Mathematics – Maverick Philosopher

Topics in Current Technical Threads

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.

Weyl’s Tiles: An Argument against Discrete Space

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 (8² + 8² =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.

Notes on Infinite Series

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 1/2ⁿwhere 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 '1/2ⁿ' 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.

Spherical Triangles as Incongruent Counterparts?

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.

Kant, Spherical Triangles, and Incongruent Counterparts

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.

Wikipedia:

In geometry, a figure is chiral (and said to have chirality) 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 be achiral.

A chiral object and its mirror image are said to be enantiomorphs. The word chirality is derived from the Greek χείρ (cheir), the hand, the most familiar chiral object; the word enantiomorph stems from the Greek ἐναντίος (enantios) 'opposite' + μορφή (morphe) 'form'.

Tom and Van: A Tale of Two Idealists

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.

For the Left, the Subject is not the Subject: Why Math is ‘Racist’

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 from Rules 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.

“But it’s exponential!”

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.

Beware of Cranks

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.

Jerking Towards Social Collapse

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.

Image credit: Frank J. Attanucci

3/14 is π Day

My best π day post

Infinity and Mathematics Education

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?

Math is Racist!

Here

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

Super π Day Approximately

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"?

Social Utility and the Value of Pure Inquiry: The Example of Complex Numbers

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 their pure 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.