What is the problem with the argument for the diagonal length of a square?

[Vinay080]

Premise 1: All straight lines have the value of length equal to the numerical value of the end point, provided the starting point of the line is assigned the numerical value zero.

Premise 2: No point can be assigned the value y.xxxxxx... or y.abcdef...(Example: 8.9999... or 8.39465..). We can either have the point with the assigned numerical value to be either y.xxx, y.xx...x, y.abcdef..g, etc, but not y.xxxxx... or y.abcdef...

Premise 3: All lines have starting and ending point, thus they have the value of length equal to the numerical value of the ending point. From the premise 2, the value must be either y.xxx, y.xx...x, or y.abcd...g, etc.

Premise 4: Diagonal of the square whose side is the unit of length, has got starting and ending point. Therefore, the length of the diagonal should be a value which can be expressed as the fraction with terminating decimal form.

By this argument, length of the diagonal (√2) seems to have fractional form with a terminating decimal form, which (I think) is not true, then what is going wrong in the argument? Or else is it that the diagonal (in this case) has no starting and ending point?

 

[micromass]

Premise 2: No point can be assigned the value y.xxxxxx... or y.abcdef...(Example: 8.9999... or 8.39465..)

Why not? This seems very arbitrary and wrong.

 

[jbriggs444]

Premise 1 defines the length of a line in one dimension -- i.e. on the x axis. It is silent about the defined length of a line whose endpoint has more than one coordinate.

Premise 2 is arbitrary but acceptable. If you want to deal only with points whose cartesian coordinates are given by terminating decimals, that's fine.

Premise 3 ignores the clause in premise 1 that requires that one endpoint being at the origin. But that's cosmetic as long as one end of the square's diagonal is placed at the origin.

Premise 4 attempts to apply premise 1 in two dimensions. But premise 1 is only applicable in one dimension.

 

[Vinay080]

No point can be assigned the value y.xxxxxx... or y.abcdef...(Example: 8.9999... or 8.39465..)

Why not? This seems very arbitrary and wrong.

Sub-Premise 1: All straight line segments have starting and ending point.

Sub-Premise 2: Lengths of certain value exist if they can be constructed by increasing the points to a certain extent.

Sub-Premise 3: Lengths of non-terminating decimal form value can't exist because they can't be constructed by increasing the number of points to a particular extent.
Ex: 1.9999...; Length of this value can't exist because, length of this value can't be constructed by increasing the number of points to a certain extent. In order to reach that value of length, first we need to achieve 1.99, then 1.99999, then 1.9999999, and so on. We don't know where to stop. If we don't know where to stop, we can't construct that line of that value.

Conclusion: From 3, we can't construct line segments of length equal to non-terminating decimal form value. So, there is no question of end point, and no question of non-terminating decimal form value.

 

[micromass]

Sub-Premise 1: All straight line segments have starting and ending point.

Sub-Premise 2: Lengths of certain value exist if they can be constructed by increasing the points to a certain extent.

Sub-Premise 3: Lengths of non-terminating decimal form value can't exist because they can't be constructed by increasing the number of points to a particular extent.
Ex: 1.9999...; Length of this value can't exist because, length of this value can't be constructed by increasing the number of points to a certain extent. In order to reach that value of length, first we need to achieve 1.99, then 1.99999, then 1.9999999, and so on. We don't know where to stop. If we don't know where to stop, we can't construct that line of that value.

You realize that $1.9999...$ is just $2$ right? So you're saying that you can't construct $2$?

Likewise, you're working in a system where you can't even construct things like $1/3$. I do not know what kind of system you are trying to model, but it seems that you can't do a lot in it.

The usual notion of constructibility involves constructing points with ruler and compass. With that system, we can definitely construct $1/3$ or $\sqrt{2}$.

 

[HallsofIvy]

VinayO80:Premise 2: No point can be assigned the value y.xxxxxx... or y.abcdef...(Example: 8.9999... or 8.39465..). We can either have the point with the assigned numerical value to be either y.xxx, y.xx...x, y.abcdef..g, etc, but not y.xxxxx... or y.abcdef...

Micromass: Why not? This seems very arbitrary and wrong.

VinayO80:
Sub-Premise 1: All straight line segments have starting and ending point.

Sub-Premise 2: Lengths of certain value exist if they can be constructed by increasing the points to a certain extent.

Sub-Premise 3: Lengths of non-terminating decimal form value can't exist because they can't be constructed by increasing the number of points to a particular extent.
Ex: 1.9999...; Length of this value can't exist because, length of this value can't be constructed by increasing the number of points to a certain extent. In order to reach that value of length, first we need to achieve 1.99, then 1.99999, then 1.9999999

No, this is NOT what "increasing the number of points" means. What you are doing by restricting points to those points whose position on the number line can be written as a terminating decimal, then you are restricting to rational numbers whose denominators, when reduced to lowest terms contain on "2" and "5" as prime factors. That is a very small part of the set of rational number, much less the set of real numbers which would be necessary to get all points on a number line.

Given that, is should be no surprise that there are many lengths you cannot get in this number system.

 

[Vinay080]

Thank you micromass and jbriggs444, I will keep your words for my further analysis.