Tyger

Looking into the eyes of a tiger, we see no mind that we can understand. And the tiger understands us not. William Blake asked a profound question in his famous poem.

For thousands of years, many people have looked around at the bewildering variety of things, and wondered. Long ago in Greece, there was an idea that everything was made of only four substances, or essences, air, earth, fire and water. This was an attempt at a minimal set of basic materials from which everything was made. Later, in order to explain some difficulties, a fifth essence was introduced. Naturally, it was called the quintessence.

There was a fundamental problem with this idea. The timing was wrong. Not enough was known about materials and the forces between them, and there was no technology for making tests of the idea. I

n the twentieth century, Einstein spent his later years trying to unify gravitation and electromagnetism. It must have seemed an attractive proposition, as both follow an inverse square law. Einstein's problem was the same as the old Greek one. Not enough was known. There were other forces. It was as if Darwin and Wallace had tried to understand evolution, knowing only the tiger and the lamb.

Another way of going wrong occurs in engineering. Some early attempts at flying used imitations of bird flight. But flapping flight was not the best way for machines. It required too much control. Perhaps powerful computers will help to make it possible. A number of early aircraft did in fact imitate the shapes of bird wings.

The important aspects of the bird, bat, or insect were actually the aerofoil and the structure. The wing needs to be light and rigid. A skin stretched over a framework does the job well. Feathers help to create an efficient wing profile, though for small and slow animals like insects the profile is not so important.

Progress in science is often made by finding relationships between different things. The last thirty years have seen a tendency for inter-disciplinary research which is in contrast to the extreme specialisation that research so often demands.

Many people believe in the unity of the world. What we cannot do, however, is to predict the structure or the behaviour of the entire universe from the basic laws of biology, chemistry, or physics..

This cannot be done for several reasons. One is that at each level, the type of organisation can be, and is, very different.

At school or at college you may learn that the exponential function and the sine and cosine functions are closely related.

How can the smooth curves that describe an explosive increase in some variable, or the fading of the light as you dive in the sea, be related to the wiggly curves that describe the oscillations of a pendulum or a suspension bridge?

They are related by the fact that they are both special cases of a more universal mathematical function. The connection is only revealed if we use the so-called imaginary numbers.

This page refers to the unification of many apparently unrelated pairs.

See also Sine and Exponential`

Sir Isaac Newton, by creating his theory of gravitation, was able to show that the fall of an apple and the orbital motion of the moon were examples of the same phenomenon. According to this theory, both objects are falling freely in a gravitational field. A maggot in the falling apple would feel weightless. Astronauts on the moon did not feel weightless, because the moon has a gravitational field.

They were indeed virtually weightless as regards the earth, but not quite, because they were slightly nearer to the earth than the centre of the moon is, and so were carried around the earth at a speed which did not correspond to their orbit around the earth. Their weight on the moon would have been slightly reduced by this, the same cause that produces tides.

The inverse square law of force could hardly have been discovered only by observations on earth, at the time of Newton, but by using the distant moon and a nearby apple, Newton had access to a wide-ranging distance scale. His theory, of course, accounted almost completely for the motions of the planets. A small discrepancy in the motion of Mercury remained, even after later scientists evaluated the effects on Mercury of other planets.

Albert Einstein, by creating his theory if general relativity, was able to show that the "fall" of an apple and the orbital motion of the moon were examples of the same phenomenon. According to this theory, both objects are following a natural trajectory in curved space-time. Like an person in an orbiting space-craft, the maggot in the "falling" apple feels no acceleration, nor does it feel any weight, because in curved space-time they are not at any instant actually accelerating in the local frame of reference.

An apple "at rest" on the surface of the earth is really "wanting" to be on such a trajectory, but the surface of the earth keeps pushing it away from that line. That is why a hard seat becomes uncomfortable during a long and boring lecture on relativity. In a freely falling lift it would not be at all uncomfortable, until the lift hit the bottom of the shaft.

As a result of this theory, it became clear that gravitational waves should exist.

These have not been observed, but the equipment so far constructed would not have been likely to detect any waves from any known source. The waves have been detected indirectly by measuring the behaviour of certain binary stars, and seeing how the orbital energy is being lost. The discrepancy in the orbit of Mercury was entirely accounted for by the general theory of relativity, as was the curvature pf light rays near the sun, and the change in wavelength of light in a gravitational field.

Albert Einstein, in creating the special theory of relativity, showed that what was called mass and what was called energy were aspects of the same thing. In other words, scales could have been calibrated in Joules, or energy could have been measured in kilograms. In both cases this would have been extremely inconvenient, because the factor connecting the two sets of units is the square of the speed of light, which is a big value in the units we use.

