One, Two, Three . . . Continuity:
Robin Robertson (1)
Abstract: The nature of the one and the many is an immemorial problem. This paper begins with Parmenides and the paradoxes presented by his disciple Zeno, then presents at some depth the mathematical concepts of limit and the continuum developed in the second half of the 19th-century by Karl Weierstrass, Richard Dedekind, and Georg Cantor. These interpretations are explicitly contrasted with C. S. Peirce's view of the nature of the continuum, and how this implies the actual existence of infinitesimals. A brief description of Peirce's concept of "one, two, three" is presented, showing how this new view on continuity completes this model of all reality, which he now termed synechism. Finally, several modern scientific examples of a similar view of continuity are presented.
Infinity is nothing but a peculiar twist given to generality.
- Peirce in a letter to William James, June 8, 1903, (CP8:268). (2)
Twenty-five hundred years ago, Parmenides of Elea argued that all is one, that there is a single underlying principle of being. In Plato'sParmenides, we hear Parmenides gently draw out the ideas of a young Socrates. Then, at much greater length, he guides Aristotles (sic) ineluctably to a final conclusion: "whether one is or is not, one and the others in relation to themselves and one another, all of them, in every way, are and are not, and appear to be and appear not to be" (Plato, nd).
In other words, as soon as we look at the relationship between the one and the many, we're in deep waters, where it's difficult to be sure of much of anything. Parmenides' disciple Zeno argued against "the many" using a series of brilliant paradoxical arguments that are as fresh today as when they were first coined. The best known is undoubtedly that of Achilles and the Tortoise. Despite the fact that Achilles was the fastest of humans, and the Tortoise the slowest of animals, if the Tortoise is given a head-start on Achilles, no matter how small, Achilles can never catch him. For, by the time Achilles has arrived at the point where the Tortoise started the race, the Tortoise has moved forward. By the time Achilles comes to that point, the Tortoise has again moved. And so on.
Since, of course, we know that in a real race, Achilles would catch and pass the Tortoise, there must be something wrong with our reasoning. And what could that be? Zeno (and we will see, Peirce) would argue that the problem is the assumption that time and space are infinitely divisible. Instead, both would say that space and time are each indivisible concepts, impossible to break into parts. As Peirce says: "All the arguments of Zeno depend upon supposing that a continuum has ultimate parts. But a continuum is precisely that every part of which has parts, in the same sense. Hence he makes out his contradictions only by making a self-contradictory supposition" (CP5.355).
Despite numerous attempts by Aristotle and later philosophers, such as the Scholastics, in essence, Zeno's arguments remained unanswered until the second half of the nineteenth century, when a new foundational base of mathematics, set theory, led to rigorously defined concepts of infinity, limit, and the continuum. Three creators of set theory were Karl Weierstrass, Richard Dedekind, and, especially, Georg Cantor. The mathematical edifice that they built has, despite cracks in its own foundation, lasted to this day.
Standing outside this group, fully aware of their developments, but looking in from his own unique perspective, was Charles Sanders Peirce [1839-1914]. His view was, like all his views, idiosyncratic. Whereas Weierstrass, Dedekind and Cantor saw the continuum as a construction of points, Peirce regarded it as an entity in itself, beyond any construction. His view of the continuum offered the possibility for a continuity throughout matter and mind, living and dead, all supposed dichotomies. Thus he offered not only an alternative view of the mathematical continuum, but a different view of the one, which provides a possibility of finally advancing past Parmenides. I beg the reader's indulgence while I first present, at some length, the traditional view of the continuum, as developed in set theory, then contrast it with Peirce's view.
WEIERSTRASS' ATTEMPT TO DEFINE INFINITE LIMITS IN FINITE TERMS
It is essentially a merit of the scientific activity of Weierstrass that there exists at present in analysis full agreement and certainty concerning the course of such types of reasoning which are based on the concept or irrational number and of limit in general.
- David Hilbert (Struik, 1948).
If mathematics only dealt with finite numbers, there would have been no need for set theory. There would also have been no calculus. When Newton and Leibniz jointly created calculus late in the seventeenth century, they provided mathematicians and scientists with the most powerful mathematical tool ever created. Calculus enabled scientists to quantify both their observations and their theories to an extent hitherto inconceivable. But calculus depended on infinite and infinitesimal processes, and mathematicians of the time had only the vaguest of ideas what infinite and infinitesimal processes might actually mean. During the 18th century, mathematicians were largely satisfied to extend calculus, without questioning its foundation. As the 18th century gave way to the 19th century, mathematicians began to question that blithe attitude. By the second half of that century, they were ready to attack the issue head-on. In order to understand the approach they took, we need to begin with a short summary of what calculus is and does.
Imagine an irregular closed figure drawn on a piece of paper. Purely as an example, picture something like a lumpy circle. How can we find the area occupied by that shape? We can start by drawing a rectangle that is as small as possible yet fully contains the figure. It's simple enough to determine the rectangle's area, which is the product of its length times its height. In mathematical terms, we can call the area of the rectangle an upper limit on the area of the figure; that is, by using the area of the rectangle as an approximation to the area of the lumpy circle, we can insure that the true area must be smaller than our estimate.
