Relativity and the mathropologist

TL;DR – The laws of physics are the same for all observers because nature couldn’t care less what coordinate system we use.

I recently met someone who had some slight misconceptions about the principle of relativity. “Is it true that time and space are relative concepts?” “That contemporary events for you may not be contemporary for me?” “Does it mean that time, space and all of physics are just social constructs, like morality, truth or moistness?” Indeed, understanding of relativity is not absolute.

A friend of mine always insisted that Einstein proved that “everything is relative”. He would do so when arguing that he was not, in fact, two hours late but on time with respect to a different time-zone. What the principle of relativity actually states is just that “all systems of reference are equivalent with respect to the formulation of the fundamental laws of physics.”

Galileo Galilei was, in fact, the first one to bump into this idea. He was on a ship, in a cabin below deck with no windows. He thought they had been traveling for hours but, when he went up, he realized they had been anchored at the harbor the whole time. He was very angry as he was late: he needed to drop something from the tower of Pisa. But then it hit him: the laws of physics were such that traveling at constant speed or staying still looked the same within the cabin. Which is excellent for plane travel.

Einstein had a similar experience. He was stuck in an elevator with no windows and a force was pushing him to one side. And he couldn’t tell whether it was centrifugal force (like when you curve too fast in your car) or it was gravity (maybe the elevator was tilted). But then it hit him: the laws of physics were such that accelerated motion without gravity or constant speed with gravity looked the same within the elevator. I may remember some details wrong… But the point is: you can’t be a great scientist if you suffer claustrophobia.

Now, the idea that the laws of physics look the same under any reference frame may seem like a suspiciously big coincidence. Indeed, some people are awestruck and ponder on these types of things. But when something like that happens, I always feel the answer is probably something very obvious staring you right in the face. Which, by the way, I’ve been told is very impolite. So, let’s see what we can learn with another daring and pusillanimous thought experiment.

Mathropology

You are a mathropologist. What is mathropology? You tell me… It is a field of study that you yourself founded: you are the first one! From what I understand, you study the impact of social norms and constructs in the practice of mathematics. The word is a portmanteau between mathematics and anthropology.

What you wonder is whether cultural conventions of the different nations around the world have a very significant impact on the results that the applications of mathematical ideas, methods and/or techniques have on different calculations and/or proofs within the disciplines of math itself or as applied to various scientific branches. For example: what type of influence has the articulation of the sounds used to identify a particular mathematical notion on the inferential process used to construct a proof and/or calculation upon said mathematical notion?

Your detractors say you hide in convoluted prose, and would paraphrase your field as: “does two plus two equal four in every language?” Maybe because that was the title of your first study… But that kind of derogatory description does not demoralize you: it fills you with determination.

Your second attempt was inspired by East Asian architecture. The use of curved three sided shapes in the construction of towers with a repeated substructure that gradually diminish in volume, made you confident that in those cultures geometry must have taken a different turn. You were convinced that, for them, the sum of the internal angles would be less than 180 degrees, making those “hyperbolic cultures” instead of the Euclidean ones of western tradition. Unfortunately, you learned that, especially when drawing on a piece of paper, even those cultures do exhibit a preponderance of Euclidean traits. They too seem to believe in the Pythagorean Theorem. A shame for your research.

Your third attempt came about when you discovered that in Italian the word used for the sine function is “seno”, which also means breast. Just to be clear: the one women have, not the one that chickens have… I always found it kind of weird and confusing to eat chicken breasts, given that they are not mammals. Anyway, you were certain that, because of this coincidence, trigonometry in Italy would have deeply sexualized results. That triangles would have different properties as calculated by Italian men as opposed to Italian women. Unfortunately, the only effect you could see was childish giggles among students. A shame for your research.

After decades of research, you are ready to publish your final result: a 1,250-page volume in which you state your “principle of mathematical relativity”. It states that “all languages, all cultures, all nations and all points of view are equivalent with respect to the formulation of the fundamental rules of mathematics”.

Relativity of objectivity

In the same way that language or dietary restrictions play no role in mathematical results, coordinates and units should play no role in fundamental physical results. We can call a distance 1 meter or 3.28084 feet: it’s still the same distance. Naturally, the first unit makes a lot more sense, since feet are really parts of your body. I never understood why the United States of America still uses the British Imperial System: I thought they left the empire a long time ago.

The idea that names don’t matter is not true in all disciplines. Trying to pass the “increased surveillance and decreased liberty act” was hard so it was renamed to “patriot act” and got overwhelming support. I hear politicians are thinking of renaming “global warming” to “global cozying”.

Some units and reference frames, though, may simplify the description for a particular problem. For example, 1 liter of water weighs exactly 1 kilogram. Now, this is not the expression of some deep physical truth. The units were defined like that for convenience. For another example, choosing a Sun-centered reference frame makes the description of the solar system a lot easier. But it’s not “more true” than an Earth-centered reference system, which is in fact more convenient when we want to describe the orbit of the artificial satellites. The trajectories in space and time, which are the actual physical entities, are always the same: it’s the description that changes.

If we are looking for actual fundamental physical concepts and relationships, these cannot depend on the particular description. So the principle of relativity should not be at all surprising. The hard bit to wrap your head around is that space and time coordinates are part of our description, so they are not fundamental physical concepts. The speed of light is what is fundamental. But “the invariance of the speed of light” is a separate principle, which together with the “principle of relativity” gives you the “theory of special relativity”. So we should be talking about it separately. I don’t like it when my principles touch.

“Principle of relativity” is really a misnomer because it makes you focus on the different descriptions: time is relative, space is relative, moistness is relative and so on. It should really be called “principle of objectivity”: the fundamental laws of physics describe objective entities and their relationships, so they do not depend on the particular units or reference frame used. And that’s not just an opinion: it’s my opinion.