Paradox of Motion: Physics & Math Explained

[mathphys]

I see your point. It was just that your proof put in my mind the idea of the real line (because of the 'we can order ' (<--specially this one , it is and interesting issue how we can ) and 'midpoint') and Dedekind cuts. Althought that is not high school euclidean geometry.

 

[Eyesaw]

Assume that there exists a line segment with only finitely many points on it.
Then we can call that line segment L.
Since L has only finitely many points on it, and we can order the points on L, we can find two points with no other points between them.
Then we can call such a pair of points P and Q.
We know from high school geometry that we can construct the midpoint between any two points.
Then, there is a point that is half-way between P and Q.
Let's call that point R.
R cannot exist, because there are no points between P and Q. But we've proven R exists. This is a contradiction, so our initial assumption is incorrect.
Therefore, all line segments have infinitely many points lying on them.

Your starting assumption is that there is a line segment with finite
points on it. Then you introduce an assumption (high school
geometry of constructing the midpoint between any two points)
that contradicts your starting assumption to prove your
starting assumption is contradictory. That is a fallacy.

That is, your definition of a line segment in the first assumption
is one that has finite points. Your "high school geometry's
construction of the midpoint between any two points" relies
on a definition of a line segment as one that has infinite
points on it. If you use two opposite definitions for a line
segment, you will obviously prove the two definitions contradict
each other.

 

[Hurkyl]

Your starting assumption is that there is a line segment with finite
points on it. Then you introduce an assumption (high school
geometry of constructing the midpoint between any two points)
that contradicts your starting assumption to prove your
starting assumption is contradictory. That is a fallacy.

No, it's called a proof by contradiction. Even if you think it's a trivial proof, it is still not a fallacious proof.

Incidentally, there are geometries whose lines have finitely many points, and yet any pair of points has a midpoint.

And finally, I have never seen a setting where the fact that a line has infinitely many points was part of the definition, rather than proven as a theorem from the axioms.

 

[mathphys]

You are right, it is just that in the context of euclidian geometry the premise that there is always a midpoint between two arbitrary points P, Q on a line segment implies that there are infinitely many points between them. That is what i found disconcerting as part of the proof.

Nevertheless, lines and segments are abstract undefined objects in eucliden geometry as you mention. If you take the existence of the mid point as an axiom for objects called lines (or a consequence of more fundamental axioms), a property they have to satisfy, then it is clear(by your proof) that and object with a finite number of points cannot satisfy this 'line axiom', and hence is not a line.
That was the argument or intention of your post, i guess. But notice that just because you proved this way that an object with a finite number of points is not a line, that does not imply that a line has infinitely many points, that is independently implied by the midpoint premise.

Just starting from the midpoint argument , one may conclude that there are infinitely many points on a line, and this stablishes a necessary condition for sets of points to be 'considered' as lines, in euclidian geometry. Although not a sufficient one. That is way i associated inmediately with the real line.Nevertheless, again, when we said things as sets, and order, we are going out of the scope of euclidian geometry.

 

[Dmstifik8ion]

This is my first post so please feel free to chastize me and put me on the straight and narrow path; but

I sense a wide spread problem here involving the human perspective.
Reality (space/distance) is not made of points. Points are attributed to specific places (locations) in reality to enable our conceptual grasp and understanding of aspects and relationships in reality.

Might my insight and perpsective be better applied elsewhere and/or better appreciated in another area of this forum. Thank's for any comments and assistance.

 

[VazScep]

Yes, I'll say vague -- I've never been impressed by any presentation of Zeno's arguments.
Compare with, say, the Liar's paradox, in which I can formally derive a contradiction:
If P is the statement "P is false", then we have:
P = T or P = F
P = T --> P = F
P = T --> P = T and P = F
P = F --> P = T
P = F --> P = T and P = F
P = T and P = F
This is a real paradox. It, and other similar paradoxes, are the reason why the usual formal logic is designed in such a way that statements cannot refer to themselves (even indirectly!)

I know this is from several months ago, but:

Statements can refer to themselves. In theories of arithmetic, you can code all statements as Gödel numbers, so that statements are essentially capable of making assertions about other statements. A theorem known as the Fixed Point Theorem then shows that for any expressible property p, there is a statement which says "This sentence satisfies property p".

The liar's paradox is avoided because the property of being a true statement is not expressible in theories of arithmetic. This is known as Tarski's Theorem.

 

[michealsmith]

doesnt it just past each point in an infintesmal amount of time

 

[Ubern0va]

I understand his problem quite well, because I've thought of it while sitting in Math class a fiew times, but quickly dismissed it via the anthropic principle :P