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If, like our lumpy circle, the figure is quite asymmetric, we can improve our estimate by drawing two rectangles of different sizes next to each other, which together cover the whole figure. By calculating the area of each rectangle and adding the two, the upper limit will be smaller than before, closer to the actual area of the figure. If we increase the number of rectangles, our estimate will get better and better. The limit will get smaller and smaller, and approach the actual area more and more closely. With a thousand rectangles, the limit would be so close to the actual area that for all practical purposes, we could consider it to be identical. However, it's important to realize that it would still not be exact.
Now make a leap in thought! Imagine that we extend the number of rectangles endlessly. If we had some way of calculating the limit of this process by adding up an infinite number of such areas, the limit would no longer be merely an approximation to the area of the figure; it would be exactly the area of the figure. That is just what calculus does. Despite the fact that Newton and Leibniz were able to show how to calculate such limits, no one knew exactly what they were.
In the mid-19th century, German mathematician Karl Weierstrass [1815-1907] asked what do we really mean when we say that the sum approaches a limit? Simply that the sum of the rectangles can be made to differ from the limit by as small a quantity as we desire. More explicitly, if any desired difference is named, it is possible to pick a sufficient number of rectangles that their difference between their areas and the limit will be less than that quantity.
For example, say the limit is calculated to be 25 square inches using calculus. Then pick some very tiny number for the desired difference: say one thousandth of a square inch. Weierstrass says that we can find a larger enough number of rectangles that their sum will be within one thousandth of a square inch of 25 square inches. We'll say, just an example, that it takes 250 rectangles to do this. If we decide to pick a still smaller difference, say one millionth of a square inch, perhaps it might take 5,000 rectangles. But no matter how small the difference desired, it is possible to come up with a sufficient number of rectangles such that the sum of their areas differs from the limit by less than that difference.
The significance of Weierstrass' method is that he was able to rigorously define limits without ever mentioning infinity!
Weierstrass then used an extension of the same technique to discuss irrational numbers. As early as the 6th century B.C., Pythagoras had discovered that SR(2) (i.e., the square root of 2) could not be expressed as the ratio of two counting numbers (i.e., 1, 2, 3, …). Over time, mathematicians had come to realize that there were a huge number of such numbers, which they called irrational--not a ratio--in contrast to so-calledrational numbers like ½, or 23/47, …; i.e., fractions formed by taking the ratio of two whole numbers. Though irrational simply meant "not rational," and hence had no pejorative meaning, it was also true that the term was singularly appropriate because of the degree of unease it caused Greek mathematicians.
Since irrational numbers could not be expresses as a simple ratio of integers, mathematicians were forced to use mathematically complex equations involving infinite series in order to calculate their value. This meant that they presented the same sort of problems involving infinity as calculus. Weierstrass got around this problem using a similar technique to the one that was so successful in dealing with limits. He defined an irrational number using a set containing a sequence of rational numbers that approached the actual value of the irrational number as a limit.
For example, the SR(2) which caused Pythagoras so much trouble, can be expressed in decimal notation as 1.414213… (Here our 3 little dots "…" have to do double duty, and mean not only that there is no end to the decimal expansion, but also that neither is there any definable pattern to it.) Weierstrass would define the square root of 2 as the set {1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, …}. (3) If we pick some tiny difference as we did with limits--say 1/104 (i.e., one ten-thousandth)--we know that the 4th member of the set (i.e. 1.4142) is less than 1/104 from the actual value of the SR(2). Similarly for a desired difference less than 1/106 (i.e., one millionth), we have only to move to the 6th member of the set.
Once again, as with limits in calculus, the need to deal with infinity is avoided and Weierstrass manages to deal only with finite numbers. It is important to stress that Weierstrass didn't say that this set--{1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, …}--approached the irrational number as a limit (that is, "at infinity"); what he actually said was that the set itself was the irrational number. Though this might seem an insignificant difference, he was shifting to an important new frame of reference: set theory.
What is a set? A set is merely a collection of things of the same kind; e.g., the set of books about logic is composed of all books about logic; the set of proofs for the existence of God is composed of all such proofs. Each item of the given kind is called a member of the set. It's important to grasp that the set is not the same as the members--the set is the collection, the assemblage, not the things assembled. Weierstrass was considering sets of numbers, but sets are not limited to numbers. Sets can include anything whatsoever.
Weierstrass' definition of the irrational numbers was important because it shifted the emphasis from an infinite series to a set where an irrational number could be defined to any degree of precision, with only a finite number of members of the set. However, clever as this technique was, many mathematicians objected to it as a trick, since any set which defined an irrational number still had an infinite number of members, regardless of the fact that Weierstrass was able to limit his discussion to at most a finite number of the set's members. These mathematicians were never to be satisfied by any subsequent improvement on Weierstrass' technique. The shift to set theory, however, opened the door to a new, exciting world for most mathematicians, and prepared the way for still more clever attempts at describing the number line. Drawing on the concepts of set theory, both Georg Cantor and Richard Dedekind (who we should note were contemporaries with C. S. Peirce) developed new definitions of irrational numbers. Dedekind's technique, called the Dedekind Cut, has become traditional.
