1) Potential versus actual infinity.

2) Are there mathematical sets?

3) Does mathematics need a foundation in set theory?

4) Is there irreducibly plural reference, predication, and quantification? If yes, does plural quantification allow us to avoid ontological commitment to sets?

5) Discreteness, density, and continuity at the level of number theory, geometry, and nature (physical space and physical time)?

6) Phenomenal versus physical space and their relation. Homogeneity and continuity in relation to both. Lycan's puzzle about the location of the homogeneously green after-image. Wilfrid Sellars and the Grain Argument. It was (6) that got us going on the current jag.

The above topics which we have recently discussed naturally lead to others which I would be interested in discussing:

7) How is it possible for mathematics to apply to the physical world? Does such application require a realist interpretation of mathematics? (See Hilary Putnam, Philosophical Papers, vol. I, 74.)

8) Zeno's Paradoxes. Does the 'calculus solution' dispose of them once and for all?

9) The 'At-At' theory of motion and related topics such as instantaneous velocity.

10) Mathematical existence.

11) The Zermelo-Fraenkel axioms, their epistemic status, and the puzzles to which they give rise.

12) How the actual versus potential infinity debate connects with the eternalism versus presentism debate in the philosophy of time.

Is physical space, the space of the natural world, continuous or discrete? If composed of space atoms, then discrete. The Weyl Tile argument (WTA), however, seems to show that physical space cannot be discrete or 'quantized' and therefore must be continuous. This is relevant to our ongoing debate about potential versus actual infinities. For if physical space cannot be discrete, then it must be at least *compact* (the lowest grade of continuity), where "A series is called *compact* when no two terms are consecutive, but between any two there are others." (Bertrand Russell, *Our Knowledge of the External World*, Norton, 1929, p. 144.) But if between any two points in space there are others, then there are infinitely many others, so that any line segment will be composed of an actual infinity of points.

But before we return to the question of actual infinities we need to get clear about the WTA itself. The *nervus probandi* lies in the following quotation from Hermann Weyl, *Philosophy of Mathematics and Natural Science*, Princeton UP, 1949, p. 43:

If a square is built up of miniature tiles, there are as many tiles along the diagonal as along the side; thus the diagonal should be equal in length to the side.

Take a gander at the chess board below. Consider the right triangle the sides of which are a1-a8 and a1-h1, and the hypotenuse of which is the diagonal a8-h1. The sides and the diagonal are each eight squares long. Count 'em and see. But this flies in the face of the theorem of Pythagoras. If the sides are each eight units in length, then the hypotenuse is equal to the square root of (8^{2} + 8^{2} =128), which is not 8, but the irrational 11.313 . . .

The question this curious fact raises is whether physical space can be quantized, i.e., whether there are space atoms. If so, space is discrete as opposed to continuous. It may help to bear in mind that the above array is a mere model in continuous space of discrete space. So it will do no good to object that if space atoms are squares, then the theorem of Pythagoras hold for them. Space atoms are not squares: they have no shape at all. But I am getting ahead of my story.

We need to define our terms. Space is *discrete* just in case every finite extended spatial region is composed of finitely many *atomic* spatial regions. That amounts to saying that every finite extended region of space is composed of finitely many space atoms, where 'atom,' as its etymology suggests, implies indivisibility. You cannot 'split' a space atom because such atoms are inherently 'unsplittable.' A space atom is thus an individual that has no proper parts: it is a mereological atom. A non-atomic region of space is then a mereological sum of space atoms. Note that every space atom, precisely because it is an atom, is an unextended region of space. It's an itty-bitty unextended bit of space itself, not of something in space. Space atoms are not in space; they compose space.

Now for the argument:

1) The theorem of Pythagoras is not true (or even approximately true) of discrete space.

2) The theorem of Pythagoras is true (or approximately true) of actual space. Therefore:

3) Actual space is not discrete.

To understand this argument, you have to understand that nothing rides on how small the tiles/squares are. Glance back at the chessboard. Consider the small right triangle in the bottom left corner of the board. Opposing sides and hypotenuse all have a length of two units. So it doesn't matter how small the space atoms are. No matter how small the squares, the hypotenuse remains equal in length to the other two sides.

You will be tempted to think of the array of tiles/squares against the backdrop of continuous Euclidean space for which the Pythagorean theorem holds. Thinking in this way, you will imagine that no matter how small you make the tiles, the diagonal will be longer than the sides. You have to resist this temptation to understand the 'Weyl tile' (vile tile?) argument. For if there are space atoms, then they have no shape and hence no different dimensions in different directions. As Wesley C. Salmon puts, "In discrete space, a space atom constitutes one unit, and that is all there is to it. It cannot be regarded as properly having a shape, for we cannot ascribe sizes to parts of it -- it has no parts." (*Space, Time, and Motion*, U of Minnesota Press, 1980, p. 66)

I have found K. McDaniel, "Distance and Discrete Space," *Synthese* (2007) 155: 157-162, very helpful. He has an argument against the WTA which I may discuss in a subsequent post.

The resident nominalist writes,

Your post generated a lot of interest. What I have to say now is better put as a separate post, rather than a long comment. Feel free to post.

1) Plural reference provides a means of dealing with numbers-of-things without introducing extra unwanted entities such as sets. Even realists agree that we should not have more entities than necessary, the disagreement is about what is ‘necessary’.

BV: We agree that entities should not be multiplied beyond necessity, i.e., beyond what is needed for explanatory purposes. The disagreement, if any, will concern what is needed.

2) Using plural quantification we can postulate the existence of an infinite number-of-things. We simply postulate that for *any* number-of-things, there is at least one other thing. That gives a larger number-of-things, which itself is covered by the quantifier ‘any’, hence there must be a still larger number-of-things, etc.

BV: You give no example, so let me supply one. Consider the series of positive integers: 1, 2, 3 . . . n, n + 1, . . . . Given 1, we can generate the rest using the successor function: S(n) = n + 1. I used the word 'generate' since it comports well with your intuition that there are no actual infinities, and that therefore every infinity is merely potential.

3) In this way we neatly distinguish between actual and potential infinity. Using plural quantification, we can prove that there is no plural reference for ‘all the things’. For that would be a number-of-things, hence there must be an even larger number-of-things, which contradicts the supposition that we had all the things.

BV: Your argument is rather less than pellucid. Here is the best I can do by way of reconstructing your argument:

a) If the plural term, 'all the positive integers,' refers to something, then it refers to a completed totality of generated integers. But

b) There is no completed totality of generated integers.

Therefore, by

modus tollens,c) It is not the case that 'all the positive integers' refers to something.

Therefore

d) There is no actually infinite set of positive integers.

If that is your argument, then it begs the question at line (b). One man's *modus tollens* is another man's *modus ponens*. If the above is not your argument, tell me what your argument is. So far, then, a stand-off.

4) In this way we also avoid the pathological results of Cantorean set theory. If there is a set of natural numbers, then this is also a number, but it cannot itself be a natural number, so it is the first ‘transfinite number’. The nominalist approach avoids such weird numbers.

BV: But surely polemical verbiage is out of place in such serene precincts as we now occupy. You cannot *shame* Cantor's results out of existence by calling them 'pathological' or 'weird.' Most if not all working mathematicians accept them, no?

5) The problem for the nominalist arises when in trying to explain the sum of an infinite series, e.g. 1 + ½ + ¼ + ⅛ … The realist wants to argue that unless this series is ‘completed’, we don’t have all the members, so the sum will amount to less than 2.

BV: Note that the formula for the series is 1/2^{n }where *n* is a natural number with 0 being the first natural number. Recall that any number raised to the zeroth power = 1. (If you need to bone up on this, see here.)

Question for our nominalist: what does '1/2^{n}' refer to? Can't be a set! And it can't be a property! Does it refer to nothing? Then so does '1-1/2n.' How then explain the difference between the two formulae (rules) for generating two different infinite series?

Or more simply, consider *n*. It is a variable. It has values and substituends. The values are the natural numbers. Only the ones we counted up to, or generated thus far? No, all of them. The ones we have actually counted up to in a finite number of countings, and the rest which are the possible objects of counting. The variable is a one-over-the-many of its values, and a one-over-the-many of its substituends, which are numerals, not numbers. Numerals bring in the type-token distinction. And so I will ask the nominalist what linguistic types are. Are they sets? No. Are they properties? No. What then?

6) It’s a difficult question for the nominalist, but here is my attempt to resolve it. Start with the notion of non-overlapping parts. Two non-overlapping parts have no part that is part of the other. Then there can be a number of non-overlapping parts such that there is no other such part, i.e. these are ‘all’ such parts.

BV: OK.

7) Then suppose we have a method of defining the parts. Start with a line of length 2. Note that the nominalist is OK here with the existence of lines, because lines are real things and not artificially constructed entities like ‘sets’. And suppose we can divide the line into two non-overlapping parts of equal length, i.e., a part of length 1, and another part of the same length.

