快猫短视频

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鈥檚 patterns

So who is right? Well, there is much that is attractive in the Platonist
point of view. It鈥檚 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鈥檚 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鈥檚
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 鈥渦nreasonable effectiveness鈥� 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 鈥淕od ever geometrises鈥�. 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鈥檚
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 鈥渄iscover鈥� for finding things that already
exist in the physical world. Columbus discovered America鈥攊t was already
there, but neither he nor anyone else where he came from knew it was鈥攁nd
David Livingstone discovered the Victoria Falls. The word 鈥渋nvention鈥� 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鈥檚 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鈥攑eople becoming aware
that there is something new in their world.

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鈥檚 famous 鈥渃ollective unconscious鈥濃�攖he 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
鈥渃ollective鈥�? 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 鈥渢he pool鈥� 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鈥檚 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鈥攍ike Jung鈥檚 mystical telepathic
unconscious鈥攊nto 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鈥檚 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鈥檛 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鈥檝e 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鈥� heads, why is it so
鈥渦nreasonably effective鈥�? 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鈥檚 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 鈥溍塿ariste Galois鈥� you鈥檙e doing Galois theory, which shows
that you can鈥檛 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鈥檚
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鈥攖he 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. 快猫短视频s 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鈥檚 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鈥檚 the reality check in maths? Well, the deeper we delve into the
鈥渇undamental鈥� 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 鈥淗arvard law of animal behaviour鈥濃�攊n carefully
controlled laboratory conditions, animals do what they damned well
please鈥攊s more appropriate than Newton鈥檚 laws of motion. But the problem
here could be a difference of scale. Quantum physics tends to be applied to
simple arrangements of matter鈥攁 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鈥攂ut 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 鈥渆mergence鈥�, 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鈥檛 do the calculations to check that because they鈥檙e too messy and
too lengthy.

You could argue that this makes the whole question academic: it doesn鈥檛
matter whether this kind of mathematical basis exists for biology, because even
if it does exist, it鈥檚 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鈥檚 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鈥�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 鈥減rimordia鈥濃�攖iny
lumps of cells from which the interesting features of plants
develop鈥攂ecome 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鈥檚
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.

It鈥檚 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鈥檛
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鈥檛 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鈥檓 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鈥檇 understand
plasma dynamics a lot better than we do, though I suspect we wouldn鈥檛 have any
idea how they did it. But I doubt that they would have anything like
Pythagoras鈥檚 theorem. There are few right angles in plasmas. In fact, I doubt
they鈥檇 have the concept 鈥渢riangle鈥�. 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.