THE DEDEKIND CUT
Dedekind came to the conclusion that the essence of the continuity of a line segment is not due to a vague hang-togetherness, but to an exactly opposite property--the nature of the division of the segment into two parts by a point on the segment (Boyer, 1968, p. 607).
Dedekind struggled with the same sort of problem addressed by Weierstrass, but focused on a different concept: the mathematical continuum. In mathematics, the continuum is another name for the number line; the unbroken line we are used to seeing, which has "0" in the middle and extends out indefinitely to the right with positive numbers, to the left with negative numbers.
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But, though we draw numbers on it, the continuum itself is a line; hence a geometric concept, involving space. For example, Zeno's paradox of Achilles and the Tortoise can be pictured on the continuum. Alternately, the view of the mathematical set theorists like Weierstrass, Dedekind and Cantor (but, we will see, not C. S. Peirce, as we see below), is that the continuum is also the set of allreal numbers (i.e., both rational and irrational numbers), and is, hence, also an arithmetic concept. In this view, every spot on the number line corresponds to either a rational or irrational number; going in the other direction, every rational or irrational number has its place on the number line. (4)
Rather than the whole continuum, let's simply take a chunk of it, say the part between 0 and 1 (any part will do as we'll see later in discussing Cantor's theory of transfinite numbers). According to the view of Weierstrass, Dedekind, and Cantor, every place on that line segment is either a rational or an irrational number. As you'll recall, the rational numbers are the numbers that represent fractions; hence each has a unique decimal identity. In between lie the irrational numbers, which cannot be expressed by an patterned decimal expansion, either finite or infinite. That's the problem with identifying irrational numbers. We showed Weierstrass' solution where he defined each irrational number to any degree of accuracy by a finite set of rational numbers. In contrast, Dedekind proposed a solution based in a mind experiment (Dedekind, 1888) (5)
Dedekind asked us to imagine a blade with an infinitely thin blade (i.e., it has no thickness at all) which can be used to cut the continuum into two segments. Since every point on the line corresponds to either a rational or an irrational number, the blade has to have hit a point corresponding to a number. And since the blade has no thickness at all, that point has to end up in either the left or the right segment. Since it doesn't matter, we'll assume that it always ends up in the leftmost segment. Since all the points in the left segment precede all the points in the right segment, every number in the left segment has to be less than every number in the right segment. So every number - rational or irrational - can be defined by cutting the number line into two segments in this way. And every such cut - which is traditionally called aDedekind Cut - corresponds to a unique number.
Of course, it is implicit in Dedekind's solution that every point on the line interval is either a rational or an irrational number. As we'll see, Peirce strongly disagreed with this assumption.
PEIRCE'S APPROACH TO THE DEDEKIND CUT
. . . my notion of the essential character of a perfect continuum is the absolute generality with which two rules hold good, first, that every part has parts; and second that every sufficiently small part has the same mode of immediate connection with others as every other has (CP4.642).
Though Peirce never directly discussed the Dedekind Cut, he was aware of his work and did express a view of the continuum that was in marked contrast. In this section, I'm going to use Peirce's view of the continuum to produce a reductio absurdum argument against the Dedekind Cut.
Clearly, both Weierstrass and Dedekind assume that the continuum, as represented by a line, is composed of individual points (each point of which can be expressed by either a rational or an irrational number). Therefore, in Peirce's view, since there are no holes in the line, and since the line is supposedly made up of points, even before we cut the line in two, we must be able to exactly identify the cutting-point. That is a clear consequence of each point having an individual identity.
But then, because the blade has no width at all, it must be possible to divide the cutting-point into two points, one in the left region {A}, and one in the right {B}. We can do this, because, in Peirce's view, any "point" on the continuum must be like a raindrop which the cut divides into "two ideal rain drops, distinct but not different" (6) (CP4.311). Dedekind ends up with the point in either {A} or {B}(we chose {A} for simplicity) because he views points as the "atoms" of the line. Peirce insists that there are no "atoms," only "parts," which can be divided endlessly.
After the cut, in Peirce's view, we can now see an ordering that wasn't there before; that is, the rightmost point of {A} has to be earlier in sequence (i.e., to the left) than the leftmost point of {B}. But, on the other hand, if we rejoin the two parts, the two points have recombined again into a single point. Obviously an impossibility: a point can't both be an indivisible "atom" and also capable of being divided into further "atoms." Hence, the line is not made up out of points; it is only the act of making the cut which creates an ordering of points of a line. Or put another way,points don't exist except when we create them.
In Peirce's words: "the points on a line not yet actually determined are mere potentialities" (CP3.568)."Mere potentialities." Hence, for Peirce, the only way to create points is to identify them by some act of distinction. Yet, no matter how often we create points by making such distinctions, we can never exhaust the points on a line, since there are always unlimited sequences of points between any two points we identify.
If there is room on a line for any multitude of points, however great, a genuine continuity implies, then, that the aggregate of points of a line is too great to form a collection: the points lose their identity; or rather, they never had any numerical identity, for the reason that they are only possibilities, and therefore are essentially general. They only become individual when they are separately marked on a line; and however many be separately marked, there is room to mark more in any multitude. (CP7.209)
HOW DISTINCTIONS CREATE WORLDS
. . . when one element cannot even be supposed without another, they may ofttimes be distinguished from one another. Thus we can neither imagine nor suppose a taller without a shorter, yet we can distinguish the taller from the shorter. I call this mode of separationdistinction. (CP1.353).