BV: You shouldn't say that sets are artificially constructed. After all, you think numbers are artificially constructed, no? They are artifacts of counting. Your beef is with *abstract* objects, not artificial objects. Sets are abstract particulars. You oppose them for that reason. As a nominalist you hold that everything is a concrete particular. (Or am I putting words in your mouth?)

Second, you are ignoring the difference between a geometrical line and a line drawn with pencil on paper, say. The latter is a physical line, which is actually a 3-D object with length, width and depth. In addition to its pure geometrical properties, it has physical and chemical properties. It is a physical line in physical space. The former is not a physical line, but an ideal line: it has length, but no width or depth. Ideal lines are not in physical space. Suppose physical space, the space of nature, is non-Euclidean. Then Euclidean lines are obviously not in physical space. But even if physical space is Euclidean, Euclidean lines would still not be in physical space.

8. So the proposition “2 = 1+1” says that a line of length two can be divided into two equal non-overlapping parts. Then suppose that we divide the second part into two equal parts. Thus “2 = 1 + ½ + ½” says that the line can be divided into three non-overlapping parts, one of length 1, and the other two equal. Do the same again, thus 2 = 1 + ½ + ¼ + ¼. And again and again!

BV: An obvious point is that the arithmetical proposition '2 = 1 + 1' is not about lines only. It could be about a two-degree linear cool-down of a poker. (I am thinking about Wittgenstein's famous poker-brandishing incident.) It could be about anything. Two pins. An angel on a pin joined by another.

Besides, "2 = 1 + 1" cannot be about the non-overlapping parts of a particular line, the one you drew in the sand. It is about a geometrical line, which is an ideal or abstract object. The theorem of Pythagoras is not about the right triangle you drew on the blackboard with chalk; it is about the ideal right triangle that the triangle you drew merely approximates to.

9. It is clear that for *every* such division, the parts ‘add up’ to the same number, i.e. 2.

10. Then consider what the proposition “2 = 1 + ½ + ¼ + ⅛ … “ expresses. Surely that *every* such series, however extended, has a sum of 2. Do we need the notion of a ‘set’? No.

BV: I don't see how this answers the question that you yourself raised in #5 above. What makes it the case that the series you mention *actually* has a sum of 2? The most you can say is that series potentially has a sum of 2. The Cantorean does not face this problem because he can say that there is an actual infinity of compact fractions that sums to 2. No endless task needs to be performed to get to the sum.

Over the last 24 hours I have been obsessing over Kant's spherical triangles. He claims that they are incongruent counterparts. Now I understand how a hand and its mirror image are incongruent counterparts. (A right hand's mirror image is a left hand.) But it is not clear to me how Kant's spherical triangles are incongruent counterparts. Supplement the above diagram with a second lower triangle that shares its base (an arc of the equator) with that of the upper triangle and whose sides are two arcs whose vertex is the south pole.

David Brightly's comment is the best I received in the earlier thread. (He works in Info Tech and I believe he has an advanced degree in mathematics.) He writes,

Not clear to me either, Bill. Why does Kant resort to spherical triangles? [To show the existence of incongruent counterparts.] Consider first two right triangles in the plane with vertices (0,0), (3,0), (0,4) in triangle A and (0,0), (3,0), (0,-4) in B. In plane geometry A and B are considered congruent, not by translation or rotation in the plane but rotation out of the plane ('flipping') with their shared edge as axis. Now think of these triangles on the sphere with edges of length 3 along the equator and those of length 4 on a meridian. The lower triangle cannot be flipped into congruence with the upper---it curves 'the wrong way'. Congruence on the sphere is more restrictive than congruence in the plane. But they are mirror images of one another in the equatorial plane. Likewise, Kant's isosceles triangles cannot be flipped into registration. Has he just overlooked that they can be slid on the sphere into alignment?

As Brightly quite rightly points out, "The lower triangle cannot be flipped into congruence with the upper --- it curves 'the wrong way'." That was clear to me all along. My thought was that if you rotate the lower triangle through 180 degrees so that its southern vertex points north, it would fit right over the upper triangle. I think that is what David means when he writes, "they can be slid on the sphere into alignment."

In other words, the lower triangle needn't be rotated off the surface of the sphere with the axis of rotation being the common base, it suffices to slide the triangles into alignment and thus into congruence along the surface of the sphere.

Therefore: Kant's spherical triangles are not incongruent counterparts or enantiomorphs.

Now David, have I understood you? I am not a mathematician and I might be making a mistake.

Buckner demands an argument from incongruent counterparts to the ideality of space. But before we get to that, I am having trouble understanding how the 'spherical triangles' Kant mentions in the *Prolegomena to Any Future Metaphysics*, sec. 13, are incongruent counterparts. Perhaps my powers of visualization are weak. Maybe someone can help me.

I understand how a hand and its mirror image are incongruent counterparts. If I hold up my right hand before a mirror what I see is a left hand. As Kant says, "I cannot put such a hand as is seen in the glass in the place of its original; for if this is a right hand, that in the glass is a left one . . . ." (p. 13) That is clear to me.

Now visualize a sphere and two non-plane 'spherical triangles' the common base of which is an arc of the sphere's equator. The remaining two sides of the one triangle meet at the north pole; the remaining two sides of the other at the south pole. The two triangles are exact counterparts, equal in all such internal respects as lengths of sides, angles, etc. They are supposed to be incongruent in that "the one cannot be put in place of the other (that is, upon the opposite hemisphere)." (ibid.) That is not clear to me.

Imagine the southern triangle detached from the sphere and rotated through 180 degrees so that the south vertex is pointing north and the base is directly south. Now imagine the southern triangle place on top of the northern triangle. To my geometrical intuition they are congruent!

So, as I see it, hands and gloves are *chiral* but Kant's spherical triangles are not.

In geometry, a figure is

chiral(and said to havechirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to beachiral.A chiral object and its mirror image are said to be

enantiomorphs. The wordchiralityis derived from the Greek χείρ (cheir), the hand, the most familiar chiral object; the wordenantiomorphstems from the Greek ἐναντίος (enantios) 'opposite' + μορφή (morphe) 'form'.

Top of the *Substack* stack.

...................................

Tony Flood comments (12/23):

This was enjoyable on so many levels. There's irony in labeling these gents "idealists" (I know the sense in which you meant it) since Marxists considered theists like Merton metaphysical "idealists," but and how could any mathematician, even a Marxist one, be anything but an idealist when it comes to the reality of numbers? Your historical vignette is rich and your comparison and contrasts apt.

I know that Karl Marx occupied himself with the foundations of analysis (calculus), but I don't know whether or not he wrote anything about the philosophy of mathematics. To answer Tony's question with a question: Why couldn't a Marxist take a nominalist tack and simply deny the existence of numbers and other mathematical items?

Tony replies (12/24):

"Why couldn't a Marxist take a nominalist tack and simply deny the existence of numbers and other mathematical items?"

Abstractly, Bill, I have no idea what tack Marxist materialists might take if pressed about the reality of numbers, e.g., what (and "where") they are (Plato's problem); how they're "unreasonably effective" in the natural sciences, which Marxists revere, i.e., how numbers can cause mathematical belief (Benacerraf's problem); and how numbers are knowable on the materialist/naturalist terms to which Marxists subscribe, i.e., what neural process could possibly answer to the perception of a mathematical object (Goedel's problem). I wish I could have asked Stalinist mathematician Dirk Struik (1896-2000) these questions when he and I were comrades, but I wasn't asking them then. (I'm not asking them these days, but your question stimulated memories of when I did.) Nominalism is not an integral way out for Marxists, but what grounds Marxists have for valuing integral solutions, I have no idea.

Thanks for the Wigner pdf. It gets at a question that fascinated me when I was a student of electrical engineering at the end of the 'sixties. How is it that the theory of complex numbers -- developed *a priori* in response to a purely theoretical question about the roots of negative integers -- finds application in alternating current theory?

I say 'developed,' Wigner says 'invented.' "The principal emphasis [in mathematics] is on the invention of concepts. Mathematics would soon run out of interesting theorems if these had to be formulated in terms of the concepts which already appear in the axioms." I wrote 'developed' because of my platonizing tendency to view mathematical entities -- 'entities' betrays me too inasmuch as it begs the question I am about to pose -- as *discovered* rather than *invented*. The question that my use of 'entities' begs is precisely the question whether mathematical 'items' -- a colorless, non-question-begging bit of terminology -- are made up by us (in which case they cannot be called entities or beings) or are really but non-spatially 'out there' in Plato's *topos ouranios*. My platonic drift links up with my classical theism and issues in the view that the unspeakably vast *actual* infinity of mathematical items are accusatives of divine awareness: their Being is their being-known/created by the archetypal intellect. This sort of view allows for the mediation of two extremes, a **synthesis** if you will.

