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“Why might the Pythagorean theorem exist?”

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Yesterday I answered a question in Quora about the Pythagorean theorem and I wanted to publish it as well on the blog.

The question was:

“Why might the Pythagorean theorem exist? Is it a purely an arbitrary relationship observed in nature?”

My answer was:

Hi Ari, I think this is a very interesting question. The most important advances in maths and physics started thinking “conceptually” on the bases of assumed ideas and asking “why”. Does the arbitrariness we detect in Nature exist actually or are we losing a hidden and logical and reasonable explanation of it? Are our mathematical and physical models – and calculations – a correct explanation of an apparently irrational reality? because arbitrariness is always irrational…

I’m not a mathematician nor a scientist but I’m going to tell you what I think.

I think nature is not irrational at all and if we find things – unexplained asymmetries, contradictory results, logical inconsistencies, etc – that we cannot explain in a reasonable way through the intervention of a cause and its effect, it’s because our models, our interpretations, or our assumed ideas are good enough yet. Randomness does not exist.

When it comes to mathematical calculations things get very problematic for some mathematicians. Could our operations being correct not be an accurate description of reality?

The Pythagorean theorem is related to two fundamental problems:

– The question about the irrational hypotenuse inside of the square (even inside of our referential square of root 1): how is it possible that we cannot measure in an exact way such a limited segment?
– The question about the equality of the areas built on the legs and the hypotenuse: how is it possible that if the sum of the two legs is not equal to the length of the hypotenuse, the squares build on those unequal segments are equal?

None of those questions has been logically resolved yet. Actually, most mathematicians are not thinking about them as assuming these kinds of logical inconsistencies, these lacks of unexplained anomalies are something “normal” or “natural”. Most of them are not worry about them because they only calculate or operate in an abstract way.

But you are not a less mathematical person than them by asking these questions, quite QUITE the contrary. Think by example about Bernard Riemann and all the enormous consequences his conceptual ideas had to modern mathematics and physics (his conference On the Hypotheses which lie at the Bases of Geometry did not have almost any formula).

So if we think about the apparent arbitrariness of the Pythagorean theorem we can assume that things are in that way and start to operate with the algebraic equation or we can start thinking about reasonable explanations of what we are measuring.

I think the first thing we should start thinking about is, are we working on a flat space that is invariable? Are we drawing all our segments and squares on a same and static plane? Mathematicians have assumed so far an affirmative answer to those (maybe yet unformulated) questions.

If we take two segments of length 1 and build two squares of area 1 on them, we are working on a flat static space on a unique and static plane with the coordinates X and Y. But when we trace the diagonal inside of one of those squares to build the square of area 2, what did we do then? Did we add a new Z coordinate on that static and unique plane, or did we displace 45 degrees our Y coordinate towards the right or left side? If we did such a displacement, as we still have the XY fixed coordinates on a fixed plane for the squares of area 1, the consequence is that we are expanding the space and the plane we are working on, or that we have created a new plane, the Z plane – displaced with respect to the XY plane – on that same static space and that such a displaced Z plane is ruled by a primary referential segment that is not a segment of length 1, because by displacing the plane we increased the length of the referential segment 1.

So now we have to referential metrics – the “rational” and the “irrational” ones, that cannot be compared directly as if they were on a same quadratic plane. If we try to compare directly those different referential segments or their derivative ones, we are going to get always a non-integer quantity because they are ruled by different referential metrics. (When we measure the space we need to start by an arbitrary referential length, our hand, our feet, etc, took by convention. But we are going to transform such initial quadratic or rational reference when we displace our initial segment in a circular way keeping fixed the originary XY coordinates.).

But, if our referential segments are not comparable why the areas we built on them are equal? Well, I think we are assuming they are equal but maybe they are only coincident, which is not the same than being equal.

When we take a primary referential segment to measure linear lengths we do an abstract act because we did not take account of every single point or line inside of that segment, we simply said the space of such a segment is going to have a specific and referential value, i.e. the value 1. But at the center of such a segment is going to be a very concrete point which is the center of symmetry of the segment.

When we build our referential square to measure areas based on our referential segment, we also make the abstraction of considering the space inside of that square has an are of a specific and referential value, in this case the vale 1. That square has also a specific center of symmetry.

Are we sure the centers of symmetry of the squares 1 and 2 are the same to say those areas are identical?

Anyway, how is it possible that those squares are even equivalent? I think that if they are equivalent is because their elements are equivalent. But how many legs and hypotenuses are we considering when comparing those areas? eight legs and four hypotenuses of the two squares 1, and four legs and two hypotenuses of the square 2 are not equivalent. But are we sure we are working with such amount of elements only? Why do not consider the in and out – or the right and left – sides of the legs and hypotenuses as different elements? In that case, the 16 legs of the two squares 1 are equivalent to the 4 hypotenuses of the square 1, and the eight hypotenuses of the squares 1 are equivalent to the eight sides of the square 2.

pythagorean_triangles_0001

These are only suggestions to show you that maybe we are not interpreting the measures we do because we are not being aware of all the interveners elements. You can thing others by yourself if you will.

John Milnor, in 1956, showed that “two manifolds that are equivalent at the level of continuous geometry could be different in an essential way at the level of differentiable functions (see the beginning of the book “Gravitation” by Misner).

I think the squares of the Pythagorean theorem can be considered as manifolds (take a look if you will at this article about Riemann geometry and his concept of manifold: http://web.ics.purdue.edu/~plotn…).

I think here differentiable manifolds are those that can be obtained by using differentiable operations, that is to say by using differential calculus (integration or derivation. But any calculation is actually about addition and subtraction).

We cannot create or “derivate” the square of area 2 by calculating it – derivating it – from the legs of the squares 1 – but we can get it by transforming in a “smooth” continuous way the segment of length 1 – expanding it.

So I think it could be said that the pithagorean square of area 2 is a non differentiable manifold.

I think the areas we wet in that way are equivalent in the sense of being coincident spaces. But are different because their centers of symmetry are related to different referential metrics.

These are suggestions trying to understand what I think mathematicians have not yet understood. There are other fundamental and logical inconsistencies in maths and physics you will find out by yourself if you keep pulling that Ariadna’s thread trusting on your reason and asking the magical question: WHY?

 

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Written by also65

octubre 13, 2018 at 5:31 pm

Publicado en Uncategorized

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