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CHRISTMAS DIVERSIONS: Paradoxes of Relativity from (relatively speaking) Down Under - Gravity and Parallel Lines - plus Achilles and the Tortoise

Troubled pre-amble



Well, Christmas has come around yet again, and still no-one has offered an explanation of the "equivalence" paradox (see below). Could be no-one has read it yet. Some of my pages are hard to find and recently Google ran a 4-month blackout on virtually the entire site. Can't say I blame them; it's such a diffuse, poorly linked set of pages.

tsunami damage
Disaster strikes our region, leaving stunned minds in the same state of desolation as the Aceh shoreline. While water
kills untold thousands in Asia ...
SA bush fire
...on South Australia's Eyre Peninsula fire consumes land and livestock, as well as people and their homes

The year has ended like no other - in appalling devastation and misery for thousands upon thousands of families throughout the Asian region. The toll from the tsunami has climbed over the 100,000 mark (Update: 225,000 as at 20/01/05) and thousands more are staring into the face of death as the squalor left in its wake turns into a hot-house of disease. It's going to be a testing time ahead for everybody, but especially for those unlucky enough to be the victims.

Unlucky? Already there are people saying it was an "act of the Almighty", "proof that the end of the world is drawing nigh" or "God's punishment for those causing trouble in Aceh". What gruesome cancer of the mind drives these people to believe in such an obscene, malevolent, inhumane demon, who hands out punishment at random because of the supposed misdeeds of a few? This is unspeakable nonsense of the most revolting kind. Luck (or rather, misfortune) has indeed played its part, but one wonders whether prudence and wise government could have drastically reduced the damage and desolation. The most important thing anyone expects from their government is the maximum possible security. No government can prevent earthquakes and tsunamis (or cyclones, droughts and bushfires), but they can tell us how best to avoid their fury and, in most cases, provide advance warning of impending danger.

Tragedies such as this embroil us in perhaps the greatest paradox of all. Thanks to technological progress we know quite well how most of them happen, yet we still feel mentally devastated when they do happen. And though it seems as if nature has sucked the human spirit from our bones, yet somehow we manage to renew that spirit and muster the resources to bring us through, if we are victims, or help to rebuild lives and infrastructure, if we are observers.

Anyway, let's try to put aside all this wretchedness and consider some more lighthearted paradoxes.

No doubt everyone who reads this will have heard of Einstein's famous Twins paradox, or perhaps one of it's more improbable offspring such as the tale of the travellers who go on a round trip of the universe and return to earth before they've been born. But here are two quite different "paradoxes" that may be less familiar. In fact I've never come across the first one in print, although, admittedly, this might be because it's all too easily explained away by the gurus of Relativity. If so, I wish one of 'em would show me how! As for the second (not really a paradox, just a persistent pest), it has been widely debated, "explained" and rammed down our throats, but there are still some, such as myself, who believe we're being positively hoodwinked. For this enigma I therefore offer some explaining (and extensive complaining) of my own it will certainly not appease everyone. Finally, I've recently added a "solution" to the ancient paradox of Achilles and the tortoise.

1. Ouch! - "What if Newton's apple had kept on falling?"

No, this is not the puzzle about what happens to gravity at the centre of the Earth. It's about the so-called "equivalence" principle the alleged parity of the forces due to gravitational fields and the forces due to acceleration, as supposedly demonstrated by the well known lift experiment (or, as you Americans call it, the elevator experiment). Well, here's another experiment, which we all perform most of the time. And we Aussies tend to perform it with considerable enthusiasm around this time of year!

You're more or less at rest on the surface of the Earth, right? Here, you are subjected to a gravitational force equivalent to your mass times the acceleration due to gravity, which is approximately 9.8 ms-2. But according to relativity theory, if you were none the wiser, you might just as well be in empty space, ensconced in a capsule which is accelerating uniformly at 9.8 ms-2 in a straight line. Under those conditions you would experience the same force as you do sitting around here on Earth. It's easy to calculate, however, that if in fact you were in that space capsule, after only 354 days you would be travelling at the speed of light (about 3 x 108 ms-1). And after only a short lifetime you would of course be travelling at many times the speed of light. Which is impossible, so they say.

