The infinite sequence 1, 1/2, 1/3, 1/4, … is commonly described as “getting closer and closer to 0”. This fits with the mental representation (metaphor) students have of the sequence as something whose successive entries are revealed or created over time.
People who are used to abstract math have another handle on that sequence: It is the function whose value at each positive integer n is 1/n. Stated that way, the sequence is being pictured as existing all at once (another metaphor). It is a completed infinity or actual infinity. Completed infinities occur all over modern math and mathematicians rarely remark on them. But some people outside math, in particular philosophers, get all hot and bothered about it.
I have not found a good reference on the internet to the idea of completed infinity. The article on actual infinity in Wikipedia gives lots of reasons against the idea but few reasons in favor of it. Many other articles on the internet also mostly discuss the opposition to the idea.
The phrase “the function whose value at each positive integer n is 1/n” is a definite and clear description of the sequence. Mathematicians have proved all sorts of statement about the sequence using classical logic; it is a Cauchy sequence converging to 0, for example. These statements all seem to be consistent with each other and with other parts of math. To put this in another way, we have a clear syntax for talking about this sequence and others, and we have built up a fund of statements about it that all hang together.
As always when talking about anything, we use metaphors and images when talking about this sequence. The image of the sequence as getting closer and closer to zero is one, which we could call the Xeno image. The image of the sequence as existing all at once is another, let’s say the actual-infinity image.
Neither image says anything about physical reality. The sequence is of course represented physically in our brains by arrangements of neurons or some such thing, but in no way does that imply that every entry is represented in our brain. What is represented in the brain are properties of the sequence, the metaphors and images we have mentioned (and others), relationships with similar sequences, and so on, what the math ed people call our schema of the concept. (See [1] and [2].)
I propose that the trouble philosophers and students have with the actual-infinity image occurs because there is a physical arrangement in a brain that serves to recognize big bunches of individual things. (Compare this post.) It may be triggered, for example, if you look out of a high window and see a big crowd of people standing in the street. You have a person-recognizer in your brain and you also have a big-bunch recognizer in your brain, which works with the person-recognizer to identify a crowd of people. (I talked about a similar idea, the I-saw-this-before recognizer, here).
A smart person soon realizes that there is a difference between a crowd of people and an infinite sequence, namely that the crowd is large but finite whereas the sequence is infinite. Well, that is true. So what? The crowd may be finite but you still can’t hold an image of each individual in the crowd in your head. The finite-infinite difference is indeed a difference, just as the idea that one is composed of physical objects (people) and the other is composed of mathematical objects (numbers). You can reason about both the crowd and the sequence, make discoveries about them, and so on.
Why is there a difficulty? Perhaps it is because a device in your brain that is usually used for bunches of physical objects is now being triggered by a bunch of abstract objects. That can be disconcerting.
But: You don’t have to get all upset about something just because it disconcerts you.
Now, then, no one ever needs to worry abou
t actual infinity again. (Fat chance).
REFERENCES
[1] Ed Dubinsky and Michael A. McDonald, APOS: A Constructivist Theory of Learning
in Undergraduate Mathematics Education Research
[2] David Tall, Reflections on APOS theory in Elementary and Advanced Mathematical Thinking.
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Mark Bridger gave me permission post this response:
Your blog entry seems, if anything, to reinforce Aristotle’s assertion that there are no completed infinities. Basically, you are saying that we don’t “image” anything other that relatively small finite sets. In fact, you end on a note which brings to mind Charles Lamb’s famous line: “Nothing puzzles me more than time and space; and yet nothing troubles me less, as I never think about them.”
I also don’t understand your example involving the sequence of reciprocals. The definition of this sequence as a function is quite Aristotelian: The nth term is simply 1/n. No infinities of any kind intervene here. A function f is simply a rule or a procedure which, for each admissible “input” x, constructs or “associates” a certain “output” f(x). We only need to write out the rule. This rule may be iterative or recursive or in closed form.
Perhaps you think of a function as a set of ordered pairs, and want to claim that this set is infinite. But how do you know that this set exists? In all versions of set theory that I know of, you have to have an “axiom of infinity.” Of course, I think that this is an awful definition of function; in fact, hardly anyone uses it anymore as far as I can see.
