Questions on a numberphile video

  • #1
picketpocket826
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The above two video from numberphile are trying to motivate real analysis (I think?). The latter continues on from the former.

Presenter's argument goes kind of like this:

He first considers the real number line and talks about measuring distance between numbers

4-3=1

Then he talks about measuring the length of points and takes 3.5 as an example.

3.5-3.5=0

Then supposes that if you removed the point 3.5, the length of 4-3 stays the same and the number line remains unchanged.
(He doesn't actually say this next part but it's implied)
However, if you remove the length between 4 and 3, then there is indeed a change to the number line.The he states this inconsistency almost broke maths.
The major problem here being that there is no "next point" after 3.5

So my question:

1. Is this a valid argument (that motivates real analysis) ? I can't help but not be convinced but it is interesting. Brady (guy interviewing) argues if he removes someone's feelings their size wouldn't change and likewise if he removes points from the number line it shouldn't change in size either since points are abstractions. Although there is a kind of counter to this argument - feelings do have a physical substrate so removing them would at least change a person's weight if not their size.

2. At one point he says that this line of thinking led to the invention of the computer. I've never heard anything of this before. Is this true in some indirect sense? How? I cannot seem to find anything about this on google. I thought the computer was invented to crack codes.
Edit: I realised he kind of explains at the end.
 
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  • #2
I prefer the constructive perspective like
https://www.physicsforums.com/insights/counting-to-p-adic-calculus-all-number-systems-that-we-have/
or
https://www.physicsforums.com/insights/yardsticks-to-metric-tensor-fields/
instead of removing something I cannot draw from another something that I cannot draw.

The second video is very clear: our concept of a point breaks down in reality. Reality does not know points, they are dimensionless. Yet, my stopwatch halts at only one position. A famous example is the Banach-Tarski-paradox:
https://www.physicsforums.com/threa...-about-lebesgue-measure.1054643/#post-6933788
https://en.wikipedia.org/wiki/Banach–Tarski_paradox
We cannot draw a number line or a point. What we draw is a "representation" as said in the video. All we do is three-dimensional, but a line is one-dimensional, and a point is zero-dimensional. It is easier to fool ourselves with two-dimensional circles, squares, and triangles because we cannot see that our paper isn't flat and a mountain area in reality.

You can prove a lot of funny things in mathematics that are true but hard to imagine: Hilbert-hotel, Sierpinski triangles, Cantor sets, you can map a line onto a square, i.e. fill the entire square by drawing a line (that has zero width, only length), etc.
 
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  • #3
Well, since there are different levels of structure, they need to specify which aspects of the Real line remain unchanged by removing 3.5. Cardinality remains unchanged, but the new space once 3.5 is removed is disconnected, so the removal does not preserve the topological structure, i.e., ##\mathbb R-{3.5}## is not homeomorphic to ##\mathbb R##, for one.
 
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  • #4
picketpocket826 said:
The above two video from numberphile are trying to motivate real analysis (I think?).
What is he actually trying to say/motivate in these short videos? Has he actually mentioned that (in some other videos)? He probably thinks about some well-known ideas from math, but since he is conscious of talking to a layman audience he doesn't state them too explicitly, because the audience doesn't know them, and instead resorts to these kind of vague hand-waving from which only confuses you more and doesn't give a clear picture.
The only thing I can glean from this is that he's trying to hint how naive ideas about measuring sets and the properties of these measures lead to contradictions in measure theory (the way out of which is to concede that some sets of the real line can't have measure, if you want measurable sets to work nicely).
 
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  • #5
BTW, these non-measurable sets are more exceptional than usual( despite that they can be found in any set of nonzero measure), and somewhat contrived. Maybe someone who knows more Physics can tell us if they show up often in real life.
 
  • #6
fresh_42 said:
Yet, my stopwatch halts at only one position.
Not really. Not unless its momentum is infinitely spread out. :cool:

Edit:
And if its momentum is all over the place, it hasn't really stopped, has it?
 

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