Axioms Part 3: Modification of Descriptive Axioms and Axiomatic Sets: The Example of Physics

Descriptive axioms, even of the most fundamental kind that define the existence of everything including ’empty’ space, are subject to modification over a historical period through cultural development. It, therefore, seems wise to historicize our more general view of what an axiom represents and think of them as assumptions (or declarations) made at a particular time during the development of systems of thought in particular academic disciplines.

One of the interesting features of science is that the axioms not only develop over time but they are also modified so that some of the axioms (or assumptions) within a set can be abandoned or modified and axioms can be added.  Although Newton reasoned that there was a universal force of gravity that applied to everything from falling teacups to the orbit of the earth around the sun, he was mystified by how that force could transmit its effect to another body through the void. Was there some kind of invisible chain holding the earth in orbit? It was not until Albert Einstein developed the geometric description that space-time could be warped by mass that there were the first glimmerings of an explanation. As we will see below this naturally led to the development of new axioms that were not present in classical particle dynamics. At the same time, this caused us to realise more about what assumptions were implicit in the earlier descriptions of classical physics.

As a result of the revolutions in 20th-century physics brought about by the development of Special and General Relativity, Quantum Mechanics and Quantum Field Theory, the axioms of classical particles mechanics were violated. Indeed some of the axioms of classical mechanics only arise because we now realise that there are more possibilities than previously understood in the period when classical mechanics was being formulated. How, for examples could anybody change the idea of how quickly time passed.  Why would one even want to think that the passage of time could vary depending on circumstances? What was wrong with a 3-dimensional idea of space? The list below illustrates a few of the changes to some of the core beliefs (or axioms) in physics.

In Quantum Field Theory small particles do not have mass in the sense we normally think about it (in relation to large objects). Higgs particles are said to have the property of interacting with the Higgs field.”The more strongly a particle interacts with the Higgs field, the more massive it is.”

a) Massless ‘particles’ exist such as the electromagnetic photon.

b) In quantum physics, very small particles cannot be described as having a definite location in space, only a possible distribution within space as defined by a probability.

c) Trajectories of motion do not exist at the quantum level although the idea of ‘spin’ is present. ( ‘Spin’ in this context might not refer to the classical spinning of large objects that we can easily observe.)

d) Three-dimensional space has been modified in relativistic thinking to include 4 dimensions of the space-time continuum  (Although time does not have the same properties as the spatial dimensions since there is no stillness of time from the point of view of a human observer). In String Theory, there are even more mathematically described dimensions.

 e) Time, or at least its passage, is no longer relativistically invariant but varies according to the velocity of a particle within a reference frame.

f) Space is no longer regarded as rigid but is instead distortable by bodies with mass according to General Relativity.

g) Energy is no longer continuously variable but is quantised into discrete levels by an amount determined by the Plank Constant.  (The fact that the value of the Plank Constant is so very tiny gives rise to the notion in classical mechanics that energy is non-quantised. This simplification is accurate to an extremely high degree of precision. In other words, for many practical purposes, quantisations are so small that they are both undetectable and computationally irrelevant. The size of the Plank Constant also tells us that the wavelength of large objects such as the human body is completely insignificant, although technically all objects have a wavelength defined by their quantum mechanical wavefunction.)

h) Strangely, pairs or groups of particles can be described as being quantum entangled, so even the fundamental ontological concept of discreteness needs to be modified. What we now think of as location is relegated to the metaphysical notion of ‘local realism’.

i) On the minute scale of the ‘Plank Length‘ (or greater) space might be ‘granular’ rather than continuous.

Quantum theory has been a marvellous development in 20th-century physics that has revolutionised human understanding of the nature of light and the mechanics of extremely small particles. As a result of this theory, we have come to readily accept that very small things such as macromolecules, atoms, subatomic particles and photons can be observed to behave in very strange ways, by our everyday standards of perception. It has become axiomatic that such apparently strange properties exist.

The famous 2-slit experiment shows that small particles can easily be shown to have wave properties.
The wave properties of larger bodies are too small to be detected in this way and are irrelevant to our understanding. In the earlier axiomatisation of classical physics, the phenomenon of particle interference would have made no sense.

Nevertheless, there are very severe limitations of quantum theory and the current mathematics used in its formulation.  The possible distribution of a single electron in relation to the nucleus can be understood analytically using solutions to the Schrodinger wave equation. However, there is currently no analytical solution that describes the distribution of electron waves in complex atoms, although numeric approximations can be calculated using supercomputers.

Modern physics is of necessity highly mathematical and can only be properly understood in mathematical language. Things have now got to the stage however where theorists make claims based on the elegance of the mathematics rather than observations of nature.  There has even been a debate in theoretical physics about whether or not a physical theory can be ‘proved’ by mathematical conjectures and proofs alone; in other words, the axioms arise out of the maths. I think we should resist such claims because mathematical descriptions are only a valid representation given the meta-mathematical or non-mathematical axioms, not the other way around.

Steve Campbell
Glasgow, Scotland
2017, 2019

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