Nature's Numbers

Nature’s Numbers

Ian Stewart

This entertaining and very readable little book describes the relevance of mathematics for describing and appreciating nature.  In the process it supplies quite a lot of curious information, some of which feeds into serious issues if you really want it to, but most of which is purely entertaining and easily accessible.   It points out that mathematics is not simply about numbers, but also about shapes, patterns, regularities, transformations, and evolutions.  There is nothing here to intimidate those with a maths phobia, a cuff about the ears for so called pragmatists who only want to hear about so called “applied” or “useful” mathematics, and a stern reproof for those scientists too arrogant to acknowledge the contribution mathematics makes to their fields.

A few signs of the book’s age are amusing but do not affect its quality at all.  “But the next time you go jogging wearing a Walkman, or switch on your TV, or watch a videotape, pause for a few seconds to remember that without mathematics none of these marvels would ever have been invented.” [p72]

 It can be amusingly gross: “Nature prefers the icosahedrons above all other viral forms: examples include herpes, chickenpox, human wart, canine infectious hepatitis, turnip yellow mosaic, adenovirus and many others.”  [p82]

Mathematics is not only about numbers.

“It’s time we started pulling the bits together,  Because only then will we truly understand nature’s numbers – along with nature’s shapes, structures, behaviours, interactions, processes, developments, metamorphoses, evolutions, revolutions...” [p150]

“Notice that this approach again changes the meaning of “solve.” First that word meant “find a formula.”  Then its meaning changed to “find approximate numbers.” Finally, it has in effect become “tell me what the solutions look like.” In place of quantitative answers, we seek qualitative ones. ... Why did people want a formula in the first place? Because in the early days of dynamics, that was the only way to work out what kind of motion would occur.  Later, the same information could be deduced from approximations.  Nowadays, it can be obtained from theories that deal directly and precisely with the main qualitative aspects of the motion. ... For the first time we are starting to understand nature’s patterns in their own terms.” [p59]

It is suitably irate about popular notions of ‘usefulness.’  

“... two of the main things that mathematics is for: providing tools that let scientists calculate what nature is doing, and providing new questions for mathematicians to sort out to their own satisfaction.  These are the external and internal aspects of mathematics, often referred to as applied and pure mathematics. (I dislike both adjectives and I dislike the implied separation even more.) “ [p17]

“Engineers designing a bridge are entitled to use standard mathematical methods even if they don’t know the detailed and often esoteric reasoning that justifies these methods.  But I, for one, would feel uncomfortable driving across that bridge if I was aware that nobody knew what justified those methods.  So, on a cultural level, it pays to have some people who worry about pragmatic methods and try to find out what really makes them tick.” [p18]

“One of the strangest features of the relationship between mathematics and the “real world,” but also one of the strongest, is that good mathematics, whatever its source, eventually turns out to be useful.” [p18]

“...it is ...very tempting to assume that the only useful part of mathematics is applied mathematics; after all, that is what the name seems to apply... But it gives a very distorted view of the origins of new mathematics of practical importance. Good ideas are rare, but they come at least as often from imaginative dreams about the internal structure of mathematics as they do from attempts to solve a specific, practical problem.” [p62]

I was interested in a brief comment on mathematical reasoning.

“A great deal of work in philosophy and the foundations of mathematics has established that you can’t prove everything, because you have to start somewhere; and even when you’ve decided where to start, some statements may be neither provable nor disprovable.”

Textbooks of mathematical logic say that a proof is a sequence of statements, each of which either follows from previous statements or from agreed axioms – unproved but explicitly stated assumptions that in effect define the area of mathematics being studied. This is about as informative as describing a novel as a sequence of sentences, each of which either sets up an agreed context or follows credibly from previous sentences. Both definitions miss the essential point: that both a proof and a novel must tell an interesting story.”  [p39]

A nice aside about the nature of creativity:

“ ... some of the emotional experiences of the creative mathematician: frustration at the intractability of what ought to be an easy question, elation when light dawned, suspicion as you checked whether there were any holes in the argument, aesthetic satisfaction when you decided the idea really was O.K. and realized how neatly it cut through all the apparent complications. Creative mathematics is just like this - ...” [p43]

A long account of symmetry and the breaking of symmetry is capable of informing many debates about cosmology.  Sadly I have abandoned such internet debates as a waste of my time.

