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Saturday, October 28, 2006
The impossibility of circles: a proof
Below the fold is a proof of the impossibility of circles. It’s not mine; it was invented by John Sladek; you can find it in his hilarious 1984 collection The Lunatics of Terra. Here’s the thing; I don’t see its flaw. If somebody who can see the flaw would be so kind as to point out where its reasoning goes wrong, I’d be very grateful. It’s doing my head in, a little.
Proof: that there are no circles
1. It is impossible to ‘square the circle’, ie to construct a square with the same area as a given circle using only compasses and straight-edge. To do so would involve constructing a length pi, which cannot be done.
2. For the same reason, it is impossible to ‘triangle’ the circle, or ‘pentagon’ the circle, or construct a figure if any number of sides n equal to a given circle.
3. It is also impossible to construct a figure of 0 sides.
4. From number theory, whatever is true of the number 0, and when true of some number n also true of n + 1, is true of all numbers.
5. Therefore no figure of any number of sides can be constructed equal to a given circle.
6. A circle is itself a polygon of an infinite number of sides (and infinity is a number).
7. Therefore a circle cannot be copied by another circle; all circles must be of different sizes.
8. In this figure, a big circle contains two smaller circles A and B. which meet at its centre. But (by 7) they cannot be the same size.
9. Therefore the centre of the big circle is not in its centre, or in other words, it is not a circle.
10. Since the same figure can be drawn for every circle, there are no true circles.
Comments
The parenthetical in 6 is yer problem; e.g. by 4 & 6, n+1 > n, n+1 ≠ n. But infinity+1 = infinity. (By the same logic all circles must have zero perimeter, since each side of the polygon of an infinite number of sides has length zero, and any number times zero is zero.) HTH.
Thanks: but “(By the same logic all circles must have zero perimeter, since each side of the polygon of an infinite number of sides has length zero, and any number times zero is zero.)“ Surely each side of the polygon of an infinite number has an infinitesimal size (x/infinity), not zero size?
I think I see what you’re saying. The core of the proof is the idea that ‘you can’t square, pentagon, hexagon etc a circle ...’ is a sequence that continues up to a polygon with an infinite number of sides, ie a circle. Clearly you can circle a circle; so where is the line drawn? At what point is it possible to ‘polygon’ a circle, if you see what I mean?
6 leapt out right away at me, too. Most such paradoxes boil down to division by zero or to treating “infinity” as if it’s a finite number. It’s not.
Specifically, there is no “polygon of an infinite number of sides”. We can formulate an infinite series of polygons which increasingly approach a circle as its limit, but at no point in that series will you find a polygon which is a circle.
so where is the line drawn?
Obviously you can’t draw the line at any natural number, because you can generate the next natural number by adding one. Equally obviously you can’t draw the line at “the natural number before infinity”, because there is no such number.
Conclusion (as Ray above): circles simply don’t lie on this sequence.
"circles simply don’t lie on this sequence.”
So are you all saying that the statement ‘a circle is a polygon with an infinite number of sides’ is simply not true? I’ve come across it in various places. Ditto ‘a straight line is an arc of infinite radius’. If that’s not true it would be interesting; nay surprising.
I have to say that saying ‘circles don’t lie on this sequence’ sounds to me like saying ‘Achilles simply doesn’t catch up to the tortoise’. Isn’t it truer to say that Achilles catches the tortoise at the end of the infinite sum that defines his coverage of the distance between them? And that similarly, circles do indeed lie on this sequence of increasingly faceted polygons ... at the end? Converging infinite sequences do end after all; which is to say, they can be summed. (And we’re not, here, talking about a diverging infinite sequence, after all).
I’ve pondered, and I think I see it now. We can restate Sladek’s ‘proof’ as follows.
1. It is impossible to square the circle.
2. It is impossible to ‘pentagon’ the circle.
3. It is impossible to ‘hexagon’ the circle.
4. It is impossible to ‘septagon’ the circle.
5. It is impossible to ‘octogon’ the circle.
6. Statements 1 - 5 are the beginning of an infinite sequence which can be described as follows: ‘it is impossible to create a polgyon of x sides of equal area to a circle’, where ‘x’ can always be replaced with ‘x+1’
7. A circle is a polygon with an infinite number of sides.
8. Logically, therefore, it is impossible to ‘circle’ a circle, ie to create a circle of the same size as a circle. From this all sorts of absurdities follow.
I think the flaw is as follows: 1 to 5 are all true, but they are progressively less true; which is to say, as you add more and more facets to your polygon it becomes possible to approximate the area of a circle with greater and greater accuracy. It is true that you can never create a polygon of x sides that precisely equals the area of a given circle, but with more and more sides to your polygon you get closer and closer. And at infinity (the circle) you arrive!
Or to put it another way: calculus.
It’s still misleading, at least, to say “at infinity you arrive”. Iterated addition is not the path to the infinite.
"Iterated addition is not the path to the infinite.”
I disagree. Achilles does actually arrive at the point where he overtakes the tortoise; it’s not a figure of speech. The half plus a quarter plus an eighth plus a sixteenth ... and so on, all iterated additions ... does sum to 1.
From number theory, whatever is true of the number 0, and when true of some number n also true of n + 1, is true of all numbers.
It doesn’t seem to me that a general proof that “when anything is true of some number n it is also true of n + 1” has been given. Instead there are (alleged) proofs for 0, 4, 3, and 5. Not for 1, and no general proof.
And infinity is not n+1 for any n.
S/b “when this is true of some number n it is also true of n + 1”
The perimeter length of the increasing polygon is a converging sequence; the number of sides isn’t. It is not accurate to say that 1-5 are “progressively less true"--something is either true or it isn’t. Calculus is irrelevant here--this is pre-calculus mathematics. Infinity (and the infinitesimal) are really just mathematical heuristics, and as someone earlier said, are at the core of almost all maths paradoxes. It’s a useful tool, but can lead to confusion. What I’m saying (like the others) is, you can think of a circle as a polygon with infinite infinitesimal sides (as a way of getting your head round it), but if you start thinking of those infinite sides as actual entities in a polygon, ie. a quantity, you’ll end up with problems like this.
To be fair, John, Sladek doesn’t say “when this is true of some number n it is also true of n + 1”; he says “when this is true of some number n and it is also true of n + 1 then it is true of all numbers.” That’s fair enough, isn’t it?
“And infinity is not n+1 for any n.” That’s true of course; but infinity is a number.
Well, I think that he didn’t establish the general principle that “when this is true of n it is also true of n+1”. He found some “n"s and “n+1"s of which it was true, but didn’t generalize it.
And I think that because infinity is not an n+1, it isn’t a number in the sense of the number-theory rule.
This is as far as I can go. I’ll sit and watch now. This is a question which actually does have a simple answer, I’m sure.
Oh, no it isn’t.
The half plus a quarter plus an eighth plus a sixteenth ... and so on, all iterated additions ... does sum to 1.
Equivocation with regard to “iterated addition”. One doesn’t reach infinity by adding a number to a number. Your situation differs in that one has an infinite sequence [.5**x | x <- N] and you’re summing over that. (Though I challenge you to point out where that sequence actually does add up to one, and for what value of x .5**x is zero.) Sure, the limit as x approaches infinity of the sum of .5**i for 1 <= i <= x is one. But the limit as x approaches infinity of x is infinity. That doesn’t mean that you can get to infinity (or one) that way.
Actually, John makes a good point about the induction here. You’ve nowhere actually shown that, given that we can’t construct a regular n-gon from a circle, we can’t construct a regular (n+1)-gon. Instead, what you’ve said (in steps 1 and 2) is that we can’t construct any regular n-gons for a circle, because doing so would involve constructing a line of length pi. N+1’s inconstructibility has nothing to do with n’s inconstructibility, you’ve just got a reason that’s supposed to apply for all n.
But to circle the circle, we don’t need to construct a length pi.
Circles are the regulative ideal of regular polygons.
Square this:
re: “One doesn’t reach infinity by adding a number to a number.”
Depends what you mean by ‘infinity’. Infinity isn’t an abstract or a Platonic ideal; it’s a descriptor; as for instance in the phrase ‘an infinite sequence.’ There are various kinds of infinite sequences. One doesn’t ‘reach’ infinity (or, to put it better, reach the end of) the series ‘1+2+3+4+...’ But one does reach the end of one + a half + a quarter etc. That’s why Zeno’s paradox isn’t really a paradox, and why Achilles does overtake the tortoise; as we all know he does. The sequence of polygons with 4, 5, 6 ... sides is of the latter kind, and it is summed—not potentially, but actually—in the circle. Sladek’s paradox is no more a paradox than Zeno’s; it’s a problem that is solved by something unknown to Zeno, bu which Sladek presumably knew about: differential calculus.
re: the tyre. I concede the field. You have beaten me, sir.
Sorry Conrad: missed you there (sometimes the comments feature of the Valve software plays peculiar tricks).
You say: “The perimeter length of the increasing polygon is a converging sequence; the number of sides isn’t.“
This is like saying ‘the series “half plus a quarter plus an eighth...” is a converging sequence; the numbers in the denominator portion of the fractions that constitute the series isn’t ...’
It is not accurate to say that 1-5 are “progressively less true"--something is either true or it isn’t.
What a black-and-white fundamentalist view of truth you have.
Calculus is irrelevant here--this is pre-calculus mathematics.
I disagree. Sladek’s paradox here (as I say above) is actually a variant of Zeno’s paradox, and you need calculus (or at least the understanding that some infinite series can sum in real time) to understand that.
Infinity (and the infinitesimal) are really just mathematical heuristics, and as someone earlier said, are at the core of almost all maths paradoxes. It’s a useful tool, but can lead to confusion. What I’m saying (like the others) is, you can think of a circle as a polygon with infinite infinitesimal sides (as a way of getting your head round it), but if you start thinking of those infinite sides as actual entities in a polygon, ie. a quantity, you’ll end up with problems like this.