**Thesis**: math items exist in themselves in splendid independence of ectypal intellects (whether human, Martian, angelic, whatever). **Antithesis**: math items do no such thing; they are the conceptual/linguistic fabrications of ectypal intellects such as ours. And now my mind drifts back to Hartry Field's nominalistic *Science without Numbers*, circa 1980, the gist of which is that science can be done without ontological commitment to any so-called abstract entities. There are some very smart nominalists and they are hard to beat. Shooting from the hip, I say Field 'out-quines' Quine.

But here's a thought. Suppose Wigner is right and *mathematica* are inventions by us, which is to say that they are conceptual/linguistic fabrications that do not refer to anything real anywhere, whether in Plato's heaven or on Aristotle's earth. Would that not make the problem of the applicability of mathematics to the physical world utterly insoluble?

There is a Kantian-type solution, but then you have to take on board the Kantian baggage.

It looks like I have, willy-nilly this Christmas eve, added a log to my aporetic fire in support of my metaphilosophical thesis that the central problems of philosophy, though obviously meaningful, *pace* the later Ludwig, are all of them *absolutely insoluble* by intellects of our constitution. Insofar forth, I am mightily impressed by the thesis of the infirmity of reason. The Fall had noetic consequences.

Below: Raphael, *The School of Athens* depicting Plato gesturing upwards, as if to the *mundus intelligibilis* and Aristotle downwards as if to the *mundus sensibilis*.

It has often been noted that for the Left, the issue is not the issue. David Horowitz:

As President Obama’s political mentor, Saul Alinsky, put it in

Rules for Radicals: “One acts decisively only in the conviction that all of the angels are on one side and the devils are on the other.” Here is another statement fromRules for Radicals: “We are always moral and our enemies always immoral.”The issue is never the issue.The issue is always the immorality of the opposition, of conservatives and Republicans. If they are perceived as immoral and indecent, their policies and arguments can be dismissed, and even those constituencies that are non-political or “low-information” can be mobilized to do battle against an evil party. (emphasis added)

"The issue is never the issue." The issue is the gaining and maintaining of power so as to "fundamentally transform America." For example, if leftists (Democrats in U. S. politics) were really concerned about the spread of COVID-19, they would not open the borders to illegal aliens as the Biden administration has now done. Whatever concern they have about the spread of disease is trumped by considerations of how the problem can be exploited to enhance their power. Power first, public health second, if that. Never let a crisis go to waste; that is, never let it go unexploited for ideological leverage. And now a further step left: *never let a crisis end*.

It occurred to me the other day that something structurally similar explains the absurd claim that mathematics is racist. No one believes this, not even the most febrile of leftists, just as no one believes that a serious health crisis will be unaffected by allowing disease-carrying illegal aliens to flow into the country in great numbers unchecked and unvetted.

So why do so many on the Left say that math is racist? Because the subject is not the subject. The subject is not mathematics, a discipline about as far removed from ideological taint as can be imagined, but the supposed 'systemic racism' of American society. There is no such thing, of course, but no matter: invocation of this nonexistent state of affairs is useful for the promotion of the leftist agenda just as he inefficacy of masks and the uselessness and outright deleteriousness of lock-downs is no reason not to make use of masks and lock-downs and draconian rules to further the destruction of the American republic as she was founded to be.

Peggy Noonan advertises her ignorance in this opening sentence:

This coronavirus is new to our species—it is “novel.” It spreads more easily than the flu—“exponentially,” as we now say—and is estimated to be at least 10 times as lethal.

Why is it "novel"? It *is* a form of flu, and it is *not* unique in spreading exponentially.

Noonan seems to think that 'exponentially' is some newfangled buzzword. Not so. It has a precise mathematical meaning, and the Wuhan Flu -- to use my preferred politically incorrect moniker -- is not unique in spreading geometrically (exponentially) as opposed to arithmetically.

If you have forgotten, or have never learned, the difference between arithmetic and geometric progressions, Dr. Math has a simple and clear explanation for you.

It starts like this:

The four impossible “problems of antiquity”—trisecting an angle, doubling the cube, constructing every regular polygon, and squaring the circle—are catnip for mathematical cranks. Every mathematician who has email has received letters from crackpots claiming to have solved these problems. They are so elementary to state that nonmathematicians are unable to resist. Unfortunately, some think they have succeeded—and refuse to listen to arguments that they are wrong.

Mathematics is not unique in drawing out charlatans and kooks, of course. Physicists have their perpetual-motion inventors, historians their Holocaust deniers, physicians their homeopathic medicine proponents, public health officials their anti-vaccinators, and so on. We have had hundreds of years of alchemists, flat earthers, seekers of the elixir of life, proponents of ESP, and conspiracy theorists who have doubted the moon landing and questioned the assassination of John F. Kennedy.

Thanks to 'progressives,' our 'progress' toward social and cultural collapse seems not be proceeding at a constant speed, but to be *accelerating*. But perhaps a better metaphor from the lexicon of physics is *jerking*. After all, our 'progress' is jerkwad-driven. No need to name names. You know who they are.

From your college physics you may recall that the first derivative of position with respect to time is velocity, while the second derivative is acceleration. Lesser known is the third derivative: jerk. (I am not joking; look it up.) If acceleration is the rate of change of velocity, jerk, also known as jolt, is the rate of change of acceleration.

If you were studying something in college, and not majoring in, say, Grievance Studies, then you probably know that all three, velocity, acceleration, and jerk are vectors, not scalars. Each has a magnitude and a direction. This is why a satellite orbiting the earth is constantly changing its velocity despite its constant speed.

The 'progressive' jerk too has his direction: the end of civilization as we know it.

Time for a re-post. This first appeared in these pages on 18 August 2010.

.........................

A reader writes,

Regarding your post about Cantor, Morris Kline, and potentially vs. actually infinite sets: I was a math major in college, so I do know a little about math (unlike philosophy where I'm a rank newbie); on the other hand, I didn't pursue math beyond my bachelor's degree so I don't claim to be an expert. However, I do know that we never used the terms "potentially infinite" vs. "actually infinite".

I am not surprised, but this indicates a problem with the way mathematics is taught: it is often taught in a manner that is both ahistorical and unphilosophical. If one does not have at least a rough idea of the development of thought about infinity from Aristotle on, one cannot properly appreciate the seminal contribution of Georg Cantor (1845-1918), the creator of transfinite set theory. Cantor sought to achieve an exact mathematics of the actually infinite. But one cannot possibly understand the import of this project if one is unfamiliar with the distinction between potential and actual infinity and the controversies surrounding it. As it seems to me, a proper mathematical education at the college level must include:

1. Some serious attention to the history of the subject.

2. Some study of primary texts such as Euclid's *Elements*, David Hilbert's *Foundations of Geometry*, Richard Dedekind's *Continuity and Irrational Numbers*, Cantor's *Contributions to the Founding of the Theory of Transfinite Numbers,* etc. Ideally, these would be studied in their original languages!

3. Some serious attention to the philosophical issues and controversies swirling around fundamental concepts such as set, limit, function, continuity, mathematical induction, etc. Textbooks give the wrong impression: that there is more agreement than there is; that mathematical ideas spring forth ahistorically; that there is only one way of doing things (e.g., only one way of constructing the naturals from sets); that all mathematicians agree.

Not that the foregoing ought to *supplant* a textbook-driven approach, but that the latter ought to be *supplemented* by the foregoing. I am not advocating a 'Great Books' approach to mathematical study.

Given what I know of Cantor's work, is it possible that by "potentially infinite" Kline means "countably infinite", i.e., 1 to 1 with the natural numbers?

No!

Such sets include the whole numbers and the rational numbers, all of which are "extensible" in the sense that you can put them into a 1 to 1 correspondence with the natural numbers; and given the Nth member, you can generate the N+1st member. The size of all such sets is the transfinite number "aleph null". The set of all real numbers, which includes the rationals and the irrationals, constitute a larger infinity denoted by the transfinite number C; it cannot be put into a 1 to 1 correspondence with the natural numbers, and hence is not generable in the same way as the rational numbers. This would seem to correspond to what Kline calls "actually infinite".

It is clear that you understand some of the basic ideas of transfinite set theory, but what you don't understand is that the distinction between the countably (denumerably) infinite and the uncountably (nondenumerably) infinite falls on the side of the actual infinite. The countably infinite has nothing to do with the potentially infinite. I suspect that you don't know this because your teachers taught you math in an ahistorical manner out of boring textbooks with no presentation of the philosophical issues surrounding the concept of infinity. In so doing they took a lot of the excitement and wonder out of it.

So what did you learn? You learned how to solve problems and pass tests. But how much actual understanding did you come away with?

There is nothing so stupid, destructive, and inane that a 'liberal' won't embrace it.

