I spent about 4 months, on and off, and finally finished reading this great book. I have a dual purpose: (a) I wanted to quickly recover my knowledge in math and physics I acquired during my prior physicist career, and (b) I wanted to see if I could apply anything I learnt from here to the machine trading models as introduced in the book "Forecasting and Timing Markets: A Quantitative Approach." I really enjoyed this book and indeed found a lot of similarities and dissimilarities between building mathematical models for interpreting the real physical world and building models for forecasting market.
Here is a summary of what I have found out to be very applicable and useful:
(1) p.7: What laws govern our universe? How shall we know them? How may this knowledge help us to comprehend the world any hence guide its actions to our advantage? ... Eventually, even the much more complicated apparent motions of the planets began to yield up their secrets, revealing an immerse underlying precision and regularity.
(2) p.18: Fig. 1.3 Three 'worlds' - the Platonic mathematical, the physical, and the mental - and the three profound mysteries in the connections between them. ... everything in the physical universe is indeed governed in completely precise detail by mathematical principles. ... all actions in the universe could be entirely subject to mathematical laws.
(3) p.28: Euclid's first postulate effectively asserts that there is a (unique) straight line segment connecting any two points. His second postulate asserts the unlimited (continuous) extendibility of any straight line segment. His third postulate asserts the existence of a circle with any centre and with any value for its radius. Finally, his fourth postulate asserts the equality of all right angles.
(4) p.45: We are to think of a light, straight, stiff rod, at one end P of which is attached a heavy point-like weight, and the other end R moves along the asymptote.
(5) p.67: The system of complex numbers is an even more striking instance of the convergence between mathematical ideas and the deeper workings of the physical universe.
(6) p. 109: What about the places where the second derivative f''(x) meets the x-axis? These occur where the curvature of f(x) vanishes. In general, these points are where the direction in which the curve y = f(x) 'bends' changes from one side to the other, at a place called a point of inflection.
(7) p. 115: Armed with these few rules (and loads and loads of practice), one can become an 'expert' at differentiation without needing to have much in the way of actual understanding of why the rules work! This is the power of a good calculus.
(8) p.151: Air, of course, consists of enormous numbers of individual fundamental particles (in fact, about 10^20 of them in a cubic centimeter), so airflow is something whose macroscopic description involves a considerable amount of averaging and approximation. There is no reason to expect that the mathematical equations of aerodynamics should reflect a great deal of the mathematics that is deeply involved in the physical laws that govern those individual particles.
(9) p. 211: It seems that Nature assigns a different role to each of these two reduced spin-spaces, and it is through this fact physical processes that are reflection non-invariant can emerge. It was, indeed, one of the most striking unprecedented discoveries of 20th-century physics (theoretically predicted by Chen Ning Yang and Tsung Dao Lee, and experimentally confirmed by Chien-Shiung Wu and her group, in 1957) that there are actually fundamental processes in Nature which do not occur in their mirror-reflected form.
(10) p. 217: For example, the configuration space of an ordinary rigid body in Euclidean 3-space is a non-Euclidean 6-manifold.
(11) p.223: As in Sec 10.2, we have the notion of a smooth function (Phi), defined on manifold M.
(12) p.388: He (Newton) had originally proposed five (or six) laws, law 4 of which was indeed the Galilean principle, but later he simplified them, in his published Principia, to the three 'Newton's laws that we are now familiar with.
(13) p. 390: It is remarkable that, from just these simple ingredients (Newton's formula GmM/r^2), a theory of extraordinary power and versatility arises, which can be used with great accuracy to describe the behavior of macroscopic bodies (and, for most basic considerations, submicroscopic particles also), so long as their speeds are significantly less than that of light.,
(14) p. 392: Galileo's insight does not apply to electric forces; it is a particular feature of gravity alone.
(15) p. 410: We shall also begin to witness the extraordinary power, beauty, and accuracy of Einstein's revolutionary theory.
(16) p. 412: The geometries of Euclidean 2-space and 3-space are very familiar to us. Moreover, the generalization to a 4-dimensional Euclidean geometry E^4 is not difficult to make in principle, although it is not something for which 'visual intuition' can be appealed to.
