From whence do we get the impression that some mathematical models are closer to physical and spatio-temporal intuition, more down to earth and intimately tied to concrete applications, while others hover close to the heights of allegedly less useful 'abstract-nonsense' ? The real numbers and differential equations, these are seen as tied to dynamic-geometric intuition and of vast applicability and interest in engineering and science. But abstract algebra and category theory appear to have no direct relevance to applicable mathematical models or to their kind of concrete geometric-dynamic intuition. We present here a few short speculations.
Perhaps the abstract models do capture fundamental levels of reality (maybe even more fundamental than the so-called concrete spatio-temporal ones) and what is required is first of all the methodic development of a special kind of intuition or cognition to grasp the planes of reality which they model (maybe something like Goethe's method is called for?). And then further models are required which can effect a mediation or transfer between these two level, or at least allow a continuity or gradual deformation between concrete spatio-temporal reality and higher levels of reality.
A beautiful illustration of such a meditation and construction is furnished by the double fibration in Penrose's twistor theory which allows a mediation and transfer between physically significant objects on Minkowski space and abstract cohomological objects on the complex algebraic variety $\mathbb{P}^3$.This operation allows solutions to conformally invariant differential equations on spacetime (e.g., Maxwell's equations, Yang-Mills, or linearized gravity) to be identified exactly with Cech or Dolbeault cohomology classes on specific regions of twistor space. This appears to have been Penrose's attempt to give a geometric (topological) interpretation of quantum non-locality.
Another illustration is the theory of $\mathcal{D}$-modules which presents a mediation between derived categories (and monoidal categories) and concrete models of systems of partial differential equations.
The idea of Kant's schematism of the pure concepts of the understanding can be interpreted as finding the mediation between logic and geometry (including mathematical physics). This is of immense contemporary significance as we find versions of Kantian schematism in topos theory, homotopy type theory and in areas involving monoidal categories in algebra, geometry and physics and in linear logic and computer science.
An objection can be raised that our concrete allegedly intuitive mathematical models are themselves quite abstract but are only perceived as immediate and intuitive in our present cultural context by a deliberate forgetting of the complexity of their cognitive-historical past and genesis. For instance real numbers are constructed from the rationals by quite abstract, cardinality increasing, procedures involving equivalence classes and identification. And the same goes for the concepts of continuity, differentiability and so forth. Surely the ancient Greeks had other forms of intuitive perception and concreteness in their geometry, physics and engineering. We answer this objection by our arguments that the fundamental dynamic and geometric-topological intuitions of modern science were in fact identical to those of ancient Greece. It is the modern foundations in terms of Dedekind cuts and $\epsilon$s and $\delta$s that can be critiqued from a philosophical and logical point of view and alternative foundations (locales, realizability topoi, synthetic differential geometry, homotopy type theory) can be defended which are also more aligned to the theory of space and change found for instance in Aristotle's Physics.
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