§2a. Now we’re really starting. There are two ways to trace out the context of Cantorian set theory. The first is to talk about the abstract intertwined dance of infinity and limit throughout math’s evolution. The second is to examine math’s historical struggle with representing continuity, meaning the smooth-flowing and/or densely successive aspects of motion and real-world processes. Anyone with even the vaguest memory of college math will recall that continuity and ∞/limit are pretty much the fundament of the calculus, and might also recall that they have their general origins in the metaphysics of the ancient Greeks and their particular matrix in Zeno of Elea (c. 490–435 BCE, who died with his teeth literally still in the ear of Elea’s despotic ruler Nearchus I (long story)), whose eponymous paradoxes pulled the starter-rope on everything.
A few Attic facts at the outset. First, Greek math was abstract all right, but it had its roots deep in Babylo-Egyptian praxis. There is no real difference, for the Greeks, between arithmetical entities and geometric figures, between e.g. the number 5 and a line five units long. Nor, second, are there any clear distinctions for the Greeks between mathematics, metaphysics, and religion; in many respects they were all the same thing. Third, our own age and culture’s dislike of limits—as in ‘a limited man,’ ‘IF YOUR VOCABULARY IS LIMITED, YOUR CHANCES FOR SUCCESS ARE LIMITED,’ etc.—would have been incomprehensible to the ancient Greeks. Suffice to say they liked limits a lot, and a straightforward consequence of this is their distaste for/distrust of ∞. The Hellenic term to apeiron means not only infinitely long/large but also undefinable, hopelessly complex, the that-which-cannot-be-handled.1
To apeiron also and most famously refers to the unbounded natureless chaos from which creation sprang. Anaximander (610–545 BCE), the first of the pre-Socratics to use the term in his metaphysics, basically defines it as “the unlimited substratum from which the world derived.” And the ‘unlimited’ here means not only endless and inexhaustible but formless, lacking all boundaries and distinctions and specific qualities. Sort of the Void, except what it’s primarily devoid of is form.2 And this, for the Greeks, is not good. Here’s a definitive quotation from Aristotle, that font of definitive quotations: “[T]he essence of the infinite is privation, not perfection but the absence of limit.” The point being that in abstracting away all limits to get ∞ you are throwing the baby out as well: no limit means no form, and no form means chaos, ugliness, a mess. Note thus Attic Fact Four, the ubiquitous and essential aestheticism of the Greek intellect. Messiness and ugliness were the ultimate malum in se, the sure sign that something was wrong with a concept, in much the same way that disproportion or messiness was impermissible in Greek art.3
Pythagoras of Samos (570–500 BCE) is crucial in all kinds of ways to the history of ∞. (Actually it’s more accurate to say ‘the Divine Brotherhood of Pythagoras’ or at least ‘the Pythagoreans,’ because ∞-wise the man was less important than the sect.) It was Pythagorean metaphysics that explicitly combined Anaximander’s to apeiron with the principle of limit (= Gr. peras) that lends structure and order—the possibility of form—to the primal Void. The Divine Brotherhood of P., who as is well-known made a whole religion of Number, posited this limit as mathematical, geometric. It is the operations of peras on to apeiron that produce the geometrical dimensions of the concrete world: to apeiron limited once produces the geometric point, limited twice produces the line, three times the plane, and so on. However odd or primitive this might seem, it was extremely important, and so were the Pythagoreans. Their peras-based cosmology meant that the genesis of numbers was the genesis of the world. The D.B.P. were, yes, legendarily eccentric, as in their seasonal rules about sex or Pythagoras’s pathological hatred of legumes. But they were the first people to regard, and revere, numbers as abstractions. The centrality of the number 10 to their religion, for example, was based not on finger- or toe-factors but on 10’s status as the perfect sum of 1 + 2 + 3 + 4.
The D.B.P. were also the first philosophers explicitly to address the metaphysical relation between abstract mathematical realities and concrete empirical realities. Their basic position was that mathematical reality and the concrete world were the same, or rather that empirical reality was a sort of shadow or projection of abstract math.4 Moreover, many of their arguments for the primacy of number were based on the observed fact that purely formal mathematical relationships had striking implications for real-world phenomena, a famous example being how the D.B.P. abstracted the Golden Mean 
which solves to roughly 
from seashells’ whelks and trees’ rings and promulgated its use in architecture. As mentioned, some of these math/world connections had been known to earlier cultures like the Egyptians—or maybe rather ‘used by them’ would be better, since the Egyptians had zero interest in what the connections actually were, or meant. A couple more examples. In practice, the Egyptians had used what we now call the Pythagorean Theorem in engineering and surveying along the Nile; but it was Pythagoras who made it an actual Theorem, and proved it. Plenty of pre-Greek cultures also played music, but it was the D.B.P. who discovered the concepts of the octave, the perfect fifth, etc., by observing that certain musical intervals always corresponded to certain ratios in the lengths of plucked strings—2 to 1, 3 to 2, and so on. Since strings were lines and lines were geometric/mathematical entities,5 the ratios of strings’ lengths was the same as the ratios of integers, a.k.a. rational numbers, which happen to be the fundamental entities of Pythagorean metaphysics.
Etc. etc., the point being that the D.B.P.’s attempts to articulate the connections between mathematical reality and the physical world were part of the larger project of pre-Socratic philosophy, which was basically to give a rational, nonmythopoeic account of what was real and where it came from. Maybe even more important than the D.B.P., ∞-wise, is the protomystic Parmenides of Elea (c. 515–? BCE), not only because his distinction between the ‘Way of Truth’ and ‘Way of Seeming’ framed the terms of Greek metaphysics and (again) influenced Plato, but because Parmenides’ #1 student and defender was the aforementioned Zeno, the most fiendishly clever and upsetting Greek philosopher ever (who can be seen actually kicking Socrates’ ass, argumentatively speaking, in Plato’s Parmenides). Zeno’s arguments for Parmenidean metaphysics took the form—again as mentioned—of some of the most profound and nutcrunching paradoxes in world history. In support of these crunchers’ relevance to our overall purpose, here is another nice B. Russell quotation:
In this capricious world, nothing is more capricious than posthumous fame. One of the most notable examples of posterity’s lack of judgment is the Eleatic Zeno [ … ], who may be regarded as the founder of the philosophy of infinity. He invented four arguments, all immeasurably subtle and profound, to prove that motion is impossible, that Achilles can never overtake the tortoise, and that an arrow in flight is really at rest. After being refuted by Aristotle, and by every subsequent philosopher from that day to our own, these arguments were reinstated, and made the basis of a mathematical renaissance, by a German professor, who probably never dreamed of any connection between himself and Zeno.
For the record, Parmenides’ metaphysics—which is even wilder than the D.B.P.’s, and in retrospect seems more like Eastern religion than Western philosophy—is describable as a kind of static monism,6 and Zeno’s Paradoxes (of which there are really more than four) are accordingly directed against the reality of (1) plurality and (2) continuity. For present purposes we are concerned with (2), which for Zeno takes the form, as Russell mentions, of regular physical motion.
