The issue is not so much the mathematical definition itself, which I have acknowledged is adequate for most practical purposes. It is the widespread misconception that what most mathematicians call a continuum--anything isomorphic with the real numbers--is indeed continuous, and thus has the property of continuity. We seem to agree that it is not and does not. — aletheist
Perhaps Peter Lynds' essay — jgill
The real numbers cannot fulfill the conditions of a proper definition of "continuity". Real numbers produce a sequence of contiguous units. Contiguity implies a boundary of separation between one and another. This boundary must produce an actual separation between one number and the next, to allow that each has a separate value. This is contrary to "continuity" which is the consistency of the same thing.
So mathematicians have created a term, "continuum", which applies to a succession of separate units, allowing that each is different, so there is something missing in between them, and that "something", which is the difference in value, is unaccounted for. Therefore "continuum" means something completely different from "continuity".
The rational numbers are an attempt to account for this "something", the difference in value, which exists between the reals. This an attempt to create a true continuity. However, the irrationals appear, and foil this attempt. So mathematics still does not have a continuity. — Metaphysician Undercover
Note please that I'm only saying what a mathematical continuum is. I'm not addressing any of the many philosophical objections there could be to calling the real numbers a continuum, But mathematically, the reals are a continuum and the rationals are not. — fishfry
That's right, the existence of irrationals really throws a wrench into the rational number line. Where do those irrationals exist in relation to that line? — Metaphysician Undercover
The issue is not so much the mathematical definition itself, which I have acknowledged is adequate for most practical purposes. It is the widespread misconception that what most mathematicians call a continuum--anything isomorphic with the real numbers--is indeed continuous, and thus has the property of continuity. We seem to agree that it is not and does not. — aletheist
The irrationals fill in all the holes in the rationals. I already illustrated this with a sequence of rationals that approaches the point sqrt(2) but there's a hole there instead of a point. The irrationals fill in those holes. — fishfry
But the rationals fail to be Cauchy-complete. For example the sequence 1, 1.4, 1.41, ... etc. that converges to sqrt(2), fails to converge in the rationals because sqrt(2) is not rational. There's a hole in the rational number line. — fishfry
I could drill the math down a lot more but should probably wait for encouragement, and if none is forthcoming I should leave it be. I don't think you're curious about the math at all. You just want to throw rocks. But why? People uninterested in chess don't spend their lives hating on chess. They just ignore it. You think math is bullshit? Maybe you're right. Maybe it is all bullshit. The thing is why do you keep repeating the point over and over as if we haven't all heard you already? And as if we all don't already understand the point? — fishfry
Fair enough, thanks.I don't think there is a widespread misconception. I think there's a widespread lack of interest in the question; and among those who are interested, some degree of agreement that the real numbers don't express everything we think must be true about a continuum. — fishfry
:lol: Not likely. I thought the thread needed some comic relief. — jgill
That's right, I'm not curious about that math — Metaphysician Undercover
If your attitude is that these foundations were built by the greats, therefore there are no weak points, (appeal to authority resolves fundamental problems), then I think you are in need of God's help. — Metaphysician Undercover
A length that is irrational comes into play when you have a length that is the "smallest" length as the right sides of the triangle. The irrationals are not imaginary numbers. They simply go on forever, within a limit. — Gregory
Your math is wrong. — fishfry
I've been told that if I truly understood calculus, I would see how there is no contradiction in something spatially being finite and infinite at the same time. I suck at math so I could be the stupid one in the conversation :( — Gregory
Blatant contradiction is not the real problem though, rather ambiguity and vagueness, such as the difference between "continuum" and "continuity", the definitions of "object" and "infinite" are the real problem. — Metaphysician Undercover
So your argument on this thread is that there is not a contradiction in math, but that it's incomplete? — Gregory
Most mathematicians and physicists do quite well without contemplating such issues. — jgill
Violins can create an infinity of sounds with infinitesimal changes. — Gregory
My ears tell me sound is infinite, when I study music. There is an infinity within the limit of the highest and lowest frequency. — Gregory
My ears tell me sound is infinite, when I study music. — Gregory
Thanks for this. I would add that the same is true of the Planck time, since it is defined as the duration required for light to travel the Planck length in a vacuum.To be clear, since this is a common misunderstanding: The Planck scale is the point at which our theories of physics break down and may no longer be applied. It does NOT mean the world itself is quantized. Below the Planck length we simply do not know and can't even speculate, because our physics no longer works. — fishfry
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