⚫ and ⚫⚫ makes ⚫⚫⚫ (physical quantity)

1 + 2 = 3 (human number)

each of the ‘equations’ describes the same relationship; each is based on the same principles

each expresses inviolable laws which govern the combination of identical entities into groups

natural numbers are physical constants

archive of research documenting the connections between number theory and physics, showing that the connections are

a)well-documented

and

b)non-trivial

 

http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/surprising.htm

Peter Woit, University of Columbia, ‘Towards a Grand Unified Theory of Mathematics and Physics’.

A paper which outlines some of the already established connections between number theory and physics, acknowledges that there are as yet no satisfactory explanations for these connections, and concludes it is an area which deserves much more attention.

maths is so useful for physics because it’s developed from the same fundamental principles which govern the physical universe

early humans didn’t invent systems of number, then discover they were useful for measuring and quantifying the physical world

early humans developed systems of number from observation of how the physical world is naturally organised: counting systems are based on measuring and comparing the physical properties of different groups of the same or equivalent physical objects

for *any* type of equivalent physical objects

the quantity we call 1
put with the quantity we call 2
creates the quantity we call 3

this is a physical absolute, according to the law of conservation of energy

numbers and numerical relationships are not merely abstract concepts, they are developed from the same fundamental principles as physical laws

Pedro L. e S. Lopes, Jeffrey C. Y. Teo, and Shinsei Ryu, ‘Topological strings linking with quasiparticle exchange in superconducting Dirac semimetals‘, Physical Review B, 95, 235134, (https://journals.aps.org/prb/abstract/10.1103/PhysRevB.95.235134)

Abstract:

“We demonstrate a topological classification of vortices in three-dimensional time-reversal invariant topological superconductors based on superconducting Dirac semimetals with an s-wave superconducting order parameter by means of a pair of numbers (NΦ,N), accounting how many units NΦ of magnetic fluxes hc/4e and how many Nchiral Majorana modes the vortex carries. From these quantities, we introduce a topological invariant, which further classifies the properties of such vortices under linking processes. While such processes are known to be related to instanton processes in a field theoretic description, we demonstrate here that they are, in fact, also equivalent to the fractional Josephson effect on junctions based at the edges of quantum spin Hall systems. This allows one to consider microscopically the effects of interactions in the linking problem. We therefore demonstrate that associated to links between vortices, one has the exchange of quasiparticles, either Majorana zero modes, or e/2quasiparticles, which allows for a topological classification of vortices in these systems, seen to be Z8 classified. While NΦ and N are shown to be both even or odd in the weakly interacting limit, in the strongly interacting scenario one loosens this constraint. In this case, one may have further fractionalization possibilities for the vortices, whose excitations are described by SO(3)3-like conformal field theories with quasiparticle exchanges of more exotic types”.

Toby Crisford and Santos, Jorge E. (2017), ‘Violating the Weak Cosmic Censorship Conjecture in Four-Dimensional Anti–de Sitter Space’, Physics Review, Letters 118, 181101. ((https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.181101))

Abstract:

“We consider time-dependent solutions of the Einstein-Maxwell equations using anti–de Sitter (AdS) boundary conditions, and provide the first counterexample to the weak cosmic censorship conjecture in four spacetime dimensions. Our counterexample is entirely formulated in the Poincaré patch of AdS. We claim that our results have important consequences for quantum gravity, most notably to the weak gravity conjecture”

Zhi-zhong Xing, Ye-Ling Zhou (2014) ‘Geometry of the effective Majorana neutrino mass in the neutrinoless double-beta decay‘, Arxiv .