Zeno’s basic argument against the reality of motion is known as the Dichotomy. It looks very simple and is deployed in two of his most famous paradoxes, “The Racetrack” and “Achilles v. the Tortoise.” The Dichotomy later gets used and discussed, with all sorts of different setups and agendas, by Plato, Aristotle, Agrippa, Plotinus, St. Thomas, Leibniz, J. S. Mill, F. H. Bradley, and W. James (to say nothing of D. Hofstadter in Gödel, Escher, Bach). It runs thus.7 You’re standing at a corner and the light changes and you try to cross the street. Note the operative ‘try to’. Because before you can get all the way across the street, you obviously have to get halfway across. And before you can get halfway across, you have to get halfway to that halfway point. This is just common sense. And before you can get to the halfway-to-the-halfway-point point, you obviously have to get halfway to the halfway-to-the-halfway-point point, and so on. And on. Put a little more sexily, the paradox is that a pedestrian cannot move from point A to point B without traversing all successive subintervals of AB, each subinterval equaling 
where n’s values compose the sequence (1, 2, 3, 4, 5, 6, …), with the ‘ … ’ of course meaning the sequence has no finite end. Goes on forever. This is the dreaded regressus in infinitum, a.k.a. the Vicious Infinite Regress or VIR. What makes it vicious here is that you’re required to complete an infinite number of actions before attaining your goal, which—since the whole point of ‘infinite’ is that there’s no end to the number of these actions—renders the goal logically impossible. Meaning you can’t cross the street.
The standard way to schematize the Dichotomy is usually:
(1)In order to traverse the interval AB you must first traverse all the subintervals 
where n = 1, 2, 3, 4, 5, 6, ….
(2)There are infinitely many such subintervals.
(3)It is impossible to traverse infinitely many subintervals in a finite amount of time.
(4)Therefore, it is impossible to traverse AB.
It goes without saying that the interval AB doesn’t have to be a very wide street, or even a street at all. The Dichotomy applies to any kind of continuous motion. Dr. G. in class used to like to run the argument in terms of the DUI-like movement of your finger from your lap to the tip of your nose. And of course, as anybody who’s ever successfully crossed a street or touched his nose is aware, there has got to be something fishy about Zeno’s argument. Finding and articulating that fishiness is a whole other matter. We have to be careful, too; there’s more than one way to be wrong. If you’ve had some college math, for instance, it may be tempting to say that the Dichotomy’s step (2) conceals a simple fallacy, namely the assumption that the sum of an infinite series must itself be infinite. You might recall that step (1)’s 
is simply another way to represent the geometric series 
and that the correct formula for finding the sum of this geometric series is 
where a is the series’ first term and r is the common ratio, and that here a is 
and so is r, and 
in which case it appears that streets can be crossed and noses touched with no problem, and thus that the Dichotomy is really just a tricky Word Problem and not a paradox at all, except maybe for civilizations too crude and benighted to know the formula for summing a geometric series.
Except this response won’t do. Leave aside for the moment whether it’s technically correct. What matters is that it’s trivial; it represents what philosophers would call an impoverished view of Zeno’s problem. For whence exactly 
as a formula for summing this geometric series? I.e., is the formula just a bit of lawyerly semantics designed to define certain paradoxes out of existence, or is it mathematically significant in Hardy’s sense of ‘significant’? And how do we determine which it is?
Weirdly, the more standard classroom math you’ve had, the harder it’s going to be to avoid answering in an impoverished way. Such as, e.g., validating 
by observing, in the best Calc II tradition,8 that the relevant geometric series here is a particular subtype of convergent infinite series, and that the sum of such a series is defined as the limit of the sequence of its partial sums (that is, if the sequence s1, s2, s3, …, sn , … of a series’ partial sums tends to a limit S, then S is the sum of the series), and that sure enough, w/r/t the above series, 
works just fine … in which case you will once again have answered Zeno’s Dichotomy in a way that is complex, formally sexy, technically correct, and deeply trivial. Along the lines of ‘Because it’s illegal’ as an answer to ‘Why is it wrong to kill?’
The trouble with college math classes—which classes consist almost entirely in the rhythmic ingestion and regurgitation of abstract information, and are paced in such a way as to maximize this reciprocal data-flow—is that their sheer surface-level difficulty can fool us into thinking we really know something when all we really ‘know’ is abstract formulas and rules for their deployment. Rarely do math classes ever tell us whether a certain formula is truly significant, or why, or where it came from, or what was at stake.9 There’s clearly a difference between being able to use a formula correctly and really knowing how to solve a problem, knowing why a problem is an actual mathematical problem and not just an exercise. In this regard see yet another part of the B. Russell ¶ on Zeno,10 this time with emphases supplied:
Zeno was concerned, as a matter of fact, with three problems, each presented by motion, but each more abstract than motion, and capable of purely arithmetical treatment. These are the problems of the infinitesimal, the infinite, and continuity. To state clearly the difficulties involved, was to accomplish perhaps the hardest part of the philosopher’s task.
And 
fails, without a great deal of context and as it were motivation, to state clearly the difficulties involved. Stating these difficulties clearly is, in fact, the whole and only difficulty involved here (and if you can now feel the slight strain and/or headache starting, you’ll know we’re in Zeno’s real territory).
First, to save at least 103 words, have a refresher-type look at the following two rough graphs, one of the divergent sequence11 for 2n and the other of the convergent sequence for 
Exhibit 2a(1)
Exhibit 2a(2)
One of the real contextual difficulties surrounding the Dichotomy was that the Greeks did not have or use 0 in their math (0 having been a very-late-Babylonian invention, purely practical and actuarial, c. 300 BCE). One could therefore say that since there was no recognized number/quantity for the converging sequence 
to converge to (q.v. Exhibit 2a(2)), Greek math lacked the conceptual equipment to comprehend convergence, limits, partial sums, etc. This would be true in a way,12 and not wholly trivial.
Less trivial still is the aforementioned Greek dread of to apeiron. Zeno was the first philosopher to use ∞’s black-hole-like logical qualities as an actual argumentative tool, viz. the Vicious Infinite Regress, which even today gets used in logical arguments as a reductio-grade method of proof. Example: In epistemology, the VIR is the easiest way to refute the common claim that in order to really know something you have to know that you know it. Like most VIR proofs, there’s an evil-edged fun to this one. Let the variable x denote any fact or state of affairs precedable by the expletive ‘that,’ and restate the original claim as (1) ‘In order to know that x, you must know that you know that x’. Since the whole expletive phrase ‘that you know that x’ qualifies as a fact or state of affairs, in the proof’s next step you can simply expand the denotation of x so that now x = [you know that x] and then substitute it into the original claim, resulting mutatis mutandis in (2) ‘In order to know that [you know that x], you must know that you know that [you know that x],’ the next wholly valid x-extension of which then yields (3) ‘In order to know that [you know that you know that x], you must know that you know that [you know that you know that x],’ and so on, ad inf., requiring you to satisfy an endless number of preconditions for knowing anything.
