Do Martians Do Mathematics (And Do They Believe in God)?

Jean-François COLONNA
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CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France

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Abstract: Without always realizing it, we live "immersed" in Mathematics. Despite this, we still remain ignorant of their profound nature. It is a transcendent question whose complete or partial answer can only come from outside, for example, from an unlikely encounter with extraterrestrials.

Mathematics are everywhere!

Thus, in everyday life, without them, there would be no GPS. Let's take a moment to focus on what is now indispensable to many of us. The GPS [01] was developed about forty years ago at the initiative of the Pentagon. At its core lies a fundamental problem of geometry: how to determine the position of a point P knowing its distance to N reference points? In a plane (N=3), it is sufficient to calculate the intersection of 3 circles, and in space (N=4), it requires 4 spheres. But how can this be applied to the surface of the Earth, and specifically, how can the distance between two points A and B be obtained? It "suffices" to measure the propagation time of an electromagnetic signal from A to B. And as always, the devil is in the details! The objective is, of course, to achieve high precision, so it is necessary to use extremely accurate and stable clocks: provided by Quantum Mechanics, in which Mathematics play a prominent role. But that's not all: physicists had fortunately warned the military at the Pentagon about relativistic effects. Indeed, since the satellites are in motion, their clocks experience time dilation from Special Relativity [02], but being at a higher altitude and therefore subjected to a weaker gravitational potential than at ground level, this time they are faced with an inverse phenomenon of General Relativity [03]. Once again, Mathematics play an essential role: without them, quite simply, the GPS would not function correctly!

The situation is the same in the most fundamental research to such an extent that one can (and must!) consider Mathematics as a virtual optical instrument that reveals new perspectives on Reality every day. We will remember, for example, the gravitational waves [04] predicted by Albert Einstein in 1915. A mathematical prediction, but their actual detection required a century of efforts (a priori seemingly desperate due to the expected weakness of the signals...) since it was not until 09/14/2015 that they were observed for the first time at both LIGO interferometer sites in the USA. Once again, Mathematics played an essential role, both in the design of these instruments [05] and in the analysis of signals with an incredibly low amplitude, for which the contribution of French mathematicians Thibault Damour and Yves Meyer was essential.

Unfortunately, this omnipresence seems to be ignored by many people, especially our government officials. Just remember the recent and unfortunate reforms of the baccalaureate, particularly concerning Mathematics, to be convinced of this. And that is a great pity because, indeed, pursuing Mathematics (especially Applied Mathematics) guarantees an exciting life, employment, and also being useful in these times when reindustrialization, energy efficiency, nuclear power, and other related areas are gaining momentum!

But despite these successes and ubiquity, their profound nature is still unknown to us. We still do not know whether our mathematicians are "inventors" or "explorers." I am certain that the answer to this question is beyond our reach without "external" assistance, as is the case with many other transcendent aspects [06]!

Under these circumstances, could "Martians" help us answer this question? The question may seem absurd, considering the uncertainty surrounding their existence. In 1950, during a lunch with several colleagues from the Los Alamos laboratories (USA), Enrico Fermi (Nobel laureate in Physics in 1938) expressed surprise that, given the age of the Universe, Earth had not yet been visited by extraterrestrials (assuming they possess curiosity and a drive for expansion, like us). This inquiry is now known as the "Fermi paradox", and I can provide ten possible and non-exclusive explanations for it. But what does this have to do with the nature of Mathematics?

Even though I am convinced that our Earth is not the only cradle of life in the Universe, the encounter with "Martians" seems both highly improbable and undesirable due to the associated risks. However, if one of those spacecraft, which science fiction has familiarized us with, were to arrive on Earth and its occupants were benevolent, allowing for dialogue to be established, it would present a magnificent opportunity to question them, particularly about their understanding and "mastery" of the Universe: "Do you, like us, engage in Mathematics?" [07]. In the case of a positive response, our mathematicians would most likely be explorers, and the debate might be settled [08], even though the "Martians" themselves might not truly know what Mathematics truly are. But what if the answer is negative? What a disruption it would be to learn that such understanding and mastery (given that they have reached us) could be achieved without our sets, without our numbers, without our beloved equations... How could one imagine, for example, not relying on counting and yet identifying, say, the laws of invariance and then reaching the stars? Would we be capable of comprehending these truths, with our minds inevitably limited and conditioned by our terrestrial environment? The answer is certainly negative.

But unfortunately (or fortunately?), there is little chance that we will witness such an encounter and finally have the profound nature of Mathematics revealed to us.

  • [01] - Global Positionning System.
  • [02] - Albert Einstein (1905).
  • [03] - Albert Einstein (1915).
  • [04] - These are "ripples" in spacetime caused by cataclysmic events, such as the collision/merger of two black holes or two neutron stars.
  • [05] - LIGO, not forgetting VIRGO in Europe and other sites currently under construction elsewhere in the world.
  • [06] - Some examples: "Why is there something rather than nothing?", "Is there a Creator of all things, and if so, who created it?", "Where is the Universe?", "What is the Big Bang?", etc...
  • [07] - Other questions, equally relevant, could be asked, particularly: Do you believe in God?
  • [08] - These two conditionals are here to emphasize, on the one hand, that it is impossible to exclude the possibility of inventing the same "tools" on both sides of the chasms of the cosmos. On the other hand, it should be noted that mathematicians could be both "explorers" (of the most fundamental structures) and "inventors" (of higher "levels"), as Leopold Kronecker wrote in 1891: "God made the integers, all else is the work of man".

  • Copyright © Jean-François COLONNA, 2023-2023.
    Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2023-2023.