Achilles and the Tortoise

Miscellaneous History
41

I agree that mathematics is a concise, deductive system and the paradox can be resolved within itself, but that would be in a vacuum. We are relating math to reality. The truth is that there is no such thing as numbers outside of what we write on paper and in our minds. Mathematics is a subjective system, and the truth is that while Zeno was attempting to point out that change is an illusion, he inadvertently discovered that math doesn’t really exist, and neither do measurements. When I say they don’t exist, I mean physically. There is no measurement outside of our perspective.

It’s similar to the dollar bill, as Rothbard references in What Has the Government Done to Our Money? He says maleable currency is great for society because it can be fractionalized whereas if I wanted two eggs and only had one huge wheel barrow, I wouldn’t get anywhere. In that case, 0.5 oz of gold certainly exists, but its fractionalized value is completely subjective. Example: ever tried to buy a chicken’s eggs by offering the chicken 0.5 oz of gold? I’m willing to bet no transaction would happen because economics is a subjective construction that defines supply and demand and nothing more. We all simply know the rules to the game, so-to-speak, so we succeed at playing it. Zeno is simply inadvertently pointing out that it is all a game period.

42

Equivocation is inherent in the notion that a line is “made” of points that are themselves just collections of properties, or in the relation of such constructs to anything physical - depending on how exactly the math is phrased. Instantaneous velocity is another oxymoron born out of equivocation. Which basic term is being equivocated on depends on the actual definitions used, but I am suggesting that if you look for it, it will be there in every case (if you can ever get solid definitions out of a mathematician). But I bet you’ve never looked, perhaps because you trust that scientists cannot err about something so basic and agreed-upon. That view does not have a very good track record, though.

“Point to” as in show someone which aspect of their experience is being referred to by a word, and do this unambiguously enough that you can be reasonably sure everyone understands the same thing by the word, or close enough for the current context (but then you must be very careful when changing contexts). This may be hard to do, but that’s a fact of life: communication is hard. Shirking the responsibility to communicate a theory clearly is shirking science and mathematics all together. However, my point here is not to deplore those who engage in this fanciful practice, but to point out that this is neither rigorous nor efficient.

43

Praxeology as expounded by Mises had solid definitions at its core, because readers can generally understand which aspect of their experience is being referred to (pointed to) by words like “felt uneasiness” and “pain”. Just as the tribal elders would explain that diseases are caused by gods beyond all human experience, mathematics also purports to reference the unexperienceable, such no-size points, instantaneous velocities, and infinitely thin lines.

Staying home and praying to the gods works sometimes, provided the appropriate rituals are followed that accidentally help cure the disease. Similarly, math gets away with these unexperienceable entities only because because of the notion that these things must be intuitively experienceable by those higher-up mathematician demigods, and because this witchcraft ends up working by eventually stumbling upon the right set of fudges to patch over the basic slipperiness underneath. Naturally, a whole mess of paradoxes are left in the wake of these circumlocutions.

If even a social science like economics can be made rigorous, at least at its core, why give math a free pass? It is after all quite possible to visualize dots, lines, and numbers (as groups of dots, for instance). These are all quite experienceable and unambiguous in the relevant ways.

44

Dave, with all due respect, I really don’t understand why you’re being so vague here. It almost seems to me like what you’re really trying to do is insinuate that you have a vastly greater amount of knowledge here than me, and that you think it’s in my best interest for me to shut up and go away rather than presumably embarrass myself. However, I won’t embarrass myself, because no matter how much knowledge you throw at me, I won’t feel embarrassed. So please, go ahead and make your position crystal-clear.

45

Snap!

46

Autolykos,

I am not saying you should go away. Indeed I am the one who left the field, returning only as a personal favor to you.

OK, in detail:

Right - a point is an abstract mathematical construct. It has zero extension in any dimension under consideration. Therefore, any extension can be said to consist of an infinite number of points. Yet the extension is (presumably) finite. How does this make sense?