(IYI The VIR is so powerful a tool that you can easily use it to annoy professional competitors or infuriate your partner in domestic conflicts, or (worse) to drive yourself crazy in bed in the morning over, e.g., any kind of relation between two things or terms, like when we say that 2 and 4 are related by the function y = x2, or that if clouds cause rain then clouds and rain stand in a causal relation. If you consider the idea in the abstract and ask, w/r/t any relation, whether this relation is itself related to the two terms it relates, the answer is inescapably yes (since it’s impossible to see how a relation can connect two terms unless it has its own relation to each one, the way a bridge between two riverbanks has got to be connected to each bank), in which case the relation between, say, clouds and rain actually entails two more relations—viz. those between (1) clouds and the relation and (2) rain and the relation—each of which latter relations obviously then entails two more on either side, and so on, ad inf… . which is not a fun or productive abstract path to venture down in the A.M. at all, especially since the geometric series of relations here is divergent rather than convergent, and as such it’s connected to all kinds of especially dreadful and modern divergent series like the exponential doublings of cancer, nuclear fission, epidemiology, & c. Worth noticing also is that hideous divergent VIRs like those above always involve the metaphysics of abstractions, such as ‘relation’ or ‘knowledge’. It’s like some fissure or crevasse always opens up in the move from particular cases of knowing/relating to knowledge/relation in abstractus.)
Zeno himself is almost fetishistically attached to the divergent VIR and uses it in several of his lesser-known paradoxes. Here is a specifically anti-Pythagorean Z.P.13 contra the idea that anything can really be in a particular location, which in simplified form is schematized:
(1)Whatever exists is in a location.
(2)Therefore, location exists.
(3)But by (1) and (2), location must be in a location, and
(4)By (1)–(3), location’s location must itself be in a location, and …
(5)… So on ad inf.
This one’s rather easier to see the gears of, since the true Russellian difficulty here is some slipperiness around ‘to exist’. Actually, since ancient Greek didn’t even have a special verb for existence, the relevant infinitive is the even slipperier ‘to be’. On purely grammatical grounds, Zeno’s argument can be accused of the classic Fallacy of Equivocation,14 since ‘to be’ can have all sorts of different senses, as in ‘I am frightened’ v. ‘He is a Democrat’ v. ‘It is raining’ v. ‘I AM THAT I AM’. But you can see that pressing this case will (once again) lead quickly to the paradox’s deeper questions, which questions here are (again once again) metaphysical: what exactly do these different senses of ‘to be’ mean, and in particular what does the more specialized sense ‘to exist’ mean, i.e. what sorts of things really do exist, and in what ways, and are there different kinds of existence for different kinds of things, and if there are then are some kinds of existence more basic or substantial than other kinds? & c.
You’ll have noticed that we’ve run up against these sorts of questions a dozen times already and we’re still 2,000+ years away from G. Cantor. They are the veritable bad penny in the Story of ∞, and there’s no way around them if you don’t want just a bunch of abstract math-class vomitus on transfinite set theory. Deal. Right now is the time for a sketch of Plato’s One Over Many argument, which is the classic treatment of just these questions as they apply to the related issue of predication.
You might recall the O.O.M., too, from school, in which case relax because this won’t take long. For Plato, if two individuals have some common attribute and so are describable15 by the same predicate—‘Tom is a man’; ‘Dick is a man’—then there is something in virtue of which Tom and Dick (together with all other referents of the predicate nominative ‘man’) have this common attribute. This something is the ideal Form Man, which Form is what really, ultimately exists, whereas individual men are just temporal appearances of the Form, with a kind of borrowed or derivative existence, like shadows or projected images. That’s a very simplified version of the O.O.M., but not a distorted one—and even at this level it should not be hard to see the influences of Pythagoras and Parmenides on Plato’s ontological Theory of Forms, which the O.O.M. is an obvious part of.
Here’s where the truth gets a little complicated. As seems to happen a lot, the complication involves Aristotle. It’s true that the first mention of the O.O.M. is in Plato’s Parmenides, but in fact what made the argument famous is Aristotle’s Metaphysics,16 in which the O.O.M. is discussed at great length so that Aristotle can try to demolish it. There’s several shelves worth of context here that we can mostly skip.17 What’s strange (for reasons that are upcoming) is that Aristotle’s best-known argument against Plato’s Theory of Forms is virtually textbook Zeno. This argument, which is usually called the Third Man, is in effect a divergent-VIR-type reductio on the O.O.M. After observing that both individual men and the Form Man obviously share some predicable quality or attribute, Aristotle points out that there must then be yet another metaphysical Form—say, Man′—that comprises this common attribute, which entails still another Form, Man″, to comprise the predicable commonality between Man′ and [Man + men], & c. & c. ad inf.
Whether or not the Third Man strikes you as a valid refutation of the O.O.M., you may well have already noticed that Plato’s Theory of Forms18 has problems of its own, like for example a conspicuous goofiness when the O.O.M. is applied to certain predicates—is there an ideal Form of left-handedness? of stupidity? of shit? Note, however, that Plato’s theory has a lot more power and plausibility when applied to any kind of system that depends on formal relations between abstractions. Like math. The conceptual move from ‘five oranges’ and ‘five pennies’ to the quantity five and the integer 5 is precisely Plato’s move from ‘man’ and ‘men’ to Man. Recall, after all, Hardy’s thrust in §1c: when we use an expression like ‘2 + 3 = 5,’ what we’re expressing is a general truth whose generality depends on the total abstractness of the terms involved; we are really saying that two of anything plus three of anything will equal five of anything.
Except we never actually say that. Instead we talk about the number 2 and the number 5, and about relations between these numbers. It’s worth it—again—to point out that this could be just a semantic move, or it could be a metaphysical one, or both. And worth it to recall both §1d’s thing about the predicative v. nominative senses of ‘length’ and ‘gram,’ and the different types of existence-claims involved in ‘I see nothing on the road’ and ‘Man is by nature curious’ and ‘It is raining’; and then to consider, carefully, the existence-claims we’re committing to when we talk about numbers. Is ‘5’ just some kind of conceptual shorthand for all the actual quintuples in the world?19 It’s pretty apparent that it’s not, or at least that this isn’t all ‘5’ is, since there are lots of things about 5 (e.g. that 5 is prime, that 5’s square root is 2.236 … ) that don’t have anything to do with real-world quintuples but do have to do with a certain kind of entity called numbers and with their qualities and relations. Numbers’ real, if strange, existence is further suggested by the way many of these qualities and relations—such as for example that 
cannot be expressed as either a finite decimal or a ratio of integers—seem like they really are discovered rather than made up or proposed and then defended. Most of us would be inclined to say that 
is an irrational number even if nobody ever actually proves that it is—or at least it turns out that to say anything else is to be committed to a very complex and strange-looking theory of what numbers are. The whole issue here is of course incredibly hairy (which is one reason we’re talking about it only in little contextual chunks), because not only is the question abstract but everything it’s concerned with is an abstraction—existence, reality, number …. Although consider too for a moment how many levels of abstraction are involved in math itself. In arithmetic there’s the abstraction of number; and then there’s algebra, with a variable being a further-abstracted symbol for some number(s) and a function being a precise but abstract relation between domains of variables; and then of course there’s college math’s derivatives and integrals of functions, and then integral equations involving unknown functions, and differential equations’ families of functions, and complex functions (which are functions of functions), and definite integrals calculated as the difference between two integrals; and so on up through topology and tensor analysis and complex numbers and the complex plane and complex conjugates of matrices, etc. etc., the whole enterprise becoming such a towering baklava of abstractions and abstractions of abstractions that you pretty much have to pretend that everything you’re manipulating is an actual, tangible thing or else you get so abstracted that you can’t even sharpen your pencil, much less do any math.