In 1903, trying to find a way to express the nature of true continuity, Peirce recorded the following parenthetical thought in his personal copy of the Dictionary. (7)
It seems to me to point to this: that it is impossible to get the idea of continuity without two dimensions. An oval line is continuous, because it is impossible to pass from the inside to the outside without passing a point of the curve (CP6.165).
This is a perfect expression of what Peirce means by points being "mere potentialities" until brought into existence by an act of distinction. In his example, in order to bring a point into existence, one has to pass from one side of a Spencer-Brown distinction to another. In a previous article (Robertson, 1999), I described how, in his Laws of Form (1979), logician G. Spencer-Brown presented a mathematical system which dealt with the emergence of anything out of the void. Laws of Form traces how a single distinction (a "mark") in a void leads to the creation of space, where space is considered at its most primitive, without dimension. This in turn leads to two seemingly self-evident "laws." With those laws as axioms, first an arithmetic is developed, then an algebra based on the arithmetic, which is formally equivalent to Boolean algebra.
When you make a Spencer-Brown distinction, there are now two things: the "mark" and the "not-mark." You've created a "space" and that's all that there is in the space you have created. The mark and the not-mark form a necessary and complementary pair without which space--any space--cannot exist. But, once you use his two "laws" to create an arithmetic, you are able to string marks together to form a series of signs that appear to be something other than the mark and the not-mark. Until you apply the two laws, these series of signs are indeterminate: you don't know whether they reduce to the mark or the not-mark. They haven't yet "collapsed" into the mark or the non-mark.
Note how similar the views of Peirce and Spencer-Brown are to the Copenhagen model of quantum mechanics. For Peirce, points and their distinctions are created by the process of marking or referencing them, and their existence and distinctions collapse when we change our frame of reference. In the Copenhagen model, reality is indeterminate until, under observation, it appears as either a particle or a wave. When observation ends, the particle or wave once again collapses into indeterminancy. In his biography of Werner Heisenberg, author David C. Cassidy discusses how, when Heisenberg was first introducing his famed "uncertainty principle," basing it on a particle view of reality, Niels Bohr argued with his younger colleague that he should instead view it within the "complementarity principle," in which both waves and particles are necessary, complementary aspects of physical reality.
That is, we know particles have energy (E) and momentum (p). Similarly we know waves can be described fully in terms of space (q) and time (t). Furthermore, we know that all causal descriptions are based in one or more of these four variables . However, each uncertainty combines possibilities of both waves and particles. In fact, it's because we combine two incompatible elements--particle and wave--into a single possibility, that we have an inherent uncertainty. Bohr thus argued that reality emerges from uncertainty by the act of observation, not by the process of discovery. "Heisenberg rose from his seat in the audience that day to confer his public approval on Bohr's interpretation of their physics. The Copenhagen interpretation was born" (Cassidy, 1992, 244).
Thus Heisenberg and Bohr, dealing with the physical world at the quantum level, and Spencer-Brown and Peirce, dealing with the most primitive levels of mathematics and logic, arrived at a point where the act of distinction creates reality out of a previously undetermined state.
CANTOR'S THEORY OF TRANSFINITE NUMBERS
Both in his time and in the years since, Cantor's name has signified both controversy and schism. Ultimately, transfinite set theory has served to divide mathematicians into distant camps determined largely by their irreconcilable views of the nature of mathematics in general and of the status of the infinite in particular (8)
Questions about the continuum and the nature of points raise issues about the nature of infinity. The third, and greatest of our trio of mathematicians who developed set theory was Georg Cantor. Cantor's theory of transfinite numbers was a major influence on Peirce, and, hence, needs to be discussed before we move further into Peirce's own approach.
Long before Cantor, mathematicians realized that there is no end to the integers; if a largest integer could be conceived, merely add one to it and it's no longer the largest. There is also clearly no end to the points on a line provided that a point is understood to be without dimension. Cantor had the brilliance to ask the seemingly silly question: Are these infinities both the same size. Lest this sound like the medieval scholastic arguments about the number of angels who could stand on the head of a pin, the reader should be aware that Cantor found a way of quantifying infinity, and thus answering his own question.
Let's return to the concept of a set. What does it mean to say that two sets have the same number of members? For example, what does it mean to say that the set of fingers has the same number of members as the set of toes? If we think deeply about it, it merely means that we can pair off members of each of the two sets and have no members left over in either set. In our example, we could place each of our ten fingers on one of our ten toes.
Notice that it wouldn't matter what finger we match with what toe as long as we were careful not to pair any finger with two toes or any toe with two fingers. Further, we could define a cardinal number, in this case 10, which would characterize the number of members in each set. Having developed the cardinal number 10, it could then be used to describe the number of members of any set which can be paired off with the set of fingers. We would describe each as having the same cardinality. This provides an unequivocal way to compare sets to see if they are the same size.