π day is 3/14. March 14th last year was called *super* π day: 3/14/15. Years ago, as a student of electrical engineering, I memorized π this far out: 3.14159. So isn't today better called *super* π day? I mean, 3.1416 is closer to the value of π than 3.1415. Am I right? Of course I am.

The decimal expansion is non-terminating. But that is not what makes it an irrational number. What makes it irrational is that it cannot be expressed as a fraction the numerator and denominator of which are integers. Compare 1/3. Its decimal expansion is also non-terminating: .3333333 . . . . But it is a rational number because it can be expressed as a fraction the numerator and denominator of which are integers (whole numbers).

An irrational (rational) number is so-called because it cannot (can) be expressed as a *ratio *of two integers*.* Thus any puzzlement as to how a number, as opposed to a person, could be rational or irrational calls for therapeutic dissolution, not solution (he said with a sidelong glance in the direction of Wittgenstein).

Finally a quick question about infinity. The decimal expansion of π is non-terminating. It thus continues infinitely. The number of digits is infinite. Potentially or actually? (See Infinity category for some discussion of the difference.) I wonder: can the definiteness of π -- its being the ratio of diameter to circumference in a circle -- be taken to show that the number of digits in the decimal expansion is actually infinite?

I'm just asking.

Many people don't understand that certain words and phrases are terms of art, technical terms, whose meanings are, or are determined by, their uses in specialized contexts. I once foolishly allowed myself to be suckered into a conversation with an old man. I had occasion to bring up imaginary (complex) numbers in support of some point I was making. He snorted derisively, "How can a number be imaginary?!" The same old fool -- and I was a fool too for talking to him twice -- once balked incredulously at the *imago dei*. "You mean to tell me that God has an intestinal tract!"

Now go ye forth and celebrate π day in some appropriate and inoffensive way. Eat some pie. Calculate the area of some circle. A = πr^{2. }

Dream about π in the sky. Mock a leftist for wanting π in the future. 'The philosophers have variously interpreted π; the point is to change it!'

But don't shout down any speaker or throw π in his face. That's what 'liberals' and leftists do, and you are a morally decent person who believes in free speech and open debate.

As a sort of 'make-up' for missing Saturday night's oldies show, here is Queen Jane Approximately.

Which Dylan song features the line "infinity goes up on trial"?

Much as I disagree with Daniel Dennett on most matters, I agree entirely with what he says in the following passage:

I deplore the narrow pragmatism that demands immediate social utility for any intellectual exercise. Theoretical physicists and cosmologists, for instance, may have more prestige than ontologists, but not because there is any more social utility in the satisfaction of

theirpure curiosity. Anyone who thinks it is ludicrous to pay someone good money to work out the ontology of dances (or numbers or opportunities) probably thinks the same thing about working out the identity of Homer or what happened in the first millionth of a second after the Big Bang. (Dennett and His Critics, ed. Dahlbom, Basil Blackwell 1993, p. 213. Emphasis in original.)

I would put the point in stronger terms and go Dennett one better. Anyone who thinks that intellectual inquiry has value only if it has immediate or even *long-term* social utility is not only benighted, but is also a potential danger to free inquiry.

One of my favorite examples is complex numbers. A complex number involves a real factor and an imaginary factor *i*, where *i*= the square root of -1. Thus a complex number has the form, *a + bi* where *a* is the real part and *bi* is the imaginary part.

One can see why the term 'imaginary' is used. The number 1 has two square roots, 1 and -1 since if you square either you get 1. But what is the square root of -1? It can't be 1 and it can't be -1, since either squared gives a positive number. So the imaginary *i* is introduced as the square root of -1. Rather than say that negative numbers do not have square roots, mathematicians say that they have complex roots. Thus the square root of -9 = 3i.

Now to the practical sort of fellow who won't believe in anything that he can't hold in his hands and stick in his mouth, this all seems like idle speculation. He demands to know what good it is, what it can *used for*. Well, the surprising thing is is that the theory of complex numbers which originated in the work of such 16th century Italian mathematicians as Cardano (1501 - 1576) and Bombelli (1526-1572) turned out to find application to the physical world in electrical engineering. The electrical engineers use *j* instead of *i* because *i* is already in use for current.

Just one example of the application of complex numbers is in the concept of impedance. Impedance is a measure of opposition to a sinusoidal electric current. Impedance is a generalization of the concept of resistance which applies to direct current circuits. Consider a simple direct current circuit consisting of a battery, a light bulb, and a rheostat (variable resistor). Ohm's Law governs such circuits: I = E/R. If the voltage E ('E' for electromotive force) is constant, and the resistance R is increased, then the current I decreases causing the light to become dimmer. The resistance R is given as a real number. But the impedance of an alternating current circuit is given as a complex number.

Now what I find fascinating here is that the theory of complex numbers, which began life as something merely theoretical, turned out to have application to the physical world. One question in the philosophy of mathematics is: How is this possible? How is it possible that a discipline developed purely* a priori* can turn out to 'govern' nature? It is a classical Kantian question, but let's not pursue it.

My point is that the theory of complex numbers, which for a long time had no practical (e.g., engineering) use whatsoever, and was something of a mere mathematical curiosity, turned out to have such a use. Therefore, to demand that theoretical inquiry have immediate social utility is shortsighted and quite stupid. For such inquiry might turn how to be useful in the future.

But even if a branch of inquiry could not possibly have any application to the prediction and control of nature for human purposes, it would still have value as a form of the pursuit of truth. Truth is a value regardless of any use it may or may not have.

Social utility is a value. But truth is a value that trumps it. The pursuit of truth is an end in itself. Paradoxically, the pursuit of truth as an end in itself may be the best way to attain truth that is useful to us.

The Big Henry offers the following comment on my post, World + God = God?

"World + God = God" is (mathematically) analogous to "number + infinity = infinity", where "number" is finite. If God embodies all existence, then God is "existential infinity", and, therefore, no amount of existence can be added to or subtracted from God's totality.The numerical concept of infinity does not comply with the rules of arithmetic addition or subtraction. Similarly, if God is presumed to be the embodiment of all existence, He does not comply with the rules of arithmetic addition or subtraction.

To supply an example that supports Big Henry's point, 8 + = . (aleph-nought, aleph-zero, aleph-null) is the first transfinite cardinal. A cardinal number answers the How many? question. Thus the cardinal number of the set {Manny, Moe, Jack} is 3, and the cardinal number of {1, 3, 5, 7} is 4. Cardinality is a measure of a set's size. What about the infinite set of natural numbers {0, 1, 2, 3, 4 . . . n, n + 1, . . .}? How many? . And as was known long before Georg Cantor, it is possible to have two infinite sets, call them E and N such that E is a *proper* subset of N, but both E and N have the same size or cardinality. Thus the evens are a proper subset of the naturals, but there are just as many of the former as there are of the latter, namely, . How can this be? Well, EACH element of the evens can be put into 1-1 correspondence with an element of the naturals.

So far the analogy holds. But I think Big Henry has overlooked the transfinite *ordinals*. The first transfinite ordinal, denoted , is the order type of the set of nonnegative integers. (See here.) You could think of as the successor of the natural numbers. It is the first number following the entire infinite sequence of natural numbers. (Dauben, 97) The successor of is + 1. These two numbers are therefore different. Here the analogy breaks down. God + Socrates = God. + 1 is not equal to .

Moreover, it is not true to say that "The numerical concept of infinity does not comply with the rules of arithmetic addition or subtraction." This ignores the rules of transfinite cardinal arithmetic and those of transfinite ordinal arithmetic. Big Henry seems to be operating with a pre-Cantorian notion of infinity. Since Cantor we have an exact mathematics of infinity.

In any case, I rather doubt that mathematical infinity provides a good analogy for the divine infinity. God is not a set!

π day is 3/14. But today is *super* π day: 3/14/15. To celebrate it properly you must do so at 9:26 A.M. or P. M. Years ago, as a student of electrical engineering, I memorized π this far out: 3.14159.

The decimal expansion is non-terminating. But that is not what makes it an irrational number. What makes it irrational is that it cannot be expressed as a fraction the numerator and denominator of which are integers. Compare 1/3. Its decimal expansion is also non-terminating: .3333333 . . . . But it is a rational number because it can be expressed as a fraction the numerator and denominator of which are integers (whole numbers).

An irrational (rational) number is so-called because it cannot (can) be expressed as a *ratio *of two integers*.* Thus any puzzlement as to how a number, as opposed to a person, could be rational or irrational calls for therapeutic dissolution, not solution (he said with a sidelong glance in the direction of Wittgenstein).

Yes, there are pseudo-questions. Sometimes we succumb to the bewitchment of our understanding by language. But, *pace* Wittgestein, it is not the case that all the questions of philosophy are pseudo-questions sired by linguistic bewitchment. I say almost none of them are. So it cannot be the case that philosophy just is the struggle against such bewitchment. (PU #109: *Die Philosophie ist ein Kampf gegen die Verhexung unsres Verstandes durch die Mittel unserer Sprache*.) What a miserable conception of philosophy! As bad as that of a benighted logical positivist.