The most relevant points with respect to all this are that the question of mathematical entities’ ultimate reality is not just vexed but controversial, and that it was actually G. F. L. P. Cantor’s theories of ∞ that brought this controversy to a head in modern math. And that in this controversy, mathematicians who tend to regard mathematical quantities and relations as metaphysically real are called Platonists,20 and at least now it’s clear why, and the term can be thrown around later.
§2b. The first really serious non-Platonist is Aristotle. What’s odd and ironic about the Zenoish VIR Aristotle runs against Plato’s metaphysics, however, is that Aristotle’s is also the first and most important Greek attempt to refute Zeno’s Paradoxes. This is mainly in Books III, VI, and VIII of the Physics and Book IX of the Metaphysics, whose discussions of Zeno will end up having a pernicious effect on the way math handles ∞ for the next two millennia. At the same time, Aristotle does manage to articulate the root difficulties in at least some of Zeno’s Paradoxes, as well as to pose clearly and for the first time some really vital ∞-related questions that nobody until the 1800s will even try to answer in a rigorous way, viz.: ‘What exactly does it mean to say that something is infinite?’ and ‘Of what sort of thing can we even coherently ask whether it’s infinite or not?’
W/r/t these central questions, you might recall Aristotle’s famous predilection for dividing and classifying—he literally put the ‘analytic’ in analytic philosophy. See for instance this snippet from Physics VI’s discussion of the Dichotomy: “For there are two senses in which length and time and generally anything continuous are called ‘infinite’: they are called so either in respect of divisibility or in respect of their extremities [= size],” which happens to be the first time anyone had ever pointed out that there’s more than one sense to ‘infinite’. Aristotle mainly wants to distinguish between a strong or quantitative sense, one meaning literally infinite size or length or duration, and a weaker sense comprising the infinite divisibility of a finite length. The really crucial distinction, he claims, involves time: “So while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility: for in this sense the time itself is also infinite.”
Both the above quotations are from one of Aristotle’s two main arguments against Zeno’s Dichotomy as schematized on pp. 49–50. The target of this particular argument is premise (3)’s ‘in a finite amount of time’. Aristotle’s thrust is that if Zeno gets to represent the interval AB as the sum of an infinite number of subintervals, the allotted time it takes to traverse AB should be represented the same way—say like 
to get to 
to get to 
to get to 
etc. This argument isn’t all that helpful, though, since having an infinite amount of time to cross the street is no less contradictory of our actual ten-second street-crossing experiences than the original Dichotomy itself. Plus it’s easy to construct versions of the Z.P. that don’t explicitly require action or elapsed time. (For example, imagine a pie the first piece of which = half the whole pie and the next piece = half the first piece and dot dot dot ad inf.: is there a last piece of pie or not?) The point: Counterarguments about sequential time or subintervals or even actual human movements will always end up impoverishing the Dichotomy and failing to state the real difficulties involved. Because Zeno can amend his presentation and simply say that being at A and then being at B requires you to occupy the infinitely many points corresponding to the sequence 
or, worse, that your ever really arriving at B entails your having already occupied an infinite sequence of points. And this seems quite clearly to contradict the idea of an infinite sequence: if ‘∞’ really means ‘without end,’ then an infinite sequence is one where, however many terms are taken, there are still others that remain to be taken. Meaning forget street-crossing or nose-touching: Zeno can run the whole cruncher in terms of abstract sequences and the fact that there is something inherently contradictory or paradoxical in the idea of an infinite sequence ever being completed.
It is against this second, more abstract and damaging version of the Dichotomy that Aristotle advances his more influential argument. This one depends on the semantics of ‘infinite,’ too, but it’s different, and focuses on the same sorts of predicative questions that arise in the O.O.M. and Zeno’s Location Paradox. In both the Physics and the Metaphysics, Aristotle draws a distinction between two different things we can really mean when we use ‘to be’ + ∞ in a predicative sentence like ‘There are an infinite number of points that must be occupied between A and B.’ The distinction is only superficially grammatical; it’s really a metaphysical one between two radically different existence-claims implicit in the sentence’s ‘are,’ which apparently the Dichotomy depends on our not seeing. The distinction is between actuality and potentiality as predicable qualities; and Aristotle’s general argument is that ∞ is a special type of thing that exists potentially but not actually, and that the word ‘infinite’ needs to be predicated of things accordingly, as the Dichotomy’s confusion demonstrates. Specifically, Aristotle claims that no spatial extension (e.g. the intercurb interval AB) is ‘actually infinite,’ but that all such extensions are ‘potentially infinite’ in the sense of being infinitely divisible.
This all gets extremely involved and complex, of course—entire careers are spent noodling over Aristotle’s definitions. Suffice here to say that the actual-v.-potential-existence-of-∞ issue is vital to our overall Story but admittedly tough to get a handle on. It doesn’t help matters that Aristotle’s own explanations and examples—
[I]t is as that which is building is to that which is capable of building, and the waking to the sleeping, and that which is seeing to that which has its eyes shut but has sight. Let actuality be defined by one member of this antithesis, and the potential by the other. But all things are not said in the same sense to exist actually, but only by analogy—as A is in B or to B, C is in D or to D; for some are as movement to potency, and the others as substance to some sort of matter …
—are not exactly marvels of perspicuity. What he means by ‘potential’ is emphatically not the sort of potentiality by which a girl is potentially a woman or an acorn an oak. It’s rather more like the strange and abstract sort of potentiality by which a perfect copy of Michelangelo’s Pieta21 potentially exists in a block of untouched marble. Or, ∞-wise, the way anything that occurs cyclically (or, in A.’s term, “successively”)—like say its being 6:54 A.M., which happens every day like clockwork—is for Aristotle potentially infinite in the sense that an endless periodic recurrence of its being 6:54 A.M. is possible, whereas the set of all 6:54 A.M.s cannot be actually infinite because the 6:54s are never all going to coexist; the periodic cycle is never going to be “complete[d].”22
You can probably see how all this is going to play out w/r/t the Dichotomy. Again, though, it’s a little tricky. The statue and 6:54 analogies won’t quite work here. Yes, the interval AB and/or the set of all subintervals or points between A and B is not ‘actually infinite’ but only ‘potentially infinite’; but here the sense in which Aristotle means ‘AB is potentially infinite’ is closer to the idea of, say, infinite precision in measurement. Which can be illustrated thusly. My eldest niece’s current height, which is 38.5″, can be fixed more precisely at 38.53″; and with a more controlled environment and sophisticated equipment it could obviously be ascertained more and more exactly, to the 3rd, 11th, nth decimal place, with n being, potentially, ∞—but only potentially ∞, because in the real world there’s obviously never going to be any way to achieve true infinite precision, even though ‘in principle’ it’s possible. In pretty much just this way, for Aristotle AB is ‘in principle’ infinitely divisible, though this infinite division can never actually be performed in the real world.