An accelerated body
Time and tide pass by this irradiated, accelerating body, while
the cosmology book assumes a "prepare for landing" role.
No signs of weight gain here! (excepting the two with their toes
in the water)

So how come it's possible for you and me to be relaxing here on our planet, flat out on the beach soaking up the sun, cosmology book in hand, apparently breaking the light barrier every 354 days? Besides, shouldn't we be getting heavier and heavier all the time apart from the middle aged spread, that is? After about 353 days acceleration at 9.8 ms-2, according to Lorenz, I ought to have increased my weight from a paltry
67 kg to almost one tonne, discounting the grog
and increased mass due to ultra-violet radiation
landing on my back. Considering the quite
considerable number of years I've spent in
something resembling this attitude, by now I ought
to weigh many multiples of infinity kg, whatever
that might mean. And my sense of time, bad
enough as it is, should be writhing in the warpiest
of warps.

So where's Newton's apple now?! Doubtless enjoying a happy subterranean existence,
bouncing back and forth like an eternal yoyo,
wormholes and all. Well, I didn't notice it come out the other side, did you, mate?

By the way, is it mere coincidence that 354 days is also the length of the average Islamic year twelve times the mean synodic period of the moon? Hey, what's going on here?

After stimulating a few grey cells, I assume the answer to this paradox goes something like this: Since everything is relative, including all measuring systems, in any given situation the concept of how long one might have been accelerating doesn't make sense. Imagine you were born on a rapidly, uniformly accelerating spaceship that had left an unknown planet a very long time ago - say, 1000 years ago by the reckoning of the inhabitants of the planet. As far as your little world is concerned it makes no difference whether you left the planet 100, 1000 or 10,000 years ago (planet time): your measurements of weight, space and time within the spaceship are governed by spaceship conditions, which we assume have remained the same ever since the ship left the planet. Similar considerations apply to beings living in a constant gravitational field such as on the surface of a planet (though it's difficult to imagine some other situation to which this is "relative", in any significant sense). Well, whether this explanation is helpful or not, it seems to leave the universe looking even weirder than before!

One wonders about the relativity of other things. For example, if the inhabitants of another planet estimated the age of the universe, using techniques similar to those used by scientists on Earth, would they come up with a different answer to us? Then there's the Twin (so-called) Paradox, seemingly fully explained, but it's a bit worrying that there are apparently diverse explanations whose equivalence is doubtful. Indeed everything about relativity (and quantum theory)is so odd it makes one wonder whether the gurus are looking at things from the right perspective.

2. A Parallelodox - "How bent can a physicist be?"

The second "paradox" concerns the widely held view that our universe conforms to a Riemannian geometry rather than the traditional Euclidean geometry. I've got no immediate problem with that, but it leads to the challenging statement, espoused by certain philosophers and physicists, that we live in a world in which parallel lines meet. How is that possible? To us unlearned laymen with no mathematical bent (hmm!) it seems a preposterous idea.

Clearly, any geometry that contains a rule stating that parallel lines meet (or do not meet) must define parallel lines in terms other than that they meet (or do not meet). Otherwise the rule would be redundant. So of course the partisans of the view that, in our universe, parallel lines meet rest their case on what might be called a "local" definition of "parallel". This is usually some version of the idea that if two lines on a two-dimensional surface can be traversed by a third line perpendicular to both, those lines are parallel. (For instance, this is the meaning of "parallel" implicit in Prof. Jeffrey Olen's treatment of the subject in his excellent philosophy primer Persons and Their World, pp 303-306.)

But is this the meaning that you and I would give to "parallel"? I think not. We would surely preclude the possibility of parallel lines meeting because it's part of our (tacit) definition of parallel lines that they do not meet. Our definition would include words to the effect that, no matter how far you were to travel along a set of parallel lines, they would always remain the same distance apart. (In fact this is very close to the definition given in my Chambers dictionary.) So it would be more plausible for us to conclude that, in our relativistic universe, it is impossible to construct parallel lines (if indeed the geometry of the universe forces us into that position). We've been hoodwinked into believing in an unreal paradox a paradox that disappears when we realise that "parallel", as used by this party, has a much looser meaning than normal. The layman's concept of parallel lines, one might say, is intrinsically Euclidean and cannot be transferred to Riemannian geometries.

