One of Zeno’s famous paradoxes is his challenge to the mathematicians view that any finite line segment can be divided by a point. If so, the resulting lines can be subdivided. Zeno’s question is then “If lines can be divided and subdivided, what would the size of the lines be after fully dividing the line?” The absurdity does not lie in the question, but in any answer a mathematician could give. If a mathematician says there are lines with sizes as an outcome, then the sum of the infinite number of lines would make the original line infinite (we said originally that it was finite), and so a contradiction. Otherwise if the result is points, or lines of zero length, the sum of the lengths of points of course would be zero (but we said the line was of non-zero length), and so another contradiction. There are many other possible answers to the question. The reason the question is so famous is that all the answers so far have been unsatisfactory or absurd. I can only guess at why this question has been so mistreated over the 2500 years since it was asked (for example, authors often put any absurdity with Zeno himself, when the questions were intended to show absurdity in his opponents. Authors also accuse Zeno of all sorts of foolish intentions for his questions, such as that motion is impossible). Zeno’s intention, however, is not in question: he was a student of Parmenides and was simply making arguments to defend his teacher’s doctrine that there was only One thing in the world. Regardless, Zeno’s question has outlasted any answer.

Adolf Grunbaum is eminent in the position that Zeno’s paradoxes are refuted by modern mathematics. In his essay “Modern Science and Refutation of the Paradoxes of Zeno”, he began by making exactly the mistaken claim that “Zeno attempted to demonstrate the impossibility of motion” (p. 165 Zeno’s Paradoxes) Grunbaum goes on to introduce briefly the common notion among mathematicians, introduced by Cantor in the 19th century, that there are different kinds of infinities and so we are faced with choosing a particular kind of infinity for the result of the infinite subdivisions of a line. He argues that the kind of infinity of points on a line is “super-denumerable” and cannot be added the way Zeno proposed. Normal addition is reserved for the familiar denumerable infinity that proceeds like the natural numbers (1, 2, 3, etc). However, the divisions Zeno proposed begins as denumerable, (one division or point, then another, etc). And the number of divisions is the same as the number of points. Grunbaum, like Cantor before him, argues that the result of this process, while denumerable at each finite stage, results in something that is not denumerable but “super-denumerable”, refuting inductive logic. As we divide we are working with line-segments of some size, and in the limiting case what results are not segments of zero size (which could be summed), but something with no concept of size whatever. This is exactly the argument that induction is false- that even though we can make repeatable observations of an object on a controlled experiment, as soon as we stop looking the object transforms into something entirely different. Before, I defined numbers as a kind of difference, what we are summing (or subtracting) includes how it is summed, so points that are “super-denumerable” are not the same as other points because summation is different for “super-denumerable” points.

But there is something deeper going on here than this mathematical play of words. A line segment can be defined by its endpoints. Indeed, the endpoints are all that is needed to make a formula for a straight line. The “stuff” between the endpoints, using Aristotle’s terminology, would be the substance. Now we have come to the question “What is a line segment?” Aristotle would say that first and foremost it is its form (or formula), which is to say, its endpoints. Aristotle added to this of course, including in any “what” also its cause or “why” (Aristotle’s Metaphysics). Such expansive thinking has long gone out of use, but Grunbaum would have us believe that the substance of a line-segment is a thing so different from its form as to be completely incommensurable- that Aristotle’s conception of being (which was a marriage of form and substance) must be utterly divided, leaving us with much deeper problems than what we had before with Zeno’s question. Do we abandon form (endpoints) in favor of substance, since the “stuff” of the line-segment would ultimately be a collection of Grunbaum’s “super-denumerable points.” In that case, what would addition or any other mathematical concept be, since we must compare endpoints to measure, and use the result to add, without these formal concepts there is little left of mathematics at all.

