## Friday, August 21, 2009

### Irrationals on the border of existence and sqrt(2)

I have been reading a lot about abstract math, what numbers really are and are not, set theoretic number theory and related. The set theoretic approach even if I did not dig into the deepest depths of it, allowed me to be able to logically justify to myself the existence and nature of numbers.

I even bothered my wife (who could not care less) about the beauty I found in the irrational number: square root of 2, what I told her is the following:
I will prove to you how beautiful is math and that we should be grateful for all the people who contributed to it along the centuries of human thinking. I will give you a calculator and you can only use to to multiply, now with no other references, find me the exact square root of 2. Of course, one would proceed to multiply 1.1*1.1= 1.21 then 1.5*1.5=2.25, hence coming to the conclusion that 1.1 < style="font-weight: bold;">

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The following reductio ad absurdum argument showing the irrationality of √2 is less well-known. It uses the additional information 2 > √2 > 1 so that 1 > √2 − 1 > 0.
1. Assume that √2 is a rational number. This would mean that there exist positive integers m and n with n ≠ 0 such that m/n = √2. Then m = n√2 and m√2 = 2n.
2. We may assume that n is the smallest integer so that n√2 is an integer. That is, that the fraction m/n is in lowest terms.
3. Then $\sqrt{2} = \frac{m}{n}=\frac{m(\sqrt{2}-1)}{n(\sqrt{2}-1)}=\frac{2n-m}{m-n}$
4. Since 1 > √2 − 1 > 0, it follows that n > n(√2 − 1) = mn > 0.
5. So the fraction m/n for √2, which according to (2) is already in lowest terms, is represented by (3) in strictly lower terms. This is a contradiction, so the assumption that √2 is rational must be false.
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One could almost argue such numbers do not really exist, in the end, they are not called crazy/irrational (and have been fought) for no reason! The way I see it is that they don't at least as written out numbers, they do exist if we set a desired precision, this is why I am liking what I call a 'computationally theoretic number theory', no idea if it exists but you get my point. By setting a precision we can work with those things. One could argue that the number exists and that it's representation is sqrt(2), but this is not a number, the way I see it is that this is a rational (existing) number combined with an algorithm (or call it a function) that can transform this into another in this case irrational number, so either we imprecisely write down a number that approximates it to a given precision or we represent it as an algortihm (sqrt) and data (2) that expand to this 'inexisting' number, this is all layman terms and layman talk, and mathematicians will laugh, but I am recording these thoughts because, since I am a bit satisfied with what I know about this now, I will stop digging and go back to the actual reason I started to look into math again, and that is to solidify my math needed for a self designed auto-didactic machine learning 'course' in my free time. Another sqrt(2) existence thought occured to me in the car last weekend, imagine you have a piece of rubber of length 1, now you take it and stretch it to length 2, did you pass by sqrt(2)? you must have, so it exists? can one measure it? again only to some precision ... (even on the atomic/quantum level). almost mind boggling, this infinity of numbers, but it also makes sense, we allowed for it the moment we allowed ourselves to have a coma and numbers after it, after that recursively you have infinities of infinities of infinities ... but all still allow me not to explode when I look at the set theoretic number theory, basically it is ordering and number of things (in layman, mathematician laugh terms) ... between 1 and 2 there is an infinity of numbers, same as between 1 and 1.1 and 1 and 1.0001 ... in any case ... back to less mind boggling and much more practical stuff, in the spirit of the way ppl have been using numbers since ages for practical matter and not even really understanding what they are. and let me reiterate, please excuse the layman :( he's just trying to make sense of it within a very limited amount of time.

* bourbaki, one of my favorite persons for math discussions, does not think this post is utter non-sense and he just pointed me to:

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Set theory is the branch of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.

The modern study of set theory was initiated by Cantor
Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor was a Germany mathematician, born in Russia. He is best known as the creator of set theory, which has become a foundations of mathematics in mathematics....
and Dedekind in the 1870s. After the discovery of paradoxes
Naive set theory

Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most contemporary mathematics....
in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms
Zermelo–Fraenkel set theory

Zermelo?Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of axiomatic set theory and as such is the most common foundations of mathematics....
, with the axiom of choice
Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if there are infinite set many bins and there is no "rule" for which object t...
, are the best-known.

Set theory, formalized using first-order logic
First-order logic

First-order logic is a formal deductive system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus , the lower predicate calculus, the language of first-order logic or predicate logic....
, is the most common foundational system for mathematics.

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Some References:
* http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Beginnings_of_set_theory.html
* http://www.absoluteastronomy.com/topics/Naive_set_theory
* http://en.wikipedia.org/wiki/Square_root_of_2#Proofs_of_irrationality
* The essence of discrete mathematics book
* ...

.