“In short, nature is symmetric because we live in a mass-produced universe ... Every electron is exactly the same as every other electron, every proton is exactly the same as every other proton, every region of empty space is exactly the same as every other region of empty space, every instant of time is exactly the same as every other instant of time. And not only are the structures of space, time and matter the same everywhere: so are the laws that govern them. ... At the instant of the universe’s formation, all places and all times were not only indistinguishable but identical.  So why are they different now?" [p84,85]

"The answer is the ... principle of symmetry breaking... The evolving universe can break the initial symmetries of the big bang...   Potentially, the universe could exist in any of a huge symmetric system of possible states, but actually it must select one of them.  In so doing, it must trade some of its actual symmetry for unobservable, potential symmetry... the important point is that the tiniest departure from symmetry in the cause can lead to a total loss of symmetry in the resulting effect – and there are always tiny departures. ... Small disturbances cause the real system to select states from the range available to the idealized perfect system." [p85, 86]

“There is so much symmetry ... in our mass produced universe that there is seldom a good reason to break all of it.  So rather a lot survives... the symmetries we observe in nature are the broken traces of the grand, universal symmetries of our mass produced universe." [p85]

“This universality of symmetry breaking explains why living systems and non-living ones have many patterns in common.  Life itself is a process of symmetry creation – of replication: the universe of biology is just as mass-produced as the universe of physics... The most obvious symmetries of living organisms are those of form – icosahedral viruses, the spiral shell of Nautilus, the helical horns of gazelles, the remarkable rotational symmetries of starfish and jellyfish and flowers.”  [p88]

“In short there is an ideal mathematical universe in which all of the fundamental forces are related in a perfectly symmetrical manner – but we don’t live in it.” [p90]

I was fascinated by an account of the way animals deploy their limbs in order to move about efficiently.  It actually has a bearing on a quite significant problem in the history of art, which was the difficulty artists had painting a realistic horse in action. The fact is that, up to the nineteenth century, nobody knew in sufficient detail how they moved their legs.

“Two biologically distinct but mathematically similar types of oscillator are involved in locomotion. The most obvious are the animal’s limbs ... The main oscillators that concern us here, however, are to be found in the creature’s nervous system... A lot of what we do know has been arrived at by working backward – or forward if you like – from mathematical models.” [p98, 99]

"Some animals possess only one gait... The elephant, for example, can only walk... Other animals possess many different gaits;  ... The seven most common quadrupedal gaits are the trot, pace, bound, walk, rotary gallop, transverse gallop and canter. ... There is also a rarer gait, the plonk, in which all four legs move simultaneously.... sometimes seen in young deer. The pace is observed in camels, the bound in dogs, cheetahs use the rotary gallop to travel at top speed.  Horses are among the more versatile quadrupeds, using the walk, trot, transverse gallop and canter, depending on circumstances.” [p99, 100].

In general, we can learn a lot about nature by thinking in mathematical language.  This is a necessary corrective, for example, to the excessive claims made for genetics, still the easy, fashionable answer to everything and hence to nothing.

 “...mathematics can illuminate many aspects of nature that we do not normally think of as being mathematical. This is a message that goes back to the Scottish zoologist D’Arcy Thompson, whose classic but maverick book On growth and Form set out, in 1917, an enormous variety of more or less plausible evidence for the role of mathematics in the generation of biological form and behaviour. In an age when most biologists seem to think that the only interesting thing about an animal is its DNA sequence, it is a message that needs to be repeated, loudly and often.” [p105]

“Plants don’t need their genes to tell them how to space their primordia: that’s done by the dynamics. It’s a partnership of physics and genetics and you need both to understand what’s happening.” [p142]

In short, Ian Stewart has the gift of being able to make mathematics accessible and relevant in an entertaining manner.  A short enough book, this was a pleasure to read and will remain on my shelves for future reference.