This method of counting by pairing the members of one set with the members of another is commonly termed the pigeon-hole technique. Cantor found that the pigeon-hole technique produced surprising results with infinite sets. For example, he discovered that there were exactly as many even integers as there were both odd and even integers. He reasoned that for every integer i in the set of all integers, he could match i with the integer 2i in the set of even integers. For example, 1 in the set of all integers would be matched with 2 in the set of even integers, 2 with 4, 3 with 6, etc. No matter what integer you named in either set, this method uniquely defines the integer it corresponds to in the other set. Thus the pigeon-hole technique proved that the set of even integers is the same size as the set of all integers. Try it if it doesn't seem reasonable. At first it will seem obvious that there are more integers than even integers, since the odd integers are left out. But that isn't the point. If the two sets can be matched without any members of either set being left over, then they have the same cardinality. The problem isn't with the method, it's with infinity itself.
Cantor could use the same logic to show that the set of numbers evenly divisible by three is also just as big as the set of all integers. In fact the particular multiple made no difference; the set of numbers evenly divisible by a billion is still just as big as the set of all integers. Going in the other direction produced the same result. The integers are a sub-set of the rational numbers (remember: the fractions.) For example, 2 can be expressed as 2/1 as a fraction; 3 as 3/1, etc. But think of all the other fractions: ½, 1/3, 357/962, etc. Surely there are more fractions than integers! Instead, Cantor used an elaboration of the pigeon-hole technique to prove that the set of rational numbers had exactly as many members as the set of integers. This was a very surprising result indeed! In fact, Cantor proved that any infinite sub-set of the set of rational numbers contains exactly the same number of members! All have the same cardinality, which Cantor termed Aleph-0. This number is commonly referred to as countably infinite, since all such sets can be paired off with the counting numbers 1, 2, 3, …
So we see that when we are dealing with infinite sets, it is possible to match the whole set with a sub-set of itself, with nothing being left out. Richard Dedekind has traditionally been credited with this lovely definition of infinite sets: "A system S is said to be infinite when it is similar to a proper part of itself, in the contrary case S is said to be a finite system (Dedekind, 1888). But Peirce had been there before Dedekind, using the same distinction in his paper "The Logic of Number" (CP3.288). Peirce also said that he had communicated this idea in a letter to Dedekind. Peirce was especially proud of this discovery, claiming that "the proposition that finite and infinite collections are distinguished by the applicability to the former of the syllogism of transposed quantity ought to be regarded as the basal one of scientific arithmetic" (CP6.114). He produced many versions of this argument in his papers. Here's one:
Every Texan kills a Texan,
No Texan is killed by more than one Texan,
Hence, every Texan is killed by a Texan (CP 3.288).
If there is a finite number of Texans, this is true, but if, on the other hand George W. Bush's dearest wish would happen and there were an infinite number of Texans, then this would not be true. For example, Texan #1 could kill Texan #2, Texan #2 could kill Texan #3, and so on. Texan #1 would remain alive. Or, if we want more living Texans, Texan #1 could kill Texan #2, Texan #2 could kill Texan #4, Texan #3 could kill Texan #6, and so on. An infinite set of Texans would still be alive at the end of this killing spree (i.e., Texans #1, 3, 5, . . . ).
Georg Cantor went beyond this and asked if every infinite set was countably infinite; i.e., is every infinite set the same size as every other infinite set? The answer to that was even more surprisingly a resounding no. Let's turn to his idea of power sets to see why there is more than one size to infinity; why there are, in fact, an unlimited number of infinities.
For any set, Cantor defined its power set as the set of all the sub-sets of the original set. Key here is that the power set is one level higher than the original set: its members are themselves sets. For example, let's take the set of primary colors: {red, yellow, and blue}. The power set would consist of all the sub-sets of this set; first taken one at a time, then two at a time, finally three at a time. (9) So the power set would be: {{red}, {yellow}, {blue}, {red, yellow}, {red, blue}, {yellow, blue}, and {red, yellow, blue}}. (10)
Another way of looking at a power set is to consider it as composed of all the relationships between the original members of a set, all the ways they can be linked together in any combination. It is easy to see that as the original set gets bigger, the power set gets much, much bigger (11) In our little example, the set of primary colors had 3 members, while the power set had 8 (including the null set as the 8th.) A set with 4 members would have a power set with 16 members, and so forth. Cantor was able to demonstrate that the power set is always of higher order of cardinality than the original set, even if the set is infinite. Thus the power set of the rational numbers has a cardinality bigger than Aleph-0, which Cantor termed Aleph-1. Aleph-1 in turn has a power set Aleph-2, and so on. So there is no biggest size for infinity, since we can always create a bigger infinity by taking the power set of the previous set.
It is simple to demonstrate that the set of real numbers is equivalent to the first power set of the countable numbers. That is, the power set of the countable numbers is the set of all countable numbers taken one at a time, two at a time, and so forth: {1}, {2}, {3}, . . . , {1, 2}, {1,3}, . . . , {2, 3}, {2, 4}, . . . , {1, 2, 3}, . . . The real numbers can be expressed in decimal form as a decimal place followed by all the combinations of the real numbers taken one at a time, two at a time, and so forth: 0.1, 0.2, 0.3, . . . , 0.12, 0.13, . . . , 0.23, 0.24, . . . , 0.123, . . . So clearly the first power set and the real numbers are the same size.