Many people don't understand that certain words and phrases are terms of art, technical terms, whose meanings are, or are determined by, their uses in specialized contexts. I once foolishly allowed myself to be suckered into a conversation with an old man. I had occasion to bring up imaginary (complex) numbers in support of some point I was making. He snorted derisively, "How can a number be imaginary?!" The same old fool -- and I was a fool too for talking to him twice -- once balked incredulously at the *imago dei*. "You mean to tell me that God has an intestinal tract!"

Finally a quick question about infinity. The decimal expansion of π is non-terminating. It thus continues infinitely. The number of digits is infinite. Potentially or actually? I wonder: can the definiteness of π -- its being the ratio of diameter to circumference in a circle -- be taken to show that the number of digits in the decimal expansion is actually infinite?

I'm just asking.

Now go ye forth and celebrate π day in some appropriate and inoffensive way. Eat some pie. Calculate the area of some circle. A = πr^{2. }

Dream about π in the sky. Mock a leftist for wanting π in the future. 'The philosophers have variously interpreted π; the point is to change it!'

**UPDATE**: Ingvar writes,

`Of course the `*ne plus ultra* pi day was 3-14-1592 and whatever happened that day
at 6:53 in the morning.
So we have one yearly, one every millennium, and one
once.

Related articles

A reader asks, "What is meant by 'closure' or 'closed under'? I've heard the terms used in epistemic contexts, but I've not been able to completely understand them."

Let's start with some mathematical examples. The natural numbers are closed under the operation of addition. This means that the result of adding any two natural numbers is a natural number. What is a natural number? On one understanding of the term, the naturals are the positive integers, the counting numbers, the members of the set {1, 2, 3, 4, 5, . . .}. On a second understanding, the naturals are the positive integers and zero: {0, 1, 2, 3, 4 . . .}. Either way, it is easy to see that adding any two elements of either set yields an element of the same set. It is also easy to see that the naturals are also closed under multiplication. But they are not closed under subtraction. If you subtract 9 from 7, the result (-2) is not an element of the set of natural numbers.

Now consider the squaring operation. The square of any real number is a real number. So the reals are closed under the operation of squaring. But the reals are not closed under the square root operation. The square root of -4 cannot be 2 since 2 squared is 4; it cannot be -2 either since -2 squared is 4. The square root of -2 is the complex number 2i where the imaginary number i is the square root of -1. The square roots of negative numbers are complex; hence, the reals are not closed under the square root operation.

Generalizing, we can say that a set S is closed under a binary operation O just in case, for any elements x and y in S, xOy is an element of S. In Group Theory, a set S together with an operation O constitutes a group only if S is closed under O.

Now for some philosophical examples. Meinongian objects (M-objects) are not closed under entailment. The M-object, the yellow brick road, although yellow is not colored even though in reality nothing can be yellow without being colored. M-objects are incomplete objects. They have all and only the properties specified in their descriptions. So we say that the properties of M-objects are not closed under property-entailment. Property P entails property Q iff necessarily, if anything x has P, then x has Q.

What goes for M-objects goes for intentional objects. (On my reading of Meinong, an M-object is not the same as an intentional object: there are M-objects that are not the accusatives of any actual

intending.) Suppose I am gazing out my window at the purple majesty of Superstition Mountain. The intentional object of my perception has the property of being purple, but not the properties of being colored or being extended even though in reality nothing can be purple without being both colored and extended. Phenomenologically, what is before my mind is an instance of purple, but not an instance of colored item. What I see I see as purple but not as colored.

Now consider:** If S knows that p, and S knows that p entails q, then S also knows that q.** If you acquiesce in the bolded thesis, then you acquiesce in the closure of 'knows' under known entailment. For what you are then committing yourself to is the proposition that a proposition q entailed by a proposition p you know -- assuming you know that p entails q -- is a member of the set of propositions you know.

Here we read:

. . . aren't all numbers inventions? It is not like they grow on

trees! They live in our heads. We made them all up.

The author of the quotation is introducing a discussion of the imaginary number* i *= the square root of -1. His point is that we are free to introduce this number since all numbers are inventions. So we can make up any number we like. The actual argument given is self-contradictory: The point of saying that numbers do not grow on trees is that they do not occur in nature. But if they live in our heads, then they are part of nature, because our heads ae in nature and what is in our heads is part of nature.

But let's be charitable. The argument the author is trying to give is something like this:

1. Numbers are not physical objects

Therefore

2. Numbers are mental constructions.

That this is a non sequitur should be obvious. For there is a third possibility: numbers are abstract or ideal or Platonic objects. This third possibility is of course the actual view of numerous distinguished thinkers and is seen to be plausible once one considers the difficulties with the view that numbers are mental constructions.

Note first that an abstract object is not one produced by a mental act of abstraction. For present purposes we can say that an abstract object is any entity that necessarily exists but is causally inert.

Note second that a number is not the same as a numeral. One and the same number can be represented by different numerals. Thus the same number is denoted by the Arabic '9' and the Roman 'IX.' Numerals are signs of numbers, while numbers are not. So no number is a numeral. Numerals are typically physical (marks on paper, for instance); no number is physical. Ergo, etv.

We also note that 9 in a base-10 or decimal system is equivalent to 1001 in a base-2 or binary system. When I speak of the number 9 I am referring to the denotatum of the numeral '9' as this numeral functions within our ordinary base-10 system. That denotatum is the same as the denotatum of '1001' as the latter functions within a base-2 system.

One and the same proposition can be expressed by different indicative sentences. Thus the binary sentence '1 + 1 = 10' expresses the same true proposition as is expressed by the decimal sentence '1 + 1 = 2.' But if the two sentences are both interpreted relative to the decimal system, then they express different propositions, one true and the other false.

Our question is whether numbers themselves are mental constructions, not whether numerals are mental constructions. This is connected with the question of whether mathematics is in any sense conventional. No doubt notation systems are conventional, i.e. decided upon by human beings (or whatever other intelligent critters there might be elsewhere); but it doesn't follow that numbers or other mathematical objects are.

If numbers themselves are mental constructions, then they depend on our existence for their existence. Their existence is a mental existnce in or before our minds, and thus a dependent mode of existence. (Forget about extraterrestrial intelligences for the nonce.) The same goes for the truths in which they are involved. (Thus 7 and 9 and 16 are involved in the truth expressed by '7 + 9 = 16'.) But we didn't always exist. So if numbers depend ion us, they they didn't always exist. Consider a time before any minds existed, some time after the Big Bang and before the emergence of life on earth, say.

During that interval, the speed of light and the speed of sound were the same as they are now, and during that time the former was greater than the latter, as is the case now. Let 'c' denote the speed of light in a vacuum. C is identical to some number, which number depending on the units of measurement one employs. So c = 186,000 miles/sec (approximately). In the metric system, c = 300,000 km/sec (approximately). The point is that once the system of measurement is fixed -- which of course is conventional -- then some definite number is the SOL. Similarly with the speed of sound, SOS. Now

1. SOL > SOS

is true now and was true at the time when no humans existed. Of course, at that time the concept or notion or idea *greater than* (taken in its mathematical sense) did not exist since concepts (notions, ideas) cannot exist except 'in' a mind. ('In' here not to be taken spatially.) But the mathematical relation picked out by '>' existed.

For if it did not, then (1) could not have been true at the time in question. And the same goes for the relational fact of SOL's being greater than SOS. That fact obtained at the time when no minds existed. So its constituents (the numbers and the greater than relation) had to exist at that time as well.

Therefore, mathematical objects cannot be our mental creations.

Returning to a discussion we were having back in August of 2010, I want to see if I can get Peter Lupu to agree with me on one point: *It is not obvious or compellingly arguable (arguable in a 'knock-down' way) that there are infinite sets.* Given my aporetic concerns, which Peter thoroughly understands, I will be satisfied if I can convince him that the italicized sentence is true, and therefore that the thesis that the infinite in mathematics is potential only is respectable and defensible and has never been shown definitively to be false. Let us start with a datanic claim that no one can reasonably deny:

1. There are infinitely many natural numbers.

If anyone were to deny (1) I would show him the door. For anyone who denied (1) would show by his denial that he did not grasp the sense of 'natural number.' The question, however, is whether from (1) we can validly infer

2. There is a *set* of natural numbers.

If there is such a set, then of course it is an infinite set, an actually infinite set. (Talk of potentially infinite sets is nonsense as I have argued in previous posts.) So, if the inference from (1) to (2) is valid, we have a knock-down proof of actual infinity. For if there are infinite sets then there are actual infinities, completed infinities.

Now I claim that it is obvious that (2) does not follow from (1). For it might be that the naturals do not form a set. A set is a one-over-many, a definite single object distinct from each of its members and from all of them. It should be obvious, then, that from the fact that there ARE many Fs it does not straightaway follow that there IS a single thing comprising these many Fs. This is especially clear in the case of infinitely many Fs.

