Modifications of the Fundamental Theorem of Calculus

Canonical treatments of classical mechanics, quantum mechanics, cosmology, and even string theory motivate a discussion of the mathematical foundations of universal logical treatments of abstract concepts in a scientific basis. What this means is that the basic assumptions and certainties of structures in rational thinking must relate to a new approach to objective truth. According to Penrose, there are three fundamental worlds or paradigms that are nested within each other: physical universal building blocks, mental thoughts, and mathematical abstractions. Essentially, the physical world creates the mental world and the mental world creates the mathematical world. Thus, an objective conatus of ideas involving the physical and cascading mental and mathematical world must emerge from a system of equations that in all practical circles creates reality. The interesting thing is that reality is evolutionary, constantly changing, and modified according to principles of physics. It is the natural order that determines the mathematical constructs and not the other way around. It is naive to think that the mathematical world is somehow immune to the physical array of spacetime, particle, and dynamic treatment. This, therefore, leads to the result that the observational principle guides the new mathematical foundations of string theory, unified physics, and measurement.

Basic foundations of mathematics are built upon things like symmetry and reproductions of consistent form. The interesting thing is that the universe in inherently unpredictable and incapable of being replicated. From the no-cloning theorem of quantum mechanics to the dynamism of evolution in biology, physicalism portrays a special quality of distinguishability that regards every particle, every point in spacetime, and every event in each frame of reference as characteristically new and different from the last. Correcting the fundamental theorem of calculus, which is a description of fundamental dimensional analysis and a description of the mathematical world, means turning logical causation or logical formulations into a kernel of truth from physicalism. As such, the mathematical world, as described by Penrose again, is a part of the physical world. Interestingly, Murray Gell-Mann, the creator of quarks, believed that each particle state in the grand partition function describing the ensemble of canonical variables is distinguishable. Further, the infinitely complex is equal in information content to the infinitely simple. I imagine that this implies a cyclic model theory of the universe, one of creation and annihilation. Symmetry does play a role and understanding the conservation laws requires some expertise in translational, rotational and temporal invariance, however, the key part of reforming mathematics is defining fundamental physical phenomena and then relating them to mathematical formalism. Classical mechanics, to contain inflation and cosmological variables, must discuss inflation then.

A very good description of the basis of randomness and turbulence is the Riemann zeta function, which of course is not the only way to describe turbulence. There are non-linear differential equations, Fibonacci sequences, and quantum algorithms based in such things as momentum dependent momentum uncertainty. Modern computer science creates a time stamp description of states, each string corresponding to a unique identifier, charted and mapped to a transcript that motivates further discussion. The accounting in a Lagrangian mechanical system, something that describes the remnant energy of a system, must contain within it - inflation, turbulence, and a description of the multitude of states of the ensemble. Requirements thus necessitate a Riemann zeta term attached to the integration constant, something that was traditionally thought of as sacrosanct. According to this new formulation, a calculation becomes dependent on the ecology, the environment, and the variables of spacetime and particle physics that are its history in the string geodesic landscape. String theory is built from geodesics, or great arcs of particle tracks that are traced through time. The evolution of time is usually though of as a linear sequence of events, but abstraction of time from relativity discusses such things as time turbulence and dynamism in the laws of quantum mechanics which determine physical structure. This goes back to the physical world creating the mathematical world - of course through some mental neural networks.

To return to the modification of calculus from physical reality, means orienting or aligning physical phenomena to an analogous string geodesic landscape which evolves according to stochastic laws. Randomness and free will are a tangential discussion, so we'll leave that for another time. The critical point is understanding how the calculations are dependent on spacetime and part of the inflationary and evolutionary universe. Things evolve, change, oscillate, move chaotically, and bifurcate in the history picture - the one of frames of reference from each distinguishable particle. The most accurate description of basic distinguishability, or a basis of randomness, is the prime patterning in the Riemann zeta function. Implications thus create a calculus of a "time-stamp" or a tracing with a Riemann zeta prime or a sequential inflationary state weighting function associated with the calculation. Cumbersome as this may be, one must account for the physical spacetime, quantum foam, and inflationary state of variables in the ensemble that sequentially builds the nested mental world and incremental mathematical description of the world.