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Mill was educated at home by his father, who started him on Greek so early that he couldn't remember a time when he wasn't studying it. He was reading the Greek classics by the age of 8, and writing articles for the press by the age of 16. When he was 14 he spent a year in France with friends of his father and learnt French fast enough to attend University lectures in the sciences. Mill did not consider himself of exceptional ability, holding that anyone else with the same opportunity could have achieved as much, a striking example of the once fashionable discounting of heredity epitomised in Locke’s doctrine of the blank slate, a doctrine that still has echoes in contemporary thought.
Last Autumn I re-started by going back about 100 pages, and having more or less sorted out that material I managed to finish a week or two before Easter.
Penrose’s tome - I shall not repeat its monstrous title, is very strange. Apart from its great length it is quite exceptionally obscure. Although it has the great merit of presenting materials mathematically, the potential advantages of that bravery are largely unrealised because of the eccentric way the Mathematics is presented. Penrose clearly wanted to write a ‘popular’ book and presents the mathematics casually in an eccentric and non-rigorous way. I find that even when the material is already familiar to me, it is often hard to recognise what I already know in what he says and when he touches on any Mathematics I’ve half forgotten or, worse, have never known, I find his presentation completely baffling, and have to consult university text books, which always turn out to be much easier to follow. Even worse, Penrose also makes mistakes. For instance he misdefines normal subgroup, making nonsense of his subsequent use of the concept. Someone coming to the book with no prior knowledge of the material would be completely lost.
However, struggling through the volume was for much of the time an interesting puzzle, and Penrose has some intriguing ideas. I rather like the idea that Galileo replaced an Aristotelian concept of space-time as a Cartesian product of time and space by treating it as fibre bundle of spatial fibres over a temporal continuum. I shall look for an opportunity to drop that thought into a conversation.
Penrose has given me an inkling of how quantum theory contradicts general relativity. It is possible to imagine the celebrated Shrodinger cat experiment so that the arrival of a particle at a receptor switches on a mechanism that moves a large mass. That mass might therefore be consider to exist in the superposition of two quantum states. A quantum state is a solution of a partial differential equation in which differentiation is respect to time. However, according to general relativity, space-time is affected by the presence of mass, so that for each position of the large mass there is a different space time. Hence there is no time common to the two positions of the mass, which contrary to quantum theory, cannot represent dissolutions of the same equation.
I have also detected a serious error in Penrose’s argument. From time to time he compares infinite sets, asserting some to me much greater than others, when they are in fact of the same cardinality. In particular he does not realise that, for finite n, the set of points in an n dimensional space has the same cardinality as the set of points on a line. He seems also unaware that, if Z is infinite, and k a finite integer, kZ has the same cardinality as Z. (see for example page 883)
Penrose is sceptical about string theory, suspecting it is a blind alley. He considers that far too much research is devoted to it, to the almost complete exclusion of work on alternative theories, and that research is entirely theoretical. String Theory has yet to suggest any experiment that might be used to test it.
There are several mathematical errors. Leavitt misdefines the Riemann zeta function, and wrongly states the four colour theorem, making it affirm that any plane map requires 4 colours, whereas the theorem actually states that some plane maps require four colours (very easily shown) but no map requires more ( the hard part).
There are also errors about people in Cambridge. E. M. Forster is referred to as being in residence in King’ when Turing went up to Cambridge in the early 1930’s, whereas Forster, although a fellow, did not take up residence until after his mother died in 1945.
Noting that Turing expressed belief in telepathy in an article in Mind, Leavitt expressed astonishment that Turing can have come to believe anything so absurd, and wondered what the readers of Mind might have made of that. Those questions are easily answered.
The basis of Turing’s belief will almost certainly have been the work on trance mediumship done by various Cambridge dons in the 1920's and 30's, and the celebrated Soal Shackleton Experiments that convinced many previously skeptical people, until they were shown in the 1980’s to be the result of fraud by Dr. Soal, who has simply altered the records after the experiment took place. So far as readers of Mind were concerned, many would not have thought Turing’s opinion at all remarkable because they would have shared it. C. D. Broad, at the time one of the professors of Philosophy at Cambridge, was also one of the leading members of the Society for Psychical research. Turing will almost certainly have met him.