But from Logic 101 we know that an invalid argument can have a true conclusion. So, despite the fact that (2) does not follow from (1), it might still be the case that (2) is true. I might be challenged to say what (1) could mean if it does not entail (2). Well, I can say that however many numbers we have counted, we can count more. If we have counted up to *n*, we can add 1 and arrive at *n *+ 1. The procedure is obviously indefinitely iterable. That means: there is no definite *n* such one can perform the procedure only *n* times. One can perform it indefinitely many times. Accordingly, 'infinitely many' behaves differently than 'finitely many.' If something can be done only finitely many times, then there is some finite *n* such that *n* is the number of times the thing can be done. But 'infinitely many' does not require us to say that that there is some definite transfinite cardinal which is the number of times a thing that can be done infinitely many times can be done. For 'infinitely many' can be construed to mean: indefinitely many.

On this approach, the naturals do not form a single complete object, the set N, but are such that their infinity is an endless task. The German language allows a cute way of putting this: *Die Zahlen sind nicht gegeben, sondern aufgegeben.* In Aristotelian terms, the infinity of the naturals is potential not actual. But if you find these words confusing, as Peter does, they can be avoided. A wise man never gets hung up on words.

Now if I understood him aright, one of Peter's objections is that the approach I am sketching implies that there is a last number, one than which there is no greater. But it has no such implication. For the very sense of 'natural number' rules out there being a last number, and this sense is understood by all parties to the dispute. There cannot be a last number precisely because of the very meaning of 'number.' *Every* natural number is such that it has an immediate successor. But from this it does not follow that there is a *set* of natural numbers. For 'has an immediate successor' needn't be taken to mean that each number has now a successor; it can be taken to mean that each number at which we have arrived by computation is such that an immediate successor can be computed by adding 1.

But Peter has a stronger objection, one that I admit has force. His objection *in nuce* is that potential infinity presupposes actual infinity. Peter points out that my explanation of what it means to say that the naturals are potentially infinite makes use of words like 'can.' Thus above I said, "however many numbers we have counted, we *can* count more." This 'can' refers either to the abilities of men or machines or else it refers to abstract possibilities of counting not tied to the powers of men or machines.

Consider the second idea, the more challenging of the two. Suppose the universe ceases to exist at a time t right after some huge but finite n has been computed. Now n cannot be the last number for the simple reason that there cannot be a last number. This 'cannot' is grounded in the very sense of 'natural number.' So it must be possible that 1 be added to n to generate its successor. And it must be possible that 1 be added to n + 1 to generate *its *successor, and so on. So Peter could say to me, "Look, you have gotten rid of an actual infinity of numbers but at the expense of introducing an actual infinity of unrealized possibilities of adding 1: the possibility P1 of adding 1 to n; the possibility P2 of adding 1 to n + 1, etc."

The objection is not compelling. For I can maintain that the unrealized possibilities P1, P2, . . . Pn, . . . all 'telescope,' i.e., collapse into one generic possibility of adding 1. P1 is the possibility of adding 1 to n and P2 is the possibility of adding 1 to the last number computed just before the universe ceases to exist.

What I'm proposing is that 'Every natural number has an immediate successor' is true solely in virtue of the sense or meaning of 'natural number.' Its being true does not require that there be, stored up in Plato's Heaven, a completed actual infinity of naturals, a* set *of same. Since I have decidedly Platonic sympathies, I would welcome a refutation of this proposal.

A reader writes,

Regarding your post about Cantor, Morris Kline, and potentially vs. actually infinite sets: I was a math major in college, so I do know a little about math (unlike philosophy where I'm a rank newbie);

on the other hand, I didn't pursue math beyond my bachelor's degree so I don't claim to be an expert. However, I do know that we never used the terms "potentially infinite" vs. "actually infinite".

I am not surprised, but this indicates a problem with the way mathematics is taught: it is often taught in a manner that is both ahistorical and unphilosophical. If one does not have at least a rough idea of the development of thought about infinity from Aristotle on, one cannot properly appreciate the seminal contribution of Georg Cantor (1845-1918), the creator of transfinite set theory. Cantor sought to achieve an exact mathematics of the actually infinite. But one cannot possibly understand the import of this project if one is unfamiliar with the distinction between potential and actual infinity and the controversies surrounding it. As it seems to me, a proper mathematical education at the college level must include:

1. Some serious attention to the history of the subject.

2. Some study of primary texts such as Euclid's *Elements*, David Hilbert's *Foundations of Geometry*, Richard Dedekind's *Continuity and Irrational Numbers*, Cantor's *Contributions to the Founding of the Theory of Transfinite Numbers,* etc. Ideally, these would be studied in their original languages!

3. Some serious attention to the philosophical issues and controversies swirling around fundamental concepts such as set, limit, function, continuity, mathematical induction, etc. Textbooks give the wrong impression: that there is more agreement than there is; that mathematical ideas spring forth ahistorically; that there is only one way of doing things (e.g., only one way of construction the naturals from sets); that all mathematicians agree.

Not that the foregoing ought to *supplant* a textbook-driven approach, but that the latter ought to be *supplemented* by the foregoing. I am not advocating a 'Great Books' approach to mathematical study.

Given what I know of Cantor's work, is it possible that by "potentially infinite" Kline means "countably infinite", i.e., 1 to 1 with the natural numbers?

No!

Such sets include the whole numbers and the rational numbers, all of which are "extensible" in the sense that you can put them into a 1 to 1 correspondence with the natural numbers; and given the Nth member, you can generate the N+1st member. The size of all such sets is the transfinite number "aleph null". The set of all real numbers, which includes the rationals and the irrationals, constitute a larger infinity denoted by the transfinite number C; it cannot be put into a 1 to 1 correspondence with the natural numbers, and hence is not generable in the same way as the rational numbers. This would seem to correspond to what Kline calls "actually infinite".

It is clear that you understand some of the basic ideas of transfinite set theory, but what you don't understand is that the distinction between the countably (denumerably) infinite and the uncountably (nondenumerably) infinite falls on the side of the actual infinite. The countably infinite has nothing to do with the potentially infinite. I suspect that you don't know this because your teachers taught you math in an ahistorical manner out of boring textbooks with no presentation of the philosophical issues surrounding the concept of infinity. In so doing they took a lot of the excitement and wonder out of it. So what did you learn? You learned how to solve problems and pass tests. But how much actual understanding did you come away with?

Morris Kline, *Mathematics: The Loss of Certainty*, Oxford 1980, p. 200:

. . . when Cantor introduced actually infinite sets, he had to advance his creation against conceptions held by the greatest mathematicians of the past. He argued that the potentially infinite in fact depends on a logically prior actually infinite. He also gave the argument that the irrational numbers, such as the square root of 2, when expressed as decimals involved actually infinite sets because any finite decimal could only be an approximation.

Here may be one answer to the question that got me going on this series of posts. The question was whether one could prove the existence of actually infinite sets. Note, however, that Kline's talk of actually infinite sets is pleonastic since an infinite set cannot be anything other than actually infinite as I have already explained more than once. Pleonasm, however, is but a peccadillo. But let me explain it once more. A potentially infinite set would be a set whose membership is finite but subject to increase. But by the Axiom of Extensionality, a set is determined by its membership: two sets are the same iff their members are the same. It follows that a set cannot gain or lose members. Since no set can increase its membership, while a potentially infinite totality can, it follows that that there are no potentially infinite sets. Kline therefore blunders when he writes,

However, most mathematicians -- Galileo, Leibniz, Cauchy, Gauss, and others -- were clear about the the distinction between a potentially infinite set and an actually infinite one and rejected consideration of the latter. (p. 220)

Kline is being sloppy in his use of 'set.' Now to the main point. Suppose you have a right triangle. If two of the sides are one unit in length, then, by the theorem of Pythagoras, the length of the hypotenuse is the sqr rt of 2 = 1.4142136. . . . Despite the nonterminating decimal expansion, the length of the hypotenuse is perfectly definite, perfectly determinate. If the points in the line segment that constitutes the hypotenuse did not form an infinite set, then how could the length of the hypotenuse be perfectly definite? This is not an argument, of course, but a gesture in the direction of a possible argument.

If someone can put the argument rigorously, have at it.

In an earlier thread David Brightly states that "The position on potential infinity that he [BV] is defending is equivalent to the denial of the principle of mathematical induction." Well, let's see.

1. To avoid lupine controversy over 'potential' and 'actual,' let us see if we can avoid these words. And to keep it simple, let's confine ourselves to the natural numbers (0 plus the positive integers). The issue is whether or not the naturals form a set. I hope it is clear that if the naturals form a set, that set will not have a finite cardinality! Were someone to claim that there are 463 natural numbers, he would not be mistaken so much as completely clueless as to the very sense of 'natural number.' But from the fact that there is no finite number which is the number of natural numbers, it does not follow that there is a *set* of natural numbers.

