My post on infinities and the accompanying video generated some interesting discussions and illustrated the difficulties that people have the idea of ‘operational definitions’ in science and mathematics. While scientists and mathematicians, like everyone else, use everyday language to communicate with one another, they are well aware that language contains traps in the form of implicit meanings and hidden concepts that can lead to ambiguities and even paradoxes in unfamiliar situations. These arise because our intuitive concepts are developed from our experience with the everyday world and while they may work well there, problems can arise when they are extended to regions beyond our immediate experience.
For example, we all have an intuitive sense of what time and distance mean and think we have a good grasp of how to measure them. But those intuitive ideas lead to problems when we are dealing with objects that travel at high speeds. Albert Einstein introduced the operational definition of time as the readings on standardized and synchronized clocks, and distance as the difference between two readings on standardized rulers. These operational definitions correspond to our everyday notions when applied to everyday phenomena but lead to seemingly weird results (such a time dilation and length contraction) when we go outside that range. There is no sense railing against this seeming insult to our ‘common sense’ because common sense cannot be expected to hold in uncommon situations. We cannot pick and choose which results to believe and which to reject. The only alternative is to create an alternative system that also agrees with observed phenomena in all the ranges.
Similar problems between the intuitive and the operational occur in mathematics. The mammoth exercise that Bertrand Russell and Alfred North Whitehead expounded in their Principia Mathematica was a heroic attempt to get around this problem by draining mathematics of all words that might lead one astray, to make it purely axiomatic and symbolic, reducing it to a system of strings of symbols and well-defined formal manipulation rules of those symbols that allowed for no ambiguity.
Without going to such austere levels, it is possible to reduce the ambiguities by using the mathematical equivalents of operational definitions, which consist of well-defined sequences of operations that lead you to a given result. For example, it may seem obvious for people to think they know how to determine if two containers have the same number of objects or if one has more by simply counting or comparing. We have many ways of doing so and for finite numbers of objects, all those methods give us the same result.
But mathematicians know that the concepts of the ‘same’ number and ‘more’ and ‘counting’ can lead to ambiguities in unfamiliar situations, such as with infinite numbers of things. In order to overcome this problem, they adopt what are effectively operational definitions for how to count that allow you to conclude if two sets have the same number of items or not. These operational definitions give results that coincide with our everyday intuitions in everyday situations involving finite numbers of things (as they must) but can lead to surprising results when applied to novel situations for which we really have no intuition, like infinities.
As long as the results we get are consistent and do not lead to paradoxes and contradictions, mathematicians are willing to live with what seem to the lay person like surprising, counter-intuitive, or even outrageous results, the way scientists are quite comfortable with the idea that the ‘length’ of an object depends on how fast it is moving. Laypersons are advised to just grit their teeth and accept them, unless they can come up with alternative schema for counting that are also consistent and free from internal contradictions.
As was explained in the video in the previous post on this topic, following Georg Cantor, we say that two sets of things are said to contain the same number of objects if they can be put in a one-to-one mapping with each other (i.e., each item in one set can be uniquely matched with a single item in the other) while if set A contains items that cannot be matched with items in set B, but all of set B’s items can be uniquely matched with items in set A, then we say that set A is ‘larger’ (i.e., has more items) than set B.
So far, so good. The results obtained coincide with out intuitive ideas for sets containing finite numbers of elements.
But when we start dealing with infinite sets such as the natural numbers, integers, rational numbers (i.e., ratios of two integers), irrational numbers, real numbers, and so forth, we run into some issues. This is because of our intuitive idea that all the numbers can be lined up along a straight line from minus infinity to plus infinity and drawing conclusions from that image. We then observe that whereas all integers can be expressed as rational numbers, the converse is not true. In addition, between any two integers on that line there exist a vast number of rational numbers, whereas we can find two rational numbers (say ½ and ¾) between which there are no integers. Hence it seems ‘obvious’ that there should be more rational numbers than integers, since the rationals seem to encompass the integers.
