Archives for posts with tag: number

Part of the reason that the formal, natural connections between mathematics and physics are rejected as significant by most physicists is because the the two fields are so firmly intertwined at the most basic levels that the epistemological becomes confused with the ontological.

In fact, it is not that mathematics underlies physics and it isn’t that they look the same because mathematics is only our tool to examine physics with. It’s that the same fundamental principles – laws which are both natural AND mathematical – underlie them both. But we shouldn’t be surprised at this.

Human concepts of number and maths developed from experience and observation of the real world: geometry can be developed from both theory AND by measuring the physical world; counting systems are based on measuring and comparing the physical properties of different groups of the same or equivalent physical objects.

It’s not so much that either maths or physics underlie the other, it’s more that they develop from the same first principles, and are parts of the same thing. There’s only a single ‘sum total of existence’: you can define the mathematical concept 1, physically. And the first law of thermodynamics can be expressed as 1a+2a=3a.

Issues of efficiency act as constraints on how mathematical complexities develop naturally from first principles into a physical spacetime. These constraints are the difference which leads to a separation of human abstract mathematics from physical mathematicality.

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Quantum Mechanical Derivation of the Wallis Formula for

Further evidence that the connections between mathematics and physics are non-trivial.

the quantity of things we call 1

put with the quantity of things we call 2

will always make the quantity of things we call 3

physical quantities are not the result of human number systems

human number systems are the result of our perception of physical quantity

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.

What seems to be the central reason preventing mathematicians and physicists from acknowledging the significance of the connections between their respective fields is a fundamental, but understandable, misunderstanding about the nature of number systems. Numbers are considered to be abstract inventions of pure human imagination, and mathematical formulations are seen therefore as being based in nothing more than abstraction. As a consequence, the idea that number theory and physics are formally connected is rejected – even as the evidence for formal connections between the two fields is being examined and analysed as profound mystery.

This is a crucial mistake.

Number systems are at their most basic level, founded on the concretely physical property of quantity. Numerical relationships are quantitative relationships, and quantitative relationships are at the heart of physics.

(My bold)

“We extend the Hopf algebra description of a simple quantum system given previously, to a more elaborate Hopf algebra, which is rich enough to encompass that related to a description of perturbative quantum field theory (pQFT). This provides a mathematical route from an algebraic description of non-relativistic, non-field theoretic quantum statistical mechanics to one of relativistic quantum field theory. Such a description necessarily involves treating the algebra of polyzeta functions, extensions of the Riemann Zeta function, since these occur naturally in pQFT. This provides a link between physics, algebra and number theory. As a by-product of this approach, we are led to indicate inter alia a basis for concluding that the Euler gamma constant γ may be rational”.

Duchamp, Gérard H E, Hoang Ngoc Minh, Allan I Solomon, and Silvia Goodenough. “An Interface between Physics and Number Theory.” Journal of Physics: Conference Series 284.1 (2011): 17.

Quoting the same page as the last post, watch how it sums up the resistance in the scientific community, against maths and physics being formally linked. This resistance leads to complete cognitive dissonance, where evidence for formal connections is explicitly referenced followed immediately by the denial that any formal connections are possible.

“…the surprising thing is that often some newly discovered abstract formulation in mathematics turns out, years later, to describe physical phenomena which we hadn’t known about previously…”

“…physics cannot be derived from mathematics alone…”

The first statement undermines the second statement.

This page by Dr Cecilia Barnbaum of Valdosta University exemplifies the problem. On the one hand she has to acknowledge that actually there are fundamental connections between formulations derived from pure mathematics, and phenomena which exist in the physical universe. On the other she claims that human concepts of number are products of pure human imagination and that they bear no relation to the physical universe.

In order to accomodate these irreconcilable positions she concludes that “the structure of the universe itself seems to be imprinted on the human mind”, a statement which, once you think about it, deliciously contradicts the assertion that “pure notions of number … do not need physics to exist”.

We live in a universe entirely born of, governed by, and structured according to a natural mathematicality*, and it should come as no surprise if human maths and natural physics can both be derived from the same first principles.

Quantity is a physical property. Human systems of number are fundamentally based on quantitative relationships. Human systems of number describe natural quantitative relationships. This is both logically reasonable and supported by evidence.

*The term ‘mathematicality’ is used to differentiate the natural laws of numerical relationships from the human descriptions of those laws, called ‘maths’ and ‘mathematics’

e + 1 = 0

an intersection of number theory and physics

number systems are based on human perception of quantity

they formalise natural, pre-existing quantitative relationships into symbolic language

they do not constitute quantitative relationships

quantitative relationships are natural laws