(17) p. 455: Einstein's famous equation E = mc^2 tells us that mass and energy are basically the same thing and, as Newton had already informed us, it is mass that is the source of gravitation.
(18) p. 462: Einstein originally introduced this extra term, in order to have the possibility of a static spatially closed universe on the cosmological scale. But when it became clear, from Edwin Hubble's observations in 1929, that the universe is expanding, and therefore not static, Einstein withdrew his support for this cosmological constant, asserting that it had been 'his greatest mistake' (perhaps because he might otherwise have predicted the expansion of the universe!). Nevertheless, ideas once put forward do not necessarily go away easily. The cosmological constant has hovered in the background of cosmological theory ever since Einstein first put it forward, causing worry to some and solace to others. Very recently, observations of distant supernovae have had most theorists to re-introduce /\ (greek lambda), or something similar, referred to as 'dark energy', as a way of making these observations consistent with other perceived requirements.
(19) p. 466: The timing of these signals is so precise, and the system itself so 'clean', that comparison between observation and theoretical expectation provides a confirmation of Einstein's general relativity to about one part in 10^14, an accuracy unprecedented in the scientific comparison between the observation of a particular system and theory.
(20) p.490: He (Hilbert) appears to have believed that his total Lagrangian gives us what we would now refer to as a 'theory of everything'.
(21) p. 503: ... it took many years for Einstein's original lonely insights to become accepted.
(22) p. 523: Heisenberg's uncertainty relation tells us that the product of these two spreads cannot be smaller than the order of Planck's constant, and we have Delta-p Delat-x >= h_bar / 2.
(23) p. 528: I denote Schrodinger evolution by U and state reduction by R. This alternation between these two completely different-looking procedures would appear to be a distinctly odd type of way for a universe to behave!
(24) p. 541: As the state of the arts stands, one can either be decidedly sloppy about such mathematical niceties and even pretend that position states and momentum states are actually states, or else spend the whole time insisting on getting the mathematics right, in which case there is a contrasting danger of getting trapped in a 'rigour mortis.'
(25) p.686 (Chapter 27 The Big Bang and its Thermodynamic Legacy): What sorts of laws shape the universe with all its contents? The answer provided by practically all successful physical theories, from the time of Galileo onwards, would be given in the form of a dynamics - that is, a specification of how a physical system will develop with time, given the physical state of of the system at one particular time. These theories do not tell us what the world is like; they say, instead: 'if the world was like such-and-such at one time, then it will be like so-and-so at some later time'.
(26) p.687: The usual way of thinking about how these dynamical laws act is that it is the choice of initial conditions that determines which particular realization of the dynamics happens to occur. Normally, one thinks in terms of systems evolving into the future, from data specified in the last, where the particular evolution that takes place is determined by differential equations.
(27) p.689: What about evolution into the past, rather than the future? It would be a fair comment that such 'chaotic unpredictability" is normally much worse for the 'retrodiction' that is involved in past-directed evolution than for the 'prediction' of the normal future-directed evolution. This has to do with the Second Law of thermodynamics, which in its simplest form basically asserts: Heat flows from a hotter to a cooler body. ... This procedure of dynamic retrodiction is clearly a hopeless prospect in physics. ... For this kind of reason, physics is normally concerned with prediction, rather than retrodiction.
(28) p. 760: Of course, it might indeed ultimately turn out that there is simply no mathematical way of fixing certain parameters in the 'true theory', and that the choice of these parameters is indeed such that the universe in which we find ourselves must be so as to allow sentient life. But I have to confess that I do not much like that idea!
(29) p. 850: But to take this position is to part company with one of the basic principles of Einstein's theory, namely the principle of general covariance.
(30) p. 935: ... A lot of these stem from the fact Einstein's theory is 'generally covariant' (Sec 19.6).
Finally, I have to say that I really like so many drawings in the book, which are simplistic yet stupendously expressive. Thanks Professor Penrose for sharing your knowledge and achievements of many decades, which will benefit many on this planet called Earth!
