Zeno for the computer age
If you wish to better understand Zeno's worry about the continuum, you could do worse than to consider loops in software.
Case 1: You want to loop over 10 records. You write:
for i from 1 to 10
process_record()
What could be simpler? OK, let's loop over the positive integers, finding the prime numbers:
for i from 1 to ∞
check_for_primality()
This loop will run forever, but it is a perfectly valid loop. We can even set it up to loop over all integers:
j = 0
for i from 0 to ∞
j = j - 1
print i
print j
And with only a little more trouble, we could loop over the rational numbers as well.
But what if I ask you to loop over the real numbers between, say, 0 and 1? The problem here is much worse than the loop running forever: the loop can't even get started. We could print out "0"... and then what? There is no "next" real number to which we can proceed. And note: the concept of a limit does not help with this problem at all.
And this is what Zeno was noting about motion in the continuum.
Case 1: You want to loop over 10 records. You write:
for i from 1 to 10
process_record()
What could be simpler? OK, let's loop over the positive integers, finding the prime numbers:
for i from 1 to ∞
check_for_primality()
This loop will run forever, but it is a perfectly valid loop. We can even set it up to loop over all integers:
j = 0
for i from 0 to ∞
j = j - 1
print i
print j
And with only a little more trouble, we could loop over the rational numbers as well.
But what if I ask you to loop over the real numbers between, say, 0 and 1? The problem here is much worse than the loop running forever: the loop can't even get started. We could print out "0"... and then what? There is no "next" real number to which we can proceed. And note: the concept of a limit does not help with this problem at all.
And this is what Zeno was noting about motion in the continuum.
Gene, it is true that you can write a loop enumerating all the rational numbers and cannot write a loop enumerating all the real numbers. But I don't think this has anything to do with Zeno's paradoxes. Zeno would raise exactly the same concern about the rational numbers that he would about the real numbers, namely that there is no "next" rational number after 0.
ReplyDeleteZeno might not have even had the relevant notion needed to take you to the real numbers, namely that every set with an upper bound has a least upper bound. Even the ancient Greek equivalent of Dedekind cuts, the Eudoxian theory of proportions, came shortly after Zeno. Zeno would have been aware of incommensurable magnitudes, the equivalent of irrational numbers, but he might have thought that all the points of space can be constructed by straightedge and compass. And by the way, the set of points that can be constructed by straightedge and compass can be enumerated by a loop.
"But I don't think this has anything to do with Zeno's paradoxes."
DeleteWell, you would be wrong about that.
The rest of your post makes the following error: if Zeno lacked the *technical vocabulary* to talk about X, he could not have had any intuitions about X. That is false.
Once again, I urge you to read Atomism and Its Critics.
No, Keshav would be right about that, as he usually is about math.
DeleteZeno might not have even considered a continuum, and his dichotomy paradox is actually stated in terms of rationals. You have to get half-way there, and then half the remaining distance, and then half that ... All rationals.
The turtle paradox isn't about the continuum either, since all it requires is an infinite well ordered set.
There seems nothing about a continuum in the arrow paradox, it also is about limits (does it make sense to get to a "point" in time), and as Keshav says you get that in the rationals.
Keshav's argument can be stated simply. The rationals are dense in the reals, and the "paradoxes" are stateable in Q. This is correct.
Gene, I'm not claiming that Zeno didn't have the technical vocabulary to talk about the real numbers. I am saying that he didn't have even the concepts that underlie the intuition behind the real numbers. The conception of the continuum he would have relied on would correspond to the modern mathematical notion of denseness, a property shared by the rational numbers and the numbers constructible by straightedge and compass.
DeleteThe conception of the continuum that goes beyond denseness and into what we now call Dedekind completeness was, I claim, not developed until Eudoxus of Cnidus, whose theory of proportion was adopted by Euclid in Book 5 of the Elements. I don't know of any pre-Eudoxus philosophers who conceived of the continuum in ways that go beyond denseness.
But yes, "Atomism and its Critics" sounds like a good book to read.
What many including me typically be bring up against Zeno, \sum_{n=0}^\infty 2^{-n}=2, does not need any concept of numbers beyond rationals. So I don't think Dedekind completeness and unenumerability change or add anything.
DeleteYes, but as me and many others have pointed, this sort of summation is completely irrelevant to Zeno’s point. I’m not sure how bringing up something completely irrelevant is evidence of anything.
Delete"No, Keshav would be right about that, as he usually is about math."
ReplyDeleteThe funny thing is, Ken, we are discussing a philosophical problem, not a mathematical problem.
Gene, do you think Zeno would agree or disagree with the following statement: "A continuum is a set where between any two points, there are infinitely many points in between." Would he disagree with that, saying there is more to a continuum besides that? I claim no, he would say that that's exactly the notion of a continuum that he is criticizing.
DeleteAnd if I'm right about that, then any point about the uncountabiltiy of the real numbers cannot possibly be relevant to Zeno's ideas.
A philosophical problem statable as a math problem. Your position is untenable because by insisting Keshav is wrong you are insisting on knowing what you cannot know: what was actually in Zeno's mind. What we can say is that Zeno's paradoxes are about density not cardinality, but the loop thing as toy state it is about cardinality (there are countable sets that no loop can enumerate).
ReplyDelete