Robert P. Crease

For a book about equations Robert Crease has pretty much managed to exclude equations. Yes there are the eight or so equations that he has singled out for special attention but other than this the work is very much descriptive and historical. Some of it I knew, some of it I actually knew in more detail (such as the derivation of Maxwell's equations) and some of it was very new and interesting to me.
He actually devotes quite a sizeable section of the book to a philosophical discussion about equations and how symbols were able to stand in for objects, (and eventually imaginary numbers) and how these equations could be related to the world.
The equations he has chosen to focus on are as follows......(and because I can't reproduce mathematical symbols in this review) I can do little better than to adopt the word descriptions that Crease provides.....they really are very good).
1. The Pythagorean theorem: "The square of the length of the hypotenuse of a right angled triangle is equal to the sum of the squares of the lengths of the other two sides". Pythagoras was given the credit but it's pretty certain that he learned it from his travels and studies in Mesopotamia. A Babylonian tablet has a table of examples such as the 3:4:5 that give a right angle....not quite the same as an equation however.
2. Newton's second law of motion. F=ma "A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed" Or force equals mass times acceleration. (Though apparently Newton didn't actually write this down as an equation...that was done by Euler almost a century later). Newton distinguished between the weight of a body (which would be different on the moon and on the earth...but still the same body) and its mass, which is independent of what gravity it happens to be in. .....I must admit that I learned this definition in high school: that mass is the quantity of matter contained in a body...but I could never really understand why/how this was different to the weight....yeah the teacher said stuff about spring balances etc., but I never really got it.
3. Newton's law of universal gravitation. "Gravity exists in all bodies universally, and its strength between two bodies depends on their masses and inversely as the square of the distance between their centres". There was a lot of jealousy and pettiness over who was responsible for this with Robert Hooke having some claim to setting Newton off in the right direction. Not sure if the falling apple story was the origin though it ws recounted by Newton himself to his friend William Stuckley. (Though Newton was a bit devious).
4. Euler's equation. "The base of a natural logarithm (an irrational number) raised to the power of pi (another irrational number) multiplied by the square root of negative one (an imaginary number) plus one is an integer: zero. I don't pretend to understand this but Euler's result demonstrated that imaginary numbers were at the very centre of mathematics. The numbers pi to power e, 2 to power pi, and e to power pi are all thought to be irrational but e to power ipi picks out that special place where rational, irrational and imaginary numbers mix in a way that spookily balances out to zero. Euler was responsible for the introduction of the greek symbol for pi, e for the base of natural logarithms, i for the square root of -1, f(x), sin, cos. He was truly an incredible mathematician.
5. The second law of thermodynamics: S'=S>=0. The entropy of the world strives towards a maximum. (Rudolf Clausius suggested the words: "The energy of the world is constant; the entropy of the world strives towards a maximum"). Crease tells quite a nice story of the vast cast of characters who contributed to new understandings of the nature of heat and energy culminating in this law. There is one illuminating insight which was provided by Rudolph Clausius where he suggests that there is a conservation of something (not heat, and soon called energy) in exchanges of heat and mechanical work.
6. Maxwell's equations. A complete characterisation of electromagnetism that, among other things, describes how changing magnetic fields produce electric fields; asserts there are no magnetic monopoles, describes how electric currents and changing electric fields produce magnetic fields; and describes how electric fields are produced. There are four equations (and too difficult to try and reproduce them here without the right fonts). Actually there is a much more detailed description of how Maxwell came up with his equations in the book "Fields of Force" by William Berkson.......but the current book maybe puts it into a better historical perspective. And Maxwell really developed his equations from a mechanical model (an analogy)...which he described as being similar to an orrery (model of the solar system) that when it is finished, you can often see more than you got from the pieces. I'm not sure that Maxwell ever really thought his model or equations really were the way things worked but (like the quantum equations) they gave the right answers. As Hertz (who demonstrated electromagnetic waves) said: "one cannot escape the feeling that these mathematical formulae have an independent existence and intelligence of their own, that they are wiser than we are, wiser even that their discoverers, that we get more out of them than was originally put into them". Maxwell originally came up with a large number of equations (20 in his publication of 1864 and 12 general equations in his publication 1873). These were condensed and made usable by Oliver Heaviside around 1888 and it's Heaviside's equations which are really used today.
