What is the role of physics in mathematics and the role of mathematics in physics? How does mathematical reality relate to the physical one? The significant role of mathematics in physics has been recognized for centuries. Its most crystalized form came after Newton’s Principia representing the mathematical principles of philosophy or a relatively recent articulation of physics-mathematics relationship with differing perceptions by theoretical physics [1].

Meta-relations

The common way of thinking is that mathematics is the natural language of physics. Pure Mathematics is thus suggested to appear strange to the physical world, almost alien. Because the mathematical truths exist somewhere outside of it and if we want to discover them, we must voyage outside of ‘reality‘. Looking at Mathematical Platonism in philosophy of mathematics, as it is formally defined, there are abstract objects (spatial truths) that are non-spatial-temporal, non-physical and also, non-mental. Mathematics provides true descriptions of such objects. On the opposite side we have Realistic Anti-Platonism — psychologism that views that mathematical theorems are about concrete mental objects. From this perspective, numbers and geometrical objects do exist, but not independently of the human realm and the mind. They are concrete mental objects directly related to the mind. It seems that philosophers and mathematicians have a different interpretation of the concept of mathematical reality. Either accuses the other side of dogmatism. If mathematical reality lies outside us, our role is to discover or observe it. The theorems would thus be creations that are simply notes of our observations.

To some physicists, mathematics only holds the utility for describing physical phenomena, suggesting that this is the true value of mathematics. The meaning of reality — both mathematical and physical — represents the role of the mind in perception of reality. If the mind is set to perceive the reality as unity of all numbers and systems of numeration, that is one way. If the mind perceives only integers of the unity, its perspective will be focused on a fragment, not the whole. It is a debate about the organization of the non-physical, which overlaps both mathematics and philosophy. Mathematical-philosophical mind focuses on abstract things and can be ‘freed from the tyranny of the external world’. That would also imply that mathematics, in order to be performed properly and in its purest form, has to be freed from physics. But at the same time it is expected to solve problems of physics arising from said external world.

Experimental Relations

It is said that the experiment is the only source of the physical truth. On the other hand, intuition is an instrument of mathematical creation and logic makes it rigorous.

There are two routes towards explaining how mathematical concepts are useful to fundamental physics:

1. Organizing experimental information into a law of physics creating mathematical concept

2. Begin with mathematics and find a way of how to relate it to experimental reality

The first route represent the Newton’s rules of reasoning — more applicable to the past circumstances of human society. In the light of modern complexity, the second route is preferred, for organizing such an enormous amount of information into a law of physics would take a whole other level of dedication (and applicability). This introduces another variable — time. The concept of time might be the most fundamental issue in foundations of physics and mathematics and their relation.

Time Relations

The mechanics of the matrix is based on the concept of observables discarding the representation of space-time. One of the utilities (or properties) of time is to carry out the transition of the abstract into the form. The time-operator seems to be relevant in defining the inertial frame of intuitive time, which seeks to interpret space-time as emergent based on abstract (quantum) space. This may cause a conceptual crisis when it comes to describing physical reality, because it alters the perception of what is abstract and what is physical, blurring the lines.

To perceive and to intuit time is linked to the mind that is actually the carrier of time and thus subjected to the flow of time. Time is a measure of change we perceive as linear, as becoming, as abstract-physical merge. In that sense, maybe we should consider rephrasing the original:

Thesis A: „Mathematics is the natural language of physics.“

to

Thesis B: „Physics is the natural language of mathematics.“

Because:

The mathematical formations of the mind need to find a tangible expression in the physical world. It is the law of induction and a logical extension of the original thesis (A). But foremost, it creates a link between geometry and topology with matter.

It is mystical to think that everything that appears in physical reality has a mathematical/abstract counterpart. But not everything that exists in mathematical reality has a physical counterpart. For that reason, the sum of ’things’ in the abstract reality contains more (or everything), whereas the physical realm contains a limited number of things that were translated from mathematical to physical ’language’ and not the other way around.

References:

[1] Woolf, H. (1980) Some Strangeness in the Proportion, Addison-Wesley.

Suggested readings:

Landauer, R. (1996) The Physical Nature of Information, Phys. Lett A 217, 188.

Tiwari, S. C. (2012) Symmetry and Geometry: Pursuit of Beauty in Physics, Contemporary Physics, 53, 485.

Tiwari, S. C. (2006) Time-Transcendence-Truth, IONP Studies in Natural Philosophy Volume 1.

Dantzig, T. (1942) Number the Language of Science, George Allen and Unwin Ltd.