Archives for posts with tag: number theory

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

That numbers are physical constants should come as no surprise, for the reasons outlined in this chapter of Trick or Treat, The Mysterious Connection Between Physics and Mathematics published by Springer in 2016, titled ‘Cognitive Science and the Connection Between Physics and Mathematics‘.

Abstract

“The human mind is endowed with innate primordial perceptions such as space, distance, motion, change, flow of time, matter. The field of cognitive science argues that the abstract concepts of mathematics are not Platonic, but are built in the brain from these primordial perceptions, using what are known as conceptual metaphors. Known cognitive mechanisms give rise to the extremely precise and logical language of mathematics. Thus all of the vastness of mathematics, with its beautiful theorems, is human mathematics. It resides in the mind, and is not ‘out there’. Physics is an experimental science in which results of experiments are described in terms of concrete concepts—these concepts are also built from our primordial perceptions. The goal of theoretical physics is to describe the experimentally observed regularity of the physical world in an unambiguous, precise and logical manner. To do so, the brain resorts to the well-defined abstract concepts which the mind has metaphored from our primordial perceptions. Since both the concrete and the abstract are derived from the primordial, the connection between physics and mathematics is not mysterious, but natural. This connection is established in the human brain, where a small subset of the vast human mathematics is cognitively fitted to describe the regularity of the universe. Theoretical physics should be thought of as a branch of mathematics, whose axioms are motivated by observations of the physical world. We use the example of quantum theory to demonstrate the all too human nature of the physics-mathematics connection: it is at times frail, and imperfect. Our resistance to take this imperfection sufficiently seriously (since no known experiment violates quantum theory) shows the fundamental importance of experiments in physics. This is unlike in mathematics, the goal there being to search for logical and elegant relations amongst abstract concepts which the mind creates”.

The mathematical equation 1a + 2a = 3a describes a mathematical law of quantitative relationships. It also describes 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. This is not insignificant, it suggests that the physical universe developed according to natural constraints which are purely mathematical.

If we expect advanced aliens to know the natural number sequence then humans didn’t invent numbers, we discovered them.

If numbers are a human discovery then so are their mathematical relationships.

If mathematical relationships are discoveries not inventions then they’re natural laws.

1+2=3 is a natural law.

This is not insignificant.

 

more evidence of the significance of the connections between math and physics

more evidence that mathematical principles are natural laws

 

https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/press.html

nothing can be created or destroyed, which makes 1a+1a=2a an entirely natural law

 

 

Whether it takes the form of a universe or multiverse, there’s a single sum total of ‘everything which exists’, a single ‘Existence’.

Existence = 1

We’ll never know what Existence ‘is’ or what it is ‘made of’, we can only be certain of its quantity. It is the ultimate known *and* unknown.

We don’t know what it’s made of, we only know how many of it there is.

X = 1

But we also know everything which exists is part of Existence: it has been divided, internally, into its constituent parts, over time.

So if X = 1, and everything else is a result of X being divided into smaller and smaller constituent parts over time, which can only occur according to natural laws of combination. The entirety of things which exist, must ‘add up’ to the single Existence they are parts of.

This is where human concepts of number and mathematics intersect with natural laws.

 

 

mathematics and physics both develop from the same fundamental principles

the complexities of each are emergent properties of the application of these fundamental principles to their respective entities/units of existence, real or imagined

mathematics can produce concepts and objects which are impossible in physics, but this  doesn’t disprove the existence of formal connections between the two fields, or the significance of those connections

the fact that it’s possible to creatively develop mathematical concepts which are impossible in physics, shows only that human creativity in mathematics isn’t subject to the same natural constraints as the laws of physical reality

those natural constraints on how a complex physical universe can emerge from first principles may not have been discovered mathematically yet, but they inevitably exist

if you place an object (a) with an identical object (a), together they form a group (a+a)

the ratio of the quantity of a in (a) to the quantity of a in (a+a) is
a:a+a

the ratio of the mass of (a) to the mass of (a+a) is
a:a+a

the physical ratio
a:a+a
is precisely identical to the numerical ratio
1:1+1

the physical ratio
a:a+a
is precisely identical to the numerical ratio
1:2

so
a:a+a = 1:2

human number theory and natural physical laws are based on precisely identical principles

⚫ 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