Where a dictionary proceeds in a circular manner, defining a word by reference to another, the basic concepts of mathematics are infinitely closer to an indecomposable element", a kind of elementary particle" of thought with a minimal amount of ambiguity in their definition. Alain Connes
With such a statement, the "purity of thought," is speaking to a much more schematic understanding as we discuss the sociological thinking of mathematicians and the worlds they fantasize about? While deeper in reality the thought process(meditative) was engaged at a very subtle level, associated with the energy all pervasive.
Lee Smolin :
Another wonderful spin-off is that it turns out that the charge of the electron is related to the radius of the little circle. This should not be surprizing: If the electric field is just a manifestation of geometry, the electric charge should be, too.THE TROUBLE WITH PHYSICS-Published by Houghton-Mifflin, Sep. 2006/Penguin (UK), Feb. 2007, Page 46
In "Star Shine," we start from a very large circle, but there is much to see from this circle, when we consider it's radius. We think "continuity" is somehow not involved, if we freeze this circle, and call it a discrete measure of the universe's age? Yet we know to well that the motivation of this universe from a "distant point" measure today entropically lives in the multitude of complexities?
Model apprehension is part of the convergence that Lee Smolin and Brian Greene talk about, and without it, how could we look at nature and never consider that Einstein's world is a much more dynamical one then we had first learned from the lessons GR supplied, about gravity in our world?
On page 47 of the Trouble with Physics Lee goes on to say further down the page:
Unfortunately, Einstein and the other enthusiasts were wrong. As with Nordstrom's theory, the idea of unification by adding a hidden dimension failed. It is important to understand why.
If all one had was the "cosmological view" one could be very happy about the way in which his observations have been deduced from the measures of our mechanical means, that we say that GR is very well suited.
Yet it has been through th efforts of reductionism that we have said, "hey there is indeed more depth to the views we have, that the mechanical measures are being tuned accordingly?"
The strings move in a five-dimensional curved space-time with a boundary. The boundary corresponds to the usual four dimensions, and the fifth dimension describes the motion away from this boundary into the interior of the curved space-time. In this five-dimensional space-time, there is a strong gravitational field pulling objects away from the boundary, and as a result time flows more slowly far away from the boundary than close to it. This also implies that an object that has a fixed proper size in the interior can appear to have a different size when viewed from the boundary (Fig. 1). Strings existing in the five-dimensional space-time can even look point-like when they are close to the boundary. Polchinski and Strassler1 show that when an energetic four-dimensional particle (such as an electron) is scattered from these strings (describing protons), the main contribution comes from a string that is close to the boundary and it is therefore seen as a point-like object. So a string-like interpretation of a proton is not at odds with the observation that there are point-like objects inside it.
While energy is being exemplified according to the nature of the particles we see in calorimetric design, what said that the energy here is not topologically smooth in it's orientations? Even we we move our views to the quantum regime.
Maybe having solved the "Continuum Hypothesis," we learned much about Einstein's inclinations?
The surface of a marble table is spread out in front of me. I can get from any one point on this table to any other point by passing continuously from one point to a "neighboring" one, and repeating this process a (large) number of times, or, in other words, by going from point to point without executing "jumps." I am sure the reader will appreciate with sufficient clearness what I mean here by "neighbouring" and by "jumps" (if he is not too pedantic). We express this property of the surface by describing the latter as a continuum.Albert Einstein p. 83 of his Relativity: The Special and the General Theory
Even Einstein had to add the "extra dimension" so we understood what non-euclidean views meant in a geometrical sense. I again refer here to Klein's Ordering of Geometries so one understands the schematics and evolution of that geometry.