Zeno’s subdivisions could be placed in an increasing order as in a sequence used in the axiom of completeness- the axiom that “distinguishes the real numbers” (Abbot, Understanding Analysis). The problem becomes that ordered in this way the limiting “super-denumerable point” is unlike the other points in the sequence, which are endpoints of line-segments. A “super-denumerable point”, as Grunbaum states, has the property that “no point is immediately adjacent to any other.” (p. 169 Zeno’s Paradoxes) In other words, there are only endpoints/there are no zero-length segments. Perhaps Grunbaum is claiming that Zeno’s infinite process of division is not plausible, since to fully subdivide we would need all the points between any two points on the original line to be there as finished divisions. But that would reject the axiom of completeness (and consequently the real number system), where infinite divisions of this kind happen, eventually creating a zero-length segment (a least upper bound to an infinite increasing sequence is an adjacent point, if it were not adjacent to the sequence of points, it would not be a least upper bound). If Zeno’s division is somehow plausible, but somehow without creating adjacent points, what follows is pure nonsense: the resulting points are not “zero-length segments” because zero-length segments require adjacent endpoints in a way that the segment is of zero length. Thus, the ideas of a “zero-length segment” and a “super-denumerable point”, according to Grunbaum’s line of reasoning, must be totally different things. The result of Zeno’s subdividing is then something neither with size, nor of zero size, or perhaps it is both, but it doesn’t matter anymore, we will just call it a “super-denumerable point”, or “linear Cantorean continuum of points” (Grunbaum p. 169 Zeno’s Paradoxes).

At this point an appropriate question is: “What purpose does the mystical belief in super-denumerable points have?” The mystical desire to control oneself has an obvious purpose of freedom and control, but the super-denumerable is not defensible on logical grounds. It would behoove proponents of the view to explain why we should get worked up about it. My speculation is that thinkers become frustrated with how little they end up knowing as a result of thinking, and so the motivation is for teachers of math to feel we know a lot, and be able to say what we know to students. Support for belief in super-denumerable points is that knowledge justifies itself, whether it makes sense or not. Another defense is that this type of math is part of academic culture. The reason math is so important to research is another embarrassment: that math makes sense while other cultures do not. Now, the culture of number is freed from sense and can expand and take on an inclusive attitude to other views.

Aristotle’s substance, at least one of his definitions of it, was a subject that could not be predicated. Another definition was form or essence, and form and substance are deeply connected. In searching for substance, predicates divide the subject. “The human is a man” (or “Woman”) is a division of human. Metaphor, in this view, is destructive to a search for substance. It expands words: “Humans are stars” makes a mess of things, only adding possible predicates. Unless you search for the intersection of “human” and “star”, in which case you are dividing both. Regardless, seeking knowledge of substance is a process of division- so too with a process of division of line segments. And what is the result of this infinite division, this search for substance in the excised world of pure form that is mathematics? It is merely the division itself – the point – which is what we started with when we were looking for substance. What is our point? Or is it changed into a super-denumerable point? Or some other kind of point or division of a line? What is this scalpel? Have I used it violently in searching for it?

“Logic is the calculation of distinction” (Kauffman), and arithmetic is the calculation of degrees of distinction. All I mean is the degree of difference between 3 and 5 is 2 (5-3). All number can be seen as a kind of difference. Normally in textbooks on number you start with addition, but you could use subtraction instead of addition to build and define numbers, the intuitive understanding is numbers as difference (distance) between edges. Zeno is credited with finding absurdities in addition of numbers and magnitudes (G.E.L. Owen “Zeno and the Mathematicians p 153 Zeno’s Paradoxes).

When there are subjects such as two different oranges, the degree of difference is not well known, but people still have a sense that two different oranges are not as different as, say, a raven and a writing desk. Arithmetic requires that the degrees of difference are measurable in some way. One way to measure (or order) difference is to use category. Borges Essay on the Analytical language of John Wilkens ought to be enough to understand that categorizing everything is impossible, but take for example the latin names for animals- it is possible to use this categorization to measure a degree of difference between animals. The degree of difference between two species of the same genus is merely 1, but two species that are not the same genus would have a greater degree of difference, equal to height of the lowest category that has both species in common. But such a categorization of everything is illogical, a good reference of why is Plato’s The Sophist, in which it is argued that categorization of “everything that is” would have at its highest the category “being”, but then would require that “not-being” be a sub-category of “being.” I explain this to show how absurd it would be to believe we could measure exactly the degrees of difference between everything with number. “Who’s doing that?” Pythagoras, according to legend, was ready to murder people to defend the belief that “Everything is number. (see previous post on real numbers) It is also absurd to ignore our feeling that there is a degree of difference.

