Thank you Daniel for reading my essay in the first place and for the compliments. However, I would like to clarify that a vital message of this essay is the inexactness of practical expressions of the ideal thoughts of pure mathematics. Please note that ``number line'' is the geometric representation of the thought of real numbers. While the thought is ideal and arbitrarily accurate, the expression is not. When you draw a line with your pencil, your expression is only accurate up to the extension of your pencil tip. But, only with that inaccuracy of the object of expression that you can express. Same is the case for measurement or comparison e.g. if you do not see the cross-wire of the eye piece you can not make measurements in a physics lab, although the thickness of the cross-wire, that lets you see, itself serves as the basic irremovable error -- the immeasurable lets you measure. If you do not accept this limitation of the accuracy in the expression (or measurement), you can not express (or measure). The same is the case for language as well -- you and me can communicate only up to the limitation of our knowledge in English -- if you use a word that I do not understand, then communication fails. And, without expression you can not use your thought in for practical purpose e.g. if I have not written this essay to express my thoughts, you would not be reading this. This, interestingly leads to the other crucial issue i.e. relational existence. This essay does not exist if you do not read it, it does not have a value unless you find a value in it. Even if you find value in it, it is subject to the way you interpret it or relate to my expressions of thought, where the premises are English language, mathematics and symbols.
So, I may write that, if I have favored some viewpoint in this essay, it is practical reasoning that incorporates inaccuracy of expressions and inexact measurement. This is why I have talked about `practical' numbers. Have you thought what or how you count when you say `I have five fingers'. I see immense confusion in this statement if I want give formal reasoning and try to do exact mathematics, because each one of my fingers are different from each other (e.g. by virtue lengths, breadths, fingerprints, wrinkles on the skin, etc.). It seems to me that I am adding like
[math] $1_a+1_b+1_c+1_d+1_e.$ [/math]
But I do not know how to make sense of this. For me, in practice, I use the fact that the fingers are both identical and different (contradiction!) according to need. I perceive of the fingers as different, but express their counting as being identical so as to do arithmetic (sum) by disregarding information on purpose e.g. something like
[math]$1+\delta_a+1+\delta_b+1+\delta_c+1+\delta_d+1+\delta_e=5+\delta~\ni \delta=\sum_{i=a}^e\delta_i$[/math]
where $delta$ is simply ignored for practical purpose. Looking into those different $delta$-s serves a different practical purpose i.e. if one chooses or decides to ask what does a finger look like and starts investigating what is the meaning of a finger.
In a nutshell, the aim of this essay is to convey the role of contradictions in practice and in mathematical science (e.g. how the issue of seeing a point leads to some interesting consequences from standard geometric calculus that is plagued with incomplete statements made by Cauchy).