The extension is not finite in the sense of of containing finitely many points. What is finite is a numeric value we assign to a particular set of points. For example, to the the infinite set of points between the point we call 4 and the point we call 5 we assign the number 1. To make sure we do not confuse this 1 with the point we call one on the number line, we call it Length 1. Length is thus a function from a set of points to the set of numbers. Note that many functions can be assigned to the same set of points. For example, the set of points on the number line between 4 and 5 has Length 1, but Area zero [and Volume zero].

I think it makes sense if one stops thinking of a line, area, volume, etc. as being composed of points at all.

I find it hard to see how you can do this. Consider the line connecting zero to one. Is not the point 1/2 on the line? Of course it is, as are all the other points between zero and one. This is the point of view Euclid and everyone else has adopted that I was referring to.

One can instead think of these things as separations or relations between/among points. Indeed, the very notion of location implies a relation to something else.

This sounds like it contains the germ of the idea of function I mentioned above, plus extraneous stuff [e.g that a line is not composed of points] that has been rejected by Euclid and everyone else, as above.

Geometry isn’t even necessary to think about this. How many rational numbers are there between 0 and 1? Infinitely many. Yet the extension between 0 and 1 is said to be finite. Is there a paradox here? I don’t see how.

Indeed, given the explanation above.

But Zeno’s Paradox is about something else. He is not talking about points at all, but rather about intervals of finite length. All his paradoxes notice that in some way a finite number, like 1, can be “sliced” into an infinity of other finite numbers. Thus an interval of length 1 can be sliced into the following infinite number of finite intervals: 1/2, 1/4, 1/8, 1/16 etc. And yet 1 is not the sum of these numbers, because summation can only be performed on a finite amount of numbers.

He parlays this observation into the various paradoxes he lays out.

The reason I was short is because all this, as well as the refutation of some ridiculous notions tossed around in this thread as if they were brilliant new pearls of wisdom, is standard material in a second or third year math course in a university. Which means there is a lot to explain with a lot of prerequisites on the one hand, and on the other hand nothing original or new to someone who studied math. Which is why I referred you to invest a year or so of your time studying the stuff. The topic is usually called something like “construction of the real numbers from the integers”. Anyone to whom that phrase doesn’t ring a bell [=Oh, yes, I remember we did that in some course] is missing the background needed to discuss this topic intelligently, [as evidenced by the contributions of some people to this thread].

47

Another equivocation: between “on the line” and “is part of the line.”

Disclosure: Pure mathematics was my major, I did graduate research in number theory as well. I failed to question the sneaky stuff at the time, but apparently so did just about everyone because the holes are not even well-hidden.

48

Pure mathematics was my major

Impossible. Unless you got an F in every course.

Another equivocation: between “on the line” and “is part of the line.”

Absurd.

49

With all due respect, your last paragraph seems to belie your benign intentions.

I didn’t mean that the extension is finite in the sense of containing finitely many points. I meant that the extension has a beginning and an end. An infinite extension would have no beginning and no end.

Yes, of course length can be considered a function from a set of points to the set of numbers. But here you’re necessarily talking about infinite sets of points. It makes more sense to me to consider the notion of extension to be more basic than the notion of a point, in which case the latter is derived from the former. Note also that extension covers both length and distance.

I don’t see how it’s so hard. Instead of having to consider an infinite set of points, you only have to consider two, in this case 0 and 1.

With all due respect, why should I care whether something has been rejected by Euclid and “everyone else” per se? Either something makes logical sense or it doesn’t, right?

Yet an infinite geometric series is said to result in either infinity or in a finite number. If the latter case holds, the series is said to be “convergent”. I’m sure you already know this though.

One way around Zeno’s paradox is to point out that each sub-interval can also be broken down into an infinite series of intervals of finite length. Ultimately, you have an infinite sequence of intervals each with “infinitesimal” (i.e. zero) length. This is essentially the calculus problem of integration.

As I indicated at the beginning of this post, I find it hard to not interpret this paragraph as basically saying, “Shut up, go away, and spend the next year or so studying the construction of the real numbers from the integers and related math subjects, because right now you’re too stupid to discuss this topic.”

50

I can’t believe that vive la insurrection and I are the only people on this thread that resolved the paradox so far.

51

So you say…

52

It makes more sense to me to consider the notion of extension to be more basic than the notion of a point, in which case the latter is derived from the former. Note also that extension covers both length and distance.