Are there ever two differences that are the same? Is the difference between 3 and 4 the same as the difference between 4 and 5? 3 is 75% of 4, and 4 is 80% of 5, so it is easily argued that the difference between 4 and 5 is different from the difference between 3 and 4. Since it is easily argued that any two differences in arithmetic (numbers) are different, we have the result “not-equals does not equal not-equals”, at least any time not-equals is used between a different pair of numbers. This means that even in arithmetic, the different differences are not perfectly comparable. For example, the difference between 3 and 4 is a strange thing- not exactly a matter of a single measure of degree. Even if we call the difference between 3 and 4 “1”, we can’t allow ourselves to lose this particular “1”‘s context as a relationship between 3 and 4, and not a relationship between 4 and 5. Now remember we started with the idea that a number was a difference, what I have just argued is that number is strange and not exactly a matter of degree. What a number is depends on its context where it is applied; it depends on the particular things and the difference between these things that number attempts to measure. That differences are different can be applied to both number and magnitude, for magnitude it would be the observation that a line could not end at a point (a point is a nonsensical pairing of a sizeless nothing with a position), but must end someplace in a “surrounding” as the ancient Greek understanding went, or in context with its environment; because of this, no magnitude can be the same. Further, different magnitudes are only partially comparable, since their surroundings cannot be ignored in the act of measuring- the surroundings are required to find edges.

It has already been shown that any logical calculation of difference must be empty of any subject, the same goes for number. As soon as you start believing the numbers you speak have some “content”, that you are “saying something,” you are in error. But I just argued that number requires application to particular subjects before it can be used to describe the difference between two things, which is how number gets defined here. The result is that number is undefined- we can’t know what number is. Number, just number without any theories, are like real words; they have a nuanced meaning well beyond what analysis texts describe, and they have a paradoxical relationship with their object. Any use of technical language, such as the distinction between magnitude and number, only inflates my term “Number,” since it was inflated by looking at the different differences that must be included in a concept of Number.

“…the very attempt to make formal languages is fraught with the desire that each term shall have a single well assigned meaning. It cannot be! The single well-assigned meaning is against the nature of language itself. All the formal system can actually do is choose a line of development that calls some entities elementary (they are not) and builds other entities from them. Eventually meanings and full relationships to ordinary language emerge.” (Kauffman 2001) http://www.math.uic.edu/~kauffman/Peirce.pdf

Number is not what we define it to be in college mathematics; numbers are words with many philosophical, poetic, and mystical associations. I have shown that we cannot escape number’s relationship to the larger world of culture and language, number, with only a little bit of skeptical thought, cannot remain merely a matter of degree.

For example the word “1”, a single vertical line, is among the simplest symbols available, leaving a connotation that we are talking about something elemental. I think Western mysticism would have “1” be associated with Fire (not the modern scientific notion of fire, but the more poetic notion of Fire as one of the four elements). The first sign of the zodiac is Aeries, and it is believed that fire is the force of creation that has to come first. However the Thai symbol for “1” is a spiral (“spira” is Latin for Air). I hope dismissing these mystical associations, whether they make sense or not, is more troubling now to Serious People.

“In a restricted context, one may manage without being engulfed by the language as a whole, and this is indeed the game played by a mathematician (or Humpty Dumpty! [3]) who would have words mean what he wants them to mean in a special context. The cost to Humpty Dumpty is well known; the cost to the mathematician is the emergence of paradox…” (Kauffman 2001 p. 106) http://www.math.uic.edu/~kauffman/Peirce.pdf

What is the rhetoric of paradox? In my experience as a math student, paradox is generally left out. If it is presented, it is presented with as few words or discussion as possible. (in physics, for example, people just said the words “Instantaneous Velocity” and then looked at you with vacant wide eyes, waiting for you to adopt the same stare as though you understood, but Zeno’s Arrow paradox refutes it) When paradox is presented it is usually pretended that the paradox is solved in a more advanced treatment of the subject, outside the scope of what is put before the student. In the case of Zeno’s paradoxes and the trouble they cause to calculus:

“Perhaps the reader shares the widespread feeling that they are mere anachronisms that can, at best, befuddle undergraduates who have not taken any calculus yet. Their utility, on this view, continually diminishes as calculus comes to be ever more commonly taught at the high school level. As mathematical sophistication becomes more universal, one may feel, Zeno’s paradoxes will serve only to show how mathematically naive were the Greeks of the fifth century B.C. No evaluation could be further from the truth…” (Zeno’s Paradoxes 2001 Ed. Wesley C. Salmon, Preface) The effect of Zeno’s paradoxes on mathematics, while it is not positive (G.E.L. Owen, “Zeno and the Mathematicians”), still leads to a “high level of philosophical discussion”. (Salmon)

It is well and good that mathematicians such as Kauffman are aware of the deep trouble paradoxes stir up in mathematics, but why should this knowledge, this trouble, not be central to a students education in math?

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.