Unfortunately, our objection also needs to be sheltered against another concept of parallelism. It seems the devotees of the meeting-parallel-lines (MPL) view are in the minority. Most mathematicians would not define "parallel" in the way suggested by this clan; they would say, rather, that parallel lines are precisely lines that never intersect. They would therefore agree with us that there are no parallel lines in a Riemannian universe (the NPL view). But they would not agree that parallel lines are necessarily equidistant along their entire length. Because their definition of "parallel" is looser than ours, they still feel comfortable talking about parallel lines in the context of non-Euclidean geometries. This mode of speech stems from the original troubles with Euclid's fifth postulate and mirrors the subsequent division of non-Euclidean geometries into (1) Riemannian or elliptic (involving the axiom that through any point in a plane no line can be drawn parallel to a given line) and (2) Lobachevskian or hyperbolic (through any point in a plane there can be drawn more than one line parallel to a given line). In the latter geometry "parallel" lines diverge instead of converging (though some texts say that in this geometry parallel lines simply don't exist).

Don't know about you, but I can't live with either of these notions of parallelism. To me it seems just as farfetched to say that parallel lines can diverge as to say they can intersect. The current tide of opinion that the real world conforms to a geometry in which the NPL view seems slightly less offensive than the MPL view is no consolation. Things might have turned out differently, and might still do so. Our quarrel is not with current cosmological theory but with the abuse of the word "parallel". To keep things simple, however, I'll continue to pitch my argument chiefly against the MPL view. For, even though MPL appears to be internally inconsistent (see addendum), it embodies a technique for explaining relativistic space which is more interesting and more intuitive than the NPL approach, and which I wouldn't hesitate to use myself only I'd refrain from talking about parallel lines meeting! (again see addendum).

It's easier for me to think of the lines in, say, an infinite double helix as conforming to the concept of parallelism than it is to admit lines that intersect into this class. But of course there are other reasons why I wouldn't call them parallel the lines aren't straight and the helix isn't two-dimensional. Still, I wouldn't complain too much if a topologist referred to these lines as parallel. After all the double helix can be untwisted into a structure to which you and I could apply the terms "flat" and "straight" unflinchingly. You can't do this with the Einstein/Riemann universe.

Railway line
What kind of train runs on this railway? The lines do meet in the distance, don't they!?
(Photo: Main railway at Ilfracombe, near Longreach, Queensland)

Occasionally someone may conjure up a sophisticated (but usually half-baked) argument intended to demonstrate that the concept of real parallel lines that meet is not entirely inconsistent with the concept of real lines that are everywhere the same distance apart. Perhaps, in reality, it's not possible to pursue the lines far enough to determine whether they really meet. It might require an infinite amount of energy, or something. Or perhaps a real observer, a kind of railway worker in hell, owing to some strange relativistic effect such as shrinkage, would find that no matter how far he went along the track he would always find he had just as far to go to reach the alleged point of intersection. Or perhaps his measurements of the distance between the tracks, or the right angle extended between them, becomes increasingly warped, so that although to him the lines appear to remain the same distance apart, in reality they are getting closer and closer.

"Explanations" such as these, however, seem to me to be thoroughly evasive and unhelpful to either side, merely transferring the burden from one concept to another while at the same time creating even more paradox. (The local observer is allowed to retain the gist of his Euclidean concept of parallel lines but gets inextricably bogged when he tries to apply it). We can always remonstrate with these sophists by further refining our own definition of "parallel lines" so that it becomes tied to a thoroughly Newtonian framework. For it seems to me the authors of peculiar arguments like these are really claiming (given a few supplementary background assumptions) that the statement that the lines meet is equivalent to the statement that the observer cannot in fact walk their whole length because it requires too much energy, or that his measurement system must shrink or that right angles reduce to zero if he walks far enough, etc. And of course we would emphatically deny this equivalence. This is not to say we would discourage the attempt to find alternative explanations, no matter how weird. Somewhere there must surely be lurking a better way of handling the paradoxes of Relativity than we have at present.

Cricket call
You're joking, how could that be out? The ball was surely following an illegal geodesic. It's just not cricket!

The root cause of the parallel lines mess appears to be the gurus' insistence on calling a line "straight" when any normal person would call it "curved". It's not hard to see why they call it "straight": in a Riemannian space, or any other kind of space, a line (ie a "straight" line) is the shortest distance between two points. It just doesn't look the shortest distance in the diagrams they draw. But if you ask them why their straight lines look curved, they're inclined to answer that the Riemannian space itself is curved. This is when your hackles start to rise. You'd think any normal person, using language in a nice way, would be justified in asking "Curved relative to what?". And you'd expect a nice answer like "Curved relative to something straight!" But no these weirdos' definition of curvature is completely introverted, spurning any reference to the common external world and deriding the language of the people who inhabit it. It's just not cricket!