Again, I disagree. This, to me, sounds like saying ‘you can say there’s a place where Achilles actually overtakes the tortoise if you like, but it’s just heuristics.’ We can ask this question: ‘what would a polygon with an infinite number of facets look like?’ And we can answer, with absolute precision: ‘like this ... O’ That’s not an approximiation to the answer, that’s the answer itself.
But one does reach the end of one + a half + a quarter etc.
Really? When? Does it come after you’ve added 1/128? Or maybe after you’ve added 1/5096? The function you’re thinking of is f(x) = 1 - (1/2)**x (x constrained to integer values). f(2) = 3/4 = 1/2 + 1/4; f(3) = 1/2 + 1/4 + 1/8, etc. The notion of an asymptote is of interest here.
If you think that you actually do reach 1, you probably also think that you can write all the digits of pi in one minute on a one-inch piece of paper by the following method: first, in half a second, write a half-inch “3” on the paper; then, in a quarter of a second, a quarter-inch “1”; proceed apace.
Anyway, John (I elaborated this a bit) has already pointed out that the supposed inductive proof is not inductive, and fails already based on the rule given in the first premise.
"Infinity” isn’t the biggest number; it’s off the scale of finite numbers.
If you studied more theology, you would not be fooled by this problem. Pseudo-Dionysius or Anselm could help you.
(or at least the understanding that some infinite series can sum in real time)
This reminds me of something a professor said when talking about hardness of problems: life is polynomially bounded.
Sladek’s paradox only makes sense if you buy its supposed inductive form. But the argument can be restated more transparently like this:
1. In order to construct a regular polygon with the same area as a given circle, one must construct a line of length pi.
2. There are no compass-and-straight-edge constructions of lines of length pi.
3. Therefore one cannot circle the circle.
But that doesn’t make sense. If one has a circle of radius 1, one does not need to construct a line of length pi to make another circle with its area. One needs to make a line of the same length as a given line—the radius—which is possible. I don’t understand why you think this is a complicated thing and involves calculus, since it obviously doesn’t, and people have been drawing circles since togas were in style.
(Obviously if the radius of the circle happens to pi you’ll have ended up constructing a line of length pi, but what’s actually disallowed is a ratio involving pi, so this ends up being kosher.)
Ben: “Really? When? Does it come after you’ve added 1/128? Or maybe after you’ve added 1/5096?“
You’re saying, in effect, that Achilles never does pass the tortoise. For ‘when does he do it? After he’s covered one minus 1/5096 of the distance?’ But he does pass the tortoise! Or do you disagree?
Adam K: ‘“Infinity” isn’t the biggest number’
I agree. Nowhere do I anywhere say anything so stupid as ‘infinity is the biggest number’.
“it’s off the scale of finite numbers...”
It’s off the scale of 1 + 2 + 3 + .... On the other hand it is the scale of half plus a quarter plus an eighth etc.
“If you studied more theology, you would not be fooled by this problem.”
I’d like to think I wasn’t so much fooled as temporarily baffled by this problem; and that here I stumbled on the solution. Maybe if I’d been reading Anselm I’d have got there faster; but not if reading Anselm had lead me to believe that infinity is always ‘off the scale of all numbers’.
On the other hand it is the scale of half plus a quarter plus an eighth etc.
WTF does this mean? There’s a one-to-one correspondence between the natural numbers and this sequence.
1 -> 1/2
2 -> 3/4
3 -> 7/8
4 -> 15/16
.
.
.
n -> (-1 + 2**n)/(2**n) (= 1 - 1/(2**n))
Same scale.
(Incidentally I don’t see why you keep ignoring the solution that cuts to the quick in favor of your own meliorist/asymptotic nonsolution.)
The calculus solution to Achilles and the tortoise, incidentally, isn’t discrete; it involves the real number system. There’s no such thing as calculus confined to the naturals. (AFAIK; I’m not a mathematician, but it would be, like, totally surprising, d00d.) To the best of my knowledge there are no regular 3.5-gons. Here are some solutions to the Achilles puzzle; if you cogitate on them super hard, you’ll see that they don’t provide a useful analogy.
1 doesn’t equal “infinity” just because an infinite series sums to 1. If that were the case, then every number would be infinity.
You’re the one who’s wrong here, yet you seem to be treating others as though they’re idiots.
I ought to be reading Schiller or some stuff about Hesperus and Phosphorus (had you heard that they’re the same?) but I can’t resist it when it happens to me.
Achilles doesn’t pass the tortoise. Not in the time allotted to him in the problem. The tortoise has, what, a ten-meter lead, say, and Achilles runs at ten times his speed. So after (say) one second, Achilles is at 10 and the tortoise 11; then 11 and 11.1, then 11.1 and 11.11, etc.
What interesting thing can we say about this? Not only is the distance by which the tortoise leads always decreasing, but so is the time we’re considering for each cut—first one second, then a tenth, then a hundredth, etc. An infinite number of increasingly smaller (in the right way, of course, the harmonic series isn’t like this) values can sum to a finite number.
But here’s the thing. If we just consider the first 11 and 1/9th seconds of the race, Achilles doesn’t overtake the tortoise. And if the race were defined in terms of time—who can get the furthest in 11 and 1/9th minus &epsilon seconds, say—Achilles would be the loser. Of course the race isn’t defined like that and we have no reason to consider the distances travelled in terms of this odd sequence of times in which travelled. We can just say, well, after two seconds, how far are they? Achilles is at 20 meters and the tortoise at 12. Just because we can construct an infinite series whose sum is the amount of time it will take Achilles to overtake the tortoise doesn’t mean that Achilles himself has to take an infinite number of steps.
This Achilles/tortoise thing is a total red-herring. You seem unwilling to entertain any other arguments, though, even though you’re wrong. My comment above isn’t even the first time I pointed out that there’s a 1-to-1 correspondence between your the 1/2, 3/4, etc series, and the natural numbers; however, since you completely ignored me the first time, I doubt you’ll attend any better this time.
11 and 1/9ths seconds, above, should be 1 and 1/9ths seconds.
and &epsilon should be ε. Teach me not to preview.
Infinity isn’t a number in the usual sense, of course, though it is possible to treat it as one to some extent. However, many paradoxes arise in this manner, because the things that are true for ordinary numbers just aren’t true any more for infinities. Therefore, asserting the principle of induction (4) and then asserting that infinity is a number (6) is quite wrong, if one intends to apply the former to the latter. (Induction does have an infinite version, but this isn’t it.)
Thats the main problem with the argument and, if one is being exceedingly generous (in allowing infinity to be treated as a number), the mistake is then (7), which does not follow from the rest.
One could carry on, I suppose, and note that (8) doesn’t follow from (7) either, since the claim that infinite polygons have a different number of sides doesn’t obviously imply that they have different areas. The idea is probably that area is a function of the number of sides....but one would need to show the formula for an infinite number of sides first.
Just to add that the above is probably overly generous. Saying that a circle is an infinite polygon would really require a great deal of justification to see how far it really is a descriptive statement. As far as I’m aware, statements that like that in math tend to be more heuristic than precise. I’m not saying it can’t possibly be made precise, but a rigorous exposition would probably remove certain equivalences between the finite and infinite case.
Even if we let “infinity” be a number (i.e., the “largest” natural number, or the limit of 1, 2, 3, ...), there’s still a difference between this “countable” infinity, which the “proof” seems to be talking about, and the nondenumberable infinity of points in a circle, which it seems also, i.e. equivocally, to be talking about. I think.
(Enlighten us, o mathematicians, lest we make your heads hurt even more than they do already.)
What Armando said.
It might be interesting to think about how familiarity with calculus makes people more susceptible to fallacious arguments involving infinity. Calculus gives us a non-contradictory way of defining a limit of a converging infinite sequence, and this limit usually turns out to be “sane” in the sense that it often shares the properties of the sequence members that we’re interested in. This builds expectations, and so one expects that a sequence of 1,2,3… have a “limit” which would be the “number infinity”, and one wants then to be able to work with it just as easily as one’d work with a point that is a limit of a converging sequence of other points. But it doesn’t work.
Adam Kotsko: “You’re the one who’s wrong here, yet you seem to be treating others as though they’re idiots.”
ben wolfson: “You seem unwilling to entertain any other arguments, though, even though you’re wrong [...] once you completely ignored me the first time, I doubt you’ll attend any better this time.”
Well, this thread on math has at least comfirmed one point: Adam Kotsko + any other denizen of The Weblog + The Valve = tedium.
Really, this is final proof that to you, the topic doesn’t matter. You could be arguing about math with an English prof, for god’s sake, with all involved knowing little and having less at stake, and you’ll still follow the same predictable course: insistance that you’re right, petulance when this isn’t acknowledged (rightly or wrongly, who cares), finally, the flamewar that you so desperately have been wishing for.
Stop throwing temper tantrums and grow up.
True story: on receiving notification that RP had commented on this thread, I had some sort of negative thought the exact import of which now escapes me. It was something like “at last, now [something will happen]”. Maybe it was that now we can have the flamewar that, actually, I haven’t been wishing for at all. (I’ve been wishing for an acknowledgement of the points I made, which seem to me worthwhile.)
I can only assume that that “you” above isn’t actually directed at me, since I barely ever comment here at all, and I actually am commenting specifically because of the individual topic and discussion. Neither Kotsko nor I have thrown any temper tantrums; Kotsko isn’t being petulant (unless, of course, you’re set on interpreting him as petulant).
This probably does prove, though, that the topic, whatever it may be, doesn’t matter to you, so long as it provides an excuse to accuse Adam or Anthony of something despicable. Like, say, pointing out that someone incorrect is incorrect. I mean, Christ, Rich, I usually (mentally; I don’t usually take part in the comments here, remember) side with you against Anthony, but you really are pretty disagreeable.
Finally, the word is spelled “insistence”.
I really think that the intellectual world could do with much less comity, and the stalwart folk at The Weblog have understood my message.
Seriously, I think that Valve commenters are unnecessarily cautious and respectful. Why? It’s just liberal arts bullshit anyway, and it will all come out in the wash.
Herakleitos blamed Homer for saying: would that strife might perish from among gods and men! For then, said he, all things would pass away.
