Thursday, September 1, 2011

Alain Connes

This time when reading Karl Sabbagh's "Dr. Riemann's Zeros" I came across an interesting quote from Alain Connes on P.205. In commenting on the relationship between Geometry and Algebra he states:

'It really is fantastic step' he said, to understand that the square of a number - which is just a geometrical square - and the cube, which is just a geometrical cube - can be added together, even though you would say, "But one has dimension the length squared and the other the length cubed" and you would never add things which have different dimensions. So algebra is an amazing achievement, and once you have formulated things in an algebraic terms then they take on a life of their own.'

Algebra of course is incredibly important. However the price that has been paid in terms of such rational abstraction is that a basic form of reductionism is involved. In other words we can see clearly in geometrical terms that 2-dimensional is qualitatively distinct from 3-dimensional reality. However in algebraic terms this qualitative distinction is quickly lost with variables interpreted with respect to their mere quantitative meaning.

Indeed it is rather ironic that Connes in speaking about the Riemann Hypothesis in an a later quote on p.208 states:

"It is probably the most basic problem in mathematics, in the sense that it is the intertwining of addition and multiplication' Connes said. It's a gaping hole in our understanding, because until we really understand it we cannot say that we understand the line. Even the line itself is extraordinarily mysterious.'

May I suggest once again that the very reason why the link between addition and multiplication seems so intractable is precisely because the qualitative nature of variable transformation - necessarily involved in all multiplication - is formally ignored in present Type 1 Mathematics.

So properly understood there are both Type 1 (quantitative) and Type 2 (qualitative) aspects to all mathematical interpretation.

The Riemann Hypothesis in fact is a key statement regarding the relationship as between these two aspects.

Likewise with the line, Type 1 Mathematics, due to the same lack of a qualitative dimension, one cannot properly distinguish finite (discrete) from infinite (continuous) notions. So once again though the infinite notion is qualitatively distinct from the finite, in Type 1 interpretation it is necessarily reduced quantitatively in a finite manner!

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