What's wrong with the following proof ?

Question

1.-Lines are composed by points and only by points.
2.-The lenght of a point equals zero.
3.-The lenght of a line equals the sum of the lenghts of the points it contains.
4.-Therefore the lenght of a given line always equals zero.

Answer 1

thw wrong one is the (3): the length of a line is not the sum of the length of the points it contains, because of the (2) and because a line contains infinite points that you can't sum even if thay'll have a length. The length of a line is the distance between its two vertex, and this is a measurable quantity.

Answer 2

There is not much wrong with your argument. I would say that if your second statement is true, then the rest would be true also. However, a definition for a point does not exist so the second statement is not true. Thus when you talk about the length of a point, what exactly do you mean? Do you mean each point is a circle of 0 radius (this is what the academics believe)? If this is what you mean, then your argument is true, otherwise it is false.

Distance is generally meaningless unless we think in terms of "distance units". For example, suppose our unit of measurement is inches; in this case each point will have a radius of 1/2 inch or a diameter of 1 inch. Therefore we could measure the length of line of points in inches or parts thereof.

Unfortunately in Real Analysis, points do not have any extent, i.e. a point is a circle with 0 radius. This unsound notion gives rise to illogical mathematics: Frechet's identity of indiscernibles states that any two numbers x and y are equal if the distance d(x,y) between these points is 0. Metric space theory is problematic because numbers are treated as points (that have no extent!) on a number line. This gives rise to other false notions such as 0.999... = 1 or a rectangle being composed of a union of circles. Both these statements are entirely false.

You are well on your way to discovering true mathematics. Continue to think for yourself. Do not believe a word your lecturer tells you because chances are he/she is probably just telling you what he/she has been taught.

Locomexican: What are you saying? That a line does not contain infinitely many points? According to real analysis it does. Points do have a dimension - they have a location relative to some origin. Lines have length. In fact the point-line relationship is unclear at times because each is defined in terms of the other. For example, any point that is not the origin can be defined by a length as well as a location even though its length may not be unique. To illustrate this further, just think of the locus of points all equi-distant from the origin - these form a special kind of line...

Answer 3

This proof is interesting. I think the flaw lies within statement number 3. Does the length of the line equal the sum of the lengths of the points it contains? I don't think so. Especially since a line is actually composed of an infinite number of points. And even if a point were measureable, then couldn't we say that point is also made of points, so it is in fact actually a line?

A similar paradox exists. When I was a sophomore in high school, my teacher really enjoyed geometry, as did I. He posed a very similar scenario using our idea of a circle. The circle is also made up of an infinite amount of points.

Answer 4

The problem is 3, just because the the points do have a length of 0, the length of the line is not nessecerily eqaul to the sum of the points' lengths but it is the sum of the distance between the points that is equal to the line's length.

Answer 5

Well if you add finitely many 0s you get 0 but if you add infinitely many 0s you do not necessarily get 0. Well uncountably infinite 0s may not add up to 0 in the sense you just described anyway... That was the thought behind the discovery of measure theory. You can't measure lengths, areas, volumes just by adding up the lengths, areas etc of points, you will always get 0. Measure theory concerns under what conditions can we measure things and when do we end up with a measure of 0 and etc. Instead you need to use a special form of addition called integration to deal with these problems. I know this is a little sketchy but it is the general idea.

Answer 6

The problem consists in part three, because a line is made up of all those points, but the length of line is determined by the distance from the first point and the last point, not necesarily the length of all the points on the line.

Answer 7

Other answerers have mentioned xenos paradox, and they are right.

Lets say your line L into n equal length pieces. The length of one of the n pieces would be L/n, lets call that value d. Multiplying d by n will give you L.

Thats an algebraic proof, offering no limitations whatsoever about your values for L, n, or d.

What you are doing is taking a limit. That means you are saying, as n goes to infinity, for constant L, d goes toward zero. Thats not the same as zero.

Infinity is a very strange idea, and its just as bizarre to divide by infinity over infinity as it is to divide by zero over zero. Thats where the wierdness kicks in. What you are really doing is dividing L by infinity, then not multiplying it by infinity.

The limit as n approaches infinity of (n/n)*L is what? The n's divide out, dont they. The limit is really L.

If you dont want to divide out the n's you could use L'Hopitals rule, and take the derivative of them with respect to n. That gives you the limit approaches (1/1)*L which is still L.

Look at it geometrically, if you keep the smaller segments in order, no matter how many times you break a line into two segments, its still going to add together to make the same line.

There is more than an infinite number of points in a line. From the proof above, we can cut a line into any n number of points. Lets say that we are going to divide it by the number of integers from zero to positive infinity. Thats dividing it by infinity, but it doesnt take up the whole line. Between each two adjacent integers, there are an infinite number of real numbers. Each segment of d which is length L divided by infinity, still can be further divided again by infinity and still not equal zero.

That supports the idea that the number of points required to make a straight line of length L is at least infinity times infinity. But we could set d to equal L /(infinity * infinity) any recursive number of times, and still not have cut the line into lengths such that n*d doesnt equal L. That implies that the number of points into which any line could be composed is greater than or equal to infinity to the power of infinity. Thats cardinality, talking about the comparisons in sizes of infinitely sized sets. Oops. Im going on a rabbit trail. Your assumption about zero length is not totally correct.

Answer 8

Part 3 you assume without proof that the length of a line is the sum of the lengths of the points.

The distance between two end points of a line segment is the length of the line segment, it is a definition that you do not prove.

Answer 9

It does not make physical sense to talk of a length of a zero-dimensional object (a point). It is not that a point has zero length/width/height, it is that it has no length/width/height, it has no dimensions at all....it is just a point.

Since points have no length, you cannot add up all the lengths of each point in the line (consisting of a set of infinitely many points).
Length is just the distance between two points.

Answer 10

Zero-point data facilitate mathematical calculations, they are not meant to refer to the actual physical length of a "point." Physicists do have problems with zero-point data when it comes to merging quantum theory with relativity theory, which is one of the reasons Superstring Theory came into being. It does away with the "zero point" problem.

The answer to your question lies in studying quantum weirdness. You could say you can never get from point A to point B because you will have to cross an infinite number of halfway points. But, obviously, one CAN get from point A to point B, because at a certain point, you do make a "quantum leap" across the "infinities."

A "quantum leap" over an infinite series of points gets you to a physical measurement of the length or your line. That line may be 1 inch long, or it may be 1.498735923847. . . inches long.

Hope that helped.