The Valve = Comity Central?
Them’s fightin’ words, John Emerson, fightin’ words.
Yeah, you’re not wishing for a flamewar at all, “Ben”. Let’s check The Weblog: gee, one hit. Oops, here’s a more recent one, written well before your strange idea above that a flamewar might be upcoming. It’s completely in jest, I know.
And you really think that there’s nothing petulant about Adam Kotsko saying that Adam Roberts is treating others as if they’re idiots—because he has the gall to repeatedly disagree, apparently—and your own “I doubt you’ll attend any better this time”? I really hope that you’re trying for a flamewar, because your lack of self-knowledge is pretty sad otherwise.
Oh, Rich, shut the motherfuck up—Adam Roberts is just fucking wrong. Your thesis doesn’t work in this single case. So just shut the fuck up.
Adam Roberts is just fucking wrong! Hey, Adam K! When all you said was that I was wrong, I disagreed. But to say that I’m fucking wrong ... well obviously now I see the error of my ways.
You think I’m wrong because (to quote you) “1 doesn’t equal “infinity” just because an infinite series sums to 1. If that were the case, then every number would be infinity.”
I won’t use the f-word in reply, so there’s no need for you to take my point: but I certainly never said that 1 is equal to infinity. Had I said that I’d be wrong. But I didn’t. I said that the infinite series half plus a quarter plus an eighth ... etc., sums to 1. And it does. Putting the word ‘infinite’ and the number ‘1’ in the same sentence does not mean I’m saying that infinity equals one.
Ben said: “(Incidentally I don’t see why you keep ignoring the solution that cuts to the quick in favor of your own meliorist/asymptotic nonsolution.)
Ben’s solution is:
Sladek’s paradox only makes sense if you buy its supposed inductive form. But the argument can be restated more transparently like this:
1. In order to construct a regular polygon with the same area as a given circle, one must construct a line of length pi.
2. There are no compass-and-straight-edge constructions of lines of length pi.
3. Therefore one cannot circle the circle.But ... if one has a circle of radius 1, one does not need to construct a line of length pi to make another circle with its area. One needs to make a line of the same length as a given line—the radius—which is possible. I don’t understand why you think this is a complicated thing and involves calculus, since it obviously doesn’t, and people have been drawing circles since togas were in style.
This is true. But the nub of the paradox ... well, it’s not really a paradox, I agree: and Ben’s explanation is a very elegant way of describing how one can circle a circle; but the sticking point for Sladek, I take it, is: four-sided polgyons cannot be drawn to equal the area of a cirlce, nor five-sided, nor six-sided; nor n-sided. So at what number ‘n’ does this become possible.
This seems to me a version of Achilles and the tortoise, ie akin to asking the distance between the runner and the animal is always 1/n**2, so at what value of ‘n’ does he actually overtake the tortoise? It’s not a question that can be answered with a number, any more than a finite number can be substitued for n in the n-sided polygon example. But with each larger n Achilles gets nearer and the polygon area-approximation gets closer. It’s an infinite series, but it can be summed.
So; the quick to which your ‘solution that cuts to the quick’ cuts is the question ‘how do we circle a circle?’ It doesn’t answer the question ‘what is the value n for the n-sided polygon that can be constructed so as precisely to equal the area of a circle.’ For that I prefer the answer that n = infinity; but if you don’t like to think of a circle as a polygon with an infinite number of facets, then I can see you wouldn’t go for that. Perhaps we can agree to disagree?
Ben also said: There’s no such thing as calculus confined to the naturals. (AFAIK; I’m not a mathematician, but it would be, like, totally surprising, d00d.) To the best of my knowledge there are no regular 3.5-gons.
That’s right; there are no 3.5-gons. But I wasn’t positing any. I was positing a series n > n+1 sided polygons that goes on forever.
Here are some solutions to the Achilles puzzle; if you cogitate on them super hard, you’ll see that they don’t provide a useful analogy.
I followed the link; to the wikipedia article on Achilles and the tortoise. The I cogitated a bit.
So: the series of increasingly-sided polygons approximates to the area of a circle with increasing accuracy (just as Achilles grows increasingly closer to the tortoise). But it’s an infinite series! But, no, that’s alright, because it’s a converging infinite series. From the link you kindly provided:
An infinite geometric series is an infinite series whose successive terms have a common ratio. Such a series converges if and only if the absolute value of the common ratio is less than one ( | r | < 1 ). Its value can then be computed from the finite sum formulae.
I like that last bit so much I’ll repeat it: “Its value can then be computed from the finite sum formulae”.
I could add: I’m gobsmacked that this trivial little post has generated such heat. Gobsmacked and intrigued.
Yeah, you’re not wishing for a flamewar at all, “Ben”.
I’m glad at least one person has believed my claims that “ben wolfson” is not my real name. The evidence you adduce that I have it in for you, Rich, is frankly rather weak.
Adam, at the moment I don’t trust myself to be sober/awake enough to read your comment comprehendingly, so for the moment I pass it by in silence.
I’m gobsmacked that this trivial little post has generated such heat. Gobsmacked and intrigued.
I’m not followed this little comity-fest closely enough to be properly gobsmacked by it, but I am intrigued at the way some converstions just blow up. Why? In the case of discussions of, say, Theory, or religion, it’s easy to see why they might get out of control. The subjects are complicated and controversial, inherently murky, and people have a lot invested in them. But squaring the circle?
As I mentioned in another thread, I’m on a trumpet discussion list and there some subjects often generate out-of-control discussions. Discussions of two well-known trumpeters—Wynton Marsalis and the late Maynard Ferguson—can sometimes get quite heated, but discussions of other players almost never do.
And then there’s equipment, the subtleties of trumpet manufacture: Does goldplate make the tone warmer than silverplate? Do heavy valve caps help the intonation? And here’s one, what about cryogenic freezing? The idea is to immerse the horn in liquid nitrogen for 20 minutes or so to “realign the molecules,” resulting in a trumpet that plays better. There’s a bunch of phenomena which one listmember have come to be perjoratively labeled as trumpet voodoo. The question that generates all the heat is whether or not any of these things have any noticeable effect on the trumpet’s sound or playability. Those discussions often get to the point where the list moderator stops them.
But why? Physical tweaks for trumpets is not an earth-shattering matter, though it obviously is of some importance to the musicians involved. Can’t really see an paradigm differences here that get in the way of rational discourse. Though, in a way, what happens is that the discussion of this or that treatment or implement or design feature becomes a discussion of “rational science” (saying doesn’t work) vs. the musician’s ear (saying that I can hear the difference). And that discussion looks rather familiar, doesn’t it.
Bill: you put your finger on it. That’s the interesting thing here, not the ostensible rightness or fucking wrongness of my thinking on infinity. (For on that matter, honestly, who cares?)
Maybe people take ‘infinity’ to be a code word for God, and get worked up as in the same way they might if the subject under discussion were theological? But then again, my other post on infinity actually brought God into it, and nobody seemed to get so cross then.
Ben: Just to say, should you find the time to read my comments (and I don’t necessarily expect that you’ll be minded to) I’m not trying to wind you up. I suspect that our disagreement is founded on differing attitudes to infinity. Some mathmos believe infinity to be quite outwith reality; others believe infinity to be part of reality. For some the statement eg ‘a straight line is an arc of infinite radius’ is a meaningless statement; for others a good way of describing an straight line. Let’s say you’re an example of the former and I of the latter, school. I’m not trying to convert you to my way of seeing infinity. I respect your view of the matter, even if I don’t share it. ‘Let’s agree to disagree’ is a sincerely intended statement, not a rhetorical trick. (Unless you think my view of the matter is so pernicious that it’s liable to corrupt and deprave innocent children, or something. In which of course you’d decline my offer to agree to disagree).
It’s not surprising if that bastard Marsalis makes people mad.
What’s curious about the Marsalis discussions, John, is that it’s difficult to take a moderate position. Some people think he’s the Devil’s Own while others think of him as the Second Coming—which is how Ken Burns presented him in that treacly jazz documentary of his. If you try to say that, yes, he’s a superb trumpet player but. no, he’s not an innovator of consequence, people at both ends of the polariztion pile on.
I’m with the late Lester Bowie.
Marsalis seems to represent the final fossilization of jazz, when it becomes a subsidized form of official entertainment where you go to show your pearls. Classical music is in terrible shape because the people who go to it don’t like anything written after about 1880, but it limps on with public money. A pre-1880 Golden Oldies orchestra would do very well.
Adam R.: “I could add: I’m gobsmacked that this trivial little post has generated such heat. Gobsmacked and intrigued.”
As I wrote above, the ostensible subject of the post has nothing to do with it. People used to call these the Theory wars, as if the flamers cared about Theory, when what they really care about is group-oriented flames.
Look at Adam K.’s apparent logic above. Adam Roberts is supposedly not just ordinarily wrong, but just fucking wrong, therefore Adam Kotsko can voice annoyance at his *behavior*, saying that he’s treating people like idiots. Can you picture what kind of person would say this to, say, some prof in the theology lounge opining about entomology? “Hey, you’re just wrong that ants hibernate in the winter, stop treating us like idiots”. And ben wolfson, meanwhile, has decided that while Adam Roberts responded to his comments, he didn’t respond up to ben’s standards: “You seem unwilling to entertain any other arguments [...] however, since you completely ignored me the first time, I doubt you’ll attend any better this time.” Can you picture what kind of person would say that to the aforementioned prof?
Not that I think that they are actually such twits as those comments, out of context, would imply; you have to take Adam Kotsko’s obvious annoyance at his history of prior argument with Adam Roberts into account, and the recent post on the Weblog that I link to above, which functions as an all clear signal for the group.
I mean, Adam K.’s actual attitude is clear, and it has nothing to do with infinity. Let’s see, what’s the most recent hit on my name at the Weblog—here it is:
“But even blogfights aren’t grabbing me like they used to. Against my better judgment, I’ve been having a back-and-forth with Rich Puchalsky, but my heart’s not in it and I think we both know it. How will I procrastinate without blogfights, though? Do I seriously want to go the rest of my life without feeling that pointless anger? Without being deeply annoyed at people I’ve never met?”