Furthermore, as we've already encountered, the mathematics/physics tribe want to talk about endless different kinds of curvature, ranging from doughnuts to multi-dimensional hyperbolic surfaces, complete with wormholes enough to turn your stomach into a Klein Bottle. To me a curve is a curve, whether it's convex, concave or like a cobra tied in knots. Would it be too audacious to ask the pundits why, if their wretched curves are so special, they still talk indiscriminately about curves and curvature? And why they still refer to the lines in these universes as being lines, ie straight lines? This is not only linguistic theft of the most offensive kind, it's plain deceit. Some of them, I'd suggest, are even deceiving themselves. Why can't they maintain a strictly mathematical terminology not to mention a code of conduct for example, by calling a "line" (in a non-Euclidean space) a "geodesic" (or something else, if their use of this word angers the geographers).

OK, words change their meaning over time and it's easier to say "line" than "geodesic". This would be all very well if there were no risk of ambiguity. But in the present case everyone outside the sect appears to be thoroughly confused so confused that many people think the idea of parallel lines meeting is a real paradox rather than (as it kind of looks to me) a purely verbal one.

Or is it purely verbal? What's really going on here? Somewhere along the line (damn!) the convention has developed whereby a line (geodesic) drawn within one of these exotic spaces can be called a straight line, but the space itself will then have to be called "curved". The buck has been passed up the line (stuff it!), so to speak, ie the meaningfulness problem has been shifted from the figures that can be drawn in the space to the space itself, as if no one's going to notice it sitting up there. It becomes a background presupposition, virtually a set of a priori conditions, just as Euclidean space represents a set of a priori conditions for everything we do in the local real world. This might seem not unreasonable, when you consider the gains in linguistic simplification. However, it's possible to imagine that this buck passing could go on ad infinitum. Suppose we're considering a two-dimensional non-Euclidean space, S. We'd call a geodesic in S a straight line, but we'd say S itself is curved. But now let's imagine we could put S (or some part of it) into a three-dimensional universe where S is regarded as being "flat" ("straight in two dimensions", if you like, relative to this three-dimensional universe). Suddenly, we're calling a space "flat" which just a moment ago we called "curved". And so on. This is a linguistic experiment only; I've no idea whether it makes topological sense. It doesn't matter much, because in practice nobody pursues this series. They stop just at that point when there's no reason to go any higher up the ladder (assuming it's possible to go higher).

And this is how it is with our Einsteinian universe. If you ask the pundits what's ouside the universe, they're inclined to reply: Don't be a fool there's no outside, no space, no time, just absolutely nothing. But, yes, our universe is curved! We're calling it curved even though, with a bit of imagination, we could reasonably call it flat. Curved space is supposed to be our latest a priori presupposition or should I say "brain-washing".

The problem with this notion of "curvature" is just that it is internally defined there's nothing external to which it relates, no body of conventional meaning into which it fits. It has no connection with the intuitive idea of curvature ("curved relative to something straight") and must stand on its own legs, whilst seeming to have no legs to stand on. In the end it seems just as nonsensical as the extraordinary vacuum it inhabits "no outside, no space, no time, just absolutely nothing". The ultimate paradox.

The point is this. Although at some level of the game it's easy enough to offer purely verbal explanations of the geometrical "paradoxes" of relativistic and other peculiar spaces, there comes a point usually sooner than later when any further explaining is simply denied us. To put it bluntly, we're politely told to get lost, to stop trying to imagine the unimaginable. It's a rebuff which everyone with any philosophical spirit should defiantly repel even though, for the time being, they might feel they lack the ammunition to deal with it.


I mentioned earlier that the MPL view apparently has a serious flaw: relative to any universe with an elliptic or hyperbolic geometry, the "local" definition implicit in MPL is internally inconsistent. Here's why. Presumably we can take it for granted that every line has some length (by definition a line connects two separate points), and that it makes no sense to speak of parallel lines that have no length. Now try to draw two parallel lines, as short as you like, on a curved surface (a surface that cannot be transformed into a Euclidean plane without distortion). To do this you will need to draw a line traversing, and perpendicular to, one of the lines and make the second line perpendicular to this (or use some equivalent procedure). But since (by definition) the lines have length, if they are parallel it should now be possible to draw a second traverse perpendicular to both lines and the same length as the first traverse. This is impossible.

So why do I sympathise with MPL rather than NPL? Well, for one thing, if "parallel lines" are to be admitted as a feature of hyperbolic geometry it seems logical to allow them in elliptic geometry too. Conversely, if we want to exclude them from these geometries, it seems logical that we can remove them from both with one stroke of the pen.

