The Identity Function Paradox

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Here is a paradox I found in a simple math problem. You know the graph of x = y? This is called the identity function. For example, find "1" on the x-axis and correlate it with "1" on the y-axis. The entire graph will simply be a straight line at a 45 degree angle.

Now we know the difference between 2 and 1 is 1, right? But what is the difference between the location "1" and the location "2" on the ACTUAL graph? Clue: it isn't 1. Because of the pythagorean theorem it will actually be the square root of 2. This means, in a way, that there are more numbers on the graph than there are on either the x- or y-axis.

Thoughts?

The same cardinality.

This isn't a paradox. Its a relationship.

A line is simply a collection of an infinite number of colinear points. If I tell you that there are two lines, line A and line B, each with two end points. Now suppose I tell you that for every point Pa on line A, there is a corresponding point Pb on line B, such that if the distance between Pa and the starting point of line A is N, and the distance between Pb and the starting point of line B is MN, where M is some constant, I haven't described a paradox, I've only described a relationship.

Also, from linear algebra, we know that a 2-dimensional coordinate space can be defined in terms of any two arbitrary vectors, so long as those vectors are not co-linear. Cartesian space is defined by two vectors, one from the origin heading horizontally to the right, one unit, and the other from the origin heading vertically upwards, one unit. But you could basically have any two vectors defining the space. Do that, and you'll see that whereas in Cartesian terms we might express a point as (1,1), the corresponding point in another vector space might be expressed as (4.7678,-2.43). Its just a different way of describing the same point

By the way, the Pythagorean Theorem (or something analogous to it) extends into n-dimension space. If you have three dimensional cartesian space (x,y,z) and then look at the line between the origin and (1,1,1), you'd find that despite the fact that you've only advanced one unit on each axis, your point is now a distance from the origin of the square root of 3, not the square root of two.

In fact, this helps explain it a different way - if you are only in one dimensional space, then THAT is the true identity. I.e. if we only have one axis, X, and we advance 1 unit from the origin along X, then the line from the origin to that point is also only one unit. But as soon as you expand things to two dimensions, now you are mapping single dimensions to two dimensions and things are no longer a 1:1 relationship.

The basis for mathematics is the theory of sets. And the question you ask can be better understood in it and made simpler. For example, think of the set of natural numbers {0,1,2,3,4,...}. Comparing the set of natural numbers with the set of even numbers {0,2,4,6,8,...} we are tempted to say that the set of natural numbers is bigger. And so they thought for more than 2000 years, until the 19th century. Georg Cantor then defined that two sets had the same "number of elements" (or cardinality) if they can by put into a one-to-one correspondence by a function (that is, if there's a bijection between the two sets). Acording to this definition, the natural numbers and the set of even numbers have the same number of elements: you just define the bijection f(n)=2n. The same can be said with your case of the identity function. The identity is a bijection between the each set of points of the lines. So this is not a paradox.

But don't be discouraged. Discoveries need a certain boldness to be made. Don't lose that. But also bear in mind that literaly thousands of geniuses learned what you've learned so it's very unlikely that you find a paradox so easily.

Indeed there is always a NxN mapping between a point and a distance from origin.

You have the same number of points in a line.

Consider two circles of different radii, C1 and C2, centered on the origin of the X-Y plane.

Without loss of generality, assume that C1 has a larger radius than C2.

Choose any point on C1. Draw a line segment from that point to the origin. You will see that it crosses C2 at one point. Furthermore, for any other line segment from any other point on C1 through the origin, it will not cross C2 at the same point as the other crossing. Thus, for every point in C1, you can map that point to a unique point on C2.

Similarly, choose any point on C2 and draw a line segment from the origin, through that point on C2 and on to C1. Once again, no other line segment going from the origin through any other point on C2 will cross C1 at the same point. Thus, every point in C2 can be mapped to a unique point on C1.

Thus, C1 and C2 have precisely the same number of points.

You can also do the same with two line segments of different lengths. For example, if you have a line L1 with length 1 and a line L2 with length 2, begin measuring from the left side of each line. That is, L1 will be the segment [0,1] and L2 will be the segment [0,2]. Define a function f: L1->L2 by f(x)=2x. Then f will map every point of L1 to a point on L2. Thus, for every point on L1, there is a unique point on L2 that corresponds to it and therefore L2 contains at least as many points as L1. Similarly, the inverse function g: L2->L1 given by g(x)=x/2 (i.e. f(g(x))=x and g(f(x))=x) will map every point in L2 to a point on L1. Thus L1 contains at least as many points as L2. Therefore, L1 and L2 contain precisely the same number of points.

