As is his wont, my eight-year-old son had been reading one of his science books in bed last night, and in particular reading about temperatures at the earth’s center. Telling me about it this morning – as, still bleary-eyed, I made my coffee – sparked in him the following chain of thought: If the earth has a center, then that center has to have a center, and that center has to have a center as well, and so on to infinity. But then, he reasoned, the earth would have to be infinitely big, and this would be absurd.My son is very bright and I am used to hearing such thoughts from him, but I was nevertheless mildly startled that he had, however inchoately, more or less stumbled upon the second horn of the dilemma posed by Zeno’s paradox of parts, which (at least on one reading) goes like this: If the world consists of a plurality of things, then the parts that make it up either have no size or they do have size. If they have no size, then nothing has size, since adding together parts with no size can never result in a whole with any size. But if the parts do have size, then they can be divided into parts of smaller size, and those parts into yet smaller parts, and so on to infinity. And in that case the number of parts is infinite. But a thing with an infinite number of parts would be of infinite size. So, if there is a plurality of things in the world, then things either have no size or they have infinite size; and either suggestion is absurd. So the world does not consist of a plurality of things.
This was a teaching moment, so I used it to launch into an impromptu mini-lecture on Zeno’s paradoxes and Aristotle’s response in terms of the distinction between actual and potential infinities. My son loved it. (He also liked that, while moving my coffee cup back and forth through the air to illustrate motion, I managed to spill coffee all over my shirt and pants.)
But now I’m worried. Learning about Zeno and the other Pre-Socratics is what turned me on to philosophy. So, I may have just set my son on the path to becoming a philosopher. Does that count as child abuse?
(Bonus link: Here is a comical comic book treatment of Zeno’s paradoxes. Unfortunately, I can’t show it to my son, for reasons that will be obvious if you read it!)
The pre-Socratics for me too. Fascinating stuff - James Chastek's exhortation to listen to Duane Berquist here was spot on.
ReplyDelete"We should think about the wise words of the first philosophers. For they contain the seeds of wisdom. And the importance of a seed should be judged, not by its size, but by that to which it gives rise. Moreover, the one who
considers something from its beginning is apt to get the best understanding of it. If, then, we consider carefully the fragments we have from the first philosophers, we shall better understand philosophy."
Provides fascinating context for Aristotle's act/potency resolution of the apparent paradox of change.
It’s not child abuse. But if you catch him reading David Hume in his bed, stern measures may be called for.
ReplyDeleteWell according to the Fundie New Atheists on him any world view violates his "rights". You should let him develop his own world view. Of course if teaching philosophy or religion to a child is wrong then the only smart thing to do is not teach kids anything since odds are you will infect them if indirectly with a world view.
ReplyDeleteThis explains why most public schools are the way they are.....I'm just saying.....
Wow, your son is really smart. For this particular instance I think it might be relevant to bring up Hilbert's Hotel. Wouldn't it be possible for the Earth to have infinite parts but be finite in size? It's like walking halfway to something and each time walking half the distance you just walked. There is a finite distance but you will never get there.
ReplyDelete"But a thing with an infinite number of parts would be of infinite size."
ReplyDeleteThat is just false.
Did you ever take calculus? Or any science classes at all?
You can sum an infinite series and the result is a finite number.
This is really basic high school stuff.
http://en.wikipedia.org/wiki/Geometric_series#Zeno.27s_paradoxes
Did this guy ever take calculus? Because the way I remember it (and I'm only a few credits away from a BS in mathematics), everything is done using limits - the value a f' tends towards as dx approaches zero, or what have you. That seems to be pretty much ceding the field to Aristotle's assertions about actual and potential infinities.
DeleteSince potencies are real, the potential infinite is also real, so there's a fact of the matter about what the answer would be if we could sum the infinite series. But since actual infinities are impossible, the potentially infinite series is only ever potential, so we can't actually perform the summation, and have to engage in all this mucking about with limits instead.
Once you know the method, it's easy to think that you're actually, say, adding up infinite numbers of infinitely small things when you do an integral. You're not. What you're actually doing is calculating what the sum would have been had you been able to perform the infinite division required to perform it. Given the way that the fundamental notions of calculus are defined, this is all you ever can be doing.
Incidentally, Feser himself seems to be under similar misconceptions about the nature of calculus, based on the way he deals with this sort of objection in the relevant passages of Aristotle's Revenge.
Anon, how did you miss the part where Ed said he was explaining one reading of Zeno's paradox of parts? Where did he say he agrees that "a thing with an infinite number of parts would be of infinite size"?
ReplyDeleteAnd why is it that almost every time I encounter someone on the internet who mentions "calculus" or "physics," he demonstrates in the very same sentence that his reading comprehension skills are pathetic?
Anon-
ReplyDelete"Zeno's arguments are often misrepresented in the popular literature. That is, Zeno is often said to have argued that the sum of an infinite number of terms must itself be infinite - that both the distance and the time to be travelled are infinite. However, Zeno's problem was not with finding the sum of an infinite sequence, but rather with finishing an infinite number of tasks"
http://en.wikipedia.org/wiki/Zeno%27s_paradoxes
Yes, we all know that brilliant high school students like yourself discuss Zeno's paradoxes in your calculus classes. Hopefully your reading comprehension, along with your sense of humility will improve if you ever plan to get to college.
Did you ever take calculus? Or any science classes at all?
ReplyDelete"or"??
Anon, calculus isn't a considered science.
Ed,
ReplyDeleteWhere did you get that comic strip about the Pre-Socratics?
