OK, dat is duidelijk. Ga ik dat in gedachten houden bij het lezen over de flowing point. Een eerste gedachte die bij me opkomt is dat (1,3) en (3,2) betrekking hebben op coördinaten, en we het in dit deel van de uiteenzetting nog niet hebben over de dimensionele wereld van ruimte.
Ik wou dat ik dit soort passages moeiteloos kon lezen:
Maar het stimuleert me om het uit te zoeken. Beter laat dan nooit.The standard formula is: eix = cos x + i sin x. However, we are now going to write ei(cx) = cos(cx) + i sin(cx), but with c = 1, so that we have not actually changed Euler’s Formula at all. We shall of course equate c to the speed of light (or, rather, the speed of sinusoids), and we shall ensure that it always equals 1, no matter the frequency, by writing 1 = fλ (as in c = fλ, with c = 1), where f is frequency and λ is the wavelength. No matter what f is, we can always pair it with λ = 1/f, ensuring that c = f x (1/f) = 1. We now have a speed associated with Euler’s Formula, and it’s the same for every instance of the formula, regardless of how much we change the frequency.
That’s why the speed of light is an absolute. (And note that we have a speed of light for the cosine wave and a speed of light for the sine wave).
In physics, you will see expressions such as f(t) = a sin ωt, where t is time, a is amplitude and ω is angular velocity. It’s vital to realise that you cannot use such expression in terms of the immaterial ontological mathematical world outside space and time to which we have been referring. You can use time only when it has been defined and derived. You can’t just assume it. So, forget all about time and space references in relation to the fundamental equations of existence.
Dit soort passages, ook weer over een "stromende punt", fascineren me véél meer dan de abstracte presentatie van mijn wiskundeleraar vroeger:
Ook jullie bezwaren stimuleren me om de begrippen beter te begrijpen. Twijfel is goed. Mijn hernieuwde interesse in de boeken en de besproken onderwerpen was een onverwacht gevolg van mijn besluit om op dit forum een account aan te maken. Ik ben jullie dankbaar.What is the Planck constant really all about? Consider a point flowing round the circumference of an Euler circle. If the point had zero speed, it wouldn’t flow at all. It would have no dimensions. If the point had infinite speed, on the other hand, it would be everywhere on the circle at once, and thus have the complete dimensionality of the circle. However, if its speed is neither zero nor infinity then its speed must be a finite number between zero and infinity, and this number must be part of a complete and consistent mathematical system, just like π and e. The number in question is c, the speed of light, or, to be more precise, the speed of sinusoids in the frequency domain outside space and time. This is strictly a mathematical number. It’s not “scientific” at all. In natural units, it is simply “1” (since real numbers and imaginary numbers are perfectly balanced).
All sinusoids in the frequency domain travel at exactly the same speed, and that means that the flowing point in every sinusoid has exactly the same size. This is a tiny 1D number, and is none other than the ontological Planck constant. Once again, this is a number of pure mathematics, not of science.
The flowing point is the basic particle and is the smallest possible size of a particle. The famous problem of wave-particle duality is resolved by the fact that the flowing point follows a wave trajectory (the wave aspect), while the flowing point itself is a localised energy packet (the particle aspect). The higher the frequency of the wave, the more energy the flowing point carries.
Providing you analyse wave-particle duality mathematically rather than scientifically, there is no contradiction. The very essence of a sinusoid is to have all of its energy concentrated in a tiny 1D package called the flowing point. While the 1D packet is in the frequency domain, its net effect is dimensionless (as the packet flows round the Euler circle, it passes through as much negative territory as positive territory, so we can say that its 1D nature changes from positive to negative, and thus has a resultant of zero).
When the 1D packet enters spacetime, it now has a dimensionality in relation to space and time, and this, in fact, is none other than the basis of the 1D string loops of M-theory. M-theory is just a clunky spacetime heuristic superimposed over ontological mathematics. We defy any M-theorist in the world to prove us wrong! No matter what science says or does, we can always underpin it with analytic mathematics, and show how all of the concepts of science are simply clumsy versions of, and approximations to, underlying mathematical concepts and entities. There is no such thing as a science not grounded in pure ontological mathematics, the queen of the sciences.