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 As a returning reader... 
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Joined: 03 Feb 2009, 01:43
Posts: 1
Post As a returning reader...
Having recently finished reading "The Emperor's New Mind," I was very excited to purchase "The Road To Reality". Thus far, it has proven to be as interesting and eye-opening a read as TENM. The prose is clear and concise, the maths beautifully presented and the ideas impressive.
I am particularly intrigued by the notion of the genuine existence of mathematical concepts beyond human consciousness. These constants were found by us, not created... and I'm having to force myself not to sneak a peek ahead to find out what the "Axiom of Choice" might be!
Thank you, Mr. Penrose. Your work is among my greatest intellectual joys.

Christine


03 Feb 2009, 01:56

Joined: 29 Mar 2009, 18:16
Posts: 15
Post Re: As a returning reader...
Having recently finished reading "The Emperor's New Mind," I was very excited to purchase "The Road To Reality".
ME. I reasoned as you did.

Thus far, it has proven to be as interesting and eye-opening a read as TENM.
ME. RTR is much more ambitious than TENM.


The prose is clear and concise, the maths beautifully presented and the ideas impressive.
ME. RTR can be clear (I wish this were more often true). Concise is an adjective I cannot apply to a book
nearly 1100p long. As for the beauty and elegance of his presentation, I defer to mathematician and
theoretical physicists. That said, I prefer that math be set out as first order theories, although I hasten to add
that many working mathematicians, and all but a handful of theoretical physicists, do not think that way.


I am particularly intrigued by the notion of the genuine existence of mathematical concepts beyond human consciousness. These constants were found by us, not created...
ME. Penrose is a fairly ardent Platonist. I am excited by the radically anti-Platonic approach of Lakoff and Nunez, who argue that mathematics is entirely invented by humans for human purposes. We can never know what nature is truly like; we can only talk about how we humans perceive nature. Mathematics helps us organize and communicate those perceptions. In short, maths is created not found.

Math as a man-made symbolic language has been spectacularly successful, albeit almost entirely the preserve of boffins. The culture of mathematics is in large part to blame, by making math texts too difficult and abstract, and by teaching only to bright students.


and I'm having to force myself not to sneak a peek ahead to find out what the "Axiom of Choice" might be!
ME. Metamathematics, axiomatic set theory, and ontological issues posed by set theory and mathematics
are not Penrose's strong points. The axiom of Choice is one of those things that has perplexed set theorists.
I do not see why theoretical physicists need bother with it.

Set theory is the theory of collections, and nature consists of finite collections of elementary particles. Setting aside the dark matter and energy that have come to exercise cosmologists, nature consists of packages of up and down quarks called protons and neutrons, and of electrons, photons, and neutrinos. (The other particles exist only in high energy environments and decay rapidly.) Moreover, I was astounded to discover that the visible universe contains only about 10^80 protons and the same number of electrons. And about 10^90 photons. In my view, these orders of magnitude are manageably finite. I am skeptical of the multiverse hypothesis, but if it is true, it is conceivable that there are more universes than there are electrons, protons, and photons combined in our universe! Konrad Zuse, Edward Fredkin, and Stephen Wolfram, and others have argued for a Theory of Everything grounded in finite maths.

One could easily suppose that set theory could contribute to physics on the deepest level, and perhaps other levels as well. In fact, it has not; sets whose members are concrete objects are just not all that interesting. Things get a lot warmer when the notion of set is applied to infinitely many abstract objects... Mathematics and physics do not yet share a common ontology. Tegmark thinks that we will one day see a grand unification of pure maths and theoretical physics, and he has convinced Paul Davies. I have a lot of time for Tegmark, but his bold idea is a research programme, not conventional wisdom.

I wish I could tell you where to turn to learn more about set theory from the point of view of physics and metaphysics. Mary Tiles wrote an undergrad text on the philosophy of set theory. It's cheap, because Dover prints it, and I own it. But I cannot really recommend it. Keith Devlin's heart is in the right place, but I cannot vouch for his The Joy of Sets.


Last edited by roger desmoulins on 06 Aug 2009, 07:54, edited 1 time in total.

26 Apr 2009, 08:41

Joined: 07 May 2009, 16:45
Posts: 62
Post Re: As a returning reader...
"Metamathematics, axiomatic set theory, and ontological issues posed by set theory and mathematics are not Penrose's strong points."

No offence, but who the *** are you to say that metamathematics and axiomatic set theory are not the strong points of the Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford?