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An erudite but failed effort, far from self-contained: ostensibly targeted at the 'educated layman'
Reviewed in the United States on April 25, 2018
Let me start off by saying (its relevance will soon be revealed) I have a bachelor of science in mathematics and a master's in computer science (and I wasn't in the bottom 50% of the class), and after the first 300 or so pages (out of 1200) the math in this book (and it's at least 40% or more straight math, not text, and often without text explaining the math) is way above my head and is left often undefined in the text. The author doesn't even do the courtesy of pointing the reader to textbooks where these concepts, such as pseudo-Riemannian geometry and anti-de Sitter spaces and Seiberg/Gromov-Witten manifolds, are defined and can be learned.The book fails in its promise and purpose to be a self-contained guide to the current mathematical- or theoretical-physical understanding of the universe. Required prerequisites: understanding of linear algebra (Lie, Poisson, Frobenius, Ricci calculus), scalars and higher rank TENSORS (and MORE tensors), several varieties of noneuclidean geometry (Minkowski, de Sitter, Riemann), scalars, topology and n-manifolds, group theory (Lie groups), gauge theory, etc., or the willingness to learn these from expensive secondary sources, because Penrose will not teach you them here and the arguments of the book are incomprehensible without them. Without them, one would be reduced to skimming the 20% of the book that is text (especially the final chapter, which is comprehensible to any semieducated layman) and taking the author's word for the rest of it. Just about the only thing he explains in full is twistor theory (his own invention).It is far from accessible to the layman (I have postgraduate training in math and I was a good student and its inaccessible to me), and to grasp the concepts in this book, I'd have to spend probably a year of free time and a thousand or more dollars in secondary sources (if I bought them used and cheap). I bought this book to get a $20 overview (like Collier's 'A Most Incomprehensible Thing' for the theories of relativity [I prefer the original 'invariance'], which was technical but self-contained and comprehensible; reading that is the only thing that gave me any knowledge at all of tensors, which this book is chock full of): what I got was in essence a 1200 page bibliography without the authors being noted and without the important works being starred.This is a very ambitious book which fails utterly in execution.The author goes from explaining what complex and irrational numbers are and why they are useful (this is freshman high school math) in the introduction to pseudo-Riemannian geometry (this is postgraduate pure math) 200 pages later. He spends about five pages defining all of classical mechanics, and then assumes that you understand classical mechanics. This same breakneck pace is kept up throughout, which is how he manages to range from logarithms and complex numbers to doctoral-level mathematics in 500 or 600 pages. Once he goes out of the pure math and back to applied math (i.e.. physics proper) it gets a little easier but I'd still not recommend trying to tackle this book with less than a bachelor's degree in math (if you're a math nerd and keep your knowledge up) or a master's in math or physics or some other strongly quantitative discipline (if you're not), or a self-taught prodigy in pure maths.The book promises to be a self-contained guide to the best mathematical understanding of the universe we have, but it ends up more like the author just stuck the important theorems in with a minimum of explanation (he does hit almost all of them: one thing that struck me as unnecessarily erudite - showing off - and odd was the statement of Maxwell's field equations, which is mathematically simple and elegant, in terms of tensors, which are very, very difficult), so it's a complete guide if you already know all of the math (in which case you don't need the book): it's much more of a refresher and quick reference for people who already are familiar with and understand (or at one time understood) the concepts the author represents.See attached pictures (representative pages from 250-, and these are not nearly the most difficult of the equations): No, you're not the only person going 'smh, wtf' at that math. (Not to mention yet again that many of the terms are never defined in-text! The author goes from explaining that he'll have to use logarithms in the introduction, to this stuff which is Chinese to me as a math major, within a few hundred pages. It seems Penrose let his mathematical understanding [brilliance for all I can tell, I have no idea what he's saying] run away with itself after writing the preface for a book where he's apologetic about using logs and then sticking that preface on this work.)I still have to award two stars for the obvious intensity and depths of erudition which Penrose funneled in to this work, but only two because it doesn't even partially fulfill its stated purpose or self-description.*I have familiarized myself additionally with pure math concepts like number theory, group theory, field and Galois theory and combinatorics, along with stochastic calculus and linear algebra to a degree. I am even partially comprehending of tensors after reading 'A Most Incomprehensible Thing', a lay mathematicians' introduction to relativity. In other words, I'm an educated lay mathematician, and the stated target audience for this book is 'educated laymen' with AP high school or gen ed college math in general.
Reviewed in the United States on April 25, 2018
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