Adam K., that’s cute and all, but I have no interest in appearing as part of your pseudo-therapeutic narrative. Can’t you just get over this? No habitual poster on the Valve that I know of comes over and starts to pointlessly flame Weblog threads. People here, whatever you might think of the quality of their argument, are genuinely interested in discussion. Why don’t you go flame somewhere else?
Sure, jazz is over and it’s not coming back, ever. Marsalis is chief curator of the museum. And he’s become a more moving performer than he was 10 or 15 years ago.
Rich, I’m going back to my old policy of never talking to you.
I could add: I’m gobsmacked that this trivial little post has generated such heat. Gobsmacked and intrigued.
Why? The formula is simple and well-established enough: First, express ingenuous curiosity; then, forever after refuse to accept any sincerely proferred answers. Wouldn’t you be irked by a student who treated you that way? ("I’m sorry, Dr. Roberts, but I’m just not convinced....")
Or would you just flunk them with a sigh of relief?
Myself, I wasted some time yesterday morning drafting a second reply before I realized it would more fun to move to a math site and explain who wrote Shakespeare’s plays.
I would like to echo what Ray is saying, but also add that such behavior is especially frustrating coming from a person who once had a series of posts about how people or books were wrong (italics in original). So seeing all the contortions Adam R. was going through in order to avoid conceding a point to anyone (apparently developing a whole new branch of set theory so that he wouldn’t have to accept Wolfson’s obviously correct solution)—well, it’s frustrating, to say the least.
I would speculate on Adam R.’s motives, but the only people who are allowed to be psychologized in Valve comment threads are me and Anthony.
"Adam” is an ill-starred name, no? First bringing sin into the world, and now this.
the only people who are allowed to be psychologized in Valve comment threads are me and Anthony.
For the record, I welcome offers of free therapeutic analysis. All Bay Area psychologists covered by our insurance have long since been fully booked by rich people. (And god help anyone who needs a dermatologist for health reasons....)
What’s nice about the therapy Rich offers is that you know that it’s not all just academic for him—he’s out in the trenches, devoting his life to real activism. It gives his psychological assessments a “real world” credibility that most personality analyses in a blog comment context sadly lack.
Sure, jazz is over and it’s not coming back, ever.
Tell it to Ignaz Schick.
Argh. Okay, I’m only a historian who’s twenty years out of high school geometry, but look, just because you can’t construct a polygon with pi-fractional side lengths using a ruler and compass doesn’t mean you can’t construct a circle. That’s what the compass is for.
Never mind all this argufying about Achilles’ tortoise and Zeno’s paradox and whether infinity is a number (and which regular Valve commenter is behaving more badly)—by starting with a system that includes a tool for constructing circles, Sladek’s whole proof assumes what it’s trying to disprove.
(Isn’t there a real mathematician in the house? Damn two cultures… C.P. Snow must be is spinning in his grave...)
your problem is the interaction of 4 and 6. there are plenty of perfectly good theories that involve infinite numbers, but the traditional principle of mathematical induction applies to all natural numbers, excluding any infinities. (there is a principle of transfinite induction, but it’s inapplicable in this case.) of course, the notion that a circle is a polygon with infinitely many sides is, at best, in need of clarification, but before we get there, proving that 0 has a property, and proving that if n has the property then so does n+1, isn’t sufficient to prove that infinity has the property. (consider that 0 is finite, and that for all n, if n is finite then so is n+1, but that infinity is pretty clearly not finite.)
I’m a real mathematician, David, or at least what passes for one in this crowd. At any rate, it’s my only academic credential.
OK, in honor of David Moles, who I admire, and in hope of keeping Sokal the hell away from here, I will again play the gullible sincere idiot and I will post my drafted second reply:
I can easily believe that Adam has encountered statements like “a circle is a polygon with an infinite number of sides” and “infinity is a number”, but [rant about pop science goes here].
Some branches of mathematics include concepts called “transfinite numbers”, true. Those concepts do not follow the rules of normal arithmetic and they are not compatible with construction of a polygon.
The circle is the limit of the polygon sequence, true: that is, the series of polygons is bounded by the circle. However, by definition it’s a hard border. No matter how long you draw polygons, you’ll never find yourself drawing a circle.
And Achilles wouldn’t have overtaken the tortoise if he’d actually been condemned to take a series of infinitely diminishing steps one at a time. Luckily for Achilles, and (given how bad-tempered Achilles was) for the tortoise as well, Zeno provided a misleading description of the race. Since the series takes place across infinitely diminishing intervals, it converges within a finite time.
Postscript: David, it’s true that Sladek’s sub-Zeno paradox is trivial to circumvent via circumferancing, just as Aristotle found it trivial to refute Zeno and Dr. Johnson found it trivial to refute Bishop Berkeley. But the assigned problem is to refute them on their own terms—deductively rather than empirically.
It seems to me that the problem lies with the fourth premise:
“4. From number theory, whatever is true of the number 0, and when true of some number n also true of n + 1, is true of all numbers.”
To be true, this should read: “..all finite numbers,” for it is clearly not true of infinity. For instance, the property “can be reached by counting either up from -1 or down from 1” is true of 0 and, when true of any particular number n, is also true of n+1. But it certainly is not true of infinity, which, virtually by definition, cannot be reached by counting.
If we leave P4 uncorrected, then the argument is valid but unsound. If we correct P4, then the argument is no longer valid. Either way, we don’t reach the conclusion.
Adam Roberts: I suspect that our disagreement is founded on differing attitudes to infinity. Some mathmos believe infinity to be quite outwith reality; others believe infinity to be part of reality.
I don’t think that’s where you and Ben disagree, if I’m understanding you both correctly. You characterize the nub of the paradox as being:
four-sided polgyons cannot be drawn to equal the area of a cirlce, nor five-sided, nor six-sided; nor n-sided. So at what number ‘n’ does this become possible.
But the point is that Sladek’s paradox rests on the implicit assumption that, if infinity exists, it must be somewhere in the sequence 5, 6, n, n+1, ... . But infinity can’t be in that sequence without leading to contradictions; so, if infinity does exist, it must not be in that sequence.
You can be an intuitionist, and deny that infinity exists, or you can be a cantorian, and deny that infinity is the successor of any number. What you can’t (consistently) do, but what you seem to what to do, is hold both that infinity exists, and that it lies in the sequence of natural numbers.
Rich, I’m wondering who these “profs” are you keep talking about? (It’s a “point” you’ve raised before - and it seems to point to an unease, if I may, with the denizens of The Weblog being “students” while many at The Valve are “profs.” That is, a strange appeal to authority.) I’m just wondering where this mythical university is where students treat professors with undue respect, professors treat students with honor, no one swears, and, when there is a dispute, the prof is given the benefit of the doubt.
Obviously, you are not required to answer because it is none of my business, but have you done graduate work in the social sciences or humanities in the English speaking world? (And that includes teaching or supervising graduate students.) I’ve been a graduate student long enough to be tired of being a graduate student, and I’ve not once seen this comportment you make constant reference to. But then, maybe a lot has changed since you were in school. Swearing at a “prof” - especially one whom you’re on a first name basis with or one whom you have frequent casual conversations on matters unrelated to specialties - is quite common everywhere I’ve been. Mind you, telling you external examiner at defence that they are “fucking wrong” is not the best idea, but then, The Valve isn’t an examination room.
(I was under the impression that The Valve thought of itself in those mythical terms of the Enlightenment: as a salon where titles are left at the door and everyone equally prone to attack.)
Craig, I’m happy to consider Adam Roberts the student in this case. I was not doing so previously due to argumentative charity; the idea of Adam Kotsko snapping to one of the undergrads in his classes “you’re answering as if we were idiots” or ben wolfson telling a student “I doubt you’ll attend any better this time” would make it even clearer what kind of person would say such a thing. It is marginally more sympathetic for a student to say this to a teacher than the reverse.
Or, for that matter, it’d be interesting to see what Ray Davis would do if he were irked by such a student. I’d guess that something like the following would be forthcoming, “Jesus Christ, haven’t you gotten it *yet*? First you ask a stupid question, then you refuse to accept my sincerely proferred answers. You must be deliberately pretending not to understand, in order to infuriate me! Well, I’ll try one last time...” We’ve probably all observed teachers like that.
But of course you can pitch this “RP thinks of us as bad teachers” in the same way as you obviously tried to pitch “RP thinks of us as uppity students”. If you want me to use the terms of a salon of equals, I’d say that these are the answers of boring, not very witty people who shouldn’t be invited back, not until they learn how to do something other than sputter angrily.
Oh, and Adam Kotsko—I wasn’t even psychologizing, I was mostly just quoting. If you write about how you want to procrastinate, and feel attracted to blogfights that you know that you shouldn’t really be in, and sign my name to the thing, then don’t be surprised if I bring it up later.
I will quietly note that Rich is, typically enough, attempting to completely redirect the conversation.
That being the case, I hereby christen Rich Puchalsky “Ricky Red Herring.” I revoke my policy of never responding to him, solely so that I will be able to use this nickname.
Rich, all the same, Adam K. and Ben aren’t here as students and Adam R. isn’t here as a professor. I don’t see what any of them would do in class or at a seminar or conference has to do with anything being discussed; viz., what is wrong with the offered proof and why the answers provided by AK and BW have not been deemed acceptable by AR. Whether AR, AK or BW are good students or good teachers - or, again, terrible students or terrible teachers - isn’t relevant to the issue at hand. And it isn’t clear why you frequently make recourse to such an argumententative strategy, except, perhaps to provoke AK or APS. Whether they learn or teach well is as relevant as whether they watch TV well, are good at shitting, or are half-assed sexual partners. (I can only assume that all protagonists in the discussion shit, fuck, and watch TV - just as all the protagonists might be found in a classroom or at a conference.)
I won’t stop using your full name, Adam Kotsko.
