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DEDUCTIVE EXTENSION. “For example: the elementary theory of equilibrium classical statistical mechanics for a strictly isolated system, which postulates the microcanonical measure on the energy hypersurface, is a definitional extension of the mechanics of the micro-constituents, once we use an underlying logic strong enough to express calculus (of many variables).” https://lnkd.in/dMm_YkR View in LinkedIn

THE KNOT. “Space and time are such crucial categories for thinking about, and describing, the empirical world, that it is bound to be ferociously difficult to understand their emerging, or even some aspects of them emerging, from ‘something else’.” https://lnkd.in/dMm_YkR View in LinkedIn

EXTENDING DEDUCTIVE REACH. “In effect all of classical pure mathematics was a definitional extension using just logical operations, of set theory. But it is equally remarkable how many examples of definitional extension there are in physics, at least once we give the reducing physical theory a suitably strong underlying logic (i.e., a rich enough set of operations) so as to ‘extend its deductive reach’.” https://lnkd.in/dMm_YkR View in LinkedIn

VALIDITY CEASES TO EXIST. “The idea that simple manifold concepts may not apply at small distances was anticipated by Riemann in his famously prophetic Habilitationschrift of 1854: “Now it seems that the empirical notions on which the metrical determinations of space are founded, the notion of a solid body and of a ray of light, cease to be valid for the infinitely small.” https://lnkd.in/dMm_YkR View in LinkedIn

BEDROCK LOGIC. “When we undertake to express such a theory in a formal language, we naturally envisage adding the theory’s distinctive physical vocabulary, and axioms governing it, to a so-called ‘underlying logic’ that is much stronger than just basic logic and set theory; i.e., to one that contains standard mathematical operations sufficient to yield the mathematical apparatus; for example, calculus, which is needed by the theory.” https://lnkd.in/dMm_YkR View in LinkedIn

THE ASSUMPTION. “Gravity is, at least prima facie, a force like the others, and these have turned out to be quantum in nature.” https://lnkd.in/dMm_YkR View in LinkedIn

MATHEMATICAL HIERARCHY. “Even quite simple physical theories such as, for example, Newtonian mechanics of point particles interacting just by gravity, are complicated from the viewpoint of logic and foundations of mathematics; simply because such theories use mathematical apparatus (for example, calculus) that is ‘high up’ in the deductive chain from basic logic and set theory.” https://lnkd.in/dMm_YkR View in LinkedIn

UNRESOLVED CONFLICT. “The fact that general relativity treats matter classically, and gravity as curvature, while our best theories of matter are quantum theories using a flat metric, is enough to show that some sort of reconciliation is needed.” https://lnkd.in/dMm_YkR View in LinkedIn

SETS AT THE HEART. “By concatenating such deductions with judiciously chosen definitions, one shows in effect that all of classical pure mathematics can in this sense be deduced from the theory of sets. This remarkable result, showing how large an expressive power can be obtained by applying this small stock of logical operations to a small initial family of predicates, is the source of the traditional philosophical restriction to considering just these logical operations.” https://lnkd.in/dMm_YkR View in LinkedIn

THE WORLD OF PROBABILITIES. “Quantum theories are conceptually very different from classical theories: in particular, the state of a system does not assign real-number values to physical quantities but only probabilities for obtaining such values if appropriate measurements are made. And, though they share with general relativity the assumption that it is appropriate to coordinatise space and time with real numbers, they do not treat gravity as curvature of the corresponding spaces: they use either Newtonian spacetime (which corresponds to a degenerate spacetime metric), or the flat metric of special relativity.” https://lnkd.in/dMm_YkR View in LinkedIn