The Road to Reality: A Complete Guide to the Laws of the Universe

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Penrose asks us to consider if the world of mathematics is in any sense real. He claims that objective truths are revealed through mathematics and that it is not a subjective matter of opinion. He uses Fermat's last theorem as a point to consider what it would mean for mathematical statements to be subjective. He shows that "the issue is the objectivity of the Fermat assertion itself, not whether anyone’s particular demonstration of it (or of its negation) might happen to be convincing to the mathematical community of any particular time". Penrose introduces a more complicated mathematical notion, the axiom of choice, which has been debated amongst mathematicians. He notes that "questions as to whether some particular proposal for a mathematical entity is or is not to be regarded as having objective existence can be delicate and sometimes technical". Finally he discusses the Mandelbrot set and claims that it exists in a place outside of time and space and was only uncovered by Mandelbrot. Any mathematical notion can be thought of as existing in that place. Penrose invites the reader to reconsider their notions of reality beyond the matter and stuff that makes up the physical world. Still though, it’s remarkable, and it will, if it’s commercially successful bring up a whole generation of young intellectuals with a grasp of some concepts (mainly geometrical) that were previously viewed as very advanced…because, let’s face it, not many of the useful concepts of modern mainstream mathematics have been brought down to the popular level (“topology is rubber-sheet geometry” isn’t a useful concept, it’s an unmotivated generalization (well…locale theory is a pointless generalization…)). A mathematical proof is essentially an argument in which one starts from a mathematical statement, which is taken to be true, and using only logical rules arrives at a new mathematical statement. If the mathematician hasn't broken any rules then the new statement is called a theorem. The most fundamental mathematical statements, from which all other proofs are built, are called axioms and their validity is taken to be self-evident. Mathematicians trust that the axioms, on which their theorems depend, are actually true. The Greek philosopher Plato (c.429-347 BC) believed that mathematical proofs referred not to actual physical objects but to certain idealized entities. Physical manifestations of geometric objects could come close to the Platonic world of mathematical forms, but they were always approximations. To Plato the idealized mathematical world of forms was a place of absolute truth, but inaccessible from the physical world. For sheer fun, find and read Penrose’s early paper on the appearence of a moving relativistic object.

The Road to Reality: A Complete Guide to the Laws of the The Road to Reality: A Complete Guide to the Laws of the

Penrose is a great and highly original guy, because of his contributions to GR, twistors, his triangle, his tiling, and so forth, but this kind of prayer is really bizarre. I have not seen the book. As a general matter of philosophy though, I very much agree with Penrose’s point of view about Kaluza-Klein. You’ve got enough trouble dealing with the metric degrees of freedom of space-time. You’re just making things worse when you add in a dynamical metric for the fibers of your principal bundle or for some internal space. This sonuds to me like a strong claim about what a candidate “fundamental theory” (if such a thing exists) *must* be able to explain. (I am not saying that you believe in the existence of such a theory, but rather pointing out what you seem to be demanding from a candidate.) Penrose’s comments about higher dimensional theories were made in the context of a criticism of string theory, so I don’t think it is unreasonable for me to discuss them in that context. If you have another context in which you want to discuss these issues, you’ll have to make it explicit. Sorry for the shift of topic, but I couldn’t resist mentioning some interesting new and recent papers:To explore the process of pursuing mathematical truth, Penrose outlines a few proofs of the Pythagorean theorem. The theorem can be stated as such, "For any right-angled triangle, the squared length of the hypotenuse [math]\displaystyle{ c }[/math] is the sum of the squared lengths of the other two sides [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] or in mathematical notation [math]\displaystyle{ a The full conception of Plato's theory of forms was not limited to only mathematical notions. Mathematics was linked to the concept of Truth but Plato was also interested in the absolute idealized forms of Beauty and Good. Beauty plays an important role in many mathematical discoveries and is often used as a guide to the truth. Questions of morality are of less relevance in this context but are critical with respect to the mental world. Moral debates are outside of the scope of this book but must be considered as science and technology progress. Penrose notes that figure 1.3 has purposely been constructed to be paradoxical in the sense that each world is entirely encompassed by the next. He writes "There may be a sense in which the three worlds are not separate at all, but merely reflect, individually, aspects of a deeper truth about the world as a whole of which we have little conception at the present time." This is my interpretation of Penrose statement. Essentially, these states have large numbers of symmetries aka Solitons. The normally postion of the mathematical community regarding highly symmetric spaces is that they are essential CHAOTIC! This is because of problems of embedding conformal manifolds into real spaces. Meaning Penrose is untrustworthy. But surely Emperor’s New Mind has already proven that. Interested in why the Lunsfords, Voits haven’t lumped him in with the celebrity stringies they so detest. So much for objectivity I guess. Penrose essentially claims that his and Hawking’s singularity theorems also apply in this higher dimensional case. If you want the details, you have to take a look at the book, although Tony Smith just posted a relevant abstract.