And there is no conversation to redirect. You, ben, and Ray insist that you’ve provided the obvious answer and that it should all be over. I’m now redirecting the phase in which you abuse Adam R. for not accepting your obvious answer—mostly because you annoy me.
The one thing I admire about you is your obvious loyalty. When you take someone’s side, you’ll defend them to the death. We’ve seen it with John Holbo, with Scott Eric Kaufman, with Adam Roberts—you’re a devoted and faithful ally. And so, Ricky Red Herring, it’s totally appropriate that you are bracketing the question of whether Adam Roberts is wrong (which he is, on an a priori basis) and shifting the focus to something where he has a chance of looking better—i.e., the relative lack of civility, use of swear words, etc., of people arguing against Prof. Roberts.
His stubbornness in refusing any correction on something he’s factually wrong about—totally beside the point! He’s being persecuted by me and my “attack dogs!” (As if that Codpiece post suddenly unleashed the floodgates of the Webloggian-LongSundayan alliance.) So typical of this crowd, treating people with such disrespect… We’ll probably make really shitty teachers someday.
Let’s keep up the conversation, though—maybe we can finally reach comment number infinity if we keep adding one and one and one…
http://en.wikipedia.org/wiki/Tortoise
argumententative strategy
“Here’s a line I’d like to try…”
Seriously, when poor tender lambkins are delivered unto me for (student) teaching, I think I’ll be able to interact with them as appropriate. Maybe not! I’ll let you, Rich, know if I get drummed out.
Alas, I’ve not been following this little battle in any detail, so I’ve not got a clear sense of the “sides,” either with respect to people or with respect to the issues. I do gather, however, that the nature of infinity is one of the points of contention, and that some people have a substantially deeper grasp of the mathematical notions than others. To the extent that infinity is at issue and that some understand it better than others, this is not, I repeat, is NOT, a salon of equals. Those who don’t grasp the math should at least consider the possibility that they might learn something by submitting to the authority of those who do know the math.
Assuming that those people are reasonably competent expositors of that understanding—an assumption that may be false—this means that the others really ought to LISTEN VERY CAREFULLY and, if what they hear doesn’t make sense, then WORK VERY HARD to change their way of thinking so that it does make sense. That’s how education works, students submit to the authority of teachers, even though it is onerous. And the teachers pledge that, if the students stick with it, it will all become clear eventually. If the teacher has no choice but to answer to every little doubt from the student at every point, then it is unlikely that learning will take place, or that it will take too freakin’ long.
To one of those who know the math, you might want to run through Cantor’s diagonal argument. I can follow a good exposition, but I’d hesitate to attempt one myself. I bring it up, however, because that’s what gave me half a clue as to how strange infinity is.
You want substance, Adam K.? OK. Here’s how Adam R. asked his question:
“Here’s the thing; I don’t see its flaw. If somebody who can see the flaw would be so kind as to point out where its reasoning goes wrong, I’d be very grateful. It’s doing my head in, a little.”
So what was Adam R. asking for? Clearly he knew that the proof was wrong and that circles can really be drawn. What he was asking for was for someone to point out a flaw *that he could understand*. So he settled on something that involves a sort of intuitive understanding of calculus and the approach to a limit. Sure, Ray jumped in early to point out the problem with 6. (and Adam K. and ben w. did also, in their less numerate ways which at times didn’t show so much knowledge of math either as with the 1 = infinity thing, and nnyhav gave a varient in comment 1). But Adam R. didn’t like that solution—it didn’t click for him. OK, this tells me that he’s not a math major. Well, I knew that.
Now, at that point you can either accept that you’ve given a correct answer and the conversation is over, or you can try to explain it some other way to Adam R. This is really an educational problem, right?
So you’re approaching it with stuff about Adam being a priori wrong, and why won’t he accept your answer, and Ray makes the explicit comparison to students who won’t accept the answer they’re given. OK, that’s poor teaching. You can certainly give up on Adam R. as a student if you want. But to pretend that he’s, like, purposefully refusing to acknowledge that you’ve scored on him? Pathetic.
Now, at that point you can either accept that you’ve given a correct answer and the conversation is over, or you can try to explain it some other way to Adam R. This is really an educational problem, right?
By my count I “tried to explain” my thinking in at least seven comments. It’s not, so far as I can tell, that Adam R. “didn’t like” any of the proffered solutions—that they didn’t “click” for him—he thought they were wrong, and proceeded to tell everyone that they were wrong, not only about the nature of infinity, but about the nature of the supposed proof as well (it’s not inductive!). Adam R makes reference to Zeno; I try to explain why that’s not relevant. He insists it is, in a way that makes it clear he didn’t actually read the comment I left saying why it isn’t; Ray says again why it isn’t.
However, as I see the wisdom of Adam’s position, which he doesn’t really carry out, of not responding to you, I will not respond to you further. Would that there were Valvular killfiles.
Not to overload everyone with all the performatives I am enacting today, but I hereby do the following:
1. Revoke the nickname for Rich, because its level of humorosity dropped off precipitously after its initial coinage; and
2. Return to my previous policy of never talking to Rich anymore.
Rich has, of course, remained banned from commenting at my blog throughout this difficult process of deliberation. If I could come up with some way of banning him from reading my blog, I would do that, too, because he keeps taking all my most tender intimate secrets and turning them against me! I feel violated. Dirty.
ben w: “he thought they were wrong, and proceeded to tell everyone that they were wrong”
Actually, no. Read Adam R.’s comments above; you won’t see a single instance of “you are wrong”. You’ll see a lot of “I disagree”. Which, I know, is horrible. He just wasn’t listening to you, or he would have agreed, obviously.
And Adam K., I’ll make you a deal; if you don’t post about threads here, or about me, on your blog, then I won’t read it.
LOL - This is a thread for the ages.
Yes, blah, the ages from four to eight.
Rich, my analogy was not to a student who was genuinely making an effort to understand and not yet succeeding, but to a student who “forever after refused to accept” what they heard. As you should know from our own interactions, I can spend a great deal of time and energy on explanation. But I don’t see any point to that expense if the other party is only in it for the fun of watching me waste my time and energy. Remember, I’ve been suckered by Adam R.’s wily European ways before....
And I can agree in principle that it’s a fun game with a long proud tradition behind it. It’s just not my game. I’m easy to pwn; all you have to do is ask.
This may be utopian of me, but I refuse to believe a phrase like “wily European ways” requires a smilie. Since we’re in Valveland, though, maybe I’d better go on....
Based on the thread up through the report of Adam’s gobsmacked state, I saw two possibilities. (I didn’t know if either was actually the case, but they’re the only two I saw.)
First, that Adam was having us up, winding us on, etc. (This wouldn’t have entered my mind if it weren’t for my “Contra” burn.)
The second was a scenario in which I’ve often played the role of exasperating student. It’s the one where someone keeps insisting on their right to an intuitive grasp of something that they haven’t put the necessary work into. At some point, it’s not enough to claim “I find this really interesting”—you have to actually do something about it. I’ve forced plenty of experts to tell me to hit the books, jack, and I’m embarrassed to say that they’ve always been right.
But I see my old friends Hotsy and Totsy. Let’s listen in:
HOTSY: “I’ve read a fascinating proof that Carlos Santana is the King of England. Can you spot the flaw? [...] 6) God ordained that a nation’s sovereignty passes exclusively through the legitimate male line.”
TOTSY: “There’s your problem. That argument was only used by certain parties at certain historical periods, and the English nation firmly decided a long time ago that other contigencies were more pressing than maintaining that partisan propaganda—for example, when maintaining it might put a follower of the Roman Church on the throne.”
HOTSY: “Odd. I’m certain I’ve heard of the ‘divine right of kings’. And what’s all this about ‘churches’? Clearly there’s only one God.”
TOTSY: “You know, I think I’ll visit Clio and talk about quantum physics.”
But, as usual with these long threads, we’ve long since left the purported subject behind. As it happens, between the “gobsmacked” remark and Rich’s “redirection”, another four explanations of the paradox were posted. Adam, did you find any of them acceptable?
Ray, I understand your point, I think (although I have no idea what your previous windup by Adam R. was supposed to have been.) But let me go through the branches of your decision tree:
1. Adam is winding you up. OK. Then don’t be wound up, if you don’t want to be. Do something else.
2. Adam is an exasperating student who wants an intuitive grasp without putting in the hard work a la Bill Benzon’s description. Well, in that case, either give up, or try to find another way to explain. If you want to lecture Adam about his study habits in this regard, you can Email him.
But neither of these branches are the ones that were being followed by people other than you, really. It was:
3. Adam is just fucking wrong and needs to be badgered and/or mocked until he admits it. You know, the bit about “Young Adam Roberts, English professor and science fiction writer extraordinaire, promised us a paradigm-shattering post—at long last, the pretension of the so-called “circle” would be shown up for the farce that it is! A figure made up of all points a distance r from a center point—it doesn’t even pass the smell test! Finally, something useful—and on a blog, no less!” —ending with the ‘just in fun’ declaration that he’s a fucking idiot.
I’m tired of that. It’s a continual attempt to turn this blog space into a pissing contest, because a couple of people are bored. If Adam R. wants to ask a question about math, let him. Hell, if someone wants to start posting pictures of their cats, go for that too. If people want to badger and/or mock, they’re going to find that their own sad public lives are fair game.
"Then don’t be wound up, if you don’t want to be. Do something else.”
But Rich, I did do something else.
Obviously, it’s hard to keep score with so many players, so let’s get explicit about it.
I posted one comment on the subject of the post, then let it drop.
I posted a new comment when a new question was opened by the topic’s author, who wrote that he was intrigued by the heat he’d generated. The score: Adam posed two questions. I offered one answer each.
Then when David begged for another try at the first question, I and three other commenters tried again. Updated score: David posed one question. I offered one answer.
Then you chose to include me in your redirection, and so I (naturally, I think) chose to respond to you.
Now, where in all this do you find my supposed desire “to lecture Adam about his study habits”? What you’re criticizing me for is answering you! If you don’t want to see me reply, simply leave my name out of your scenarios.