Einstein was brilliant, and lucky. Suppose that eclipses had been very rare, or non-existent because the moon was too small. The deflection of star-light would not have been measured. Suppose also that the orbits of Mercury and Venus had been much more nearly circular. The precession of the orbit would not have been measurable. Checking general relativity would have been greatly delayed.

Charles Darwin and Alfred Wallace developed the theory of evolution by natural selection. According to this the mutability of species, known to every breeder of plants and animals, is driven by the variable probability of reproduction of individuals. It is sometimes summarised as "the survival of the fittest".

This theory enabled people to see a unified vision of the whole of life, together with a mechanism to justify it. It answered William Blake's deep question about the tiger and the lamb. Luckily there is room for Blake and Darwin and Wallace. Whether we see the world in terms of poetry or science, we are responding to its infinite and fascinating variety. Less poetically, and less dramatically, Blake could have asked whether worms and elephants were related, or snowdrops and gannets.

Blake did indeed ask one of the most profound questions about this planet, and he asked it in a few dramatic words about a tiger and a lamb. The answer given by Darwin and Wallace does not devalue Blake's poetry, nor anyone else's.

Is not the conciseness of poetry a part of its power to stimulate the imagination, just as apparently simple ideas and equations in science have opened up new views of nature that could previously imagined? In the same way, a few themes in the hands of a great composer can open the way to enormous development.

Beethoven's Opus 131 is not exactly full of tunes, but it seems hard to imagine anything containing more music per minute.

Some people have argued that evolution by natural selection, or "survival of the fittest", does not say anything, because it is tautologous, in that "fittest" is defined as those which survive to reproduce the best.

You could, perhaps, argue that the formula E = mc² says no more than that 1 kg = 2.20462 lb. You could, perhaps, argue that 1 + 1 = 2 says nothing, in that 2 is a only synonym for 1 + 1. You could argue that when you know all the rules of chess, all possible games are immediately implied, and that they could be simulated on enough fast computers.

You could argue that given all the notes of a scale or a mode, and a list of all possible musical instruments, all possible pieces of music are implied. You could argue that all possible routes up a rock-face are already present before they are created by climbers. These suggestions are not all philosophically or logically equivalent, and they cannot be answered here.

In any case, Louis Armstrong was right about it being a wonderful world. Making theories does not make it any less wonderful.

James Clerk Maxwell, by creating his theory of electromagnetism, showed that electrical and magnetic events were actually manifestations of the same basic system. As a bonus, it became clear that electromagnetic waves should exist, and that light is an example of them.

Maxwell wrote out his equations in quite a long form, but using vector notations they can be written as four very short equations - in fact they can be made even more concise. Work by Ampère, Coulomb, Faraday, Oersted, Volta, Weber, and others, was "reduced" to these equations. Reductionism is a somewhat unfortunate word. It sounds negative.

In fact, by "reducing" a vast body of experimental observation to a small number of consistent principles, deductions can be made that would probably not have been made from the original observations.

The creation of the theory provides immense power thereafter. Nothing has been reduced: nothing has been devalued.

We wouldn't complain that a poet reduced a beautiful lake-side landscape to a few lines about daffodils and clouds, because we can still see it the way we want to, even after poem has been written.

In a very different sense, many life-forms do the same at the time of reproduction - the entire information about the parents is carried forward in only two gametes, from whose genes a selection goes forward to help in creating the new individual. From this reductionist bottleneck comes the possibility of change.

An election is rather like this. The multitude of ideas, hopes and fears of a population are summarized in simple marks on paper, which are used to determine the composition of the next set of representatives.

Throughout science, reductionism, if understood in this positive sense, is a force for creation. By creating a few ideas that explain the data from a vast number of observations, a great mind can give others the power to create new predictions and new applications to useful devices.

Look at the number of lives that have been saved by the simple ideas of bacterium and virus, and the understanding of how they operate..

In George Orwell's book, "1984", a negative reductionism was used to shrink the language and so prevent people from thinking the thoughts that the authorities wished to stamp out. Any repressive government knows that stamping out a language is a step to stamping out a culture.

Science, too, permits only limited use of words, with rather precise meaning, but with the opposite intent, namely to enable greater understanding. And what riches are conveyed by a simple word like "wave", if you have even the smallest knowledge of physics.

Scientific reduction is not like 1984, not like that at all. It is enabling, not disabling. It is creative, not destructive. Its language is liberating, not restricting.

Yes, science has laws, and restrictive ways of doing things. But you don't improve chess, tennis, football or cricket by disobeying the laws, or changing them arbitrarily.