2. So the dispute is between the Platonists -- to give them a name -- who claim that the naturals form a set and the Aristotelians -- to give them a name -- who claim that the naturals do not form a set. Both hold of course that the naturals are in some sense infinite since both deny that the number of naturals is finite. But whereas the Platonists claim that the infinity of naturals is completed, the Aristotelians claim that it is incomplete. To put it another way, the Platonists -- good Cantorians that they are -- claim that the naturals, though infinite, are a definite totality whereas the Aristoteleans claim that the naturals are infinite in the sense of indefinite. The Platonists are claiming that there are definite infinities, finite infinities -- which has an oxymoronic ring to it. The Aristotelians stick closer to ordinary language. To illustrate, consider the odds and evens. For the Platonists, they are infinite disjoint subsets of the naturals. Their being disjoint from each other and non-identical to their superset shows that for the Platonists there are definite infinities.

3. Suppose 0 has a property P. Suppose further that if some arbitrary natural number n has P, then n + 1 has P. From these two premises one concludes by *mathematical induction* that all n have P. For example, we know that 0 has a successor, and we know that if arbitrary n has a successor, then n +1 has a successor. From these premises we conclude by mathematical induction that all n have a successor.

4. Brightly claims in effect that to champion the Aristotelean position is to deny mathematical induction. But I don't see it. Note that 'all' can be taken either distributively or collectively. It is entirely natural to read 'all n have a successor' as 'each n has a successor' or 'any n has a successor.' These distributivist readings do not commit us to the existence of a set of naturals. Thus we needn't take 'all n have a successor' to mean that the set of naturals is such that each member of it has a successor.

5. Brightly writes, "My understanding of 'there is' and 'for all' requires a pre-existing domain of objects, which is why, perhaps, I have to think of the natural numbers as forming a set." Suppose that the human race will never come to an end. Then we can say, truly, 'For every generation, there will be a successor generation.' But it doesn't follow that there is a set of all these generations, most of which have not yet come into existence. Now if, in this example, the universal quantification does not require an actually infinite set as its domain, why is there a need for an actually infinite set as the domain for the universal quantification, 'Each n has a successor'?

6. When we say that each human generation has a successor, we do not mean that each generation *now* has a successor; so why must we mean by 'every n has a successor' that each n *now* has a successor? We could mean that each n is such that a successor for it can be constructed or computed. And wouldn't that be enough to justify mathematical induction?

**Addendum 8/15/2010 11:45 AM.** I see that I forgot to activate Comments before posting last night. They are on now.

It occurred to me this morning that I might be able to turn the tables on Brightly by arguing that actual infinity poses a problem for mathematical induction. If the naturals are actually infinite, then each of them enjoys a splendid Platonic preexistence vis-a-vis our computational activities. They are all 'out there' in Plato's heaven/Cantor's paradise. Now consider some stretch of the natural number series so far out that it will never be reached by any computational process before the Big Crunch or the Gnab Gib, or whatever brings the whole shootin' match crashing down. How do we know that the naturals don't get crazy way out there? How can we be sure that the inductive conclusion *For all n, P(n)* holds? *Ex hypothesi*, no constructive procedure can reach out that far. So if the numbers exist out there, but we cannot reach them by computation, how do we know they behave themselves, i.e. behave as they behave closer to home? This won't be a problem for the constructivist, but it appears to be a problem for the Platonist.

The Sarah Lee frozen pies were on sale, three for $10, at the local supermarket. I bought two, but they rang up as $4.99 each. I pointed out to the check-out girl that this was wrong, and she sent a 'gofer' to confirm my claim. Right I was. But now the lass was perplexed, having to input the correct amount by hand and brain. She had to ask me what 10 divided by 3 is. I was nice, not rude, and just gave her the answer sparing her any commentary.

(It's a crappy job, standing up eight hours per day, in a confined space, an appendage of a machine. I make a point of trying to relate to the attendants, male and female, as persons, at the back of my mind recalling a passage in Martin Buber's *I-Thou* in which he says such a relation is possible even in the heat of a commute between passenger and bus driver.)

But now I can be peevish. They learn how to put on condoms in these liberal-run schools but not how to add, subtract, multiply and divide? And how many times have I encountered pretty young things in bars and restaurants who are clueless when it comes to weights and measures? At a P. F. Chang's the other day I asked whether the beer I wanted to order was 22 oz. The girl said it was a pint, "whatever that is." This was near Arizona State and it is a good bet that she was a student there. How can such people not know that there are two pints in a quart, that a pint is 16 fluid ounces, that four quarts make a gallon , . . . , that a light-year is a measure of distance not of time, . . . .

Can we blame this one on libruls too? You betcha! A librul is one who has never met a standard he didn't want to undermine.

You many enjoy John Allen Paulos, Innumeracy. In case it isn't obvious, innumeracy is the mathematical counterpart of illiteracy.

This is wild stuff; I cannot say whether it is mathematically respectable but the man does teach at Rutgers. It is certainly not mainstream. Excerpt:

It is utter nonsense to say that sqrt 2

isNow what the hell does that mean? A rational number is one that can be expressed as a fraction a/b where both a and b are integers and b is not 0. An irrational number is one that cannot be expressed in this way. By the celebrated theorem of Pythagoras, a right triangle with sides of 1 unit in length will have an hypotenuse with length = the square root of 2. This is an irrational number. But this irrational number measures a quite definite length both in the physical world and in the ideal world. How can this number not exist? It is inept to speak of a symbol as shorthand for an ideal object since, if x is shorthand for y, then both are linguistic items. For example 'POTUS' is shorthand for 'president of the United States.' But 'POTUS' is not shorthand for Obama. 'POTUS' refers to Obama. Zeilberger appears to be falling into use/mention confusion. If the symbol for the sqrt of 2 refers to an ideal object, then said object is a number that does exist. And in that case Zeilberger is contradicting himself.

What's more, it seems that from Zeilberger's own example one can squeeze out an argument for actual infinity. We note first that the decimal expansion of the the sqrt of 2 is nonterminating: 1.4142136 . . . . We note second that the length of the hypotenuse is quite definite and determinate. This seems to suggest that the decimal expansion must be actually infinite. Otherwise, how could the length of the hypotenuse be definite?

As an ultrafinitist, however, Zeilberger denies both actual *and potential* infinity:

. . . the philosophy that I am advocating here is called

So I deny even the existence of the Peano axiom that every integer has a successor.

As I said, this is wild stuff. He may be competent as a mathematician; I am not competent to pronounce upon that question. But he appears to be an inept philosopher of mathematics. But this is not surprising. It is not unusual for competent scientists and mathematicians to be incapable of talking coherently about what they are doing when they pursue their subjects. Poking around his website, I find more ranting and raving than serious argument.

The ComBox is open if someone can clue us into the mysteries of ultrafinitism. There is also some finitist Russian cat, a Soviet dissident to boot, name of Esenin-Volpin, who Michael Dummett refers to in his essay on Wang's Paradox, but Dummett provides no reference. Is ultrafinitism the same as strict finitism?

Some remarks of Peter Lupu in an earlier thread suggest that he does not understand the notions of potential and actual infinity. Peter writes:

(ii) An acorn has the potential to become an oak tree. But the acorn has this potential only because there are actual oak trees . . . . If for some reason oak trees could not exist, then an acorn cannot be said to have the potential to become an oak tree. Similarly, if the proponent of syllogism S1 thinks that actual infinity is ruled out by some conceptual, logical, or metaphysical necessity, then he is committed to hold that there cannot be a potential infinity either. Thus, in order for something to have the potential to be such-and-such, it is required that it is at least possible for actual such-and-such to exist.

This is a very fruitful misunderstanding! For it allows us to clarify the different senses of 'potential' and 'actual' as applied to the analysis of change and to the topic of infinity. First of all, Peter is completely correct in what he says in the first two sentences of the above quotation. The essence of what he is saying may be distilled in the following principle

If actual Fs are impossible, then potential Fs are also impossible.

But this irreproachable principle is misapplied if 'F' is instantiated by 'infinity.' If an actual infinity is impossible, it does not follow that a potential infinity is impossible. For when we say that a series, say, is potentially infinite we precisely mean to exclude its being actually infinite. A potentially infinite series is not one that has the power or potency to develop over time into an actually infinite series, the way a properly planted acorn develops into an oak tree. On the contrary, it is a series which, no matter how much time elapses, is never completed. An actually infinite series, by contrast, is complete at every instant.

Consider the natural numbers (0, 1, 2, 3, . . . n, n +1, . . . ). If these numbers form a set, call it N, then N will of course be actually infinite. A set is a single, definite object, a one-over-many, distinct from each of its members and from all of them. N must be actually infinite because there is no greatest natural number, and because N contains all the natural numbers.