Much more importantly, MPL is based on a local notion of parallelism. It starts with us, with our personal outlook or "space-frame", so to speak, without regard for what might be going on in distant regions of the universe. This is surely right, both historically and psychologically. The room we are in is square, as it was square for our ancestors, and nothing's going to change that. As Kant rightly proclaimed, this is the notion of space we carry around with us on our backs. Who's to say it's not the only possible notion of space we could carry around? The space of Relativity is, exactly that, relative to our inbuilt, ineradicable squareness. The immediate inconsistencies in formulating MPL result from a kind of inwards projection of non-Euclidean "space at large" into our strictly Euclidean comfort zone, and are therefore inevitable. But the great merit of MPL is that it does, at least, start in the right place.

I believe we shall soon find reasons why our view of the world comes only through the windows of our square room reasons why our measure of all things always returns to the space we were born with, reasons why this space is three-dimensional and Euclidean, reasons to do with simplicity, rigidity, possiblity, perhaps reasons emanating from the nature of mathematics itself. For why should we imagine there's more to space than mathematics? I don't think the state of the universe a few milliseconds (or less) after the big bang is necessarily the best place to be looking for clues. Perhaps we should be looking into our minds, and especially into the soul of mathematics. That's where the most telling answers will be found.

3. Zeno's paradox - "The never ending race"

Here is one paradox that some philosophers apparently still believe to be unsolved, but which has never seemed especially paradoxical to me - I had always assumed it to have been solved many years ago. I believe I first came across both the riddle and its solution in a school calculus lesson. I don't remember which teacher explained it, but quite likely it was one Mr Blow (deceased), affectionately known as "Puffer Blow", senior maths teacher at what was then (late 50's) known as the Queen Elizabeth Grammar School (QES) in Crediton, UK. (Shortly after I left, the school became "comprehensive", a move which for inexplicable reasons I always resented, though I hated almost everything about this institution including, to a large extent, the kind of education I received there.) I well remember Puffer Blow, however, because it was he who advised me to continue with maths and physics instead of doing what I did, which was to change over to biology - a move I've often regretted. Anyway, I shall attribute this solution of the paradox to Puffer or to the other maths teacher at QES at that time - a much younger person who, as I recall, was a first-rate teacher of calculus and mechanics, but whose name eludes me (actually I think it was Brown).

The astonishing thing about "Achilles and the tortoise" is that it was invented almost 2,500 years ago by the Greek philosopher, Zeno of Elea. For those who need to be reminded of it, here it is (I can't guarantee that this is the original version, but I will in any case give another form of the problem below).

A tortoise challenges Achilles to a race. Achilles can run twice as fast as the tortoise (it's a remarkably spirited tortoise and a remarkably lethargic Achilles), so they agree to a handicap in which the tortoise is allowed to start well down the track. Off they go, and Achilles soon covers the distance to the point where the tortoise started from. At that time, the tortoise will have moved ahead by half this distance, to a new point. At the time Achilles reaches this point, the tortoise will again have moved ahead by half this distance, and by the time he reaches that point the tortoise will have moved ahead by half that distance, and so on and so on. Therefore, according to Zeno, Achilles never catches up with the tortoise because each time he reaches one of these points where the tortoise was, the tortoise has moved ahead.

But obviously, provided the race track is long enough, Achilles does in fact catch up with the tortoise and overtakes it. So what's wrong with Zeno's argument? Well, as I said, I've never really understood what's right with it, so I'll just try to explain the solution I've long been familiar with. (I'm calling it a solution for now, because I think it ought to be regarded as a solution by most people, including most logicians and mathematicians. But it doesn't solve an underlying problem that I regard as being more serious.)

Firstly, the riddle doesn't show that Achilles never catches up with the tortoise, but only that Achilles doesn't catch up with the tortoise within the time frame defined by the structure of the riddle. This is in fact a finite time, t, represented by the equation:
t = d/s + d/2s + d/4s + d/8s + .....
where d is the distance from Achilles' starting point to the tortoise's starting point and s is Achilles' running speed. This equation can be written more plainly:
t = (d/s) (1 + 1/2 + 1/4 + 1/8 + ......)
which contains the convergent geometrical series 1 + 1/2 + 1/4 + 1/8 + ...... the limit of which can be shown by calculus to be 2. Therefore when the series is infinitely expanded:
t = 2d/s.
Thus it is false logic to infer that Achilles never overtakes the tortoise. What is true is that Achilles never overtakes the tortoise within the time implied by the problem. (He draws level with the tortoise at time t = 2d/s.)