Ray, if you think that Adam is purposefully winding you up, then you must think that his second question was not a question—that he’s simply pretending to be puzzled, while knowing full well what’s going on. I think that the idea that Adam is on a campaign to wind people up is silly, given what I’ve seen of his writing, but hey, it’s possible.
If you think that his second question was honest, then you were scolding him for not realizing the effect of his bad study habits: “First, express ingenuous curiosity; then, forever after refuse to accept any sincerely proferred answers. Wouldn’t you be irked by a student who treated you that way? ("I’m sorry, Dr. Roberts, but I’m just not convinced....")"
It was a remark intended to blame Adam R. for provoking the people attacking him into attacking him, and was understood that way by everyone involved. If you don’t want to see me reply to you, don’t get involved.
I wish there was a way to tell this site’s software to not display or send by email comments of sophomoric idiots people one would put on an ignore list. Is there?
On a constructive note, Adam (R.), consider the following three experiments.
1. Achilles and the tortoise. If for simplicity we assume that Achilles has just twice the speed of the tortoise (it’s a very fast tortoise), then to overtake it, he’ll have to cover say 10 meters in 1 seconds, then 5 meters in 1/2 seconds, 2.5 meters in 1/4 seconds “and so on”... when 2 seconds have passed, Achilles has left the tortoise behind.
2. A man with a lamp. The lamp has an on/off switch. The man switches the lamp on for 1 second , then off for 1/2 seconds, then on for 1/4 seconds, then off for 1/8 seconds “and so on”... which state is the lamp in after exactly two seconds?
3. We draw a triangle inside the circle, taking 1 seconds to do it, then a square taking 1/2 seconds, then a pentagon taking 1/4 seconds “and so on”... what are we drawing as we cross the 2 seconds mark?
4. Just like 3., but nevermind the time. We draw a triangle, then a square, then a pentagon “and so on”.. what are we drawing as we “reach infinity”?
Your problem is that you think 4. or 3. is like 1.
But it’d help you to look at it more like you’d look at 2., in which you can hopefully clearly see that the answer is not defined. There is no definite state of the lamp after 2 seconds, because there’s no reason that it’d be on or off. One way to explain it is simply to say that the physical limitations make the experiment self-contradictory. One could retort: nevermind limitations, look at it as an idealised mathematical experiment! But then, from a purely mathematical point of view, the idealised man will just continue clicking ad infinitum, never reaching the two seconds mark. It requires a physical universe to say “what happens when 2 seconds have passed?”
1. gives you a clear intuition that an infinite process can converge to a sane finite value. But you do not have the right intuition of when this can happen and what it means. Mathematicians obtain that intution when they study topology (and not mere calculus, which usually doesn’t give enough of an explanation).
If you have a sequence of points (say, Achilles’ boot tracks) converging to a limit, the sequence never gets there in any sense. When you have an infinite sum 1+1/2+1/4 that “is” 2, the summing-up is never actually infinite even though we say it is for the sake of convenience. In both cases what happens is that a particular point is declared by fiat to be the limit, or the particular number is declared by fiat to be the sum; it’s not actually reached by any process. Instead, it’s recognized that the process gets us arbitrarily close to the limit/sum, never straying too far away, and that is taken as the definition.
So it isn’t automatically possible to speak of a limit of some infinite sequence of actions (like drawing n-gons, or switching a lamp, or running after a tortoise a little bit more). A limit that makes sense doesn’t exist just because there’s infinity in the distance, and the fact that “the number infinity” exists in set theory is completely irrelevant here.
For a limit to exist, you have to have a reasonable notion of a topological space, and a convergent sequence of points in it. For a limit to have physical meaning, you have to have a converging sum of dimishing time intervals, like 1 1/2 1/4… seconds, that you can “jump over” by physical fiat and declare that you’re interested in the 2 second mark. Your sequence of drawing n-gons doesn’t give you an automatically well-defined notion of a limit that you can use, and in any case, even though you could present a circle as a limit of n-gons in an appropriate sense, that wouldn’t make it a polygon in any exact sense.
You’ll just have to recognize that the Achilles metaphor is driving your intuition into waters it isn’t really supposed to reach, where it simply becomes wrong. You need real understanding of notions of topological spaces, limit points and properties of limits to realize precisely just why you can’t generalise from polygons to a circle in this case.
Anatoly’s is the fourth solid account (leaving mine aside) offered since we last heard from the post’s author. Adam, can you stop this crazy thing?
Rich, it’s with some relief and some disappointment that I finally accept Adam Kotsko’s account of you.
Before this comment thread closes, I would request a formal apology from Mr. Puchalsky for falsely attributing my post on this topic to Mr. Kotsko. While I appreciate Mr. Kotsko’s misguided generosity in allowing me to post on his blog, I hate him with every fibre of my being, and in fact, if I ever found myself somehow transmuted into him, I would kill myself in order to rid the world of one of its greatest scourges.
My stupid math (arithmetic?) question is about numbers, generally. And an intuition (certainly ignorant on my part) that numbers have a weird quasi-metaphysical (mystical?) aspect. Say, back in the cavemen days:
“There’s a rabbit.” (set of one)
“There’s another rabbit with it now.” (set of two)
“There’s another rabbit still.” (set of three)
Now these numbers for various sets of things have unusual properties on their own, aside from counting rabbits. Prime freaking numbers! Where the heck did divisible-by-one-and-themselves numbers come as a supervention to rabbit counting?
Please be kind—I’ve already identified myself as stupid and ignorant.
Ray, in that case, I have to say that your snit now is very similar to the snit with which you stopped posting at the Valve. If you want to support Adam Kotsko when he calls posters here idiots, fine, but don’t get on your high horse about the quality of the discourse here when you do so.
Ray asked me a question. It’s lose-lose for me, obviously, but I’d like to answer.
‘Do the four solutions offered satisfy me?’ I’ve gone back over this sorry thread (sorry in the sense ‘hey, I’m sorry I started this’) and, in the space specified: I can see David Moles saying you don’t need to square the circle to construct a circle, which is obviously right. Benjamin R. George saying that infinity is not a number since n+1 doesn’t add anything to n when n is infinite (which is also obviously right; although saying ‘infinity is a number’ is not saying that it is a rational, or finite number); but then there’s NB saying that Sladek’s fourth does not work when infinity is inputted, which is clearly right. There’s Ray, at more length, and much more helpfully (for me); and I reply to him below. There’s voyou saying “What [Adam R] can’t (consistently) do, but what [he] seems to what [want] to do, is hold both that infinity exists, and that it lies in the sequence of natural numbers.” But nobody would say that infinity lies ‘in’ the sequence of natural numbers (with numbers on either side of it? Of course not). But we can say that infinity is on the number line, because that’s the line that runs from minus-infinity to infinity. Saying ‘infinity is a number’ is not saying ‘infinity is the largest number’ (that would stupid); nor is it saying that infinity is a rational number. It’s just saying the infinity is on the number line.
But what really interests me is the circle question: that a circle is a polygon of infinite sides. That’s what was wittily proposed as the key proof that I’m a fucking idiot in the weblog spoof. The INFINIGON!!!
The options offered are (a) I’m maliciously stirring up trouble, repeatedly insisting something I know to be untrue just to mess with the heads of the people who have tried to explain it to me, or (b) I’m simply too dumb to see the proof offered me. To this Rich P. suggests a third possibility, that (c) I’m just a whipping boy, for whatever reason. If I persevere in my folly I shall incur wrath and if I repent I shall incur disdain. Ho hum.
Now the people who offered reasons why Sladek’s daft ‘proof’ is wrong did so kindly; they were not obliged to; they got no payment or credit for doing it; it cost them time and effort typing it in. I’m grateful. ‘Why didn’t I accept the suggestions offered?’ is surely a less pressing question than ‘why should anybody care whether I’m satisfied or not?’ But, to make the point that my disinclination to accept the reasoning offered was not motivated by sheer discourtesy or pig-headedness, I’ll go through it.
So the figure who emerges from the thread above is something along the following lines:
ADAM R: I believe it is possible to square the circle.
KINDLY COMMENTATORS: Adam, really it’s not.
ADAM R: I don’t care! I’m going to go on believing that!
Substitute ‘I believe that two and two make five!’ or ‘I believe that three and one are the same number!’ or ‘I believe that infinity is the biggest number!’ and I can see how my insistence would infuriate. But I wasn’t say any of those things. I was, actually, saying two things.
Sladek’s stupid proof included a point six that asserted ‘a circle is a polygon of infinite facets’ and ‘infinity is a number’. Most of the people who responded picked that point (either by itself or in conjunction with [4]) as the flaw in the reasoning. Most of the commentators here have said that a circle is not a polygon of infinite sides, and that infinity is not a number. Obviously once you reject [6], or [4] and [6] Sladek’s ‘proof’ collapses. Job done. Why, then, did I persevere?
I’d like to keep on thinking that a circle is a polygon of infinite sides. Now obviously if I’m wrong (as many people say) I’ll stop thinking this. But nobody has explained to me why I’m wrong. Clearly I’m wrong in their eyes; but am I flat wrong, absolutely wrong, fucking wrong, couldn’t be wronger? An answer here of ‘yes’ will duplicate what has already been written; an answer of ‘yes, because …’ will cause me publicly to recant. Provided it’s a ‘because …’ that makes sense. But there’s alternative; that this is not so much ‘fucking wrong’ as ‘well, that’s another way of looking at the matter, although not one I agree with personally.’ Because if that’s the state of play, then the vehemence and aggression of the thread here looks a little, what’s the word, disproportionate. Doesn’t it?
I’ll dilate on this a little. Ray is the only person who has provided an explanation of why a circle cannot be considered a polygon with an infinite number of sides.
The circle is the limit of the polygon sequence, true: that is, the series of polygons is bounded by the circle. However, by definition it’s a hard border. No matter how long you draw polygons, you’ll never find yourself drawing a circle.
In other words, the attempt to construct such a polygon would take an infinite amount of time. Even if we had all the time in the world we could not construct such a body, even though we would get closer and closer to it. I hope I’m not misrepresenting Ray’s argument here.