The Road to Reality: A Complete Guide to the Laws of the The Road to Reality: A Complete Guide to the Laws of the

Maybe at a later time you will speak to this in more detail? This clarifies to me the essence of your resistance to other theoretical approaches and helps to point towards more information to be look at. This is good. Anyone who wants to can look at what Penrose has to say about this stability issue, and then debate whether it makes sense or not. I’m just not interested enough in the question to spend time on this. Derivations based on string theory have a logically consistent foundation, but they only apply to special solutions in unrealistic world models, and they do not explain the simplicity and generality of the results inferred from the other methods[4, 5]…

As for objectivity, I notice that people like Schreiber, Helling, Motl and Distler criticize LQGists essentially for missing conformal anomalies (this is a clear symptom if not the cause of the problem), but they have no interest in trying to understand why 4D diff anomalies do not arise in string theory. Note here that anomalies are physical effects seen in any reasonable quantization scheme – path integral quantization of the Polyakov action also singles out 26D, i.e. the conformal anomaly does not only arise in canonical quantizion. to draw the obvious conclusion, Wilczek seems willing to entertain reservations about current attempts to join quantum mechanics and general relativity and to go out on his own looking for new ones, as in the case of this paper.

The Road to Reality : A Complete Guide to the Laws of the

I think it was fairly obvious that the anonymous poster was talking about Penrose’s claims on the classical stability of KK spacetimes encountered in the string theory literature. I think it is legitimite to dwell on what the precise objection here is. I don’t think it is acceptable to show a tendency to sweep the issue under the rug if it turns out that this particular objection of Penrose’s turns out not to be so well-founded, but put flashing banners if there is the slightest possibility that it might be a valid objection. This is not how scientists should work, though unfortunately similar tendencies prevail in both the string theory camp and the anti-string theory camp. (Penrose’s objection *might* be well founded, I still don’t understand what the precise objection here is.) Finally, I’ll draw these threads together more tightly by citing the following paper by Dowker, Henson, and Sorkin:I dont know what Wilczek thinks or what he has said about extra dimensions. But here is something that could help round out the picture. Wilczek is evidently interested in quantum gravity and has just posted this paper with Sean Robinson Now correct me if I am wrong, or if I am not even right?..but Penrose clearly leaves the doors and windows open for ‘a breath of fresh air’, a humble way to entice the reader, whatever her/his previous thoughts were, you cannot help but wonder and reason? To me, at least, it sounds very awkward when a theoretical (hep-th or math-ph) physicist dismisses math as much STheorists do nowadays… and, before this last comment starts a flame war, let me just say that i only read about Donaldson Polynomials, Knot theory, gerbes and so forth on books either by the AMS (on QFT!) or by Kauffman or Baez. Personally, i never saw a single STheorist (mainly the ‘pop’ ones) talking about those topics; in fact, in more than one occasion i have been condemned for ‘breaking up a discussion’ with such ‘mathematics’ topics.