Achilles and tortoise
"I'm sick of this race - it's gonna take forever just to write the log"
(Illustration by Kevin Pease. Used with permission.)

Secondly, you must make a clear distinction between the sum of the actual time intervals in the race - which, as we have seen, is a finite figure - and the sum of the number of steps in the riddle - which is infinite, and which, therefore, would presumably take an infinite length of time to calculate if you persisted in calculating it step by step as prescribed by Zeno. There is no connection between this hypothetical, psychological time and the real time taken to run the race (or any segment of the race).

Well, really, that's all there is to it. So where exactly is the paradox? It's surely nothing like as serious a paradox as that the number of primes is equal to the number of integers! Regardless, we'll follow the trail a bit further.

Here's another, simpler form of the riddle. A runner can never reach his goal because first he must traverse half the distance to it, then half the remaining distance, then half the remaining distance and so on ad infinitum. Again the problem word is "never", which means "not in a finite length of time". Yet time is nowhere mentioned in the premisses of the riddle (the clause after the word "because"). It is, of course, presumed that you'll infer that it takes time to traverse a distance, and then you can see the problem is just the same as before. But this version is a little more transparent, because the word "never" links more easily to the infinite number of steps implied in the problem rather than to the "real" time that the runner takes to reach his goal. Once again the paradox is largely due to confusion between a needlessly assumed psychological time (to work out the problem) and the real time intervals which in fact add up to a finite length of time.

(Note:The second version of the riddle is not too good, for this reason: if a runner can never reach his goal because first he must traverse half the distance to it etc etc, then clearly he cannot even traverse half the distance, because that involves reaching a goal too. So in a sense the riddle collapses as soon as it's stated.)

Someone might still say "What this shows is that, in the race between Achilles and the tortoise, the time t = 2d/s never elapses, and therein lies the real paradox: obviously in reality t does elapse, but the formula and the logic prove that it never does". And one could say this of any activity whatsoever, and therefore of any predetermined period of time. Consider the next one minute of your life: first you'll have to live through half a minute, then through half of what's left, then through half of that and so on. Therefore the minute will never elapse and you'll spend the rest of your life (an infinity of time) living the next minute! What's more, the same applies to the next second and indeed to any infinitesimal interval of time, so you could spend the rest of your life not living at all! This shows how extreme are the kinds of absurdity the paradox implies, but it also exposes its real weakness. For, although we could answer in much the same vein as before, it could be contended that we cannot now so confidently draw the distinction between "real" time and "psychological" time. Or, to put it rather differently, the time taken to work out the problem is irrelevant: on the one hand we have a problem about time, and on the other a geometrical series which has been invoked to solve it, but which itself has nothing to do with time. In that case it's simply wrong to say that you'll never get to the end of the next hour of your life just because the problem is posed as consisting of an infinite number of steps. The word "never" can't be applied to the steps. What about the time intervals represented by the steps? Well, they get smaller and smaller, approaching a limit of zero. You do not, so to speak, take a longer time to live through each interval than the calculated time! Time lived and time calculated are one and the same and, as we have seen, the sum of the intervals is a finite length of time - in this example, one hour.

So what still remains of the riddle? Only the maths, the infinite geometrical series. Here, at last, I acknowledge defeat. Well, first let me declare a truce. I don't think I've failed in the sense that I haven't got to the bottom of the paradox of Achilles and the tortoise. For any "normal" person would see no paradox in the series 1 + 1/2 + 1/4 + 1/8 + ......, or any other infinite series. It's just an ordinary mathematical series, they'd say, and that's that. But I do see a paradox, or rather, a mystery and a deception, and it's questionable enough for me to join battle again. First let me ask you: if you can find absurdity in Zeno's paradox but not in 1 + 1/2 + 1/4 + 1/8 + ......, doesn't that make your thinking somehow inconsistent, and, if not, is it possible that somewhere along the line you've been brainwashed? For further discussion of this topic, I'll have to direct you to section 8.3 of Six kinds of proposition. (It might help to read sections 8.1 and 8.2 first.) Essentially the problem is that you can't write out (or think) the series in full; in fact you can't even write (or think) a very large part of it. So, unless you're a Platonist, it doesn't exist. I suspect much of the blame for (an all too often arrogant) modern-day mathematical Platonism must go to Gottlob Frege (1848 -1925), the founder of mathematical logicism.

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