If you start out with a definition of length to be the difference between the numerical value of the beginning and end points [for example the length of a line from 3 to 5 is 5-3=2], then you get into difficulties. Because the set consisting of the endpoints alone [the set {3,5}] will have the same length as the set of all points from beginning to end [the set of all points from 3 to 5]. Thus, a person buying paint to paint his whole fence will need the same amount of paint as one who intends to paint only the endposts.

Point is, you cannot have a meaningful definition of length without getting into the grubby details of points.

One way around Zeno’s paradox is to point out that each sub-interval can also be broken down into an infinite series of intervals of finite length.

That does not resolve the paradox.

Ultimately, you have an infinite sequence of intervals each with “infinitesimal” (i.e. zero) length.

This is a huge confession of ignorance, AJ, I’m sorry. You never wind up with infinitesmal, or zero length.

“Shut up, go away, and spend the next year or so studying the construction of the real numbers from the integers and related math subjects, because right now you’re too stupid to discuss this topic.”

I do not think you are stupid, nor am I telling you to shut up or go away. But yes, you lack the backgriound to discuss this question.

53

I’m sorry but I don’t see how your analogy necessarily holds. Where did I say anything about creating sets consisting of the endpoints alone vs. the endpoints and the infinite number of points between them?

I’m sorry but I don’t see how.

Well how not?

You mean “Autolykos”, not “AJ”. And can you please be so kind as to once again explain yourself?

Well guess what? I’m still going to discuss it. Nothing you can say or do will keep from that. Understand? I don’t care if you think I lack the background to discuss it.

54

Well guess what? I’m still going to discuss it. Nothing you can say or do will keep from that. Understand? I don’t care if you think I lack the background to discuss it.

Sure, knock yourself out.

55

That’s right. But I’d appreciate a response to the rest of my post.

56

Sorry, you lack the background to grasp my response.

Hey, plenty of other people here know math enough to answer your q’s.

But I’ll address one point, just cause /i’m a nice guy. You were saying that 1/[2X2X2x2…] , where 2 is multiplied by itself N times in the denominator of that fraction, eventually becomes zero. But it never does. It’s an immediate result of the axiom of Archimedes, which itself is the result of the least upper bound axiom, which itself is the result of the construction of the real numbers from the integers. All of which is taught in standard math courses.

Oops. It’s an immediate result of arithmetic. say 1/A=zero for some value of A. Multiply both sides by A and get 1=0, a contradiction. Therefore there is no such A.

57

I was taught that 1 divided by infinity is equivalent to zero. How do you square that with what you posted?

58

1 divided by infinity is undefined. Whoever told you that either did not know what he was talking about, or [more likely] was telling you enough to pass a test in basic calculus that was not rigorous.

1 divided by infinity equals zero can be a very informal way of making a statement about limits, that the limit of a sequence of numbers, each of which is a fraction with one in the numerator and increasingly large numbers in the denominator, such that the numbers in the denom become arbitrarily large, is zero. But zero is never the value of any of those fractions, and infinity never appears in any of those fractions either.

59

There is no number ‘infinity’: such an equation as n=infinity is as it stands meaningless: a number n cannot be equal to infinity, because because ‘equal to infinity’ means nothing. G.H Hardy, A Course of Pure Mathematics, Chapter Four

60

I’m a math major myself, and I would like to contribute, but it’s hard to see what anyone’s really saying. Sort of reminds me of a friend of mine, who rejects the notion of infinite sets citing, “A collection of objects cannot hold infinitely many objects.” When I confronted him with

(0, 1) : = { x in R | 0 < x < 1}

Apparently to him, this was a set. He was confusing boundedness with cardinality. I’ve grown use to laymen who think they have anything useful to say about mathematics. Topics like Godel, 1/0, .999 = 1, occupy a place in their minds that far outstrips their interest to current mathematicians. But it’s clear they don’t know what they’re talking about when they remain silent on ideas more relevant to modern math like computing homotopy groups, the elegance of the Riemann-Roch theorem, or the existence of non-measurable sets as a consequence of the Axiom of Choice.