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 mathematical principles and formula which could be used to 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 more 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.