01 Aug 2009, 19:28

Joined: 29 Mar 2009, 18:16
Posts: 15
Post Re: As a returning reader...
It is quite common for working mathematicians (and mathematical physicists) to take little interest in set theory and mathematical logic. Mathematicians have told me over afternoon tea that they just can't get excited about "foundations." To many working mathematicians, fussing about set or category theory is pedantry that does not move the discipline forward.

Penrose does not convey the expressive power of first order logic, and of axiomatic set theory as a first order theory. RTR does not even state the ZFC axioms. On mathematical logic, I invite you all to peruse the texts by Mendelson, or by Boolos and Jeffrey. In TENM, Penrose lingers a bit over Godel's theorem, but I don't find it especially enlightening. The indeces to NETM and RTR do not mention Peano or Robinson arithmetic. Gregory Chaitin's name is absent from both books.

Let there be three axioms defining a least number (0 by convention) and the notion of successor of a number. Let there be 4 more axioms giving the conventional recursive definitions of addition and multiplication of numbers. If the resulting formal system (Robinson arithmetic) is consistent, it cannot be complete (so that there are true statements about RA that cannot be proved within RA), and is also incompletable (there does not exist a finite set of additional axioms that can transform RA into a complete formal system). Over the past 30-40 years, Boolos, Chaitin, Smullyan and others have devised short proofs
of these and related theorems (eg., Tarski's), applicable to even simpler formal systems. Expositions of Goedel's results (which are surely among the greatest human achievements of modern times) almost never touch on what i write in this paragraph. In this regard, Goedel Escher Bach was a great missed opportunity.

I should grant that buried in RTR's huge reference list is a bit of work on formulating slices of physics as axiomatic theories. This topic fascinates me, but I grant that very very few physicists take any interest in that approach.


01 Aug 2009, 20:16

Joined: 07 May 2009, 16:45
Posts: 62
Post Re: As a returning reader...
Well, it is obvious you are a very educated person, and talking over my head.

Your first response seemed to suggest that Penrose was out of his depths in discussing notions such as the "Axiom of Choice" because he was just a theoretical physicist, and should not worry his head about such things. Your second admits he is a mathematical physicist, however a pedestrian "working" one, and is not concerned about such matters. I wanted to say he his a brilliant Mathematician, with a capital M, who has done a lot of important work in pure mathematics, as well as physics, and who indeed is interested in the "foundations" as you call them, but would seem to take a different view on these things from you, and perhaps therefore does not discuss your favorite theorists in his book.

As far as pursuing other texts, I am still trying to surmount this one (RTR), being just now still four chapters from where he promises to talk about the "Axiom of Choice"

And what did you mean by mangageably finite?


01 Aug 2009, 22:10

Joined: 29 Mar 2009, 18:16
Posts: 15
Post Re: As a returning reader...
Manageably finite: For at least 20 years, the hardwired floating capabilities on desktop computers have ranged from at least 10^-310 to 10^310. The order of the monster group in group theory, the largest exceptional group, is about 10^53. The string theory "landscape" is hypothesized to have an order around 10^500. Given these facts, that the number of electrons and protons in the observable universe is about 10^80, and that the number of photons is about 10^90, are things I feel we can take in stride -- thus "manageable." Another way of putting it is that doing physics apparently only requires the antechamber to Cantor's Paradise: aleph null and the continuum. And if Konrad Zuse, Edward Fredkin, and von Weiszacker are right, the universe is a finite state machine!

I nominate Penrose as the greatest living British mind, and the greatest since Newton, Maxwell, and Dirac. He trained as a mathematician, and moved to mathematical physics because of his encounter with Dennis Sciama, whom I see as the greatest physics Ph.D. supervisor of the last century. I am very very warm to Penrose's disagreements with string theory, and wish his dear twistors well. That said, set theory and math logic are not his specialties, a fact his books for general readers clearly reveal. And RTR is for general readers only as a manner of speaking, though!! Until David Deutsch's 1997 book, it was not at all clear what significance, if any, math logic had for fundamental physics. Hence Penrose's weakness in the area is entirely forgivable.

Max Tegmark, in my opinion, has a vision of how the entire corpus of maths could marry physical theory, that is clearer than Penrose's. (I think that Tegmark's vision also has its flaws.) "Entire corpus" includes set theory and first order logic. Here's a link to a recent article by Tegmark that I deeply admire:

http://arxiv.org/abs/0704.0646

Elsewhere on this forum, I have a long post taking RTR to task for its weakness re Noether's theorem and particle physics, which I see as keystones in the arch of physics, as an 8 lane wide part of the "road to reality."


Last edited by roger desmoulins on 06 Aug 2009, 07:57, edited 2 times in total.