(IYI Final bit of complication: For the most part, what Aristotle calls “Number” (meaning mathematical quantities in general) apparently is potentially infinite not in the way measurement is potentially infinite but in the way the set of all 6:54 A.M.s is potentially infinite. For instance, the set of all integers is potentially infinite in the sense that there is no largest integer (“In the direction of largeness it is always possible to think of a larger number”); but it is not actually infinite because the set doesn’t exist as one completed entity. In other words, numbers for Aristotle compose a successive continuum: there are infinitely many but they never coexist (“One thing can be taken after another endlessly”).)
As a refutation of Zeno’s Dichotomy, the potential-∞-v.-actual-∞ distinction isn’t all that persuasive—evidently not even to Aristotle, whose own Third Man regressus looks like it could be dismissed as only potentially infinite. But the distinction ends up being terribly important for the theory and practice of math. In brief, relegating ∞ to the status of potentiality allowed Western math either to discount infinite quantities or to justify their use, or sometimes both, depending on the agenda. The whole thing is very weird. On the one hand, Aristotle’s argument lent credence to the Greeks’ rejections of ∞ and of the ‘reality’ of infinite series, and was a major reason why they didn’t develop what we now know as calculus. On the other hand, granting infinite quantities at least an abstract or theoretical existence allowed some Greek mathematicians to use them in techniques that were extraordinarily close to being differential and integral calculus—so close that in retrospect it’s amazing that it took 1700 years for actual calc to be invented. But, back on the first hand, a big reason it did take 1700 years was the metaphysical shadowland Aristotle’s potentiality concept had banished ∞ to, which served to legitimate math’s allergy to a concept it couldn’t really ever handle anyway.
Except—either back on the second hand or now on a third hand23—when G. W. Leibniz and I. Newton now really do introduce the calculus around 1700, it’s essentially Aristotle’s metaphysics that justifies their deployment of infinitesimals, e.g. dx in the infamous 
of freshman math. Please either recall or be informed that an infinitesimal quantity is somehow both close enough to 0 to be ignorable in addition—i.e., x + dx = x—and distant enough from 0 to serve as a divisor in derivations like the above. Again very briefly, treating infinitesimals as potentially/theoretically existent quantities let mathematicians use them in calculations that had extraordinary real-world applications, since they were able to abstract and describe just the kinds of smooth continuous phenomena the world comprised. These infinitesimals turn out to be a very big deal. Without them you can end up in crevasses like the .999 … = 1 thing we glanced at in §1c. As was then promised, the quickest way out of that one is to let x stand not for .999 … but for the quantity 1 minus some infinitesimal, which let’s call 
such that 
Then you can run the same operations as before:
yields 9x = 9 − 
in which case x still comes out to 1 − 
and there’s no nasty confabulation with 1.0.
Except of course the question is whether it makes metaphysical or mathematical sense to posit the existence, whether actual or potential, of some quantity that is <1 but still exceeds the infinite decimal .999 …. The issue is doubly abstract, since not only is .999 … not a real-world-type quantity, it’s something we cannot really conceive of even as a mathematical entity; whatever relationship there is between .999 … and 
exists24 out past the nth decimal, a place no one and nothing can ever get to, not even in theory. So it’s not clear whether we’re just trading one kind of paradoxical crevasse for another. This is yet another type of question that is totally vexed before Weierstrass, Dedekind, and Cantor weigh in in the 1800s.
Whatever you might think of Aristotle’s potential-type ontology for ∞, notice that he was at least right to home in on words like ‘point’ and ‘exist’ in the nonpredicative sentence ‘There are [= exist] an infinite number of intermediate points between A and B.’ Just as in Zeno’s Location Paradox, there’s obviously some semantic shiftiness going on here. In the revised Dichotomy, the shiftiness lies in the implied correspondence between an abstract mathematical entity—here an infinite geometric series—and actual physical space. It’s not clear that ‘exist’ is the more vulnerable target, though; there’s a rather more obvious ambiguity in the semantics of ‘point’. If A and B are the two sides of a real-world street, then the noun phrase ‘the infinite number of points between A and B’ is using ‘point’ to denote a precise location in physical space. But in the noun phrase ‘the infinite number of intermediate points designated by 
‘point’ is referring to a mathematical abstraction, a dimensionless entity with ‘position but no magnitude’. To save several pages of noodling that you can do in your own spare time,25 we’ll simply observe here that traversing an infinite number of dimensionless mathematical points is not obviously paradoxical in the way that traversing an infinite number of physical-space points is. In this respect, Zeno’s argument can look rather like §1’s three-men-at-motel brainteaser: the translation of an essentially mathematical situation into natural language somehow lulls us into forgetting that regular words can have vastly different senses and referents. Note, one more time, that this is exactly what the abstract symbolism and schemata of pure math are designed to avoid, and why technical math definitions are often so numbingly dense and complex. You want no room for ambiguity or equivocation. Mathematics, like child-measurement, is an enterprise consecrated to the ideal of precision.
Which all sounds very nice, except it turns out that there is also immense ambiguity—formal, logical, metaphysical—in many of the basic terms and concepts of math itself. In fact the more fundamental the math concept, the more difficult it usually is to define. This is itself a characteristic of formal systems. Most of math’s definitions are built up out of other definitions; it’s the really root stuff that has to be defined from scratch. Hopefully, and for reasons that have already been discussed, that scratch will have something to do with the world we all really live in.
§2c. Back for a moment to the Zeno-and-semantics-of-‘point’ thing. The relation between a mathematical entity (e.g. a series, a geometric point) and actual physical space is also the relation of the discrete to the continuous. Think of a flagstone path v. a shiny smooth black asphalt road. Since what the Dichotomy tries to do is break a continuous physical process down into an infinite series of discrete steps, it can be seen as history’s first-ever attempt to represent continuity mathematically. It doesn’t matter that Zeno was actually trying to show that continuity was impossible; he was still the first. He was also the first to recognize26 that there is more than one species of ∞. The to apeiron of Greek cosmology is pure extension, infinite size; and the integral series 1, 2, 3, … ascends and recedes toward this same kind of Big ∞. Whereas on the other hand Zeno’s Little ∞ appears to be nested amid and between ordinary integers. Which latter is naturally hard to conceive.