It is worth noting, as I have noted before, that 'actually infinite set' is a pleonastic expression. It suffices to say 'infinite set.' This is because the phrase 'potentially infinite set' is nonsense. It is nonsense because a set is a definite object whose definiteness derives from its having exactly the members it has. In the case of the natural numbers, *if* they form a set, then that set will have a transfinite cardinality. Cantor refers to that cardinality as aleph-zero or aleph-nought.

But surely it is not obvious that the natural numbers form a set. Suppose they don't. Then the natural number series, though infinite, will be merely potentially infinite. What 'potentially infinite' means in this case is that one can go on adding endlessly without ever reaching an upper bound of the series. No matter how large the number counted up to, one can add 1 to reach a still higher number. The numbers are thus created by the counting, not labeled by the counting. The numbers are not 'out there' waiting to be counted; they are created by the counting. In that sense, their infinity is merely potential. But if the naturals are an actual infinity, then they are not created but labeled.

Or consider a line segment. One can divide it repeatedly and in principle 'infinitely.' But if one does so is one creating divisions or recognizing divisions that exist already? If the former, then the infinity of divisions is merely potential; if the latter, it is actual.

Peter seems worried by the fact that no human or nonhuman adding machine can enumerate all of the natural numbers. But this is no problem at all. If there is an actual infinity of natural numbers, then it is obvious that a complete enumeration is impossible: the first transfinite ordinal omega has aleph-nought predecessors. If there is only a potential infinity of naturals, then as many enumerations have taken place, that is the last number created.

Peter seems not to be taking seriously the notion of potential infinity by simply assuming that the naturals must form an infinite set. He doesn't take it seriously because he confuses the use of 'potential' in the context of an analysis of change, where change is the reduction of potency to act, with the use of 'potential' in discussions of infinity.

But now I'm having second thoughts. I want to say that from the fact that a line segment is infinitely divisible, it does not follow that it is actually divided into continuum-many points. But what about the number of possible dividings? If that is a finite number, one that reflects the ability of some divider, then how can the segment be *infinitely* divisible? But if the number of possible dividings is a transfinite number, then it seems we have re-introduced an actual infinity, namely, an actual infinity of possible dividings. In other words, infinite divisibility seems to require an actual infinity of possible dividings. Or does it?

The Regressive Dichotomy is one of Zeno's paradoxes of motion. How can I get from point A, where I am, to point B, where I want to be? It seems I can't get started.

A_______1/8_______1/4_______________1/2_________________________________ B

To get from A to B, I must go halfway. But to travel halfway, I must first traverse half of the halfway distance, and thus 1/4 of the total distance. But to do this I must move 1/8 of the total distance. And so on. The sequence of runs I must complete in order to reach my goal has the form of an infinite regress with no first term:

. . . 1/16, 1/8, 1/4, 1/2, 1.

Since there is no first term, I can't get started.

Let us consider the 'calculus solution' to the paradox. Consider the infinite sequence

1/2, 1/4, 1/8, . . . ,1/2^{n}, . . . .

A sequence is said to be *convergent* if it has a limit. Roughly, the *limit* of a sequence is a number L such that the terms of the sequence become and remain arbitrarily close to L as we consider the terms in succession. Thus the limit of the sequence displayed above is 0, since as we run through the sequence, each term gets closer and closer to 0. The larger n becomes, the smaller is the difference between 1/2^{n} and 0. The sequence converges to 0 as its limit.

We can also speak of the convergence of an infinite *series*, where an infinite series is the result of adding all the terms in an infinite sequence. But how do we add the terms in an infinite sequence? What meaning can we attach to 'sum of an infinite sequence of terms'? In standard analysis (calculus), the sum of an infinite sequence is the limit of the sequence of partial sums:

1/2, 1/2 + 1/4 = 3/4, 1/2 + 1/4 + 1/8 = 7/8, . . . .

This sequence of partial sums converges to the limit 1. Thus the sum of a convergent infinite series is a number that can be approximated arbitrarily closely by adding up sufficiently many finite terms.

Now how is this supposed to solve the Regressive Dichotomy paradox? Since the sum of an infinite series has been defined as the limit of the sequence of partial sums, and since this sequence converges to a finite limit (1 in our example), it is intelligible how infinitely many terms in a convergent series can have a finite sum. Note, however, that this intelligibility is a matter of stipulative definition: it is just stipulated that 'sum' in the case of infinite series shall have the meaning 'limit of a convergent sequence.' And since this stipulation is intelligible, it is supposed also to be intelligible how a runner can traverse the infinity of points lying between A and B.

The rub, however, is in applying the mathematical formalism, which I cannot fault, to the physical world. Our convergent infinite series has a sum, but only in the sense that the sequence of partial sums converges to the limit 1. The series does not have a sum in the sense of some number that is actually computed by someone or some computer. But to get from A to B I must actually complete an infinite sequence of runs. There is an infinite number of acts that I must perform assuming that space is composed of points. It is not enough that I converge upon point B; I must actually arrive there. In the mathematical case, however, it is enough that the infinite series converges upon the limit 1. There is no need that one actually compute up to 1.

Although I haven't expressed this as clearly as I ought to, my sense is that the 'calculus solution,' although fine as mathematics, is no solution to the Dichotomy paradox. One can render intelligible how an infinite series of terms can have a finite sum. But this does not render intelligible how an infinite series of acts can be performed in a finite time.

REFERENCE: Wesley C. Salmon, *Space, Time, and Motion*, Chapter 2.

Much as I disagree with Daniel Dennett on most matters, I agree entirely with the following passage:

I deplore the narrow pragmatism that demands immediate social utility for any intellectual exercise. Theoretical physicists and cosmologists, for instance, may have more prestige than ontologists, but not because there is any more social utility in the satisfaction of

theirpure curiosity. Anyone who thinks it is ludicrous to pay someone good money to work out the ontology of dances (or numbers or opportunities) probably thinks the same thing about working out the identity of Homer or what happened in the first millionth of a second after the Big Bang. (Dennett and His Critics, ed. Dahlbom, Basil Blackwell 1993, p. 213. Emphasis in original.)

I would put the point in stronger terms and go Dennett one better. Anyone who thinks that intellectual inquiry has value only if it has immediate or even *long-term* social utility is contemptibly benighted, and a danger to free inquiry.

One of my favorite examples is complex numbers. A complex number involves a real factor and an imaginary factor *i*, where *i* = the square root of -1. Thus a complex number has the form, *a + bi* where *a* is the real part and *bi* is the imaginary part.

One can see why the term 'imaginary' is used. The number 1 has two square roots, 1 and -1 since if you square either you get 1. But what is the square root of -1? It can't be 1 and it can't be -1, since either squared gives a positive number. So the imaginary *i* is introduced as the square root of -1. Rather than say that negative numbers do not have square roots, mathematicians say that they have complex roots. Thus the square root of -9 = 3i.

Now to the practical sort of fellow who won't believe in anything that he can't hold in his hands and stick in his mouth, this all seems like idle speculation. He demands to know what good it is, what it can *used for*. Well, the surprising thing is is that the theory of complex numbers which originated in the work of such 16th century Italian mathematicians as Cardano (1501 - 1576) and Bombelli (1526-1572) turned out to find application to the physical world in electrical engineering. The electrical engineers use *j* instead of *i* because *i* is already in use for current.

Just one example of the application of complex numbers is in the concept of impedance. Impedance is a measure of opposition to a sinusoidal electric current. Impedance is a generalization of the concept of resistance which applies to direct current circuits. Consider a simple direct current circuit consisting of a battery, a light bulb, and a rheostat (variable resistor). Ohm's Law governs such circuits: I = E/R. If the voltage E ('E' for electromotive force) is constant, and the resistance R is increased, then the current I decreases causing the light to become dimmer. The resistance R is given as a real number. But the impedance of an alternating current circuit is given as a complex number.

Now what I find fascinating here is that the theory of complex numbers, which began life as something merely theoretical, turned out to have application to the physical world. One question in the philosophy of mathematics is: How is this possible? How is it possible that a discipline developed purely* a priori* can turn out to 'govern' nature? It is a classical Kantian question, but let's not pursue it.

My point is that the theory of complex numbers, which for a long time had no practical (e.g. engineering) use whatsoever, and was something of a mere mathematical curiosity, turned out to have such a use. Therefore, to demand that theoretical inquiry have immediate social utility is shortsighted and quite stupid. For such inquiry might turn how to be useful in the future.

But even if a branch of inquiry could not possibly have any application to the prediction and control of nature for human purposes, it would still have value as a form of the pursuit of truth. Truth is a value regardless of any use it may or may not have.

Social utility is a value. But truth is a value that trumps it. The pursuit of truth is an end in itself. Paradoxically, the pursuit of truth as an end in itself may be the best way to attain truth that is useful to us.