“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! ) 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?