Investigating Logic Behind Line Segment Lengths

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Discussion Overview

The discussion revolves around the reasoning related to the lengths of line segments defined between two parallel lines, focusing on the implications of having an equal number of points on each segment and how this relates to their lengths. Participants explore concepts from geometry and set theory, including the nature of infinite sets and the application of principles like Cavalieri's principle.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that having the same number of points on two line segments does not imply that the segments have equal lengths.
  • Others argue that the lengths of the segments depend on the angles they make with the parallel lines, suggesting that equal angles lead to equal lengths.
  • A participant introduces the Pythagorean theorem as a method to measure lengths, implying that geometric relationships matter.
  • One participant mentions the Hilbert Hotel paradox to illustrate the counterintuitive nature of infinite sets and their cardinality.
  • Another participant states that every interval on the real number line has the same number of points, but this does not relate to the lengths of the intervals without a defined metric.
  • Concerns are raised about applying real number arithmetic to infinite sets, with references to limits and their implications in calculus.
  • Some participants discuss Cavalieri's principle, noting historical flaws and the evolution of its understanding in mathematics.

Areas of Agreement / Disagreement

Participants express disagreement regarding the implications of having an equal number of points on the segments and whether this leads to equal lengths. There is no consensus on the validity of the reasoning presented, and multiple competing views remain throughout the discussion.

Contextual Notes

Limitations include the dependence on definitions of length and the nature of infinite sets. The discussion touches on unresolved mathematical concepts and the historical context of Cavalieri's principle.

  • #31
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  • #32
My mathematically profane answer: The points of the second line are bigger :smile:
If you say for any point of short(?) line there [STRIKE]exist[/STRIKE] correspond a point on the long(?) line, it means that the points of the latter are the projections of the shorter line's points. He he.
 
  • #33
mireazma said:
My mathematically profane answer: The points of the second line are bigger :smile:
If you say for any point of short(?) line there [STRIKE]exist[/STRIKE] correspond a point on the long(?) line, it means that the points of the latter are the projections of the shorter line's points. He he.

Haha. No. Points are points.

The folly in the argument is the part where he equates one infinity with another. You can't treat infinities arithmetically.
 
  • #34
Yep, as soon as you realize that the number of points has nothing to do with the length of the line, it's all very obvious.

I can see where the confusion comes from though; in real life, even if we have billions upon billions of points, say all arranged in a little line, like a line of atoms, then we can still stretch them, but the distances will actually increase between the atoms, and if we increase far enough, then we will eventually "see" the gaps. So no matter how close you approximate the infinity of points, even if you use trillions of them all bunched up, it still never has the same properties of what goes on when you have infinitely many.
 

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