As was briefly discussed in the previous article, the modern conception of space does not differ much from the pre-history era, the only difference is that moderns have, with symbolic language, codified it and think they have the upper hand in terms of understanding the universe. However, looking at the genealogy of ideas, it seems the geometric detour that physics took breaks down every step of the way, and patches things up: renormalization is used so as to tell ourselves that numbers shooting up to infinity aren’t real but require careful patchwork, with the mantra “Shut up, and calculate”.
(Disclaimer: this article does not relate to the geometry of gravitational anomalies such as singularities, but rather delves into the brief history of Cartesian geometry and its relation with quantum mechanics.)
Descartes’ Geometry and the Industrial Revolution
René Descartes is one of my favorite philosophers; in fact the famous quotation which often gets thrown around, “Dubito, ergo cogito, ergo sum”, was in fact very inspiring for me when, early on, I got familiar with philosophy. Looking at it now in retrospect, it seems the demons that Descartes was fighting were from spherical aspects of Euclid’s geometry, after so much thought experimentation to understand that by assuming he could not trust his senses, he took to abstract realms and concluded that the shape of the triangle, even if distorted in the world, would be perfect in his mind. Except, could he meaningfully distinguish between a triangle in a flat plane and abstract realms? I doubt it.
Descartes introduced, or rather codified, the primitive perception of space with an x, y, z grid on a flat plane, building on top of Euclid’s Geometry, which of course did not have such mathematical notions, although from Euclid’s work the coordinates that can be derived relate to spheres and can include r, θ and φ.
Of course, the codification of those coordinates led to a boom in technological advancements in the age of Enlightenment and the Industrial Revolution and revolutionized many domains of human life. However, a point to be noted is that advancement of technology is not indebted to a single domain of knowledge; human technology is always advancing in multiple domains, and right here Descartes’ geometry boosted the trajectory because in other domains individuals were able to reason about things and make use of them.
The trajectory of advancements went on and on. Because Descartes’ geometry was so trustworthy, subsequent mathematicians such as David Hilbert and many others built on top of it, and it makes sense in this context.
The Elephant in the Room: Quantum Mechanics at Crossroads with Geometry
To help you imagine the problem, let’s do a thought experiment which will show the issues with the geometry that physics uses. We place Hilbert, Descartes and an adult male elephant in the room, with all x, y, z 3D geometry intact, and they have to do what they do best: geometry. Suppose with some hyper-advanced tool we start shrinking the size of an imaginary elephant that is standing still; the direction in which the elephant shrinks is orthogonal to itself, and it gets visibly smaller, slowly. Now, the moment the elephant disappears out of sight completely, Hilbert geometry yields 0 until further microscopic or quantum studies can establish otherwise, while Descartes’ geometry also yields (0, 0, 0) for coordinates. Their conclusion: out of sight, out of reality. But that’s not the case: in so much as the reality we perceive with our senses is real, so is the quantum soup; just because we cannot directly perceive it does not mean it’s not real. Euclid’s invariant geometry is not bound by scales and would keep track of the imaginary elephant that is getting smaller.
A trip down memory lane: the proton particle in the 1960s was described as “point-like”, and assigning anything point-like 0D is questionable, because 0D implies a “container” that does not exist, yet that “point-like” object exists with properties like spin and more. This is essentially a very contradictory and illogical way of saying “nothing is containing something”. However, in the SLAC National Accelerator Laboratory scattering experiments, the proton was found to have sphere-like properties.
When Cartesian geometry at such small scales almost always shoots up to infinity, one way or another, and physics keeps using g-tensor for “renormalization”, instead of fixing the broken geometry they choose to keep doing the patchwork, which is dishonest with the spirit of science. The g-tensor that is used is also based on Cartesian geometry, but with many fancy variables, and its escape hatch is “coordinate invariance”, which in case they forgot is also inherent to Euclid’s geometry, the geometry they wrongly dismissed.
Einstein’s 1913 draft was really an intuitive and intellectual genius, except the distorted view of Euclid’s geometry was problematic for the framework he was originally trying to build, and in fact may have brought humans close to a unified theory of gravity had it been completed using the right geometry. However, Einstein was forced to use the Ricci tensor, which he resisted for about two years.
In conclusion, the geometry that is used by physics, which in the past 400 years helped advance humans in many different domains, is not compatible with the quantum realm and should be discarded altogether. Many different scientists, even Feynman himself, who helped formalize it really despised it and described it as a “dippy process” and “mathematical hocus-pocus”, sweeping infinities under the rug.