Having shown that power set of the countable numbers was a larger size of infinity than the countable numbers themselves, and that endless, ever larger infinite sets could be constructed by taking power sets of power sets, Cantor proposed that this was all that there was to infinity. This proposal is called Cantor's Continuum Hypotheses. Cantor's Continuum Hypothesis says simply that the power of the continuum = Aleph-1 (i.e., the power set of the countable numbers, which is equivalent to the set of all real numbers). (12)
In famed mathematician Kurt Gödel's words:
Cantor's continuum hypothesis is simply the question: How many points are there on a straight line in Euclidean space? An equivalent question is: How many different sets of integers do there exist? This question, of course, could arise only after the concept of number had been extended to infinite sets (Gödel, 1964, p. 254).
Cantor's hypothesis that the number of points is Aleph-1, which he was sure was correct, remains one of the great unresolved conjectures in mathematics. Again, quoting Gödel:
But, although Cantor's set theory now has had a development of more than seventy years and the problem evidently is of great importance for it, nothing has been proved so far about the question what the power of the continuum is…Not even an upper bound, however large, can be assigned for the power of the continuum (Gödel, 1964, p. 256).
Thus Peirce's idea of the continuum as being too rich to be exhausted by the real numbers (or in fact, by any combination of numbers), seems less strange.
INFINITESIMALS & THE DENIAL OF THE CONTINUUM HYPOTHESIS
I believe that adding up all that has been said one has good reason for suspecting that the role of the continuum problem in set theory will be to lead to the discovery of new axioms which will make it possible to disprove Cantor's conjecture (Gödel, 1964, p. 264).
Peirce was mightily impressed by Cantor's achievement and said so numerous times in his own papers; e.g., "in truth, there are but two grades of magnitude, the endless [by which Peirce meant countable infinity] and the innumerable [by which he meant the power sets]." In his early admiration for Cantor's theory, Peirce did not seem to feel that matching the continuum to the real numbers was impossible. In that earlier period of time (13), he still felt that "it is evident that there are as many points on a line or in an interval of time as there are real numbers, in all" (CP6.118).
It was only after struggling with the ideas of Weierstrass, Dedekind, and Cantor for nearly two decades that Peirce came to his own unique approach to the continuum. In a letter to Paul Carus, editor of The Monist, on Aug. 17, 1899, Peirce said that " . . . the true nature of continuity . . . is now quite clear to me." Previously, Peirce said, had been "dominated by Cantor's point of view." Now he saw that it was best not to try "to build up a continuum from points, as Cantor does" (Peirce, 1998b, p. xxii).
In Peirce's new view, the continuum is an inexhaustible source of numbers but does not, itself, consist of points or numbers! One way to visualize Peirce's argument is to imagine that first we approximate the continuum by the rational numbers; there will still be, however, "holes" between the rational numbers. We can then plug those hole with the irrational numbers. (14) But Peirce would say that even after adding the irrational numbers, there would still be holes. We can fill those holes with further sequences of numbers by taking the power sets of the irrationals. But, Peirce would insist that there will still be holes. So we can try filling those holes with still further sequences of numbers. According to Peirce, no matter how many sets of transfinite numbers we use, holes will still remain. It is in this way that we see how the continuum is considered to be inexhaustible and not just infinite. Though Peirce was probably the first to realize the basic problems with attempting to relate a spatial concept like the continuum, with any set of numbers, he wasn't alone in his belief.
As early as 1905 René Baire…suggested that Cantor's continuum hypothesis assumed the identifiability of two concepts that were intrinsically different and of noncomparable orders of magnitude…The two ideas were inherently antithetical: the nature of the continuum, regarded as the collection of all infinite sequences of integers was something totally different (Dauben, 1979, p. 269).
After producing his epochal Incompleteness Proof in 1931, Kurt Gödel spent much of his later mathematical life trying to resolve the Continuum Hypothesis. In 1940, he managed to prove that if a modified set theory, which does not include the Continuum Hypothesis, is consistent, then it will remain consistent if the Continuum Hypothesis is added as an additional axiom. In 1963, mathematician Paul Cohen was able to prove the reverse: If a modified set theory which does not include the Continuum Hypothesis, is consistent, then it will still be consistent if the continuum hypothesis is assumed to be false. In other words, between Gödel and Cohen, they had demonstrated that the Continuum Hypothesis was undecidable within set theory.
That being said, both Gödel and Cohen felt that ultimately, within some extension to set theory, the Continuum Hypothesis would be decidable, and both felt that it would be false. Gödel didn't go as far as Peirce, and at one time thought that perhaps the cardinality of the continuum might beAleph-2. Dauben says that "in [Paul] Cohen's view, the continuum was clearly an incredibly rich set one produced by a bold new axiom which could never be approached by any piecemeal process of construction" (1979, p. 269). This view sounds close to Peirce, as both felt that it was impossible to arrive at the continuum by constructing it from points, or sequences of points. The continuum simply was, and points were simply abstractions for possibilities within it.
