New Theories of Everything

Alas, no.
Over the years I have read several Barrow books: The Book of Nothing, Impossibility, Pi in the Sky, etc. Maybe even the first edition of this one (I don’t have it, and seem to remember it but very hazily, but that might be a consequence of Barrow’s writing essentialy one book under several titles, an impression of mine probably deriving from the fact that he tackles metaphisically entangled themes such as infinity, being, the nature of reality, TOEs, etc., which in my view are intimately related).

“New TOEs” is in my opinion a somewhat obscure and defective book, because (1) the first edition hasn’t been rewritten but addded to; (2) it’s an uneven mixture of dumbing down and illusory depth; and (3) Barrow has, not a golden, but a leaden (or iron, were we to follow Hesiod) pen.

(1) NOT REWRITTEN BUT ADDED TO: in page 3 he writes as if the 20th C had still to end; in his first summary of superstrings (“ss”) (p. 24) he doesn’t mention M theory, an omission which he makes good in page 32 ff., but without including the landscape problem: this is fleetingly alluded to only once in the whole book (p. 133), as contrasted to the constant references to eternal inflation and bubble Universes; there’s constant emphasis on the heat death of the Universe whereas the acceleration of expansion (pp. 130/133, oddly introduced as a “rival Theory of (almost) Everything” to ss in p. 129) is treated only once; an unclear graph extends only to 1988 (p. 170); “if ss theory manages to produce some observable prediction in the not too distant future” (p. 224); etc. etc.

(2) UNEVEN MIXTURE OF DUMBING DOWN AND ILLUSORY DEPTH. I’ll give just two examples (they take space), although there are many others: in pp. 46/50 Barrow discusses (unnecesarily in my view) the transfinite numbers -by the way there he states, mistakenly, that “the real numbers possess a higher cardinality than the natural numbers and it is denoted by … (aleph-one)”, when actually neither Cantor nor anybody else managed to prove that ‘c’, the power of the continuum, equals aleph-one, which is the cardinal of the first uncountable ordinal-. This is conceptually, for a layman, quite advanced stuff; yet elsewhere in the book he finds it necessary to define angular momentum as the total rotational energy of a body. Now, is it conceivable that a person who doesn’t know what angular momentum is will be at ease and indeed understand four pages on denumerable and non-denumerable cardinals?
The other example is in page 228, where we are told that power series expansion and the “implicit function theorem … define … what local information about the world can be deduced from global … information”, and that “Stoke’s famous integral theorem and the process of analytic continuation” are examples of the converse. Now, I know the meaning of these terms because I studied real and complex analysis in college. But a layman? For me, this information is unnecessary; for a layman (I suppose) unintelligible. So, whom is the book adressed to?

(3) BARROW HAS, NOT A GOLDEN, BUT A LEADEN (OR IRON, WERE WE TO FOLLOW HESIOD) PEN. Where to begin? At random: in p. 57, speaking of oscillating infinite series, we find the baffling statement "the limiting value of a sum must be specified together with the procedure used to calculate it".
In p. 70 the fall of a rock is described in such a confusing way that I had to spend some time figuring what he must have wanted to say so that the paragraph would not be incorrect.
In p. 79 he asserts that "the Newtonian Universe will not tolerate the consideration of an infinite space distributed with matter: this leads to an infinite aggregate of gravitational influences at any one point": what does he mean by that? does he refer to an isotropic Universe? is it a restatement of the reverse of Olber's paradox for gravity? Because Newton's answer was that, as the Universe was infinite and therefore symmetrical (around the Earth, for example) influences cancelled out. Does Barrow mean that in a Newtonian cosmology the Earth would be torn apart by infinite gravitational forces coming from everywhere, from the "space distributed by matter"? Would each one of us be sucked towards the outer limits of the Universe (but not torn apart, because the gradient wouldn't be as steep as near a black hole's central singularity?)?
In p. 226 what I assume to be Taylor's power series expansion is described with such an unusual terminology (I mean, "mathematical operation upon an input x" is perfectly acceptable, but why not say "function" when in pp.220/222 he mentions "Riemmanian geometry and tensors"; "Groups", "Hilbert spaces" and "Complex manifolds"?) and notation that I'm left in doubt as to what he wanted to convey.
In p. 227 we learn that "the world is non-local. This is the import of Bell's famous theorem". I don't doubt that Barrow knows what he's talking about, but that's not Bell's theorem. Eddington, he of verily the golden pen, would have put it differently.
Well, enough for me. Am I nitpicking? But all this from a tense-challenged (p. 98) mathematician!

There's a fourth point, but that depends on personal tastes: dwelling so extensively on time and its arrow, entropy, thermodynamics and the heat death, etc., I would have liked Barrow to have said something about the problems of recurrence, Bolzano's worries, Poincaré's theorem, etc. In the case of Wheeler-DeWitt's equation and Hartle-Hawking state, I would also have liked something said about loop quantum gravity. Idem about background independence (there's only one line about it).

The book's strong points are its emphasis on philosophy of math and phy; the clear if brief treatment of Einstein's cosmological constant; the mention of Xia's result about Newtonian mechanics (pp. 30/31, interesting because its resulting invalidation parallels GR's by the prediction of black hole singularities); the apt titles of some section headings: "The eternal golden braid", "The importance of being constant", "Goodbye to all that", etc., which, if really Barrow's, show culture and a wry sense of humour; the inclusion of all the "sexy" problems in cosmology, with the fourth point caveat and excepting the COBE and WMAP probes (but they really have little to do with the book's main thrust); and the very moderate space given to M "theory".

For me the book rates three stars, but I learned nothing new, and it wasn't particularly enjoyable, so one star less for the loss of time.