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Roger Penrose – Mathematical Realism and Platonism in Tension with Principles of Reason

March 22, 2021

A common theme in the work of Roger Penrose in theoretical physics is a strong predilection towards a mathematical form of realism, in which physical nature at the fundamental level mimics mathematical structures.

In the case of Penrose he considers structures such as fractals and those provided by complex numbers, and with a more subtle mathematical theory, you can describe his Twister theory approach to Quantum gravity.

The existence itself of singularities is a consequence of the mathematical theories and models, and in many of these areas there is a deep faith that mathematical structure and the structure of reality coincide at the most fundamental level.

It remains to be seen how correct this view is and how far it can be taken. For instance, once more it is his strong faith in following the mathematical structures where they lead, rather than being distracted by other more usual physical and mechanical intuitions about nature, that leads him to his cyclic cosmos approach with Conformal Cyclic Cosmology.

There is a potential downside to this approach which as always is this tendency to hypostatise our own models and structures for understanding reality onto reality itself. Naturally, Penrose does a good job avoiding this error in many areas.

But in one area, with his ORCH approach in quantum gravity in which there is an objective orchestrated reduction or collapse of the wave function in quantum theory under the effect of gravity, there is a tension potentially with other options. Namely, the option of following Leibnizian principles of reason, which are pursued more fully by the likes of Lee Smolin and Julian Barbour.

This leads them to the relational view of space and time. However this conflicts with Penrose’s Orch approach, for in this view the gravitational field is taken, to be a real “thing”, that causes the quantum collapse of the wave function, as a way to explain the measurement problem for macro as opposed to micro objects.

But if there is a pure scale invariance in space and time. I.e. if they are purely relational, there would be no way to distinguish a micro from a macro realm, based on space and time themselves. The only way would be if these things are absolute in some way. And this actually is the way that Einstein himself ultimately went.

Einstein, in his special theory went for a relational view originally, inspired by Mach, but then influenced by Minkowski’s Space-Time interpretation and by others, in his general theory of relativity, Space was brought back as in some way an absolute thing, for if the theory is that space curvature causes gravity, then there must be an existent “thing” that is curved.

So, I would side here with Smolin and Barbour, in thinking that space is in some key sense fundamentally unreal. And would take the inspiration here of Leibniz’ basic principles of reasoning as outweighing the reality of the mathematical structures.

Taking this further, philosophically, would lead to a discussion on what is called the hole argument, which is a criticism of the standard Einsteinian interpretation of general relativity, suggesting, it can be reduced to a relational approach.

That will be for another time, but at this point, to summarise, we have to keep in mind that in aiming at realism we have competing considerations to bear in mind. The mathematical realism of the structures we use to describe nature, and the principles of reasoning we use through which to express truths about nature.

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