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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

Macabre dance, Museu Nacional d'Art de Catalunya. View in LinkedIn

SELF-REFERENTIAL SENTENCES. “All of these sentences are "floating" in the context of the English language. They can be compared to icebergs, whose tips only are visible. The word sequences are the tips of the icebergs, and the processing which must be done to understand them is the hidden part. In this sense their meaning is implicit, not explicit.” https://lnkd.in/dmfn9qW View in LinkedIn

FORGET LOCAL GEOMETRY. “The description of the neural network in terms of Fourier components of a local field, allows to give a geometric interpretation to something intrinsically non‐local. Indeed, neurons interact non‐locally: Everybody talks to each other according to the rules determined by the synaptic connections, with no real sense of local geometry.” https://onlinelibrary.wiley.com/doi/full/10.1002/prop.201800007?casa_token=GdPRXs_tGLEAAAAA%3Ag-pxPMtkcleTYDUNK3mYGfpEzKRyc2LqQknTgFFB7dkYFDL8rDe2s0f4_0Q0oKjXD4_AEgFPGWolVTQ View in LinkedIn

QUIET ENORMITY. “However, as we shall see, despite its simplicity, it suffices for generating an exponential enhancement of memory storage and pattern recognition capacities with the corresponding entropy that obeys an area law.” https://onlinelibrary.wiley.com/doi/full/10.1002/prop.201800007?casa_token=GdPRXs_tGLEAAAAA%3Ag-pxPMtkcleTYDUNK3mYGfpEzKRyc2LqQknTgFFB7dkYFDL8rDe2s0f4_0Q0oKjXD4_AEgFPGWolVTQ View in LinkedIn

ON GRAVITY AND NEURONS. “The above synaptic connection among the neurons, in the excitatory regime, is analogous to a gravitational interaction among particles. Therefore, such a setup represents a most naive way of bringing a gravity‐like dynamics into the neural networks.” https://onlinelibrary.wiley.com/doi/full/10.1002/prop.201800007?casa_token=GdPRXs_tGLEAAAAA%3Ag-pxPMtkcleTYDUNK3mYGfpEzKRyc2LqQknTgFFB7dkYFDL8rDe2s0f4_0Q0oKjXD4_AEgFPGWolVTQ View in LinkedIn