It turns out that the most perspicuous way to represent these two different kinds of ∞ is with the good old Number Line, yet another feature of the ordinary 2nd-grade classroom.27 The Number Line is also another bequest from the Greeks, who you’ll recall treated numbers and geometrical shapes as pretty much the same thing. (Euclid, for instance, rejected any piece of mathematical reasoning that could not be “constructed,” meaning demonstrated geometrically.28) The thing to appreciate about the humble Number Line’s marriage of math and geometry is that it’s also the perfect union of form and content. Because each number corresponds to a point, and because the Number Line both comprises all the points and determines their order, numbers can be wholly defined by their place on the N.L. relative to other numbers’ places. As in, 5 is the integer immediately to the right of 4 and to the left of 6, and to say that 5 + 2 = 7 is to say that 7 is two positions to the right of 5—that is, the mathematical ‘distance’ between unequal numbers can be represented and even calculated pictorially. Even without zero or negative integers,29 and with ‘point’ being rather fuzzily defined by Euclid as “that which has no part,” the Number Line is an immensely powerful tool. It also happens to be the perfect schematization of a continuum, meaning ‘an entity or substance whose structure or distribution is continuous and unbroken,’ and as such the N.L. embodies perfectly the antinomy of continuity that Zeno proposed and no one ’til R. Dedekind could solve. For on the N.L. the following are both true: (1) Every point is next to another point; (2) Between any two points there is always another point.
Even though everyone knows what it looks like, the Number Line30 is reproduced here, starting at the 0 the Greeks didn’t have because for now it doesn’t matter:
If the Big ∞, the infinity of extension, lies at the endless right of the Number Line, the Little ∞ that Zeno exploits lies in the totally finite-looking interval between 0 and 1, which interval he reveals as containing an infinite number of intermediate points, viz. the sequence 
What’s more (so to speak), it’s clear that this infinity of 
does not actually exhaust the points between 0 and 1, since it leaves out not only convergent infinite sequences like 
etc., but the whole other infinite set of fractions 
where x is an odd number. And when you consider that each of these latter fractions will correspond to its own infinite geometric sequence via the expansion of 
—e.g. 
etc.—it appears that the finite N.L. interval 0–1 actually houses an infinity of infinities. Which is, to put it mildly, both metaphysically puzzling and mathematically ambiguous—like would this be ∞2, or ∞∞, or what?
Except it gets worse, or better. Because all the prenominate numbers are rational. You probably already know that the adjective here derives from ‘ratio’ and that the phrase ‘the rational numbers’ refers to all those numbers expressible either as integers or as ratios of two integers (that is, as fractions). This is just review, but it’s important. The discovery that not all numbers are rational was at least as hard on the Greek worldview as Zeno’s Paradoxes. And it was particularly upsetting to the Divine Brotherhood of Pythagoras. Recall the Pythagorean convictions that everything is a mathematical quantity or ratio and that nothing infinite can really exist in the world (since (peras → form) is what enables existence in the first place).
Then recall the Pythagorean Theorem. As mentioned, an interesting bit of trivia is that the D.B.P. were not the true discoverers of this theorem; it actually shows up in Old Babylonian tablets as early as 2000 BCE. One reason it’s called the Pythagorean Theorem is that it enabled the D.B.P.’s discovery of ‘incommensurable magnitudes,’ a.k.a. irrational numbers or surds.31 These numbers, which turn out to be inexpressible as finite quantities, were so lethal to Pythagorean metaphysics that their discovery became sort of the Greek version of Watergate. You will remember from childhood that the Pythagorean Theorem causes no problems with figures like the 3-4-5 right triangle of Intro Geometry, wherein the sum of the squares of 3 and 4 is a number whose own square root is rational. Understand, though, that this ‘squares of’ stuff was literal for the Greeks. That is, in a 3-4-5 triangle they treated each leg as the side of a square—
—and then added up the areas of the squares. There are two reasons this is noteworthy. The first was mentioned someplace above: while we now toss exponents and radicals around in abstractus, math problems for the Greeks were always formulated, and solved, geometrically. A rational number was a literal ratio of two line-lengths; squaring something was constructing a square and taking its area. The second reason is that by most accounts it was a plain old humble square that started all the trouble. Consider specifically the familiar Unit Square, with sides equal to 1, and even more specifically the isosceles right triangle whose hypotenuse is the Unit Square’s diagonal:
What the D.B.P. realized (probably through actual and increasingly frantic measurements) is that no matter how small a unit of measure is used, the side of a Unit Square is incommensurable with the diagonal. Meaning there is no rational number 
such that 
The quantity 
was something the D.B.P. eventually called arratos, ‘the not-having-a-ratio,’ or—since logos could mean both word and proportion—alogos, which thus meant both ‘the nonproportional’ and ‘the unsayable’.
The actual demonstration of the incommensurability of 
is another famous instance of reductio proof, and an especially nice one because it’s very simple and requires only jr.-high math. So here it is. First, for reductio purposes, assume that D and S are commensurable. This means that 
equals some ratio 
where p and q are integers with no common factor greater than 1. We know by the Pythagorean Theorem that D2 = S2 + S2, or D2 = 2S2, which if 
means that p2 = 2q2. We know further that the square of any odd number is going to be odd and the square of any even number will be even (feel free to test these out). Plus we know that anything times 2 will obviously be even. This is all the ordnance we need. By LEM, either p is odd or p is even. If (1) p is odd, there’s an immediate contradiction, since 2q2 has got to be even. But if (2) p is even, that means it’s equal to some number times 2, say 2r, so plugging this equivalence back into the original p2 = 2q2 yields 4r2 = 2q2, which reduces to 2r2 = q2, which means q2 is even, which means q is even, which means both p and q are even, in which case they have a common factor greater than 1, which is again a contradiction. (1) yields a contradiction; (2) yields a contradiction; there is no (3). So D and S are incommensurable. End proof.32
The fact that rational numbers couldn’t express something as quotidian as the diagonal of a square—not to mention other easily-constructed hypotenoid irrationals like 
etc.—was obviously destabilizing to the Pythagoreans’ whole cosmogony. The coup de grâce was apparently the discovery that their beloved Golden Mean was itself irrational, working out to 
or 1.618034 …. There’s all sorts of lurid apocrypha about the ends the D.B.P. supposedly went to to keep the existence33 of irrationals secret, which we can skip because much more important, historically and mathematically, are surds themselves. They’re important for at least three reasons. (1) Mathematically, irrational numbers are a direct consequence of abstraction. They’re a whole level up from 5 oranges or 
a pie; you don’t encounter irrationals until you start generating abstract theorems like the P.T. And please note that they’re really only a crevasse for pure math. The Egyptians et al. had run into irrationals in surveying and engineering, but because they cared only about practical applications they had no problem with treating a quantity like 
(2) Surds’ discovery marked the first real divergence of math and geometry, the former now able to manufacture numbers that geometers couldn’t actually measure. (3) It turns out that irrationals, just like Zeno’s 
s, are a consequence of trying to express and explain continuity w/r/t the Number Line. Irrational numbers are the reason why the Number Line isn’t technically continuous. Like the Dichotomy’s VIR, surds represent gaps or holes in the N.L., interstices through which the limitless chaos of ∞ could enter and mess with the tidiness of Attic math.