6. Einstein's famous equation E+MC squared. Or energy and mass can be converted into one another with the amount of energy being equal to the mass multiplied by the speed of light squared. Crease has a nice historical description of the development of this equation by Einstein. His paper fo 1905 was based on two principles: relativity and the constancy of the speed of light. (This involved doing away with the concept of an ether). But basically the 1905 paper was about the contraction factor for moving bodies. Later Einstein realised that Maxwell's equations combined with there relativity principle requires that mass be a direct measurement of the amount of energy contained in a body. But it was not until 1912 that he published the formula is its current form. It was not until 1932 that Cockcroft and Walton measured mass loss and energy gain (lithium nucleus plus proton turning into 2 helium nuclei) which was accounted for by Einstein's formula.
7. Einstein's equation for general relativity. Space-time tells matter how to move, matter tells space-time how to curve. The equations were developed over the period from about 1907 (when Einstein wondered about whether the principle of relativity applied to systems accelerated with respect to each other).....and culminated in the publication in 1916 of his General Theory paper. (He'd actually exposed this to the Prussian Academy in November 1915).
8. Schrodinger's equation (1926) ...the basic equation of quantum theory. How the quantum state of a system ---interpreted, for instance, as the probability of a particle being detected at a certain location ---- evolves over time.
I've always struggles with Schrodinger's equation. (Yeah, I've read lots about Schrodinger's cat but really struggled with the equation). Crease, has a nice historical account of how the equation developed: from a paper by de Broglie who showed that you got the right orbits if you assumed that electrons had wavelengths. This was dismissed by Derbye who said if it was a wave you needed an equation. So Schrodinger developed an equation that incorporated a "wave function" called (Greek letter) Psi. Schrodinger initially visualised this as a kind of particle fog....with waves that added up and interfered etc....but ran into problems with the maths and imaginary numbers (indicating that there would be a hidden phase of the wave function). Max Born, working with his matrix determined that the Psi function only made sense if it was treated as the probability of states. Pauli took this further, declaring the Psi represented the probability of a particle (eg an electron) being at a particular position. But this came at a cost: The Schrodinger waves, wave in space and were predictable but lost their observability. The particles were observable but lost their predictability. (Note the wave function is reset when a measurement is made...it's not a collapse..according to Crease). But the wave is a probability: it is not a thing. Schrodinger eventually, but unhappily, accepted this interpretation of his equation...as probabilities....hence the cat example about the strange implications. I thought I would look up some YouTube explanations of the equation but found that there are almost unending versions of the equation (different symbols etc.) and I'm not sure that the people purporting to explain the "wave function" actually know what they are talking about anyway....I think they have also heard about Schrodinger's cat! So, moving right along .....
9. Heisenberg's uncertainty principle. 1927. Establishing the position of a particle in a small region of space makes its momentum uncertain, and vice versa, and the overall uncertainty is greater or equal to a certain amount. Then there is the great quote from Heisenberg: "Everyone understands uncertainty. Or thinks he does". He had just graduated as a doctoral student (actually nearly failing) and quantum physics was a complete mess so he started from the perspective that when the theorists construct models based on what experimenters measure, these models are only symbols of a reality that humans cannot picture. And progress in science might mean sacrificing our claim to understand nature. Basically Heisenberg gave up on trying to make a model of what was going on and just focused on the maths...producing a matrix (which didn't commute but Heisenberg didn't know about multiplying matrices). His colleagues, Born and Jordan managed to carry out the penetrating mathematical investigation ...proving in the process that there was a relationship to Planck's constant. Then Pauli was able to apply the matrix mechanics to the test case of the hydrogen atom. Initially there was considerable conflict with Schrodinger's "wave mechanics" and it was easier to use. Then Schrodinger proved that the two systems were mathematically equivalent. Still there was conflict about what the numbers really meant. In a letter from Heisenberg to Pauli, he said " if space-time is discontinuous, and electrons flit from one state to another, it must lack velocity by definition!" It was around February 1927 when Heisenberg wrote a 14 page letter to Pauli explaining the uncertainty principle ...if you looked at a particle at absolute zero, this meant bouncing a photon off it and this would disturb the electron's position.....basically the more precisely the position is determined ,the less precisely the momentum is known and vice versa. Late in 1927 Bohr and Heisenberg came to an uneasy truce about wave vs matrix ..and Bohr explained it as "complementary ways of speaking about something of which we can have no direct knowledge". Thus the beginning of the Copenhagen interpretation of quantum mechanics. But Einstein disagreed ...."every element of the physical reality must have a counterpart in physical theory". (Sounds reasonable to me but we don't seem to have been able to attain it). As Crease explains the current situation ...the wave formulations of quantum mechanics are thus not about an ideal object nor yet a real object but something more like a a semi abstract object that admits many potential experimental realisations in becoming a real object.This semi abstract object can appear wave-like or particle-like.
I actually found this book easier to read first time than when I came back to it to write this review. There is a lot there and it's not all easy to digest. But really worth the struggle. Happy to give it five stars.… (mais)

