An axiom is a statement so evident or well-established that it is accepted without controversy or question. However, in his review of Sokal’s Hoax, Steven Weinberg has suggested that ‘none of the laws of physics known today (with the possible exception of the general principles of quantum mechanics) are exactly and universally valid’. This propels physics to the same status as biology (see my post entitled ‘Laws of biology?‘ on January 13th 2016) – in lack exactly and universally valid laws and it suggests that there are no scientific axioms.

‘Things that are equal to the same things are equal to each other’ is Euclid’s first axiom and in thermodynamics leads to the Zeroth Law: ‘Two things each in thermal equilibrium with a third are also in thermal equilibrium with each other’ (see my posts entitled ‘All things being equal‘ on December 3rd, 2014 on ‘Lincoln on equality‘ on February 6th, 2013). Thermal equilibrium means that there is no transfer of thermal energy or heat between the two things, this leads to the concept of temperature because when two things are in thermal equilibrium we say that they are at the same temperature. Last week I explained these ideas in both my first year undergraduate class on thermodynamics and my on-going MOOC. This week, I have challenged MOOC participants to try to identify other measurement systems, besides temperature, that are based on Euclid’s first axiom.

For instance, its application to mechanical equilibrium leads to Newton’s laws and from there to mass as a measure of a body’s inertia. We use Euclid’s axiom to evaluate the mass of things through a chain of comparisons that leads ultimately to the international kilogram at the Bureau International des Poids et Mesures in France. Similarly, we measure time by comparing our time-pieces to an international standard for a second, which is the duration of 9,192,631,770 periods of radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.

However, given Weinberg’s statement perhaps I can give you a harder challenge than MOOC participants: can you identify exceptions to Euclid’s first axiom?

I think I can identify one: if you calibrated two very accurate timepieces against a cesium 133 clock and then took one on a journey through space travelling at the speed of light while the other remained on Earth, when you brought the two together again on Earth they would not agree, based on Einstein’s theory of relativity, or what he called relativity of simultaneity. Now see what you can come up with!

Sources:

Steven Weinberg, ‘Sokal’s Hoax’, NY review of Books, 43(13):11-15, August 1996.

Oliver Byrne, First Six Books of the Elements of Euclid, London: William Pickering, 1847

Joseph Schwartz & Michael McGuinness, Einstein for Beginners, London: Writers and Readeres Publishing Cooperative, 1979 & Penguin Random House, 2013.

Albert Einstein, Relativity: The Special and the General Theory, (translated by Robert W. Lawson), London: Methuen & Co Ltd., 1979 & on-line at www.bartleby.com/173/

I am a little unsure but I think measurement of mass is also an exception. Since mass also changes with speed.