And it’s not just a Greek problem. Because the big thing about irrational numbers is that they can’t be represented by fractions; and yet if you try to express irrationals in decimal notation,34 then the sequence of digits after the decimal will be neither terminal (as in the rational decimals 2.0, 5.74) nor periodic (which means repeating in some kind of pattern, as in the rationals 
35 Meaning that, for example, the decimal expression of 
can be carried to 1.732, or 1.73205, or 1.7320508, or literally as long as you like … and longer. Meaning in turn that a certain definite point on the Number Line—viz. the point corresponding to that interval which, multiplied by itself, corresponds to the integer-point 3—cannot be named or expressed finitely.36
The finite interval 0–1 on the Number Line is thus even more inconceivably crowded. There’s not only an infinite number of infinite sequences of fractions, but also an infinite number of surds,37 each of which is itself numerically inexpressible except as an infinite sequence of nonperiodic decimals. Let’s pause to consider the vertiginous levels of abstraction involved here. If the human CPU cannot apprehend or even really conceive of ∞, it is now apparently being asked to countenance an infinity of ∞s, an infinite number of individual members of which are themselves not finitely expressible, all in an interval so finite- and innocent-looking we use it in little kids’ classrooms. All of which is just resoundingly weird.
There are, of course, as many ways to handle this weirdness as there are connotations for ‘handle’. The Greeks,* for instance, simply refused to treat irrationals as numbers. They either categorized them as purely geometric lengths/areas and never used them in their math per se, or they literally rationalized the use of surds by futzing with their arithmetic (example: the Pythagoreans’ eventual trick was to write 2 as 
so they could treat 
38 If their refusal to acknowledge the existence of numbers that their own mathematical reasoning produced seems kind of bizarre, be apprised that up to the 1700s pretty much all the best mathematicians in Europe did the same thing,39 even as the fabled Scientific Revolution was starting to produce all kinds of results that required an arithmetic of irrationals. Not until the late nineteenth century,* in fact, would anyone* come up with a rigorous theory or even definition of irrationals. The best definition would come from R. Dedekind,* while the most comprehensive treatment of real numbers’ status on the Line would be G. Cantor’s.
§2d.
*UNAVOIDABLE BUT ULTIMATELY IYI–GRADE INTERPOLATION
Skip the following few pages if you like, but the asterisks in the above ¶ tag stuff that historically speaking is not 100% true. To wit: A certain student and protégé of Plato known as Eudoxus of Cnidos (408–354 BCE) came very close indeed to providing a rigorous definition of irrationals, which Euclid then included as Definition 5 in Book V of the Elements. Eudoxus’s definition involves geometric proportions and ratios—which is unsurprising, given that Greek math had been confronted by irrationals in the form of certain geometric proportions that couldn’t be expressed as ratios. Following the D.B.P.’s debacle, these incommensurable magnitudes seemed to be everywhere—like consider a rectangle two of whose sides equal the diagonal of the Unit Square: how were you supposed to calculate its area? More important, how could the Greeks distinguish cases of irrational-type incommensurability from cases where you simply have different species of magnitudes that can’t be compared via ratios, like a line v. an area or an area v. a 3D volume? Eudoxus was actually the first Greek who even tried to define ‘ratio’ mathematically—
Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and the third, and any equimultiples whatever of the second and the fourth, the former equimultiples alike exceed, are alike equal to, or are alike less than, the latter equimultiples taken in corresponding order.
—the opacity of which can be mitigated by translating some of the theorem’s natural-language stuff into basic math symbolism. Eudoxus’s def. here states that, given 
and the integers a and b, 
if and only if (ap < bq) → (ar < bs) and (ap = bq) → (ar = bs) and (ap > bq) → (ar > bs). This may at first look obvious or trivial40—see for instance how it resembles the rule about cross-multiplying fractions we all learned in 4th grade. But it really isn’t trivial at all. Though Eudoxus meant it to apply only to geometric magnitudes rather than numbers per se, the definition works perfectly to identify and distinguish rational numbers from irrational numbers from immiscibly different geometric quantities, etc. Plus please notice now the way Eudoxus’s definition is effectively able to operate on a whole infinite set, viz. that of all rational numbers.41 What Eudoxus does is use random integers to specify a division42 of the set of all rationals into two subsets: the set of all rationals for which ap ≤ bq and the set of all rationals for which ap > bq. His is thus the first theorem to be able to range, comprehensively and specifically,43 over an entire infinite collection. In this respect it could be called the first significant result in set theory, about 2300 years before the invention of set theory.
It is also worth pointing out that there are probably no better examples in math of Russell’s dictum about the caprice of intellectual fame than Eudoxus and his posthumous collaborator Archimedes (287–212 BCE). The latter, granted, is anecdotally famous for his ‘Eureka!’ thing; but given our overall purposes it would be unfair not to acknowledge that he and Eudoxus more or less invented modern math, which then had to be reinvented many centuries later because nobody’d bothered to pay attention to the consequences of their results.
Probably their most important invention is known as the Exhaustion Property, which Eudoxus discovered and Archimedes refined. It was a way to calculate the areas and volumes of curved surfaces and figures, something that Greek geometry obviously had a lot of trouble with (since it’s w/r/t curves that you encounter most of the problems of continuity and irrationals). Geometers before Eudoxus had had the idea of approximating the area of a curved figure by comparing it to regular polygons44 whose areas they could calculate exactly. By way of example, see how the very largest square that can fit inside a circle functions as a crude approximation of the circle’s area—
—whereas, say, the largest octagon that can fit inside will be a slightly better approximation—
—and so on, the point being that the more sides the inscribed polygon has, the closer its area will be to the circle’s own A. The reason the method never actually worked is that you’d need a ∞-sided polygon to nail A down all the way, and even if this ∞ was merely one of Aristotle’s potential ∞s the Greeks were still stymied, for the same reason mentioned w/r/t the Dichotomy: they didn’t have the concept of convergence-to-a-limit. Eudoxus gave math just such a concept with his introduction of the Exhaustion Property, which appears as Prop. 1 in Book X of the Elements:
If from any magnitude there be subtracted a part not less than its half, and if from the remainder one again subtracts not less than its half, and if this process of subtraction is continued, ultimately there will remain a magnitude less than any preassigned magnitude of the same kind.