ADAM R: But, if you’ll indulge me: imagine that we did have a polygon with an infinite number of sides.
INTERLOCUTOR: Can’t be done. It would take an infinite amount of time to construct such a body.
ADAM R: I agree. But let’s say for the sake of argument that we had such a polygon. What would it look like?
INTERLOCUTOR: It’s a meaningless question. We could never construct it.
ADAM R: But for the sake of argument, assume that we had such a polygon. What would it look like?
INTERLOCUTOR: Well, a circle.
ADAM R: It would look a bit like a circle?
INTERLOCUTOR: Well, no.
ADAM R: So it would look exactly like a circle?
INTERLOCUTOR: I guess. But we would never have enough time to construct it.
This is why Achilles and the tortoise crops up so often in my posts. As, again, Ray says: if Achilles had to take every step iterated in the half plus a quarter plus an eighth addition then he could never overtake the tortoise. Luckily he doesn’t have to. And luckily (I’d say) we don’t have to go through all the steps of shaving edges of polygons to get to the circle. We’ve got the circle already.
Now, that’s how I see it. Others see it differently; I’m certainly not saying they’re wrong. I’m saying I disagree; that’s a statement about my state of mind. But of course maybe I’m wrong. Many things would convince me of that: for instance, explanations (or links to explanations) that taking a circle as a polygon of infinite sides leads to the following logical impossibilities; or to a philosophical explanation of the conditions under something that in every respect resembles another thing is not that thing, and the ways in which this applies to the circle. On the other hand, I wouldn’t be in the least surprised if nobody can be bothered to reply to this request, either because they’ve better things to do with their time, or else because they’ve lost faith in my ability to listen to sensible explanations.
[late addition] As I log-on to post this (sorry; rather unformed and rambly) comment I see Anatoly’s explanation. He says:
1. Achilles and the tortoise. If for simplicity we assume that Achilles has just twice the speed of the tortoise (it’s a very fast tortoise), then to overtake it, he’ll have to cover say 10 meters in 1 seconds, then 5 meters in 1/2 seconds, 2.5 meters in 1/4 seconds “and so on”... when 2 seconds have passed, Achilles has left the tortoise behind.
2. A man with a lamp. The lamp has an on/off switch. The man switches the lamp on for 1 second , then off for 1/2 seconds, then on for 1/4 seconds, then off for 1/8 seconds “and so on”... which state is the lamp in after exactly two seconds?
3. We draw a triangle inside the circle, taking 1 seconds to do it, then a square taking 1/2 seconds, then a pentagon taking 1/4 seconds “and so on”... what are we drawing as we cross the 2 seconds mark?
4. Just like 3., but nevermind the time. We draw a triangle, then a square, then a pentagon “and so on”.. what are we drawing as we “reach infinity”?
Believe me I’m not doing this simply to wind you up; but I’d say the answer to ‘2’ is that it’s fifty-fifty the light is on or off (is that what you mean by undefined?) and the answer to both 3 and 4 are ‘a circle’.
1. gives you a clear intuition that an infinite process can converge to a sane finite value. But you do not have the right intuition of when this can happen and what it means.
This may very well be true; but my lack of the necessary intuition is asserted here rather than proved. I don’t mean to sound snippy; I daresay you’re perfectly correct and I’m a fool. But if in effect you’re saying ‘you should just take it on trust from better mathematicians than you that this is so’ then my saying ‘Oh—OK’ would surely sound pretty hollow.
For a limit to exist, you have to have a reasonable notion of a topological space, and a convergent sequence of points in it. For a limit to have physical meaning, you have to have a converging sum of dimishing time intervals, like 1 1/2 1/4… seconds, that you can “jump over” by physical fiat and declare that you’re interested in the 2 second mark.
This makes perfect sense to me, and I agree with it. But isn’t a circle a topological space? Let’s make the jump you mention; the infinite n-gon on the far side is perfectly circular. Isn’t it?
Your sequence of drawing n-gons doesn’t give you an automatically well-defined notion of a limit that you can use, and in any case, even though you could present a circle as a limit of n-gons in an appropriate sense, that wouldn’t make it a polygon in any exact sense.
And here I honestly (as in; really not trying to wind you up) do not take the force of the difference between ‘you could present a circle … in an appropriate sense’ and ‘any exact sense’. If I say: ‘I can present a circle as a polygon with an infinite number of facets in an appropriate sense’ (which as far as I can tell is what I mean anyway) will people stop yelling at me? How can it be not a circle in any sense, but a circle in a presented, appropriate sense?
On the other hand, let’s imagine everybody is now so bored of this post and this overlong comment by me proves to be the last. Imagine that!
I don’t think 7 follows from 6. Why can’t a polygon with an infinite number of sides be identical to another polygon with an infinite number of sides?
Although you say you don’t think this, the way you talk about infinity makes it seem like you think it’s one number among others (and perhaps even the biggest number). The only times when it seems that you don’t think this are those time where you are officially declaring something like “Now of course I don’t think infinity is the biggest number.” If we deleted those protestations, a reasonable reader would be able to gather that you are somewhat confused on the meaning of “infinity” and the precise implications of a (Cantorian) “actual infinity.”
That is, despite your “official” stance that you are not doing this, you are in fact treating infinity as if it were among the set of finite numbers (similar to if I were to insert claims that I am showing you nothing but the deepest respect, then calling you “fucking wrong,” etc.). As someone says above, the Achilles problem seems to have led your intuition astray in this matter.
The vehemence of response stems from the performative contradiction inherent in your statements. I find this to be a persistent problem in threads to your posts—you absolutely refuse to let go of your initial interpretative frame, even after someone has decisively proven it to be inadequate, and even after you claim you are no longer working with the original frame. For instance, in the thread about transubstantiation, you continually went back to the stance that it has to be empirically disprovable, even though it’s not; or the thread on Heidegger, you insisted that Heidegger had to be straightforwardly wrong about “time” due to his apparent disagreement with Einstein, even though his approach to the concept of time is not the same as that of natural-scientific inquiry, etc., etc., etc., etc., etc.
Adam,
I, personally, am not exasperated by your refusal to accept my explanations. You certainly don’t have to take my word for anything. But, if I can’t find enough common ground for simplistic explanations to work, we’ll just have to leave it at that, because this isn’t the right venue for a college-level course in topology, right?
With that, let me have another go at the infinite polygon thing. A polygon, by definition, is a closed non-self-intersecting curve which consists of a finite number of straight line segments. A straight line segment always has positive nonzero length - a point by itself is not a line segment. An “infinite polygon” is not possible by the definition of the notion of polygon, because a polygon is explicitly posited to have only a finite number of segments (this is a very important property of polygons, that allows us to prove various properties of theirs. It’s not specifically crafted to exclude the circle. It’s also e.g. why you can’t various fractals that arise from ever-refined polygons as polygons, either).
This is what I mean by saying that you can only say that “a circle is an polygon with an infinite number of sides” metaphorically, not in any exact sense. A circle doesn’t include any line segment, however small, and it doesn’t consist of a finite number of such segments, either. Ergo, it’s not a polygon.
Now, in what sense do the inscribed n-gons approach a circle as n goes to infinity? For example, in the sense that the area of the “extra part” of the circle, the one that’s not captures inside the n-gon, grows smaller and smaller all the time, so the areas of the n-gons approach the area of the circle all the time, never reaching it. If you write out the areas of the n-gons as an increasing sequence, the limit of that sequence is going to be the area of the circle. Which does not mean that the circle is a polygon, in any sense or shape. It simply doesn’t follow. It may look “right” in a vague sort of way, it may warrant a metaphorical explanation like “an infinite polygon”, but it simply doesn’t work in any exact mathematical sense.
Now, suppose we say: OK, but that just shows that the exact definition of a polygon is too limited. Let’s say that an “infinite polygon” is a species different from an ordinary polygon, with its own definition, without the limitation of a finite number of straight line segments. But what, then, would its definition be? What would an infinite polygon, in generality, be? Well, the only thing you can definitely say about it is that you can “get it” (so it seems to you) by drawing more and more exact inscribed polygons. But a moment of thought will tell you that you could do that with any closed non-intersecting curve, not just a circle. Take an ellipse, flop it around, stretch and bump it however you want, and then draw n-gons (not perfect ones, obviously) inside it that cover more and and more of it as n goes to infinity. So if you take that as your definition, it turns out that any simple closed curve is an “infinite polygon”. Well, turns out we have a word for “infinite polygon”. It’s “curve”. It’s nothing like a polygon. The desired notion of an “infinite polygon” turns out to be not only incoherent, because it contradicts the definition of polygons, but also unnecessary anyway - we already have a word for what it would be like.
So the answer to your
And here I honestly (as in; really not trying to wind you up) do not take the force of the difference between ‘you could present a circle … in an appropriate sense’ and ‘any exact sense’. If I say: ‘I can present a circle as a polygon with an infinite number of facets in an appropriate sense’ (which as far as I can tell is what I mean anyway) will people stop yelling at me?
is: no. You’re missing the crucial difference between “I can present the circle as a limit of a sequence of polygons, in an appropriate sense” (what I’m saying) and “I can present the circle as a polygon with an infinite number of sides in an appropriate sense” (what you’re saying). The former is correct; the latter is not. The two claims are different and not equivalent. What you’re saying can’t be made to work.
Consider yet another analogy. Every irrational number has a sequence of rational numbers that converges to it, has it as its limit. For instance, for pi we could take the sequence of rational numbers 3, 3.1, 3.14, 3.141, 3.14145 and so on. pi is a limit of a sequence of rational numbers. Is it a rational number itself? No. Is it “a rational number with an infinite number of digits”? No! It’s not rational at all. Not in any sense, not in any way. Simply isn’t. In the same way, a circle can be seen as a limit of a sequence of polygons, but is not a polygon.
"Provided it’s a ‘because...’ that makes sense.”
