Victoria Falls. The word "invention" means
bringing into existence something that was not previously there. Edison invented
electric light, Bell invented the telephone.
However, when Columbus landed in America he was actually trying
to invent a new trade route to India. And Livingstone's discovery came as
no great surprise to the local inhabitants, who saw the Victoria Falls every
day. Edison would have felt as if he had invented the idea of electric lighting,
but then spent many years trying to discover how to make it a reality. So
invention and discovery both happen within a particular context-people becoming
aware that there is something new in their world.
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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, not 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 neighbour 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.
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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 subconsciousnesses. 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.
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But if mathematics resides in mathematicians' heads, why is
it so "unreasonably effective"? [Ref: P.A.M p526 {E.Wigner} ]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. Wigner's
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 world's scientists all got together and decided
that 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 Newton's laws of motion.
But the problem here could be a difference of scale. Quantum physics tends
to be applied to simple arrangements of matter-a few atoms, say. In biology,
the significant arrangements of matter are enormously more complex: there
are trillions of atoms in the human genome, and this is just one
DNA strand inside one cell of a much more complex
organism. An atom - by - atom description of a human being would involve
numbers with an awful lot of zeros. Human beings could well be behaving according
to mathematical rules-but it is mathematics so complicated that human
mathematicians cannot possibly write it down, let alone grasp what it means.
Moreover, it is mathematics whose structure is almost totally impenetrable,
for the boring reason that there is just too much information to 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 recognisable 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
Newton's law of gravity and the equations
of quantum mechanics, and also to astrology and
an obsession with the measurements of the Great Pyramid. Ironically, what
mathematics tells us about choosing
lottery numbers is that any patterns
we think we see are illusions.
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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 matter, 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, for whom everything is in constant
flux. Maybe they'd understand plasma dynamic 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's theorem.
[H2=Ö
(a2 + b2) ] 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.