The arguments that space is filled with knowable and nameable positions (real numbers) are subtle and abstract. The ultimate concept for an ultimate stratification of space is difference.

Difference is related to movement and movement to the mind. If there is no difference at all, movement is not possible (if everything is A we cannot move to B), likewise if every position on the number line is different from each other (as is claimed by naming -“filling”- each space with a real number), movement across the number line is not possible (at each point the object is at rest) Zeno is the first recorded person to ask questions that reduced early notions that motion is fully knowable with points to absurdity.

Difference is at the center of thought, and talking about it can very easily lead into nonsense. Some people are paralyzed by a fascination with difference, assuming every difference to be “real”. This is strengthened by over-precise language. “If we already have words that separate these things, then they must “really” be separate”. The arguments for real numbers began in ancient Greece, but the rational number system was shown to be inadequate for describing “everything” with the logical argument that the square root of 2 is not a rational number.

Pythagoras of Samos was a mathematician during the time of ancient greece. Called the “Father of numbers”, he wrote about philosophy and religion of numbers. He came to believe through something like a divine revelation that “Everything is number”, “Everything is measurable”. According to legend, Pythagoras’ student Hippasos was executed for his proof that the square root of 2 was not rational. Why? They did not have real numbers; at the time the proof could have been interpreted to mean that not everything was number. Here is the likely proof of Hippasos: (the method is proof by contradiction)

Claim “~p”: square root of 2 is not a rational number

Suppose “p”: Square root of 2 is a rational number

By definition of a rational number, square root of 2 can be written as r/q with r, q integers having no common factor and q not equal to 0.

r/q= square root of 2=rt{2}
AND r-squared over q-squared = r^2/q^2=2
AND r^2=2q^2
AND r^2 is even
AND r is even,  r=2k, some integer k. (This is because r^2 is even so there must be a 2 in the prime factors of r.)
AND 2q^2=4k^2
AND q^2=2k^2
AND q^2 is even.
AND q is even.  Both r and q are even
AND 2 is a common factor of r and q
AND r and q share no common factor.

This is a contradiction, so what we supposed, “p”, must be false. Therefore (by the Law of the Excluded Middle) “~p” is true and square root of 2 is not a rational number.[1]

Calling these things that may not be numbers and that may not exist “irrational numbers” is a way of “covering it up”, as Pythagoras tried to do by (supposedly) murdering his student. Why cover it? What we are touching on here is “what are words?”. What can words do and how do questions modify our words?

To ease students into belief in irrational numbers often the teacher will draw a square with sides of one unit. The diagonal is square root of 2. This is a persuasive argument for the existence of the square root of 2 as a quantity. Is it a proof?

The following proof was something I was invited to teach in a lecture on logic. It is from a published book that also had the previous proof in it. At first the following may seem unrelated to the previous proof about what is a number and what isn’t, but I will show how they are both connected to the existence of a square:

Prove that if two lines are each perpendicular to a third line in the plane then the two lines are parallel. The method of proof is by contradiction.

Want to prove: Given lines L1, L2 and L3, if L1 is perpendicular to L3 and L2 is perpendicular to L3 then L1 is parallel to L2. Suppose not: If L1 is not parallel to L2 then they intersect.

Before we know what not-parallel is (intersection?), what is parallel? If not-parallel is intersection and parallel is not-intersection, then a curved line and a straight line can be parallel, when they don’t intersect each other.

The “proof” that was published in a math textbook continues to argue that since L1 and L2 intersect, a triangle is formed with two right angles, and that this is impossible. However a triangle with two right angles is possible. The geometry on a sphere (such as the Earth we’re standing on) seems to be logically consistent (meaning no contradictions have been found). Euclidean geometry and geometry on a sphere are on equal footing in this- they are equally consistent. And, if you laid out very large triangles on the ground, the angle measure would be larger than 180 degrees. Without using Euclidean distance, what is distance on the Earth? Number does not seem to be enough to describe.

Euclidean space carries with it an assumption- that means no proof. It is called Euclid’s parallel postulate. Here it is: Given a straight line and two straight lines that intersect the first line, and the interior angles on the same side of the first line add up to less than two right angles, the two straight lines intersect. (it is the same as the attempted proof about L1, L2 and L3) Does this seem like a big assumption? It did to many mathematicians and they have been making arguments that they have “proved” it since Euclid wrote his book in 300 BC. Everyone who has claimed to prove it has made an assumption somewhere in the proof that is equivalent (the same as) the parallel postulate. In other words, mathematicians have failed to prove the parallel postulate to this day. It is a queer fact that Legendre attempted to prove it as well; there is a lot at stake in a proof of the parallel postulate, even in modern times. A successful proof would change the world. Examples of assumptions that are equivalent to the parallel postulate follow; they are all the same, even though they seem different.

