Weyl on mathematics vs. reality

From Bas C van Fraassen Scientific Representation: Paradoxes of Perspective

p. 208 “Herman Weyl expressed the fundamental insight as follows in 1934:

‘A science can never determine its subject-matter expect up to isomorphic representation. The idea of isomorphism indicates the self-understood, insurmountable barrier of knowledge. […T]oward the “nature” of its objects science maintains complete indifference.’ (Weyl 1934:19)

The initial assertion is clearly based on two basic convictions:

o  that scientific representation is mathematical, and
o  that in mathematics no distinction cuts across structural sameness.”

p. 209 “Weyl illustrates this with the example of a color space and an isomorphic geometric object. … The color space is a region on the projective plane. If we can nevertheless distinguish the one from the other, or from other attribute spaces with that structure, doesn’t that mean that we can know more that what science, so conceived, can deliver? Weyl accompanies his point about this limitation with an immediate characterization of the ‘something else’ which is then left un-represented.

‘This – for example what distinguish the colors from the point of the projective plane – one can only know in immediate alive intuition.’ (Ibid.)”

p. 210 “We seem to be left with four equally unpalatable alternatives:

o  that either the point about isomorphism and mathematics is mistaken, or

o  that scientific representation is not at bottom mathematical representation alone, or

o  that science is necessarily incomplete in a way we can know it to be incomplete, or

o  that those apparent differences to us, cutting across isomorphism, are illusory.

In his comment about immediate alive intuition, Weyl appears to opt for the second, or perhaps the third, alternative. But on the either of this, we face a perplexing epistemological question: Is there something that I could know to be the case, and which is not expressed by a proposition that could be part of some scientific theory?”



My answers

04.11.2012 08:18

Let us imagine that we have a mathematical model that isomorphic with the whole reality. Let us say that this model is before you as some computer implementation. The problem of coordination still remains. To use this model, you need to find out its particular part and relate it with reality. The model of the whole reality does not do it by itself.

05.11.2012 20:32

Yet, even after the theory has been developed (let us imagine that the science has included in its model the dark energy, the dark matter and have found a way to make GR and QM compatible), one needs to take a decision what a particular part of the theory is necessary to drive a particular innovation. Even a complete scientific theory will not do it by itself. In this sense, it will be still incomplete.

05.11.2012 22:19

>What exactly is ‘coordination’ and why is it a problem?

An analogy would be using a map. One needs for example to locate oneself in a map. This could be generalized.

Let us consider how an engineer for example uses Maxwell equations. An engineer starts with a design. This design could be described by Maxwell equations but one needs an engineer to suggest the design. Maxwell equations on their own are not enough.

>I don’t understand the problem; are you simply saying the model of reality is not reality itself?  That seems rather trivial.

In a way. A scientific model is after all a representation. And a representation is

p. 21 “Z uses X to depict Y as F

Hence even a complete scientific theory does not contain “Z uses”. This remains somehow outside of even a complete Theory of Everything.

In a way it is trivial, I agree. Yet, it seems for example Hawking in his Grand Design does not agree with such a trivial observation.