Room to wiggle
Onderhavige blog entry bestaat uit de korte samenvatting en de inleiding van een heel wat langere paper, getiteld "Wiggle Room". Mensen wier interesse is gewekt door dit korte fragment, kunnen me steeds mailen op firstname.lastname@example.org voor het bekomen van het volledige werk. Het werk is gericht naar mensen met zowel een filosofische en/of wiskundig-natuurkundige achtergrond. Alvast bedankt voor een mogelijk beloftevolle uitwisseling.
Up until now mathematicians and physicists are having a really hard time to reconcile the extremely small with the extremely big. It is to say, both domains are understood fairly well and can rely on rigorous mathematics in order to describe their inner workings. Yet, when one tries to interchange the math between the theories, calculations quickly become nonsensical, verging off into infinity and producing unwanted singularities. The big and the small are nevertheless both part of the same reality, so it seems odd –to say the least– that they would somehow require incompatible mathematics. That, in itself, does not make sense. While it is understandable that quantum mechanics and general relativity necessitate different approaches, one would at least expect there to exist a way to consistently translate one into the other. However, decades of research has resulted in one failed attempt after the other, which leads us to ponder the question whether something might truly be lost in translation. If we were able to pinpoint what that something is and possibly even attempt to formalize it, it could prove to be an indispensable tool on the eve of a renewed understanding leading us towards a very welcome road to reconciliation.
Mathematics is a representation of reality through a formalized language and an idealized one at that. One could even call it an abstract interpretation of said reality, rendering math an approximation by default built upon our own experience of the world. That is not to say, mathematics in itself is not consistent –at least up to Gödel–, it surely is, yet it remains in a profound way a simplification of reality and a much needed one at that. For in order to start making sense of the innermost workings of the world, one has to reduce its building blocks to manageable concepts first, eventually leading to theories which do or do not stand the test of a perceived reality. Out of this we finally learn and adjust accordingly.
Mathematics actually deals in perfections not found in any real setup whatsoever. How so? Let us take a closer look at the concepts of a circle and a square through their mathematical definitions: it is not at all difficult to see that there is in fact no circle –nor a square for that matter– to be found in nature. You may be inclined at this point to take out a piece of paper and draw a circle and a square in order to prove the statement wrong. Even if most people would agree you just drew a circle and a square, what you have actually done is draw something that more or less accurately resembles the mathematical idea of both concepts. In other words, a real life circle is not the same as the mathematical concept of a circle. The latter is an idealized concept derived from the former. Needless to say, the same goes for the square. One might even go as far as to claim the mathematical circle does not exist, that is if we reserve the word existence for that which can be only be known through experience. If it does not exist, what are we to make of it? The mathematical circle can be regarded as a common denominator, that which leads to a real life circle being perceived as circle-ish. First and foremost the general definition of a circle presents itself as an unresolved relation, unresolved in that before all else it merely is. Experiencing what exists can now be rephrased as resolving what is, i.e. lending a context to it. We see two realms coming into view: the Realm of Existence (Reality) and the Realm of Being (Mathematics). As Reality may now be seen as an expression (through experience) by way of resolving Mathematics –or that which resides in our understanding (a cluster of relations)–, the realms may be paraphrased as the Realm of Expression and the Realm of Understanding.
The fact that mathematics in itself –out of sheer necessity– has to make use of expressions, gives us a first clear indication that we might be treading highly unstable ground. For, as we just saw with our circle and square conundrum, our understanding tends to slide down a slippery slope by losing itself within an expression of it. In other words the distinction between what is mathematically sound and what is perceived in a real set-up becomes blurred. As our vision becomes blurred, so does our ability to fully comprehend what is going on right in front of us. A possible way to snap into focus again might be to somehow incorporate the idea of context itself in mathematics. For if a certain context demands we perceive of a square peg as being round, it may just fit that particular hole just fine.
"Wiggle room" proposes just so through a series of onsets and ideas, some of them already rather elaborate, others mere hunches, yet all of them begging for further development. First and foremost do not expect to find resolute answers within its pages, but read them with an open mind and the willingness to pick up new ideas, maybe to later expand upon yourself. This is not a paper on prescribed reality, but rather a vigorous invitation for all readers to think for themselves. If one simple phrase within its pages sets of a train of thought that spawns an original work by just one of its readers, the paper has fulfilled its purpose.
The content equally leans on philosophy, mathematics and common sense. I do not hold a degree in the first two and I can only presume I possess a healthy dose of the third ingredient, as it has guided me along the way. A good friend, when presented with the premise of the paper, urged me to include an instruction for its readers in the introduction: to those who are foremost mathematically inclined, read on and make your way through the philosophical meanderings; to those foremost philosophically equipped, do not give up and work your way through the mathematical formalism. As always, the reader has the last word.
Einstein’s theory of Special Relativity teaches that the laws of physics are invariant under Lorentz transformations. The equivalence between inertial and gravitational mass subsequently led to the formulation of General Relativity. The present paper argues that in order to further our understanding of the hitherto most elusive inner workings of the universe –bringing together the extremely small and the extremely big– another principle tends to make its presence felt: the fundamental indistinguishability of space and time through the use of a Euclidean metric signature (4,0). Moreover, it is pointed out that the more familiar Minkowski space-time of heterotic signature (3,1) may be retrieved as a particular limit of the Euclidean variety, avoiding any unwanted singularities. A highly speculative philosophical re-evaluation of contemporary mathematics shows a new paradigm shift might be lurking on the horizon: the notion of a universal Contextual Relativity.