But we run into problems when we try to use that intuitive method to compare the rationals with the irrationals. For one thing, neither number can be expressed as the other. In addition, it turns out that between any two rational numbers, there exists an irrational number and between any two irrational numbers there exists a rational number. So is the set of rationals larger than the set of irrationals, smaller, or the same? There is no intuitive way of determining this.
But the idea of mapping enables us to answer this question. It turns out that the set of integers can be put into a one-to-one map with the set of rationals but cannot be done with the irrationals. Hence the set of irrationals is said to be larger than the set of rationals.
One consequence of consistently applying this mapping method is that if we use it to compare the integers with the rationals, it turns out that because the set of integers has a one-to-one map with the set of rationals, it leads to the surprising conclusion that the set of integers is the same ‘size’ as the set of rational numbers. This confounds our intuitive idea of more and less derived from the number line.
While most people have little trouble with the idea that the set of irrationals is larger than the set of rationals (because irrationals are not part of people’s everyday experience and they have little intuitive sense of what they are), they tend to gag on the idea that the sets of integers and the rationals are of the same size.
But there is nothing to be done but to grit one’s teeth and accept it (as must be done with the idea that time and distance vary with each observer) as the consequence of trying to create a comprehensive and coherent system. One cannot pick and choose what results one wants to keep and what to reject. The only alternative one has is to create an entirely new comprehensive and coherent system.
And it gets worse. I hesitate to bring up the kinds of weird things that emerge as a consequence of something in mathematics known as the ‘axiom of choice’. Maybe some other time…
Well, one can do mathematics without accepting the “axiom of choice” but that really limits the results that one can get.
Example: it is taken for granted that the the principle of mathematical induction works, but that depends on the “least positive integer” (or well ordering principle) which is equivalent to the axiom of choice.
ollie, #1: Well, one can do mathematics without accepting the “axiom of choice” but that really limits the results that one can get.
Not as much as one would think. A surprisingly large proportion of the constructions and examples used in mathematics tend to be separable. In that case, dependent choice is sufficient to get most of the important results without the controversial paradoxes like the Banach-Tarski Paradox.
Actually, most mathematicians accept the Axiom of Choice so that most of the important results extend to non-separable sets. I just mention this because there are still a small number of people who still don’t accept the Axiom of Choice. I’ve even met one of them!
“Maybe some other time…”
You’re a mean guy, MS. First you make me curious and now you tell me I’ll have to wait for an undetermined period.
Fortunately I got the difference between countable and uncountable sets immediately during my education as a teacher maths and physics. Still I would have loved to have your lecture back then on this subject. This column is excellent, both its content and from a didactical point of view (and that’s something I can judge).
So please, tell me more about the axiom of choice. I promise I’ll do some reading a priori.
And to think some people just go “fucking math…” and give up at the algebra level.
*shakes head*
You should try teaching elementary ed majors. A frighteningly large number of them resist learning anything about arithmetic!
That explains a few things…
in Hilbert space, no one can hear you scream.
As I see it, the key problem is violating the assumptions underlying the understanding of the words used, yet continuing to use the same words as though nothing has changed in context. Sure, there are analogues of time and distance even when traveling near the speed of light. Are they exactly the same thing? No. The problem is in the presumption that what a thing is, even an abstract concept, can be completely separated from its environment. Realistically, we have to admit that the environment is part of what defines everything.
Infinity breaks the rules in much the same sense. When we engage in abstract reasoning that uses infinity, we have modified the context in a way that is guaranteed to break many assumptions that formed in a finite world. Concepts like size and counting don’t hold up the same way any more. There will be analogues, but the analogues will not match up exactly.
So while it’s true that by one definition the set of rational numbers is the “same size” as the set of integers, in that there’s a one-to-one mapping which can be constructed, by a different definition it is false. If we decide that the “real” test for whether an infinite set is larger were whether we can construct a mapping where every element in the “smaller” set has a match in the “larger” set, but there are still unmapped elements in the “larger”, then the set of rational numbers is greater. Every integer can be divided by one to give itself, and this is a one-to-one map between all integers and an incomplete sub-set of the rational numbers.
In abstractions we freely violate rules and assumptions we would expect to be held to in any other circumstance, and get away with it. This is mostly swept away with semantic games and poor language use. Or with fewer words, a recipe for confusion.
