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Considering the Quantum Mechanical definition of particle and its two possible ways of measurement(detecting), point-like object and wave.

Does this have implications for how we should perceive Euclidian geometrical objects like point and line(plane)?

In other words, I take it that QM challenges our perceptions of particles, by showing they exist in a degenerate form consisted simultaneously of point and/or plane(wave). Thus, this breaks the rigid barrier between point and line/plane. Does this have implications for how we should view geometric objects?

  • I've re-revised your edit (no need to edit just by adding). I'm going to reopen it, but .... I think jobermark has the right idea of why this should have no implications. You also might want to review your QM as to range in which the uncertainty happens and thus the degree to which this changes anything for normal scale objects. – virmaior Apr 26 '15 at 2:34
  • @virmaior Thanks for the re-revision! As to the exhibition of wave properties, you are right, observable de Broglie wavelength is attributed only to small mass objects with large speed; the more precise the measurement of one of the conjugate coordinates the larger the uncertainty in the other, so that their product is always smaller than half the reduced Plank constant. – Ziezi Apr 26 '15 at 9:23
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Physics never has implications for mathematics, only the other way around. Mathematics is not an experimental science, but an 'exact' one, and is only expanded, never altered, by the discoveries in other sciences.

Surely there are productive extensions of Euclidean geometry that capture the newly discovered notions of space and of matter. (Back in the 80's I have seen folks define slight variations of topology that give open sets a quantum-logic penumbra, so that the resulting points are 'sort of fat' in a way that simplifies quantum and relativistic geometry, to the degree that quantum logic ever simplified anything.) But the old system captures something so natural to human beings that it will never cease to be our common reference point for what geometry means.

From an intuitionist point of view, Euclidean geometry is part of our genetic inheritance, deeply embedded in our psychology, there to be discovered. This is why it was the perfect example of anamnesis for Plato. Folks who take a more functional or a more ideal approach to mathematics are also going to agree that this has the most immediate survival value, or that this occurs naturally so often that it must be part of underlying reality.

None of them is going to expect it to change due to discoveries in physics.

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  • I agree that the relation between Mathematics and Physics is unilateral, with the latter borrowing precisely constructed abstractions to model/represent real world observations (like Riemannian Geometry describes space-time in General Relativity). Just the more I read about QM the more it distorts some of the most basic, and as you nicely said "genetic inherent, deeply embedded in our psychology" facts about the world. – Ziezi Apr 26 '15 at 9:35
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First, it's useful to point out an interpretation of QM that brings it closer to classical mechanics; and this is the Bohmian Model in which particles move in a random walk; this is directly comparable to the Lucretian view that particles move randomly in themselves and not due to impacts ie the clinamen.

In Euclid one recalls that his definition of a point is

that which has no parts

Which directly recalls the notion of an atom as that which has no parts; parts meaning being seperable; but not distinguishable.

For Euclid, a line is not made up of points as we understand it; it is a synthetic object. Points mark the ends of a line or where one line crosses another; they are markers of position.

Hence, one can feasibly say that the first synthesis of lines and points is to conceive lines as points.

Now a particle is simply a point in motion; and a wave is a line in motion. Concieved as points, a wave is just particles moving up and down; this might conceivably called the first synthesis of particles and waves.

But what about QM? After QM, physics began to posit extended objects - vortices, strings and branes; one might say they began to explore the possibility that conceiving particles as points was the basic mistake - a mistake that can be traced to The Euclidean definition of a point.

The first atomists were clear that their atoms had extension since they were modelled as Platonic solids; so, in a sense we've returned to an older tradition.

So yes; the point is, that points aren't simply points; that they don't have extension or parts; but that they have extension.

Aristotle pointed out that the actual infinite could not physically exist; but he did admit infinite divisibility; but this isn't the infinitely small; and the contrary appears to be true; that the infinitely small does not either in any aspect, be it space, time, matter or energy, exist; doing so leads to errors.

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