A Brief Guide to the Great Equations by Robert Crease

(some reading notes)

Equations can be born from several different kinds of dissatisfactions. Some spring from a scientist’s sense that a confusing heap of experimental data can be better organized. Others arise from the feeling that a theory is too complicated and can probably be simplified, or that its parts are not fitting together properly. Still other dissatisfactions arise from mismatches between a theory’s predictions and experimental results.

‘Perhaps we should construct a whole new mechanics, of which we only succeed in catching a glimpse…in which the velocity of light would become an impassible limit.

Henri Poincaré

Ethnographers say that when two cultures interact, they do not meet all of a piece but through ‘congeners’ through which certain members of one culture look at, try to understand, and respond to the other. Congeners can include artifacts, rituals, practices, and art; fear, fascination, and exoticism usually play a role. A congener is like a little lens that allows the members of the one culture to approach the other culture in a focused way, to get an introductory grip. A congener is thus more than a symbol or logo of the other culture, but guides and disciplines curiosity and fascination into a first interaction with it.

A young physicist named Herbert Goldstein – who told me the story – was sitting next to his mentor and former colleague from the MIT Radiation Laboratory, Arnold Siegert. ‘Pais’s theory is far crazier than Ehrenhaft’s’, Goldstein asked Siegert. ‘Why do we call Pais a physicist and Ehrenhaft a nut?’ Siegert thought a moment. ‘Because’, he said firmly, ‘Ehrenhaft believes his theory.’

the Moses and Aaron model

Science critics, according to Winner and Ihde, would have an essential function. They would be there to assess the impact of science and technology on our political world (Winner) and on the human experience (Ihde). So, for example, Winner writes about the ‘politics’ of technological artifacts, while Ihde writes about the transformation of experience by instruments. The kind of criticism advocated and practiced by Winner and Ihde, in short, judges the presence of science and technology in society, and has clear moral and political dimensions.

impedance matching

there is another, complementary model for science criticism, one that involves another kind of interpretation, outlining the impact of scientific discoveries on our understanding of ourselves, the world, and our place in it.

Philoponus is also the first person known to have actually experimented with falling bodies of different weights, discovering, as Galileo would a thousand years later, that they fall at approximately the same rate.

John Philoponus (‘Lover of Hard Work’, ca. 490–570)

‘Axioms, or the Laws of Motion.’

‘Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.

‘A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.’

‘To any action there is always an opposite and equal reaction.’

In the ancient and medieval world, the exploration of physical influences among heavenly bodies, and between the heavenly bodies and objects on earth, was generally called ‘astrology.’ But we must not confuse this with the current socially acceptable form of bigotry that seems to entitle the human beings who believe in it to prejudge the character of others based solely on their dates of birth.

‘eorum omnium actiones in se invicem’, or ‘the actions of all these on each other.

when once asked how he made discoveries such as the law of gravitation: ‘By always thinking [about] them’, Newton said. ‘I keep the subject constantly before me and wait till the first dawnings open little by little into the full light.’

‘The energy of the world is constant; the entropy of the world strives toward a maximum.’ A popular formulation in simple language is: ‘You can’t win. You can’t break even, either.’

Sir Edmund Whittaker referred to ‘Postulates of Impotence’, which assert ‘the impossibility of achieving something, even though there may be an infinite number of ways of trying to achieve it.’

Maxwell’s equations described a new kind of phenomenon – the electromagnetic field – that was unanticipated by Newtonian mechanics. These equations characterized this new phenomenon completely. They also predicted something novel: the existence of electromagnetic waves that could travel through space. And the understanding of electromagnetism that grew out of these equations helped transform it from a curiosity into a structural foundation of the modern era, embodied in electronic equipment and in any device based on electromagnetic waves, including radio, radar, television, microwave, and wireless communication.

Faraday, in his mind’s eye, saw lines of force traversing all space where the mathematicians saw centres of force attracting at a distance: Faraday saw a medium where they saw nothing but distance: Faraday sought the seat of the phenomena in real actions going on in the medium, they were satisfied that they had found it in a power of action at a distance impressed on the electric fluids.

Maxwell

Maxwell’s was one of the most brilliant uses of analogy in the history of science, and it helped to bring about one of the most surprising and decisive transformations in civilization’s history.