The book was interesting to me mainly because it mentioned a number of people I knew quite well, especially Norman Routledge who was for a while my co-director of studies and sometimes invited me to his parties, and Richard Braithwaite. Forster and Pigou were also still about in my time, though I never spoke to the latter, and exchanged only a few words with the former.
This was the Millikan of the famous oil drop experiment to determine the charge on the electron. The autobiography is another of the books that have sat unread on my bookshelves for decades, but now I have at last got round to reading it I’m fascinated, not least by Millikan’s recollections of his early life in the rural mid-West, where his grandfather killed the beasts, tanned the leather, and then used it to make the family’s shoes.
Particularly strange is Millikan’s story of how he came to be a physicist.. After leaving school he enrolled in Oberlin College (which appears to have been in Ohio), where he sent his first two years studying Trigonometry, Analytical Geometry and Greek, but in his second year also took a single 12 week course in Physics, which he found disappointingly dull. At the end of his second year his Greek professor asked him to teach the introductory course in Physics the following year. When Millikan pointed out that he knew no Physics Professor Peck replied ‘Anyone who can do well in my Greek can teach Physics’. Millikan therefore taught himself physics from text books, taught the course, went on to teach more Physics and, finding the text books inadequate wrote better text ones and eventually won a Nobel prize for Physics.
I’ve had this book for many years, dipping into it quite often, but now, for the first time, I read it from beginning to end.
My past dippings concentrated on material about prime numbers in the middle and latter part of the book, and I’d completely neglected much of the material in the early chapters and suspect that is why I originally found much of the later material so hard.
I now know what Farey series are ! In the past I’d often come across references to them, but there always seemed to be other topics that were more interesting. Now I’ve found out what they are I find them quite interesting and linked more closely to the rest of Mathematics than I had expected. While not earth shaking, they provide a congenial subject for meditation while waiting at the bus stop or doing the washing up.
It is instructive to compare Hardy and Wright with Penrose. Both are quite tough, but I find Hardy and Wright more rewarding because, after moderate thought, the puzzling parts eventually become clear. The proofs offered are often more like notes with the aid of which one can construct a proof, but if I think long enough I always can, at least so far - page 281 out of 416. Behind the obscurities of Penrose’s treatment there sometimes seems to be little more than the assumption that his readers are not clever enough ever to understand.
It is said of Hardy that once, in the course of a lecture, he remarked that something was obvious, but then paused and said ‘I shall have to think about that’ and left the lecture room to ponder the point. After a few minutes he returned to the lecture theatre, said ‘Yes, it was obvious’ and resumed his lecture.
I used to think that was just a joke, until I had similar experiences myself while reading the book. Several times, when puzzling over one of Hardy’s proofs I’d be quite baffled for a while, wondering how he reached a particular conclusion - (Hardy is quite parsimonious with signposting and frequently does not say from which earlier step a particular statement follows) then, suddenly, I’d see that the puzzling statement was simply obvious, usually just equating two different ways of writing the same expression, so that it needed no justification. The Theory of numbers seems to lead to an abundance of such statements.
Edited by H. G. Alexander, this consists of ten letters, five in each direction, between Leibnitz and Dr. Samuel Clarke, printed together with a number of extracts from the works of Leibnitz and Newton.
The correspondence began with a letter from Leibnitz to the Princess of Wales criticising Newtonian Physics on theological and philosophical grounds. The Princess recruited Dr. Clarke to reply. Much of the controversy concerns gravitation and space. I plan to write an essay on space and time and hoped that the correspondence will provide useful background material. It was quite easy reading, at least compared to the Theory to Numbers which I read in parallel with it.
