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A 'simple' Physics postulation...

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T3sl4co1l:
Well, if you want debate the physicality of numbers, it's pretty clear rationals exist; that's just a matter of definition.  The rest, whatever you like.

Irrationals, maybe, but it's not like we can construct nor measure a perfect right triangle.  Even if one made a crystallographically perfect shape and counted the atoms along each side, one would find whole numbers, a mere approximation to the irrational ratio.  How should you even count the corners?  They aren't perfect right angles, there's at least one ionic radius around the edge.  Should you project the lines/planes towards their ideal intersections, or should you measure the real distances, taking account of the corner somehow?

One likely cannot even construct a 3-4-5 right triangle, as the hypotenuse is not on a crystal plane -- unless there happens to be a symmetry group with exactly such a plane, in which case merely pick any other of the infinite Pythagorean triples you can choose from and check again.


I'm fond of a little rant about "real" numbers, as there are so very few indeed that can be constructed or computed in any meaningful way -- pi and e happen to be two of the most powerful ones, showing up in a great many places.  The whole space of real numbers -- let alone just those that can merely be computed, through any arbitrary (and likely impossibly slow) process -- is truly, unsettlingly vast.  It's not just an uncountable infinity, but most -- indeed, "nearly all", well and truly defy any possible description, by any finite means, in infinite time, in the real universe-as-we-know-it.

For example, we can construct the number which is the decimal concatenation of the maximal number of steps of execution (with halting) of an N-state Turing machine.  That is, the "Busy Beaver" function.  The first few values of which are known, and which grow incredibly quickly.  The currently known bounds on BB(6) are unimaginably large, and no computer can exist in the known universe to compute it directly.  The problem in general, is a statement about the decidability problem, a problem which is provably unprovable.  Therefore, the exact value of this constant cannot be known.  And it doesn't have any handy identities to allow a shortcut; this isn't your average pi we're talking about.

See: https://math.stackexchange.com/a/462835
They give a slightly less "dense" form, using powers of two rather than concatenation.  Obviously, there are infinite ways we could encode this function/sequence as a real number, so there's already a small infinity of real numbers constructed with this uncomputable function, in part.

So, I would humbly submit that, you can give or take algebraic (irrational) or computable numbers, but "real" Real numbers, most definitely aren't!

Tim

TimFox:
Again, this was all worked out to generate rigorous definitions of "real", "algebraic irrational", and "transcendental" numbers by mathematicians years ago, especially after the rigorous development of limit theory.  Numbers are mathematical constructions that evolved from the "counting numbers" or "natural numbers" (positive integers, excluding zero) found in everyone's day-to-day life.  As Leopold Kronecker reportedly said, "God created the natural numbers; all the rest is the work of man."  (The work of man started with adding zero, then the negative integers, and the rest is history.)
An interesting detail, found by Georg Cantor, is that infinite sets such as the set of rational fractions and the set of algebraic irrational numbers can be put into a one-to-one correspondence with the natural numbers, and are therefore "countable infinite" sets.  However, the set of real numbers cannot be put into that one-to-one correspondence, and are therefore called "uncountable infinite" sets.  He assigned cardinal numbers to these infinities, but I don't know how to enter Hebrew letters into this post.
Mathematics is very useful to describe physical phenomena, but should not be confused with the phenomena themselves.  Pythagoras was correct about 3 by 4 by 5 triangles in geometry, but that does not imply that you can produce an exact 3 in by 4 in by 5 in triangle in the machine shop from tool steel made from atoms.  That triangle is still useful for human purposes:  supposedly, the ancient Egyptian tax authorities used it to produce right angles and to measure the area of peasants' fields (after each inundation) by dividing the irregular shapes into triangles for tax assessment, hence the term "geometry".

magic:
T3sl4co1l is right, real numbers are a kludge invented to justify calculus after it had been handwaved into existence :-DD

vad:
Natural numbers, integer numbers, rational numbers, real numbers, complex numbers - are all abstract concepts invented by humans for one reason or another. They are all useful for mathematical modeling of real world phenomena. The real world, on the other hand, appears to be more complex than any abstract model humans came up with so far.

dietert1:
Philosophers and psychologists say the exact opposite: It is absolutely fascinating and surprising to which extent the human brain can model and understand reality. The failure of a human brain to accomplish this in perfection or let's say even minor difficulties to do so are called mental illness, and it is more abundant than we imagine.

Regards, Dieter

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