In modern notation, this is equivalent to saying that if p is a given magnitude and r a ratio such that 
≤ r < 1, then the limit of p(1 − r)n is 0 as n tends to ∞—i.e., 
This allows you to approach, arbitrarily closely, an infinite number of sides to a polygon, or an infinite number of rectangles under a curve, each side/rectangle being arbitrarily (= infinitesimally) small, and then to sum the relevant sides/areas by the inverse of the very process by which you derived them. The Method of Exhaustion is, for all intents and purposes, good old integral calculus. With it, Eudoxus was able to prove, e.g., that the ratio of any two circles’ areas equals the ratio of their radii’s squares, that the volume of a cone is 
the volume of a cylinder with the same base and height, & c.; and Archimedes’ Measurement of a Circle uses Exhaustion to derive an unprecedentedly good approximation of π as 
Notice also the metaphysical canniness of Exhaustion’s abstract entities. Eudoxus’s method of getting infinitesimally small sides/figures into equations makes no claims about the existence of infinitely tiny magnitudes. Look at the bland language of the Elements’s Prop. 1 above. The “less than any preassigned magnitude” is particularly clever—and strikingly similar to modern analysis’s ‘arbitrarily large/small’.45 It’s basically saying that, for mathematical purposes, you can reach magnitudes that are as small as you want, and work with them. It’s this concern with method and results rather than ontology that makes Eudoxus and Archimedes so eerily modern-looking. The way their “magnitudes less than any preassigned magnitudes” are created and deployed in Exhaustion is pretty much identical to the way infinitesimals will be treated in early calculus.
Why, then, Europe had to wait nineteen centuries for actual calculus, differential geometry, and analysis is a very long story that essentially bears out Russell’s dictum. One cause is the same reason nobody thought to apply Exhaustion to Zeno’s Dichotomy: the Greeks cared only about geometry, and nobody then thought of motion/continuity as abstractable into the geometry of the Number Line. Another reason is Rome, as in the Empire, whose sack of Syracuse and murder of Archimedes around 212 BCE brought an abrupt end to Hellenic math,46 and whose hegemony over the next several centuries meant that a lot of the substance and momentum of Greek math was lost for a long time. The most efficient cause, though, was Aristotle, whose influence of course not only survived Rome but also reached new heights with the spread of Christianity and the Church from like 500–1300 CE. To boil it all way down, Aristotelian doctrine became Church dogma, and part of Aristotelian doctrine was the dismissal of ∞ as only potential, an abstract fiction and sower of confusion, to apeiron, the province of God alone, etc. This basic view predominated up to the Elizabethan era.
END INTERP.
§2e. (Continuation of §2c from the ¶ on pp. 80–81 with the interpolative asterisks in it) Here, as a sort of hors d’oeuvre, are some of the things that G. Cantor47 eventually discovered about the nested ∞s of Zeno and Eudoxus. Discovered as in not just found out but actually proved. The Number Line is obviously infinitely long and comprises an infinity of points. Even so, there are just as many points in the interval 0–1 as there are on the whole Number Line. In fact, there are as many points in the interval .0000000001–.0000000002 as there are on the whole N.L. It also turns out that there are as many points in the above micro-interval (or in one one-quadrillionth its size, if you like) as there are on a 2D plane—even if that plane is infinitely large—or in any 3D shape, or in all of infinite 3D space itself.
More, we know that there are infinitely many rational numbers on the infinite Number Line, and (courtesy of Zeno) that these rationals are so infinitely dense on the Line that for any given rational number there is literally no next rational number—that is, between any two rationals on the N.L. you can always find a third one. Of which fact here’s a brief demo. Take any two different rationals p and q. Since they’re different, p ≠ q, which means one’s bigger than the other. Say it’s p > q. This means that on the Number Line there’s at least some measurable distance, no matter how small, between q and p. Take that distance, divide it by some number (2 is easiest), and add the quotient to the smaller number q. You now have a new rational number, 
between p and q. And since the number just of plain integers by which you can divide (p − q) before adding the quotient to q is infinite, there are actually an infinity of rational points between any p and q. Let that sink in a moment, and then be apprised that even given the infinite density of the infinite number of rationals on the Number Line, you can prove that the total percentage of N.L.-space taken up by all the infinitely infinite rational numbers is: none. As in 0, nil, zip. The technical version of the proof is Cantor’s, and notice how Eudoxian-Exhaustive in spirit it is, even in the following natural-language form, which requires a little creative visualization.
Imagine you can see the whole Number Line and every one of the infinite individual points it comprises. Imagine you want a quick and easy way to distinguish those points corresponding to rational numbers from the ones corresponding to irrationals. What you’re going to do is ID the rational points by draping a bright-red hankie48 over each one; that way they’ll stand out. Since geometric points are technically dimensionless, we don’t know what they look like, but what we do know is that it’s not going to take a very big red hankie to cover one. The red hankie here can in truth be arbitrarily small, like say .00000001 units, or half that size, or half that half, …, etc. Actually, even the smallest hankie is going to be unnecessarily large, but for our purposes we can say that the hankie is basically infinitesimally small—call such a size ϕ. So a hankie of size ϕ covers the N.L.’s first rational point. Then, because of course the hankie can be as small as we want, let’s say you use only a 
size hankie to drape over the next rational point. And say you go on like that, with the size of each red hankie used being exactly 
that of the previous one, for all the rational numbers, until they’re all draped and covered. Now, to figure out the total percentage of space all the rational points take up on the Number Line, all you have to do is add up the sizes of all the red hankies. Of course, there are infinitely many hankies, but size-wise they translate into the terms in an infinite series, specifically the Zeno-esque geometric series 
and, given the good old 
formula for summing such a series, the sum-size of all the infinite hankies ends up being 2ϕ. But ϕ is infinitesimally small, with infinitesimals being (as was mentioned in §2b) so incredibly close to 0 that anything times an infinitesimal is also an infinitesimal, which means that 2ϕ is also infinitesimally small, which means that all the infinite rational numbers combined take up only an infinitesimally small portion of the N.L.—which is to say basically none at all49—which is in turn to say that the vast, vast bulk of the points on any kind of continuous line will correspond to irrational numbers, and thus that while the aforementioned Real Line really is a line, the all-rational Number Line, infinitely dense though it appears to be, is actually 99.999 … % empty space, rather like DQ ice cream or the universe itself.
Let’s each pause privately for a moment to try to imagine what the inside of Professor G. F. L. P. Cantor’s head might look like as he’s proving stuff like this.
A canny reader here may object that there’s some kind of Zenoid sleight of hand going on in the above proof, and might ask why a similar hankie procedure and series couldn’t be applied to the irrational numbers to quote-unquote prove that the total % of Line-space taken up by the irrationals is also 2ϕ. The reason such a proof can’t work is that, no matter how infinitely or even ∞∞ly many red hankies you drape, there will always be more irrational numbers than hankies. Always. Cantor proved this, too.
is irrational, which was the way Dr. G. first presented it in class.
If you’re able to see why the sum of this particular infinite series is 1.0, it’s probably going to occur to you that the above-mentioned 0.999 … 1.0 paradox isn’t really a paradox at all but merely a consequence of the fact that there’s always more than one way to represent any given number in decimal notation. In this case, the quantity 1 can be expressed either as ‘1.000 … ’ or as ‘0.999 … ’. Both representations are valid, although you need a certain amount of college math to see why. (IYI Again, if you can keep stuff in your head for long intervals and many pages, know now that in §§ 6 and 7, G. Cantor is going to make ingenious use of this technical equivalence between 1.0 and .999 … in a couple of his most famous proofs.)
where n is irrational.
really does equal 