Leibnitz objected to Newton’s assumption of absolute space and time, asserting that space cannot exist without matter because there can be no such thing as empty space. He also objected to Newton’s atomism, on the ground that it involved the supposition that different atoms of the same substance were identical. Leibnitz believed in the principle of Identity of Indiscernibles, according to which it is impossible for there to be two objects that are identical.
This is a lively book and easy reading, but the quality is uneven.
He misunderstands the so called First Cause argument for the existence of God, because he does not seem to realise that it originated with Aristotle, who used ‘cause’ in a more general sense than we do, to refer to anything that explains a change. Aristotle went on to postulate an uncaused cause, which was uncaused because it was unchanging. Christian theologians later tried to identify that supposed uncaused cause with the Christian God, a grotesque error that has inspired much confusion. Dawkins realised that something was wrong with their arguments but, apparently unaware of the background, was unable to say precisely what was wrong.
Dawkins tries to explain religious belief in terms of the theory of memes, a sort of intellectual analogue of genes. Although his remarks are interesting, I didn’t find them entirely convincing because I‘m not sure of the status of memes..
The treatment of morality is rather disappointing. He mishandles the argument that, without religion, there would be no basis for religion, and delivers himself of the absurdity:
‘Moral Philosophers are the professionals when it comes to thinking about right and wrong’ (p. 265)
Ferdinand Gregorovius Lucrezia Borgia
This is another of Father’s books, which I came upon recently while browsing through my book collection.
Lucrezia was the daughter of Luis Borgia, who eventually became Pope Alexander VI, though he was only a cardinal when he fathered her. Luis was appointed a cardinal at the age of 27 by his uncle, Rodrigo Borgia, Pope Calixtus III.
Alexander's son, Cesare, appointed a cardinal by his father when only 17 years old who was Machialelli's hero, and who employed Leonardo da Vinchi as military engineer, tried to etablish for himself a kingdom in North east Italy.
Lucrezia herself seems to have been less wicked than I had supposed and tried to stop Cesare murdering her second husband, to free her for a more advantageous marriage. Marrying Lucrezia to one nobleman after another was an important part of Alexander's and Cesare's strategy to extend their power in Italy and beyond. The numerous stranglings and poisonings were mostly organised by Cesare, with the connivance of his Papal father.
I don't think the Borgias were mentioned at all in school history - a pity, theirs was just the sort of tale young lads would enjoy.
A university Physics text book I borrowed from the City Library - it is quite rare for me to read a library book these days. Robert Mills was co-author of the celebrated Yang Mills Field Theory. He died in 1999 aged 72 - a pity. I now feel I almost understand Special relativity, including the twins paradox, and have half an inkling about general relativity too: I think I've got to the bottom of E = mc^2, it seems to be a constant of integration. I also have a clearer idea than before about Entropy as a measure of disorder. At school I was introduced to it as integral dQ/T along any reversible path, which makes it just a useful invariant.
Something quite new to me was Nöther's principle, that to every physical principle invariant under a suitable set of transformations, there corresponds a conservation law. details here.
Finally, Mills even explained why 'i' appears in Schrödinger's equation. All the other books I've read on the subject just produce the equation with no indication of how anyone ever came to think of it.
The joy of partial enlightenment was slightly diminished by several references in later chapters to the divine. Trying to estimate how long it would take a universe in a state of disorder, to reach by random changes a state of order comparable to its present state (he thought about 10^200 years) Mills seems to have been so flumuxed by the very big number that 'Good God' was the only response he could manage.
As well as brillaintly explaining its subject matter, Space, Time and Quanta, gives a fascinating and rather depressing insight into the state of American Education. Writing for first year undergraduates who planned to become Physicists or engineers, Mills felt it unsafe to assume any previous knowledge of calculus, of elementary mechanics,or even oftrigonometry, feeling obliged to explain even sine and cosines before using them. However, at least he uses Maths instead of just waffling.
The first edition of this page contained a list of all the books I’d read since setting up this web site and threatened to grow to a preposterous length. is page described only one year's reading. I’ve moved all the material about books I finished reading before the end of 2006 to this document.
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