Oliver Heaviside (1850– 1925), a self-taught electrical engineer, eccentric, and maverick (and discoverer of what was once called the Heaviside layer and now called the ionosphere), who is often called ‘the last amateur of science.’

Heaviside’s version of Maxwell’s equations were quickly and gratefully adopted by prominent electromagnetic researchers, including Hertz, and the entire scientific community converted by the 1890s.

‘anosognosia’

The Bohr-Kramers-Slater paper is a unique document in the history of science. It is renowned among historians for being both obviously wrong and strongly influential. It was strongly influential because it brought to a head the conflict between particle and wave theory. It said: This is the kind of sacrifice you have to pay in order to keep what you have. The partisans of each side were only being cautious and conservative, trying to preserve those elements of classical theory which they thought most robust. But quantum phenomena were resisting.

The Schrödinger wave waves in configuration space. The particles were observable but lost their predictability; the waves were predictable but lost their observability.

The Schrödinger equation implied a radical change in the events on the world-stage. No longer could it be assumed that, eventually, you could plug in numbers, turn the crank, and get predictions. Instead, you plugged in numbers, turned the crank, and got…probabilities.

‘The real fun of life’, he wrote, ‘is this perpetual testing to realize how far out you can go with any potentialities.’

Richard Feynman

Einstein often uses the same phrases to refer to the aims of science and religion. ‘I maintain that the cosmic religious feeling is the strongest and noblest motive for scientific research… A contemporary has said not unjustly that in this materialistic age of ours the serious scientific workers are the only profoundly religious people.’ And again, ‘The most beautiful experience we can have is the mysterious. A knowledge of the existence of something we cannot penetrate, our perceptions of the profoundest reason and the most radiant beauty, which only in their most primitive forms are accessible to our minds – it is this knowledge and this emotion that constitute true religiosity; in this sense, and in this alone, I am a deeply religious man.’

Wasn’t the lesson of Maxwell’s path to his equations that sometimes one had to abandon mechanical explanations to capture reality?

momentum (p) and position (q)

Born, Heisenberg, and Jordan entitled ‘On Quantum Mechanics II’, known to historians of physics as ‘the three-man paper.’ Its central feature is what they called the ‘fundamental quantum-mechanical relation’, the strange equation pq – qp = Ih/2πi. The paper is a landmark in the history of physics, for it is the first map of the quantum domain.

In all these discussions, there was never any question that the mathematics was correct. It was the interpretation that was at issue, and even the nature of interpretation. Bohr demanded more of interpretation than Heisenberg, and both demanded more than Schrödinger.

‘There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.’

Bohr

The Copenhagen interpretation – that somewhere beyond or beneath the macroscopic world lurks something that we cannot visualize, and that is made visualizable by an ensemble or arrangement of things whose behaviour is macroscopic – amounts to a clear, logical interpretation, and appears to be the simplest one consistent with all experimental and theoretical constraints. It is an interpretation that makes us all uncomfortable, but that is a psychological phenomenon, not an argument for or against the interpretation.

The idea of intermediate kinds of reality was just the price one had to pay. – Werner Heisenberg

The uncertainty principle is incomplete in a different sense. It is a mathematical relation, and a feature of the statistical interpretation of the wave function in quantum mechanics. It makes no reference to any underlying physical picture; there are no references to waves or particles, nor to physical experiments. It is not obvious what it refers to, except possibly the clicks of a detector. Yet it is about gaps in the world itself. These gaps are not epistemological but ontological; having to do not with our knowledge but with the world.

We can bring the strange home, and bring it home with precision. – Stephen Dunn, Walking Light:Memoirs and Essays on Poetry

It is tempting to think that there is some pre-existing structure embedded in nature that we are only discovering and translating into mathematical language – that equations are descriptions rather than interpretations or creations. But how we translate depends on the journey we have already taken, on our dissatisfactions with it, and on how we responded to those dissatisfactions.

Lynn Thorndike, ‘The True Place of Astrology in the History of Science’, ISIS 46 (1955), p. 273.

E. A. Burtt, The Metaphysical Foundations of Modern Science (Garden City, NY: Doubleday, 1954), p. 64.

James Clerk Maxwell, A Treatise on Electricity and Magnetism