If this were true, however, then what number abstractions might we assign to "the holes in the continuum"? Well, if we go all the way back to the creation of calculus by Newton and Leibniz, we find that, in addition to limits, both theorists expressly assumed infinitely small particles (though both assured other scientists that these were merely intellectual abstractions). Newton called his particles fluxions, but they are better known as infinitesimals. Infinitesimals are greater than zero, yet less than any normal positive number. In a famous quote, 18th-century philosopher Bishop George Berkeley, lampooned fluxions:
And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities . . . ? (Smith, 1959, p. 633).
In contrast, Peirce came to regard infinitesimals as something very real, in fact, as the only way to explain the continuum: " . . . the infinitesimals must be actual real distances, and not mere mathematical conceptions like SR(-1)" (CP 3.570). In this assumption, Peirce was clearly going against the main-stream of late 19th and early 20th-century mathematics, which tried to avoid infinitesimals like the plague.
In our more recent time, two different mathematical approaches have emerged which explicitly admit infinitesimals into mathematics: the surreal numbers developed by the legendary John Horton Conway (Knuth, 1974; Conway & Guy, 1996) (15), and the hyperreal numbers that emerge from the late Abraham Robinson's non-standard analysis (Robinson, 1966).
Using only two simple generative rules, Conway is able to make all integers emerge, both positive and negative, then rational numbers, then the real numbers that fit between, then all of Cantor's transfinite numbers. And it just begins there. Infinitesimals, which are the inverse of the transfinite numbers, emerge, then algebraic roots of transfinites and infinitesimals, and so on endlessly.
But Abraham Robinson's non-standard analysis has had more impact on mathematics. Robinson took the set of real numbers R, then created an expanded set R* by adding all the infinitesimals and infinites. Infinites and infinitesimals are the inverse of each other (i.e., if x is infinitesimal, 1/x is infinite, and vice versa). Anything in R*, not in R, is non-standard. All hyperreal numbers have a standard part and a non-standard part. I won't go into the details here, but what is special about Robinson's achievement is that anything proved within R*, using non-standard analysis, is automatically true in R, using standard mathematical techniques. Peirce would have been fascinated not only with Robinson's achievement, but with the fact that non-standard analysis evolved out of logic, and needed one major theorem from logic. (16)
Peirce would have been less thrilled, however, with the fact that Robinson regards hyperreal numbers merely as notational tools, not as actually existent. But then Robinson was skeptical enough to have once remarked: "I am not sure I really BELIEVE in the set of all natural numbers! (Dauben, 1995, p. 354).
ONE, TWO, THREE . . . CONTINUITY
TRICHOTOMIC is the art of making three-fold divisions. Such division depends on the concept of st, 2nd, 3rd. First is the beginning, that which is fresh, original, spontaneous, free. Second is that which is determined, terminated, ended, correlative, object, necessitated, reacting. Third is the medium, becoming, developing, bringing about (Peirce, 1992, p. 280).
Throughout his writings, early and late, Peirce was fascinated by (one might almost say "obsessed by") the concept of "three-ness", "triads," "trichotomy," "One, Two, Three," etc. He even joked about this tendency. "I am a determined foe of no innocent number; I respect and esteem them all their several ways; but I am forced to confess to a leaning to the number three in philosophy" (CP1.355).
In "A Guess at the Riddle" (CP1.354-416), which the editors of The Essential Peirce consider Peirce's "greatest and most original contribution to speculative philosophy" (Peirce, 1998b, p. 245), Peirce attempts to extend this principle to all reality, beginning with the essential concept, then extending it into reasoning, metaphysics, psychology, physiology, biology, physics, sociology (by which he means not the study of humans in groups, but the study of a "community of cells"), and theology. Perhaps the best-known of his hierarchies of triads proceeds from Quality - Relation - Representation (or Sign); then breaks Sign into Icon - Index - Symbol; then Symbol into Terms - Propositions - Arguments; and Arguments into Hypothesis (or Abduction) - Deduction - Induction.
I won't belabor these categories, as they are the core of Peirce's philosophy and logic, and will undoubtedly be treated at length by others. But there was a stumbling block implicit in the concept of "one, two, three": if the 3rd participates in both the 1st and the 2nd, it necessarily contains both within itself already. Or perhaps the 2nd and the 3rd are already contained within the 1st, and so forth. This is an inherent problem in any system which has to proceed from the ideal to the actual, via a hierarchical system. Both the Neoplatonists and the Gnostics struggled with this problem and evolved hierarchies which approximated the process. As Peirce expressed the issue in logic: " . . . were a proposition to be true up to a certain instant and thereafter to be false, at that instant it would be both true and false. (Eisele, 1976, vol. 3, p. xvii).
As Murphey expresses Peirce's problem, and its resolution through his late views on the continuum:
[Peirce] required a property characterizing unactualized possibilities which would be itself actual so that it could be observed. Yet incredibly enough Peirce found such a property in 1896 in continuity. For by his definition of the continuum--and it must be borne in mind that he regarded his definition as the only one which avoided the paradoxes of set theory--any true continuum must contain potentialities which are not only not now actualized but which are greater in multitude than any set of events which can ever be actualized (Murphey, 1993, p. 395).