What could be more exciting than a brilliantly executed try, or a beautifully timed stroke of the bat, sending he ball swiftly to the boundary? Our pleasure in a sport is not diminished by knowing the rules. On the contrary, if we don't know the rules, the whole thing is meaningless.

This is perhaps not a fair analogy with science. A thrush is beautiful whether we know anything about biology or not. And the laws of science are different from the laws of cricket. The laws of cricket evolved in order to make possible an enjoyable game.

But the laws of physics, as we know them, are made by people trying to understand something of how the world works. We can never know if they are the "right" ones, or whether they the only ones that will do the job. And to scientists, the laws may be elegant in themselves - more so than the actual events they describe.

You would never say that about a game. If you found the rule-book more interesting than the game it would be very peculiar. Perhaps the laws of physics are a little like poetry - they attempt to describe some aspect of reality, but they are interesting in their own right.

There is one big difference. Poetry, like other arts, can describe an imaginary world, real to the creator, but not objectively existing. A good artists will make us believe in, or at least temporarily accept, her or his world. But a theory, however elegant, is a nice piece of mathematics, but valueless to science if it does not agree with the facts. That is an overstatement. Many ideas have been clearly inadequate when put forward, but have led the way to better ones, by pointing people's minds in a different direction from the old one.

Bohr's theory of the atom left much unexplained, but it was a tremendous achievement, and a great step forward.

Perhaps engineering can be taken as a bridge between science and art It uses mathematics and scientifically proven techniques, but it has to solve real world problems. The engineer cannot say - "Let the weight of the vehicles be zero." And the engineer can in some cases, such as bridge building, produce objects that we laymen can appreciate, though only slightly understanding how they work.

During the first half of the twentieth century, two independent formulations of quantum theory were created. Schrödinger invented his wave equation, which enabled people to calculate a wave function which give the probability of events occurring.

Heisenberg's method used the elements of matrices to achieve the same thing. And quantum mechanics has been a tremendous unifying force, enabling people to understand and enormous range of phenomena, and to create an enormous range of engineered objects.

More recently, in the 1960s, Abdus Salam and Stephen Weinberg built upon work done by Sheldon Glashow to create the electro-weak theory, uniting the electromagnetic theory and the weak interaction theory. Like the other unifications, the electro-weak made predictions. It predicted the possibility of neutral currents, in which both the strongly interacting particles and the weakly interacting particles would not exchange electric charge. It also predicted the existence of three new bosons - the Z⁰ and the W⁺ and W^-. All were found as predicted. What is a boson? Click for Kaons

In the classical field theories of Newton and Maxwell, a mass or a charge was thought to create around it a field which would create a force on another mass or charge. Modern theories take a completely different approach, which nevertheless gives the same results in the areas of validity of the older theories, while predicting extra phenomena.

The great hope of physicists is to unify the interactions they know about - the colour force between quarks, the electromagnetic forces, the weak force of beta decay, and gravitation, which Einstein's theory says is not a force at all.

Fermat's last theorem is easy to state. It says that there are no solutions to these equations, in which A, B, C and k are integers, when k is greater than 2 -

A^k = B^k + C^k

Although the theroem looks like a single statement, in another sense it makes an infinite number of statements, because k can have an infinite number of values.

This theorem was eventually solved by a unification of two apparently unrelated areas of mathematics, elliptic functions and modular forms.

In the 1950s, Goro Shimura and Yutaka Taniyama put forward the hypothesis that there was a profound relationship between elliptic functions and modular forms, but they could not establish a universal one-to-one matching.

The proof was completed in 1994 by Andrew Wiles. In completing this proof he also proved Fermat's last theorem. How? In 1984, Gerhard Frey had "created" a special elliptic equation, on the assumption that Fermat's last theorem was wrong. Ken Ribet proved that if it existed, it could not be modular.

So if it were proved that all elliptic functions must be modular, Frey's equation could not exist, and Fermat's theorem must to be true, because if it were false the equation would exist.

It is very simple to state, but it took many years for Wiles to prove it.

You can read about this amazing story in "Fermat's Last Theorem", by Simon Singh, published by 4th Estate.

The great physicist, Sir Arthur Eddington, worked on a theory that proposed to construct basic physics from a very few simple ideas. He thought that he could produce the fine structure constant, about 1/137, from a simple equation.

He thought that it was exactly 1/137, but it wasn't. In fact, he at first fitted his theory to 1/136, which was near the measured value, and he changed his theory when the measurements suggested that 1/137 was a better value.

He could have derived this number by writing his name, Arthur Stanley Eddington, taking a = 1, b = 2, etc, adding the resulting numbers together, and dividing by 2 (the electron gyromagnetic ratio) giving 137.

The time was not right for such a grand attempt at unification.

Will it ever be the right time?