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Thoughts from Einstein, Bohr, Schrödinger, Heisenberg, Weyl, Dirac, Wigner, Feynman, Hilbert, Riemann, Russell, Whitehead, Gödel, Connes and many others!
How did Gödel prove his conclusions? Up to a point, the structure of his demonstration is modeled, as he himself noted, on the reasoning involved in one of the logical antinomies known as the "Richard Paradox," first propounded by the French mathematician, Jules Richard, in 1905 [...] The reasoning in the Richard Paradox is evidently fallacious. Its construction nevertheless suggests that it might be possible to "map" (or "mirror") meta-mathematical statements about a sufficiently comprehensive formal system into the system itself. If this were possible, then metamathe- matical statements about a system would be represented by statements within the system. Thereby one could achieve the desirable end of getting the formal system to speak about itself—a most valuable form of self-consciousness.
~Nagel & Newman
Mathematics has introduced the name isomorphic representation for the relation which according to Helmholtz exists between objects and their signs. I should like to carry out the precise explanation of this notion between the points of the projective plane and the color qualities [...] the projective plane and the color continuum are isomorphic with one another. Every theorem which is correct in the one system Σ1 is transferred unchanged to the other Σ2.
~Weyl
We begin with a piece of late-19th-century physics. The vacuum Maxwell equations for the electric and magnetic fields E and B,
have a symmetry under
that has been known nearly as long as the Maxwell equations themselves. This symmetry is known as duality.
~Witten
While a proper understanding of M-theory still eludes us, much is now known about it. In particular the various geometric results that have emerged from string theory become related in interesting but mysterious ‘dualities’ whose real meaning has yet to be discovered.
~Atiyah
The principle of duality in projective geometry states that we can interchange point and line in a theorem about figures lying in one plane and obtain a meaningful statement. Moreover, the new or dual statement will itself be a theorem -- that is, it can be proven. On the basis of what has been presented here we cannot see why this must always be the case for the dual statement. However, it is possible to show by one proof that every rephrasing of a theorem of projective geometry in accordance with the principle of duality must be a theorem. This principle is a remarkable characteristic of projective geometry. It reveals the symmetry in the roles that point and line play in the structure of that geometry.
~Kline