Abstract:

“The neutrinoless double-beta (0νββ) decay is a unique process to identify the Majorana nature of massive neutrinos, and its rate depends on the size of the effective Majorana neutrino mass ⟨m⟩ee. We put forward a novel “coupling-rod” diagram to describe ⟨m⟩ee in the complex plane, by which the effects of the neutrino mass ordering and CP-violating phases on ⟨m⟩ee are intuitively understood. We show that this geometric language allows us to easily obtain the maximum and minimum of |⟨m⟩ee|. It remains usable even if there is a kind of new physics contributing to ⟨m⟩ee, and it can also be extended to describe the effective Majorana masses ⟨m⟩eμ, ⟨m⟩eτ, ⟨m⟩μμ, ⟨m⟩μτ and ⟨m⟩ττ which may appear in some other lepton-number-violating processes”.

Paul Benioff (2005), ‘Towards a Coherent Theory of Physics and Mathematics: The Theory–Experiment Connection’, Foundations of Physics, 35: 1825.

Abstract:

“The problem of how mathematics and physics are related at a foundational level is of interest. The approach taken here is to work towards a coherent theory of physics and mathematics together by examining the theory experiment connection. The role of an implied theory hierarchy and use of computers in comparing theory and experiment is described. The main idea of the paper is to tighten the theory experiment connection by bringing physical theories, as mathematical structures over C, the complex numbers, closer to what is actually done in experimental measurements and computations. The method replaces C by Cn which is the set of pairs, Rn,In, of n figure rational numbers in some basis. The properties of these numbers are based on those of numerical measurement outcomes for continuous variables. A model of space and time based on Rn is discussed. The model is scale invariant with regions of constant step size interrupted by exponential jumps. A method of taking the limit n→∞ to obtain locally flat continuum-based space and time is outlined. Also Rn based space is invariant under scale transformations. These correspond to expansion and contraction of space relative to a flat background. The location of the origin, which is a space and time singularity, does not change under these transformations. Some properties of quantum mechanics, based on Cn and on Rn space are briefly investigated”

The top replies to this question on Quora are classics of the genre – acknowledging how fundamental the relationship is between maths and physics, acknowledging that this relationship is still not fully understood or explained, but leaving it open – failing to interrogate the implications and significances.

Nonperturbative Effects and the Large-order Behaviour of Matrix Models and Topological Strings Communications in Number Theory and Physics
, Volume 2 Number 2

Authors
Marcos Mariño (Department of Physics, CERN, Genève, Switzerland)
Ricardo Schiappa (Department of Physics, CERN, Genève, Switzerland)
Marlene Weiss (ITP, ETH Zürich, Switzerland)
Abstract:

“This work addresses nonperturbative effects in both matrix models andtopological strings, and their relation with the large-order behaviorof the 1/N1/N expansion. We study instanton configurations in genericone-cut matrix models, obtaining explicit results for theone-instanton amplitude at both one and two loops. The holographicdescription of topological strings in terms of matrix models impliesthat our nonperturbative results also apply to topological strings ontoric Calabi–Yau manifolds. This yields very precise predictions forthe large-order behavior of the perturbative genus expansion, both inconventional matrix models and in topological string theory. We test these predictions in detail in various examples, including the quartic matrix model, topological strings on the local curve and the Hurwitz theory. In all these cases, we provide extensive numerical checks which heavily support our nonperturbative analytical results.Moreover, since all these models have a critical point describing two-dimensional gravity, we also obtain in this way the large-order asymptotics of the relevant solution to the Painlevé I equation,including corrections in inverse genus. From a mathematical point of view, our results predict the large-genus asymptotics of simple Hurwitz numbers and of local Gromov–Witten invariants”

http://intlpress.com/site/pub/pages/journals/items/cntp/content/vols/0002/0002/a003/index.html

It’s indisputable that the laws of physics and the physical reality they constitute develop from the same first principles as the laws of mathematics.

What is needed is serious research into the constraints on how the laws of physics emerge from purely mathematical principles.

a rigorous counting system is, by definition, physics

hence, the natural numbers are a sequence of physical constants

Seemingly esoteric notions of thegeometric Langlands program, arise naturally from the physics