Since, for Peirce, the continuum contained all numbers that could ever be, in potentia, not in actuality, it was the perfect model for his triadic principle throughout all reality. Peirce contrasted materialism ("the doctrine that matter is everything"), idealism ("the doctrine that ideas are everything"), with synechism ("the tendency to regard everything as continuous") (17) (CP7.565). He made this quite explicit in this description of how true continuity is possible only within a triadic relationship:
A potential collection, more multitudinous than any collection of distinct individuals can be, cannot be entirely vague. For the potentiality supposes that the individuals are determinable in every multitude. That is, they are determinable as distinct. But there cannot be a distinctive quality for each individual; for these qualities would form a collection too multitudinous for them to remain distinct. It must therefore be by means of relations that the individuals are distinguishable from one another. . . . No perfect continuum can be defined by a [asymmetrical] dyadic relation [since the origin and terminus would be points of discontinuity]. But if we take instead a triadic relation, and say A is r to B for C, say, to fix our ideas, that proceeding from A in a particular way, say to the right, you reach B before C, it is quite evident that a continuum will result like a self-returning line with no discontinuity whatever. . . . (CP6.188).
With synechism, everything flowed into everything else, through an ineluctable process of 1 - 2- 3. With the concept of the continuum underlying synechism, he could confidently regard space and time as seamless. "It would be in the general spirit of synechism to hold that time ought to be supposed truly continuous" (CP6.170). With synechism, Peirce felt that he had answered Parmenides: "There is a famous saying of Parmenides, . . . 'being is, and not-being is nothing.' This sounds plausible, yet synechism flatly denies it, declaring that being is a matter of more or less, so as to merge insensibily into nothing" (CP7.569).
Unfortunately Peirce died before he could see the emergence of quantum mechanics and chaos theory. In the Copenhagen model of quantum mechanics, reality is neither a wave nor a particle, until actualized through observation. In Schrödinger's wave model of quantum mechanics, reality exists in potentia until the quantum wave collapses into physical reality. In David Bohm's model, there is an underlying wholeness, much like Peirce's continuum, in which all that can ever be exists infolded in potentia, until it is unfolded into what we know as the world (Bohm, 1980).
More recently, chaos theory has shown how throughout nature, order can give way to deterministic chaos, from which new order emerges. Illya Prigogine received the Nobel prize for his discovery that, as long as complex systems (dissipative structures) can bring in matter and energy from outside themselves, they can go through a period of instability, then emerge with a new self-organization (Prigogine, 1984).
Perhaps we can give Peirce the last word on the importance of his interpretation of the nature of the continuum, and the applicability of continuity throughout nature:
I was led at the very outset to think that one great desideratum in all theorizing was to make fuller use of the principle of continuity. My attention was from the beginning drawn to the need of looking at matters in the light of that conception, but I did not, at first suppose, that it was, as I gradually came to find it, the master-key of philosophy (Parker, 1998, xiv).
Acknowledgments: I would like to express my grateful appreciation to Robert J. Porter, Ph.D. for his careful reading and editing suggestions to successive stages of this paper. Bob is the immediate past president of the Society for Chaos Theory in Psychology and the Life Sciences.
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1. General Editor, Psychological Perspectives; P.O. Box 7226, Alhambra, CA 91802-7226; E-mail: rrobertson@pacbell.net.
2. CP Collected Papers of Charles Sanders Peirce, Volumes 1-6 edited by Charles Hartshorne, Paul Weiss. Volumes 7-8 edited by Arthur Burks. Cambridge: Harvard University Press. References in text indicate volume and paragraph number (CP1.11)
3. Throughout the rest of this paper, I'll {}'s to indicate sets.
4. The attempt to find a way to equate number and geometry is as old as mathematics, and has led to both outstanding advances in mathematics, and to deep philosophical problems. For example, in the 17th century, Rene Descartes developed analytic geometry, which translated geometric positions and shapes into numeric coordinates and algebraic equations. Early in the 19th century, the greatest mathematician of all time, Karl Friedrich Gauss, went the other direction and translated imaginary numbers (which for lack of space we won't address in this paper) into geometric positions.
5. Interestingly, Dedekind had been anticipated in this insight by the Greek thinker Eudoxus (c. 390 B.C.), as incorporated into Book V of Euclid's Elements (Kramer, 1982, p. 35).
6. A phrase Peirce used with respect to permutations, but which fits perfectly here.
7. now in the Treasure Room at the Widener Library, Cambridge, Mass.
8. Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, p. 1.
9. Actually even taken zero at a time. A special set called the null set was needed to complete set theory, just as zero is needed to complete the number line. In all of our discussions of the continuum and real numbers, we have implicitly included zero.
10. (I've used bold brackets {} to indicate that the power set is a set, and normal brackets {} to indicate that its members are sets.)
11. The power set has 2n members, where n is the size of the original set. Hence the notation 2n is used to stand for the power set of any set of cardinality n (remember that cardinality is the same as size within Cantor's theory).
12. There is also a generalized Continuum Hypothesis, but we won't address that here.
13. From roughly 1880 through the late 1890's.
14. When I studied mathematics in college over thirty years ago, we took for granted that this filled all the holes. When I first encountered Cantor's Continuum Hypothesis, at first I couldn't even understand what it might mean. How could there possibly be any other numbers?
15. most famed to the general public for creating the game of Life.
17. Peirce was always one to coin a new term, which often makes reading him a process of translation.