Sorry, Adam, but that’s the killer: You’ve declared yourself the sole arbiter of what “makes sense” based on your intuitive grasp of a bunch of different terms. That’s no way to handle a discipline. (I should know; I mishandle disciplines all the time.) Putting the words “infinity” and “polygon” together like that seems mathematically meaningless to me. Intuitively, loosely-speaking, pop-scientistically, sure, I understand what you’re trying to say. But the way you’re saying it isn’t valid in a mathematical proof, and if you make the effort to try to make it valid, I think you’ll find the difficulties vanishing away.
Similarly, I understand what someone who talks about “the divine right of kings” and “the true heir to the throne” is trying to say even though neither rigorously explains actual UK politics. The point of my spoof wasn’t that you’re a fucking idiot. It’s that so long as you insist on smashing conceptually distinct universes together, you can expect odd results.
But hey, that’s your job. You write speculative fiction; I write speculative essays. Both of those seem like good times to me. They just aren’t math.
Not totally relevant but funny anyway:
http://hewasmymathteacher.ytmnd.com/
Ray is the only person who has provided an explanation of why a circle cannot be considered a polygon with an infinite number of sides.
Not so, my Robertsy friend!
joeo, I was that kid, which is one reason I’m not an academic.
And the person sitting beside me just said she was that kid.
I guess there have been a lot of that kid. He should just take over.
I tried to skip over all flame-war posts, so someone else might have made it: the question of whether infinity is a number is besides the point. The rule “if a statement is true for 0, and if when it’s true for n we can show it for n+1” applies to the set {0, 1, 2, 3, ...}, which does not include infinity. If you want to include infinity in that set, then the rule becomes false. For example, if you include infinity, you can prove the false theorem that infinity != infinity.
Adam K. The character you’re describing is indeed an unlikeable person. I can certainly, therefore, understand why you dislike him.
It’s partly about the protocols of debate, isn’t it? If I ask a question and you provide an answer that satisfies you, am I obliged to accept your answer?
“For instance, in the thread about transubstantiation, you continually went back to the stance that it has to be empirically disprovable, even though it’s not; or the thread on Heidegger, you insisted that Heidegger had to be straightforwardly wrong about “time” due to his apparent disagreement with Einstein, even though his approach to the concept of time is not the same as that of natural-scientific inquiry, etc., etc., etc., etc., etc."
The two examples you give are interesting. I found the Heidegger discussion really useful – I mean for my personal understanding of H. – and I learnt a lot from what you said. I’m sorry if that didn’t come across in my comments, or if my habit of continuing to ask questions, probe etc. was annoying for you. I’d go so far as to say that my sense of Heidegger is fuller now (and, actually, less dismissive) in part because of what you said. In part, also because the discussion prompted me to go away and read a deal more H. and H-related writing.
On the eucharist, not so much. In that thread I was saying that there are many things about religious belief that are not falsifiable, but sometimes religious believers say things that are. I could have picked the example of, eg, millenarian prophesy (‘God has told me the world will end Dec 31st 1999’ and so on), but instead I chose the example of the bread. You explained ways (the framework of Christian belief) in which, although the bread appears to stay the same as science might analyse it, in fact the bread has undergone an actual change … which is to say, more than a metaphorical or symbolic change. I can understand the latter; and I can understand that there are people who believe that the bread changes substantively. But I still do not believe the bread does actually undergo anything other than a symbolic change. Even after your explanation. Now, two things here: one is, do the protocols of debate require me to say ‘thank you Adam, now that you have explained it I have changed my mind’? And two, out of interest: do you believe that Eucharistic bread actually (more than symbolically) changes during mass?
Ray: ““Provided it’s a ‘because...’ that makes sense.” Sorry, Adam, but that’s the killer: You’ve declared yourself the sole arbiter of what “makes sense” based on your intuitive grasp of a bunch of different terms. That’s no way to handle a discipline.“
You’re quite right; I didn’t phrase that very well. I wasn’t trying to set myself up as ultimate arbiter. I meant ‘provided it’s more than “because I say so” or “because you’re a fucking idiot“‘. I meant, in fact, the sorts of answers yourself and Anatoly were kind enough to supply.
Anatoly: “With that, let me have another go at the infinite polygon thing.“
I’m very grateful to you Anatoly for taking the time to explain so clearly. I think I understand a little more now.
Firstly, obviously if it is part of the definition of a polygon that it is a shape with a finite number of sides (as opposed to simply an n-sided shape, which is the definition I was working with) then it cannot by definition be infinite.
Secondly, now that you’ve explained it, of course I see the point about the curve. Any closed curve could be described as a polygon of infinite facets. So when Ray says above “The circle is the limit of the polygon sequence, true: that is, the series of polygons is bounded by the circle” and Ben says “Circles are the regulative ideal of regular polygons”, it might have been more accurate to say “The curve is the limit of the polygon sequence, true … the series of polygons is bounded by the closed curve”.
One final niggle, which I offer not in the spirit of pig-headedness, but because I can’t banish the voice at the back of my head: Any closed curve could be called a polygon with an infinite number of facets (except that, as you say, this contradicts the definition of a polygon). But a circle is one special example of a closed curve. So a circle could be called one of the polygons with an infinite number of facets (except that, as you say, this contradicts the definition of a polygon). Or am I still enormously wrongheaded?
joeo, Ray: I had an 8th grade math teacher who insisted that 22/7 was the exact value of pi.
Adam R., All along, one of my premises was that it would be perfectly sensible not to believe in transubstantiation. All I was saying is that, within the framework of Christian belief, you can’t get to that point just by pointing out that the senses still perceive normal bread and wine—that’s already factored into the doctrine. It’s not a matter of empirical proof. That’s what you kept insisting on, that it was a matter of empirical proof, and you seemed to be saying that I was depriving you of any means of not believing in it.
But no! I was just saying that the “empirical proof” route wasn’t going to work. The “sheer sophistry” route does work. So do the “miracles are impossible” or “there is no God” or “Christ didn’t raise from the dead” routes. Yet I kept getting this vibe from you where you were saying, “But I want to disbelieve in it for empirical reasons!” But that doesn’t work! Every Roman Catholic in the world agrees with you about the empirical properties of the consecrated host—that’s a non-starter. If you said to them, “But look, you fools! I’ve run all these scientific tests and it’s just bread and wine,” they would yawn or laugh.
(What I personally believe about transubstantiation continues to be irrelevant to the structure of the doctrine itself.)
Similarly, when you insist that something called “infinity” is on the number line and is at the end of the number line, I don’t know what conclusion I can rationally draw from that other than that Adam Roberts functionally believes that infinity is a finite number, i.e., the largest one. Your protestations to the contrary only show me that you don’t understand what you yourself are saying, much less what anyone else is saying.
In terms of finite series, infinity just means “increases without bound.” In terms of set theory, you just posit the infinite set—you don’t assemble it through iteration.
Minor technical point: you need to specify a rectifiable closed curve to consider a closed curve a polygon with an infinite number of sides. (This will include any closed curve you’re likely to think of, though.) You are free to redefine a rectifiable closed curve to be a polygon, but then certain theorems about polygon become false. One of them is that you can no longer prove anything about polygons by induction on the number of sides: this property is true solely because by the usual definition polygons have a finite number of sides.
Adam Kotsko: “Similarly, when you insist that something called “infinity” is on the number line and is at the end of the number line, I don’t know what conclusion I can rationally draw from that other than that Adam Roberts functionally believes that infinity is a finite number”
I’m going to briefly quote wiki to you, as an example of the common understanding of the teaching concept “number line”:
“Although this image only shows the integers from -9 to 9, all of the real numbers from negative infinity to positive infinity can be located on this line.”
Also, if you search this thread for the words “number line”, you will see that Adam Roberts never said that that infinity is at the end of the number line. You made that up. In fact, as your misunderstanding of the limit concept above shows, you know less math than Adam Roberts does.
Conclusion: this, like many other incidents, proves that you really are a dull, uncareful, angry, deeply boring person. You have no right to abusively pretend that you can righteously get angry at Adam Roberts because he does not think as you do.
Having closed many a thread hereabouts, perhaps I can bring this full circle ...
I work as an applied mathematician (I’m not a real mathematician, but I play one on PC). Maths are treated not as an ultimate truth but as a tool for better eliciting such from natural phenomena. I addressed the Sladek thang on the terms presented—it only resembles math, applying a lot of windowdressing (such as an unsupported induction) to cover a basic fallacy, which several identified. It’s not necessary to invoke any notion of limits for the core objection to hold, but it can be formulated that way; there is enough imprecision in the sproof to admit a variety of interpretations. The interpretation AdamR settled on may hold other dangers in a fully formal mode, but is not mistaken in its import: were I to construct a sequence of regular polygons of increasing number of sides, it would approach the circle in its limit, and the behavior of limits cannot be reliably described by attributes of elements of the sequence. The analogies suggested may not otherwise be apt, but the concept of the circle as an infinitely-sided regular polygon isn’t wrong, except formally not even wrong (btw, AR, in these terms, each side has length zero if one hews to the behaviors of the elements in the defining sequence, any infintesimal x/infinity being indistinguishable from zero for finite x—besides, the perimeter of a circle looks ‘like this ... O’, indistinguishable from zero in some fonts, and I’m getting tired of feeding you guys straight lines—oops I did it again!). Behavior at the limits is a mirror of infinity; knowledge, much less intuition, need not be perfect to be useful. Rigor has its place in gaining a deeper, more precise understanding, but does the context require it? Just because it’s maths, is discourse hermeneutered?
nnyhav: thank you.
Every square here hath here her wrecked angles wrought.
The problem is that infinity is a concept, but not a number (contrary to his claim). You can use it in the place of a number, but it is not a natural number and does not fall anywhere along the progression of natural numbers (the progression of n followed by n+1).
Mistakenly using infinity as a natural number can easily cause mistakes more obvious than the impossibility of circles: As mentioned above, infinity+1 = infinity.
Therefore infinity+1 = infinity+0.
Therefore 1 = 0, which is obviously stupid.
Really, “infinity+1” is a meaningless term—you can’t add one to infinity, by definition.