Journal: Communications in Number Theory and Physics

# Tag Archives: mathematics

# a simple theory, in summary

Max Tegmark, Garrett Lisi and others have proposed theories of everything that rest on the premise that physical reality is governed by natural laws which are *naturally* mathematical.

Historically there have been many discoveries of connections between patterns found in purely mathematical models and naturally occurring physical phenomena observed in the universe. More subtle connections between number theory and physics are being discovered. They’re directly related.

The law of conservation of energy states: the total energy of an isolated system is conserved.

That is, that the total energy of a system must be constant over time;

The energy must always add up to the same amount no matter how it is distributed within the system;

1 + 1 = 2

1 + 2 = 3

and so on.

In fact a rigorous counting system is by definition, physics.

The mathematical equation 1/2 + 1/2 = 1 describes a mathematical law of quantitative relationships. It also describes a physical absolute, according to the law of conservation of energy. *It describes, mathematically, 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.

In fact, this is a natural consequence of simple existence:

There is, without any possible doubt, a single entity of Existence which encompasses everything which exists.

Existence = 1

.

# cognitive science, number theory and physics

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‘, demonstrates how human number theory and mathematics develop from our experiential understanding of the physical universe.

This supports the arguments made on this site, that numbers are physical constants and that this is deeply significant. It has profound implications for our comprehension of the relationships between number theory and physics. It strongly suggests that the constraints on how our physical universe developed and behaves are entirely mathematical, and should be derivable from purely mathematical principles.

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”.

# natural numbers are universal

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.

# prediction

that the rate of expansion of the universe is related to the rate of increase in the complexity of the mathematical properties of the natural numbers from 1 to ∞

# a theory

Human theories of number – and the rigorous and complex mathematics which can be developed from them – have always been extremely accurate tools that our species have learned to use in its comprehension of, explanation for and exploration of the physical environment within which we live.

In ancient times, numbers and mathematical objects were considered to have mystical properties thanks to the way they could be manipulated to produce surprisingly aesthetic or useful results – such as principles and formula developed from pure mathematics which provide answers to practical, real-world problems. As science developed over the following centuries more of these connections between maths and the way things in the physical world behaved, were discovered – at increasingly deeper levels of rigorous mathematics and more accurately measured physics.

Mathematics proved to be the most accurate tool we’ve ever discovered for measuring and predicting things in the physical world we live inside, and are a part of. Not only this but over the last few decades of truly modern science, patterns produced by the most abstract of mathematics have been shown to have symmetry with patterns observed in how the physical universe works at its most finely measured detail.

There have been a few explanations suggested for the usefulness of number theory and mathematics to physics, and for the way that highly complex patterns which could be created from the interaction of purely abstract mathematical concepts, could be observed to also occur in the interactions of matter and energy in the physical world.

Physicists such as Max Tegmark have argued for a Mathematical Universe Hypothesis – that the universe is inherently mathematical in nature, and that all physical laws develop from entirely mathematical principles. There are a few different versions of this but they essentially analyse the universe as a hologram of interacting q-bits. What they fail to explain is how this occurred in the first place – why is the universe mathematical?

Peter Woit of the University of Columbia suggests (‘Towards a Grand Unified Theory of Mathematics and Physics’) that in fact mathematics and physics are convergent, and that some theory of unification might be possible.

This is a theory of everything which aims to begin that task by showing how number theory and mathematics are based on the very same natural laws of quantity and quantitative relationships as form the basis of physical reality.

This is a theory which can be explained or argued for in a variety of ways, but the key thing to note is that it is based on a substantial body of evidence, a lot of which is collated at this archive hosted by the University of Exeter.

Essentially it argues that maths is so useful for physics because it’s developed from the same fundamental principles as the natural laws 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, they 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:

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

1a + 2a = 3a (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

for *any* type of identical physical entities

the quantity we call 1

put with the quantity we call 2

creates the quantity we call 3

The mathematical equation 1a + 2a = 3a describes a natural 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.

And if we examine the nature of our physical reality, we can begin to understand how this is an inevitable result of the mere fact of existence.

Paul Dirac suggested in 1939 (http://www.damtp.cam.ac.uk/events/strings02/dirac/speach.html) that the universe can be thought of as an extremely large quantity of minimally small units of time, and that the number of units of time might be enough to describe the entirety of its physical complexity.

This theory makes a slightly different suggestion, which is that the single universe (or single multiverse) is divided by those units of time into constituent parts of existence.

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 and the fact of its physical existence. 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.

X/t = 1/t

This is a route to what Paul Dirac suggested, that “the whole history of the universe corresponds to excessively complicated properties of the whole sequence of natural numbers” (*Reposted at http://www.damtp.cam.ac.uk/events/strings02/dirac/speach.html): a single Existence, being divided into an increasing “number” of discrete q-bits, so that the complexity of the properties of the physical universe would develop in a way entirely mathematical in nature. As the potential complexity expressable by the natural number sequence increases, laws of combinations of patterns emerge, and repeated structures with predictable properties begin to form into groups of their own which also have laws of interactions unique to their particular scale.