Bobcat, here: http://wrightopinion.com/
ReplyDeleteIt was in the address of the comic strip.
I'm lame. I can't find that comic on the site. But I'll just look some more.
ReplyDeleteWhat is infinite is the arguments without conclusions we have. That in its self is interesting. Like it was planned.
ReplyDeleteI just put this blog into "age analyzer" it turns out the age of the blogger is 65-100. I think that is a good thing. It says 'this blog is too sophisticated for typical web 2.0 users'. Still how could one get to 65 because before you got to 65 you would need to be 64, but before you got to 64 you would need to get to 63.... etc.
ReplyDelete"You can sum an infinite series and the result is a finite number.
ReplyDeleteThis is really basic high school stuff."
I've run into this kind of (snide?) rebuff more than a few times in debates dealing with kalam, Zeno, first cause, etc., and I must say the main thing that bothers me about it is how it seems to conflate purely abstract summing with actual physical composition. Does the retort mean to say that in nature, as an empirical fact, there are infinitely small, infinitely thin "slices" of objects which exist in their own right? Or is it merely some Russellian condescension about how to make sense of an infinite sum? I think it is the latter but it also seems to imply the former if it is to carry any weight in a realistic metaphysical debate.
Best,
Codgitator: "it seems to conflate purely abstract summing with actual physical composition."
ReplyDeleteWell, either space is infinitely divisible (as the ancients thought, from Aristotle to Zeno — though I'm not sure about Democritus), and then calculus is the right tool to use; or else it isn't, and Zeno's paradoxes cannot arise in the first place. If points p1 and p2 are the smallest possible distance apart, then there is no puzzle about having to get halfway between p1 and p2 because there is no between. You just "jump" from p1 straight to p2. (I think that modern physics suggests a "quantised", finitely-divisible universe, but there's no conclusive science, and God could certainly have created space/matter that was infinitely divisible if He had wanted to.)
Well, I recall many years ago - in a college physics class, not high school - the day the professor demonstrated that energy consists of discrete (very small) "packets". And I immediately thought of Zeno's paradox, as did several of the other students who walked out of the classroom with me when class ended. The physical universe is not infinitely divisible.
ReplyDeleteI still think calculus is a good enough tool to use in working with the physical world. As a carpenter's steel tape probably does not measure exactly 3-4-5 when he's laying out a floor, but it's close enough.
If the earth has a center, then that center has to have a center, and that center has to have a center as well, and so on to infinity.
ReplyDeleteThe center of the Earth is a point. Centers only make sense for things with volumes (because you need an interior to have a center). Points lack volume. Therefore asking the center of the center of the Earth doesn't make sense.
Professor Feser I'm not sure how to interpret your treatment of Zenos paradox as anything but a misunderstanding of calculus. You write in your book Aristotle's Revenge "the more extended parts a thing has, however minute those parts, the larger it is. Hence if a contiuous object is made up of an infinite number of extended parts, it will be of infinite size." You go on to say that calculus doesn't solve THIS problem but merely changes the subject.
ReplyDeleteNow it's true that within some of Zenos paradox's there is some valid debate about whether or not modern mathematics actually solves them (for example the arrow needing to travel half a meter and a fourth of a meter and so on before it ever travels a meter can be looked at as a supertask and we can fit all the modern debates and controversy around that subject into Zeno) but THIS supposed paradox is certainly, if not solved with, addressed and possibly refuted by calculus and it is certainly a mathematical not a metaphysical question.
We're asking whether a thing with infinite parts must be infinitely big and the answer not only according to modern mathematics but according to logic herself is a resounding no.
Let's say that at some point in time t1 we have a half a pie, which was merely a potential for half a pie to exist at t0, become half a pie in actuality by a miracle of God. Now let's say that God, being perfect, does not want to give the world only half of a pie. Naturally God would add on another piece but in his infinite wisdom rather than making another half God has decided to add on a fourth. At t1 we have half a pie and then at t2 (we can say a second later) we have a half a pie plus a fourth of a pie. Then at t3 we have an eight added on and a sixteenth at t4 and so on ad infinitum. Now what you should notice is that as we keep going the number which we add on to the pie moves increasingly closer to zero and the size of the pie moves increasingly closer to one. Meaning there is a horizontal asymptote or a ceiling to how big this pie can get. Now you mention in your book that this is converging and converging is not summation. This is absolutely right. The series here converges on one. It does not sum to one. The issue is that that does not mean it goes on to infinity. Converging means it infinitely goes up to a finite number. It means there is a cap. It cannot be infinitely big if there is a point that it can never get over.
Just so it is worth mentioning in closing that this theoretical pie would be virtually the same as a any other pie you might see. After about 2 minutes the pieces of the pie which we add on would be about the same as a planck length (the smallest unit of measurement you can get before space-time breaks down and nothing makes sense) meaning that after about 2 minutes you would have a pie that is so close to a full pie the distinction between it and a normal pie is not even something that could physically exist let alone be measurable.
After reading through this portion of the book I would say that this idea of continuum is not as sound as it would appear. You could very easily believe in an object which not only could potentially be broken down into infinite pieces but could actually be made of infinite pieces. Now it is worth mentioning two things. One the idea that in principle any measurement could be broken down into smaller measurements is not really necessary. As mentioned before there is a planck length. There is a point where you could not in principle just divide the measurement in half again. That is to say that there is a length so small that it doesn't actually physically exist. Now the second point is that EVEN IF it was physically possible to have an infinite amount of measurable pieces of the pieces for any given object that would NOT make any given object infinitely big. There is a difference between an object growing infinitely large and an object growing infinitely closer to a certain point.