02 Aug 2009, 02:01

Joined: 29 Mar 2009, 18:16
Posts: 15
Post Re: As a returning reader...
[quote="DimBulb"]Well, it is obvious you are a very educated person, and talking over my head.

ME. Try reading Moshe Machover's 1996 Cambridge Univ Press text. I think that undergrad math logic is easier than the real analysis or differential geometry required to understand modern physics.


Your first response seemed to suggest that Penrose was out of his depths in discussing notions such as the "Axiom of Choice" because he was just a theoretical physicist, and should not worry his head about such things.

ME. Rather, the Axiom of Choice as part of the superstructure of set theory, and a part having no clear relation to physics, except that Choice is needed to prove a number of theorems mathematical physics requires.



Your second admits he is a mathematical physicist, however a pedestrian "working" one, and is not concerned about such matters.

ME. I did not intend the adjective "working" as derogatory.



I wanted to say he his a brilliant Mathematician, with a capital M, who has done a lot of important work in pure mathematics, as well as physics, and who indeed is interested in the "foundations" as you call them, but would seem to take a different view on these things from you, and perhaps therefore does not discuss your favorite theorists in his book.

ME. Penrose came to mathematical physics from geometry, especially differential geometry (DG). His intellectual godfather is perhaps the great French mathematician Elie Cartan. Hence Penrose's deep and admirable passion for GR and cosmology, fields that make stunning use of DG. I sense that Penrose has worked hard to come to terms with quantum theory and the group theory physics requires. I am less confident of his full intuitive grasp of that jewel of modern physics, quantum field theory.

DG is fairly high up the mathematical food chain, when compared to FOL and axiomatic set theory, fields where his comfort level is not all that high. Evidence: on p. 381 of RTR, Penrose cites on the foundations of mathematics Abian (1965) and Wilder (1965). Wilder's book is not very technical; the author was a point set topologist who reinvented himself as a philosophy of maths guy late in his teaching career. Abian's book was a nice undergrad text in its day, but there are better references on this topic. One I like is Michael Potter's text. Another is the 1997 ed. of Eliot Mendelson's text.

I should grant that in 2 chapters of TENM, Penrose does betray an acquaintance with some parts of math logic, specifically those parts that descends from the pathbreaking work of Alan Turing. But these parts of TENM find no echo in RTR, whose bottom line seems to be "string theory is misguided, and loop quantum gravity and twistor theory deserve more respect than they have gotten to date." Take out twistors, and you have, more or less, the bottom line of Smolin's The Trouble with Physics. Here I completely agree with Penrose and Smolin.



As far as pursuing other texts, I am still trying to surmount this one (RTR), being just now still four chapters from where he promises to talk about the "Axiom of Choice"

ME. I am elated to discover that there are people like you who are struggling with RTR on their own. That's the intellectual equivalent of climbing the Matterhorn in winter! That takes serious mental effort unless you are an advanced PhD student in maths or physics! That said, RTR is NOT for general readers, and neither you nor I are capable of "surmounting" it.

As for other texts, I feel very fortunate to have recently acquired a used copy of the Schaum's Outline for Differential Geometry.


02 Aug 2009, 02:28

Joined: 24 Sep 2009, 23:21
Posts: 1
Post Re: As a returning reader...
Quote:
To many working mathematicians, fussing about set or category theory is pedantry that does not move the discipline forward.


If that really is the case, then "many working mathematicians" are complete goofheads, since much of the work in foundations is every bit as much a triumph as, say, Wiles' proof of Fermat's Last Theorem. Moreover, foundational issues do occasionally rear their heads in "higher level" maths; the Banach-Tarski paradox being a classical example of the difficulties involved with the axiom of choice, to name just one area.


24 Sep 2009, 23:32

Joined: 29 Mar 2009, 18:16
Posts: 15
Post Re: As a returning reader...
I interact with working academic mathematicians, and they assure me that they are not much interested nowadays in math logic, pure set theory, category theory as a foundational system, philosophy of maths, etc. They prefer to think about parts of maths that are directly useful to science and engineering. I have a friend working on constructive foundations for point set topology. His colleagues take no interest in what he does, even though it constitutes a radical breakthrough. Errett Bishop used to joke that he could not imagine a constructive variant of topology. well, here it is.

The 80th birthday of Godel's landmark paper will soon be upon us, and I still don't sense that most academic philosophers and mathematicians understand it. Any formal system capable of representing Robinson arithmetic cannot be both consistent and complete. Such a system also cannot define its own truth predicate (Tarski 1936). The latter result is curiously easy to prove.


25 Sep 2009, 08:56
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