So, are you saying that infinite sets are OK because there is an axiom that says so? I’m all for the formalist game as long as we own up to the fact that that’s what we’re playing. If mathematics isn’t *about* anything other than symbol pushing, and infinity is *asserted*, then fine, let’s play. But that can’t be what what “some people” are “hot and bothered” about.
I suspect that most people are initially uneasy about infinity for a good reason: it is beyond their experience. They only accept it when they are socialized — I might use the loaded term “brain-washed” — by math “experts” or clergy-people. When you talk or hear talk of infinity enough, the term loses its force — like some of the popular “four-letter words.” However, it won’t really go away. In spite of the many attempts to ignore them, Zeno’s paradoxes simply will not be swept under the rug (see, e.g., Alper & Bridger: “Mathematics, Models and Zeno’s Paradoxes (Synthese, January 1997), e.g.).
All the nice ideas of analysis you refer to, Cauchy sequences, limits, etc, are all phrased in language that Aristotle or Eudoxos or Archimedes would be quite comfortable with. “For each epsilon we can find a delta” is fine for “potential infinities”: we don’t need a *set* of anything. You just show the rule for constructing the “deltas” from the epsilons.
Finally, an example of a kind of propaganda for completed infinities, is the incorrect assertion that Euclid proved the “infinitude” of the primes. Only in the eye of the modern beholder. What is in Euclid is a proof that, for any (finite) list of primes, it is always possible to construct a prime not on that list. As we know, the proof is short, elegant, constructive, and proves no more nor less that what it claims. Nothing to get upset or disconcerted about.
Best,
Mark
This is maybe the last post I would have expected to see from someone who wrote a book on elementary toposes. I mean, isn’t it true that some elementary toposes (finite sets?) can fail to have infinity? And isn’t it true that some of them are toposes of genuine interest?
Or maybe that’s exactly the point. Are you saying that it’s possible to naturally augment any elementary topos so that it has infinity?
This is a reply to James’ comment. He asked: “Are you saying it’s possible to naturally augment any elementary topos so that it has infinity?” Freyd’s Embedding Theorem (in Chapter 7 of Toposes, Triples and Theories) says every small topos has an embedding into a power of Set. But that is not really relevant to what I was saying, which in part is that it is OK to conceive of an infinite mathematical object as a whole that exists all at once. Thinking about them that way harnesses built-in mechanisms in our brains and the results have not turned out to be inconsistent with each other.
By the way, I should have referred to the “Zeno image”, not the “Xeno image”.
You could use loaded language like “socialized” or “brain-washed” to describe people who have no difficulties with actual infinities; you could also use language loaded in the opposite direction, such as “technically competent”. The complexity, in a roughly Kolmogorovean sense, of the sequence described in this post is small enough that the sequence can be understood and explored through theorems, even though the cardinality of the set described is Aleph-null. Because the definitions of e, π and i require only a finite number of symbols, we can show that e raised to the power of iπ is -1 without computing an infinite number of decimal places.
I’m a little puzzled as to why the integers, defined as a sequence, are not “Aristotelian” in the same sense as the 1/n sequence: the n-th term is simply n, no infinities intervening. No doubt this is why I did not go into philosophy.
Quoting Keith Devlin:
The subtlety that appears to have eluded Bishop Berkeley is that, although we initially think of h as denoting smaller and smaller numbers, the “lim” term in [the definition of the derivative] asks us to take a leap (and it’s a massive one) to imagine not just calculating quotients infinitely many times, but regarding that entire process as a single entity. It’s actually a breathtaking leap.
In Auguries of Innocence, the poet William Blake wrote:
To see a World in a Grain of Sand
And a Heaven in a Wild Flower
Hold Infinity in the palm of your hand
And Eternity in an hour
That’s what [the derivative] asks you to do: to hold infinity in the palm of your hand. To see an infinite (and hence unending) process as a single, completed thing. Did any work of art, any other piece of human creativity, ever demand more of the observer? And to such enormous consequence for Humankind? If ever any painting, novel, poem, or statue can be thought of as having a beauty that goes beneath the surface, then the definition of the derivative may justly claim to have more beauty by far.