How viable are points as a basis to understand space?
It is humbling to see how, after the demise of Pythagorean cosmology spelled by irrational numbers, we laboriously managed to prop up what was left via the infinitesimal calculus and Newtonian cosmology… Only to see it fall again for reasons that don’t seem too different: our notion of space as a continuum of points and the related conception of matter as a collection of atoms is no good.
What is a point? If one is tempted to disregard this question with a necessarily vague answer along the lines of “points are what space is made of”, that is because one sees points as fundamental. Despite the fact that the notion has been troubling for millennia, and even more so for more than a few decades by now. So, let us start with something a bit more nuanced: a point is an absolutely precise location in space. There are some issues with the absolutely precise part — surprisingly deep issues, indeed. But now we are emphasizing the fact that points are an idealization and, as all idealizations, they can be useful or not, depending on the context.
Apparently, absolutely precise locations in space did really exist to the Greeks. They visualized the points in a line as being in correspondence with the mathematical numbers they knew about: fractions. That seemed to work and was a compelling vision, for fractions are perfectly tractable, reasonable objects (so much so that we call them rational numbers), being completely determined by the two integers a and b in a/b. There is some innocuous arbitrariness in the choices of origin (point that will correspond to the number 0), orientation (half-line that will correspond to the positive numbers) and scale (point in the positive half-line that will correspond to the number 1). So, if this is to work, any two different choices of scale must be commensurable, i.e. the ratio between the lengths of any two line segments must itself be a rational number; thus, it came as a philosophical earthquake when the Pythagoreans realized that this was just not true: the diagonal of a square is not commensurable with its side length, and in fact there are “much more” irrational numbers than rational ones if one is to keep pushing forward the idea of putting a geometric line in correspondence with a number system.
Few documented events in the history of human thought have been so momentous. It even seems that we have not yet, more than two and a half thousand years later, fully come to grips with the consequence that space might not be best thought of as a set of points. The existence of irrational numbers destroyed a cosmology: a purportedly complete and coherent picture of reality. Since then, we have seen only one similarly grand cosmology tumble down: Newtonian mechanics and gravity, superseded by two recalcitrant theories (general relativity and quantum physics) that stubbornly resist all our attempts at harmonizing them. Thus, we currently have no cosmology in the sense above (complete and coherent), and meanwhile another momentous historical mathematical realization occurred that has called into question the very possibility of having one, at least within the realm of formal mathematics: Gödel’s theorem. But let us not divert.
Once fractions are proved insufficient to cover a geometric line, the absolute preciseness requirement in the notion of a point leads us inexorably to an unmanageable infinite: no general device exists that enables one to fully specify an arbitrary real number using a finite amount of information. For the most common device, decimal expansions, this phenomenon takes the following form: irrational numbers have non-terminating, aperiodical expansions, and can therefore never be fully specified in that way. Of course, other means like the use of mathematical formulas or geometric constructions enable us to effectively communicate with absolute precision some irrational numbers (like π or √2), but that does not help the fact that we confront a fundamental limitation when dealing with the concept of a point.
For infinity is out of our reach. In what sense does it even exist? Mathematicians, however, after the awesome discoveries of Cantor — the first brave soul to take infinity seriously — seem to feel very comfortable with it. “No one shall expel us from the paradise that Cantor has created for us,” said the influential Hilbert in 1925 when opposing Brouwer, who defended a style of doing mathematics (intuitionism, itself a form of constructivism) carefully designed to talk only about things that do exist, whatever that could actually mean in such a context. Constructivists did have a point, for non-constructive methods enable one to prove things that clearly distance mathematics from reality, like the possibility of dividing a sphere and reassembling it again in such a way as to end up with another, twice as big as the original (the Banach-Tarski paradox). But such paradoxes were just the price you had to pay to be able to grant existence to all points of the real line. And the good old points felt too fundamental to let go. Cantor and his contemporaries had opened up a path to fully embrace the continuum with all of its irrational points by putting infinity and the infinitesimal calculus on a sound basis, thus setting the stage for a majestic realization of the badly desired correspondence between (real) numbers and the geometric line — which, in fact, had never ceased to be at the heart of our scientific worldview, even if the Pythagorean disaster had made it a bit problematic.
But the history of Occident’s continuing attempt at rationalizing the world wouldn’t be so fascinating if that was the end of it for points. Just as Hilbert launched his proclamation of Cantor’s paradise, Heisenberg was putting quantum physics on a sound basis, by shifting focus from (necessarily theoretical) objects to (rather operational or relational) attributes. It would only take a few more years for physicists to stumble on the nonsensical infinite answers that quantum physics leads to when naively applied to fields, in a phenomenon that would take several decades to be fully understood (via renormalization theory) but was clearly related to the infinite energy required to make a measurement of absolute precision — a point-like measurement. So, basically, we just kept stumbling upon the same old problem: our ideological conception of space as being made of points leads to nonsense for all practical purposes, because of the infinity that lurks behind the notion of absolute precision.
It is humbling to see how, after the demise of Pythagorean cosmology spelled by irrational numbers, we laboriously managed to prop up what was left via the infinitesimal calculus and Newtonian cosmology… Only to see it fall again for reasons that don’t seem too different: the notion of space as a continuum of points and the related conception of matter as a collection of atoms is no good. When the time is ripe, there will be no one to expel from Cantor’s paradise: competing foundations for mathematics based on alternatives to points as fundamental objects (be it lattices of space regions definable in terms of a finite amount of information, or algebras generated by actually observable object’s attributes) are maturing. Together with them, hopefully, a collective consciousness regarding the limitations of mathematics as an approach to nature will also mature, for we are long overdue in transcending Occident’s enchantment with reason as the ultimate, supreme reality. Life is here to be lived, not to be rationally understood, modeled and optimized.