Yes, I admit these answers are true, but I still call it a paradox. Not a contradiction, but a paradox. Paradoxes have a certain subjective nature about them. Indeed, we may be able to map point n onto point n, but we cannot map segment n from either of the two axes onto segment n on the graph without loss of identity of the segment (expressed as "length"). One may object that measuring lengths or segments is not the point of identity or, indeed, any Cartesian graph function, but I would retort that the point of a mathematical model is to be consistent all-around.

But to me it seems that any paradox can be resolved by visualizing the identity function in a different way. Or, at least, call it "another consistent way" of graphing identity:

Both the x-axis and the y-axis are visualized as sine waves instead of straight lines, such that for every length of "square root of 2" on the graph, there is a corresponding length of "square root of 2" on either of the two axes. The undulation of the sine waves occurs in the z-direction (which does not itself serve as a coordinate axis).

Even then we are still left with a paradox in that the distance measured from point "t" (which is a number that happens to occur on a trough on either the x-axis or y-axis) on either of the two axes to point "t" on the graph would not be to scale measured against a similar distance involving a point "m" (which is a number that happens to occur exactly half-way between a peak and a trough on either the x-axis or y-axis). To solve that paradox, we stipulate that space has a positive warp in either z-direction.


To me, however, I would like to view the paradoxicality of the (regular) identity function as exhibiting a reality inherent to mathematics: for every number that we can identify, there will be a number that we cannot identify (assuming there's no such thing as eternal life).

I am sure there is some crappy excuse for that, just like the fact that one divided by three (which equals .3 repeating) and then multiplied by three (technically equaling .9 repeating) does not equal one, defying one of the fundamentals of math, the reversibility of division. However, this is not a "disproof" of math, since people state that .9 repeating (while not perfectly equal to one) is close enough. The is a math problem proving that, but I still think it is stupid.

The problem is that you aren't looking at it from a rigorous mathematical point of view but from a naive non-mathematical point of view. There is no paradox at all mathematically.

From a naive point of view, it would seem that a longer line segment would have more points than a shorter line segment. But as my example above show, that is simply not true. You can easily find mappings from L1 to L2 and from L2 to L1 that conclusively shows that L2 contains at least as many points as L1 and that L1 contains at least as many points as L2. Thus, they both have exactly the same number of points.

.99999... (infiinitely repeating the 9s) is exactly equal to 1.0. It is actually trivial to prove this is true.

Let x = .99999...
Then 10x = 9.99999...

Subtract the first from the second and you get

10x-x=9.99999... - .99999...

or 9x = 9. Divide both sides by 9 and x = 1.

Thus, .99999... = 1

Consider this question: Are there more positive integers than positive even integers?

From a naive point of view, you would think that there are twice as many positive integers as there are positive even integers. After all, the set A = { 1, 2, 3, 4, ... } would seem to have twice as many elements as the set B = { 2, 4, 6, 8, ... }. But in reality, it is simple to show that both sets contain exactly the number of elements.

Create a function, f:A->B, given by f(x)=2x. Note that if x and y are distinct integers then f(x)=2x and f(y)=2y are also distinct integers. Thus, for every element in A, you can find a corresponding unique element in B and so |A|<=|B|. Now consider g:B->A given by g(x)=x/2. for any two distinct elements in B, x, and y, g(x) =x/2 and g(y)=y/2 will be distinct elements in A. Thus, |B|<=|A|.

Since |A|<=|B| and |B|<=|A|, then |A|=|B|.

Another way of looking at it:

Let x = .33333...
Then 10x = 3.33333...
Subtracting, we get 10x-x = 3.33333... - .33333...
or 9x=3. Divide both sides by 9 and
x=1/3
or 1/3 = .33333...
Multiply both sides by 3 and
1 = .99999...

In this sense, the segments map on to each other qua sets-of-points, but the segments do not map onto each other qua segments. And this problem is not limited within the ideal realm but extends to the real realm as well. E.g. I can very well map every point within the cubic meter that I inhabit to a point within a cubic meter several thousand kilometers further than me from the center of the earth, but nevertheless each "cubic meter" differs due to relativity/gravity.

That mapping is of ALL rational and irrational points in the segments.

Relativity and gravity have nothing to do with any of this.

0.9 repeating _ writen 0.(9) _ is 1. Consider that 0.(9)= sum 9/(10^n) = 9/10 x sum 1/10^n-1 = 9/10 x 1/(1-1/10) = 9/10 x 10/9 = 1