1) Playfair’s axiom: “At most one line can be drawn through any point not on a given line parallel to the given line in a plane.”

2) The angle sum of any triangle is 180 degrees

Important:

3) There exists a square, or any rectangle

4) There is no upper bound to the area of a triangle

5) Every triangle can be circumscribed (a circle drawn on the vertices of the triangle)

Now we can get back to our first proof about the existence of the square root of 2. If you remember one argument was to construct a square, but the possibility of constructing a square must be assumed without proof, since it is equivalent to the parallel postulate. There is no proof that a square exists. In physics, “real” space is not Euclidean (Euclidean space is space where one of the equivalents of the parallel postulate is assumed to be true) In psychology, perceived space is not Euclidean (it is not even logically consistent)

So why is this “proof” of the parallel postulate published in a math textbook?

I’ll remind you that the proof is false and incorrectly done, and has never been successfully done in 2 thousand years. Why would mathematicians want to persuade us that space is Euclidean? Pythagoras wanted people to believe that “Everything is measurable”. Mathematicians are making arguments in the marketplace of ideas. They are drawing in fresh young minds.

Why is this proof that “irrational numbers” exist published in a math book? Why would mathematicians want to persuade us that irrational numbers exist and are numbers?

What does it mean “not rational”?   Graph paper is a picture of a Euclidean grid (many squares). Say that each of these lines are at rational positions- where in this picture is “not-rationalness?”

Kenneth Burke talks about a “terministic screen” – words are like colorful nets that attempt to capture and hold the world, without them, the world would make no sense. We see the world through a terministic screen, a collection of terms (words) that we use to determine our world. The words “terministic screen” have been chosen very carefully. There are many connections that can be made here just with the words. One use of a screen is to see things, as in a TV screen, another use is the kind of screen used to bar bugs (monsters) from getting in the house, or people from getting out of jail. Euclidean grid rejects certain ideas as, basically, illogical. These ideas are effectively invisible to one who uses the Euclidean screen to see the world.  So a screen also refers to the limitations we impose on what we use language to see(know). Do we choose these limitations? This is why words are just as likely to use you as you are to use them, if you leave them unexamined.

What are we barring from view by calling the empty spaces “irrational numbers”?

“Many of the observations are but implications of the particular terminology in terms of which the observations are made.” -Kenneth Burke

This picture of a grid assumes there are squares, which is not necessarily true, so another terministic screen could look like a bent grid, or a cloudy snarl of lines.  Buddhists call the main body of scriptures the three “baskets”. I believe they chose the word basket to describe their holy books very carefully. If rational words are the woven wicker of the basket, what is the “not rational”-ness?

The Greeks had a god to represent these empty spaces between what rational words could describe. Hermes, the “God of the Gaps”, (Palmer http://www.mac.edu/faculty/richardpalmer/liminality.html) was also a god of interpretation and a messenger. Meaning is found through interpretation and it requires some mysterious making sense of (leaping across) these gaps left by our words. To the Greeks a god of deft and lucky chance was needed so that people could understand each other and not interpret words poorly.

What does it mean for mathematicians to give names to “That which words cannot name?”  It seems that meaning and interpretation is crowded out of our terministic screen, and that our screen is so full, so thorough, that we can’t see through it and into the world anymore. Questions can re-open our investigation of the world by suggesting the “irrational” without giving it a name.

[1] This lecture was inspired when I was invited to use a book on logic to teach from that had a “proof” that Euclidean Geometry was logically necessary, and therefore the “one true geometry”. The same book also claimed to prove that square root of 2 is an irrational number. The author made a certain kind of argument that involved using negation in a deceptive way. The author grabs extra information by assuming that square root of 2 is irrational, but more importantly, that square root of 2 is a number. From the proof we have just done, we may not know what square root of 2 is. Do we know that square root of 2 exists? Many math books assume and omit discussion of this, and just teach students to repeat use of irrational numbers, but we do not know if square root of 2 exists. We started with the number 2, we tried to do some kind of operation or apply some rule to it, and we don’t know what comes out, if anything.