Published 2019, at the sweet spot where the history of ideas, the philosophy of science, and cultural critique meet. Crease’s project counterposes two groups of thinkers: those who advocated for the pursuit of science as a social good (Francis Bacon, Galileo, Descartes, Comte) and those who decried the corrosive effects of scientific culture (Vico, Mary Shelley, Weber, Husserl, Arendt).

As Crease reminds us, Vico saw that once democracy and a belief in the power of reason & science become paramount values, the social bonds created by ‘poetic wisdom’ would begin to break down in a ‘barbarism of reflection,’ where individuals can think only in the light of their own conceptual scheme.

…For such peoples, like so many beasts, have fallen into the custom of each man thinking only of his own private interests and have reached the extreme of delicacy, or better of pride, in which like wild animals they bristle and lash out at the slightest displeasure. Thus in the midst of their greatest festivities, though physically thronging together, they live like wild beasts in a deep solitude of spirit and will, scarcely any two being able to agree since each follows his own pleasure or caprice… (Vico, The New Science)

In “Politics as a Vocation,” Weber wrote that, once the scientific revolution gave rationality new power and ruthlessness, the only effective counterforce to rationalization and ‘disenchantment’ was the charismatic demagogue who stands with the people against the narrow view of technocratic ‘experts.’ The authority of science is no match for popular ‘convictions.’ Husserl, though his phenomenology was an extension of his study with the Göttingen mathematicians, came to see that the mathematization of science created the illusion that knowledge about reality was reality itself. Elevating the scientific explanation of the world (‘objective’ & ‘theoretical’) over the ‘nonscientific’ denigrates the subjective and the practical spheres of life and weakens the humanities. Those who value faith and feeling reject science as a detached, elitist practice.

Crease's thesis reaches a crescendo in the penultimate chapter on Hannah Arendt. The ‘cultural vacuousness’ described by Husserl made possible the career of a character like Eichmann, who 'owed his first job to family connections, pontificated in recycled stock phrases and clichés, borrowed then mangled ideas from others, and denied firmly established facts.' For Arendt, the crisis of antirationalism brings a deadly, sterile passivity that allows incompetents like Eichmann to dominate what’s left of the public space, ‘setting themselves up as umpires not simply of what’s allowable, but also of what’s just and what’s scientific.’ Crease points to Arendt’s essay “Truth and Politics,” where she notes the proliferation of religious and philosophical opinions—an indicator not of toleration but of confusion and disorientation, belief without a firm grounding in experience, so that ‘factual truth, if it happens to oppose a given group’s profit or pleasure, is greeted with great hostility;’ uncomfortable facts are turned into opinions, which are then opposed to other opinions. In 1971, Arendt wrote “Lying in Politics: Reflections on the Pentagon Papers,” in which she lamented the ‘nonrelation between facts and decision’: while the intelligence community had produced accurate factual reports, political leaders had lied about them. But the lie did not creep into politics by some accident, she said—

Lies are often much more plausible, more appealing to reason, than reality, since the liar has the great advantage of knowing beforehand what the audience wishes or expects to hear. He has prepared his story for public consumption with a careful eye to making it credible, where as reality has the disconcerting habit of confronting us with the unexpected, for which we were not prepared.

In the last chapter, Crease suggests some short- and long-term tactics for fighting science denial. And while the denial of climate science is his focus, reading between the lines gives The Workshop and the World an eerie resonance in the weird 2020 Real Now.… (mais)