That is, in fact, the real test for whether an infinite set is greater than OR EQUAL TO another infinite set. The problem with your demonstration is that it only shows that the cardinality of the integers is less than OR EQUAL TO the cardinality of the rationals. Not strictly less than. And here’s why:
Consider the function f(j/k) = 2^j3^k for each j/k in the rationals. This will be a mapping, to use your terms, where ever element in the rationals has an element in the integers, but there are some elements in the integers with nothing mapping to them.
Well, now we have, by your test, that the rationals are smaller than the integers, AND that the integers are smaller than the rationals.
That’s why having an injection like that only means smaller than OR EQUAL TO, because you can sometimes construct both types of mappings (injective and bijective, to use math terms) between two sets.
That’s why, EVEN by your test the rationals are equal to the integers (in fact, by a rather cool theorem that shows that less than or equal to both ways implies equal to called the Schroeder-Bernstein theorem.)
This is very interesting. Thanks! I was not aware of the Schroeder-Bernstein theorem and will look it up.
One learns new things every day from these comments…
Why Banach–Tarski paradox is called paradox, if it involves non-measurable sets?
(I haven’t understood it from the day I heard about the theorem.)
Thank you for that video…I was a math major for a while in college, and the hierarchy of infinities was what convinced me that Computer Science was more my cup of tea.
Had I experienced this lecture then, I might have held on. It’s fascinating, but when one can’t wrap one’s mind around such a basic topic, it’s decision time…
just also thought…Trig for Dummies is a great exposition of *that* topic….easier to remember, less to derive on the test….
Proving Schroeder-Bernstein (with some help and guidance) was what first convinced me I definitely wanted to be a mathematician.
Wikipedia lists it at Cantor-Bernstein-Schroeder, which is new to me, I’ve always just heard it as Schroeder-Bernstein. But the first proof there is a pretty cool one, although the path I followed was more like the second.
Also, I wasn’t meaning to YELL in my comment above, just too tired/lazy right now for HTML. That said, I think I could have made my larger point better by saying: yes, we do define things abstractly in math, and it isn’t always exactly what the colloquial meaning of terms is, but the definitions are chosen for reasons, not arbitrarily. The reason the method I responded to is not used to determine whether a set is strictly larger or smaller than another is because of what I outlined: you would then end up with two cardinal numbers, where one is both strictly greater and lesser than the other. That’s why two sets related by a bijection are considered equal. The definition is chosen because it best fits our understanding of the concept and is most useful in creating a meaningful theory.
You are correct about most mathematicians accepting AC; but many, many results are proven using transfinite induction.
I suppose that there might be a way around using induction in those proofs….but …..why?
Count me among the mathematicians that accept AC.
Just for the record, I have a modest publication record and I’ve never had to use anything other than “finite”, “countably infinite” and “uncountable”. But I am aware of the other infinities out there. 🙂
…and the hierarchy of infinities was what convinced me that Computer Science was more my cup of tea.
Heh. That was what convinced me that mathematics was far cooler than I had ever imagined. This was from a Scientific American article from the 1980s (when it was a pretty deep magazine).
If you haven’t already, check out this blog post and the discussion below it. I mostly agree with the original post, that it isn’t really a paradox, but there’s a lot of back-and-forth in the comments that’s pretty educational on the subject.
Dr. Singham, this is an *excellent* description of both the issue of mathematical infinities and also the more general issue of how technical definitions given ‘intuitive’ names can cause confusion, and how that’s sometimes just the price we pay for speaking in natural language (e.g. English, French, or whatnot, as opposed to formal language such as symbolic logic).
I really enjoyed reading this piece. It was extremely clear and made an excellent analogy with Einstein’s relativity that was very apt and (I think) understandable for most people who are at least vaguely familiar with the (intuitively) strange findings of relativity. You have shed light on what is very often an obscure and dry subject, and sparked at least my curiosity about it.
If you have a list of some of your best writing pieces, you should definitely add this one to the list. Cheers! 🙂
By the way, have you heard of the latest thorn in the side of some creationists/fundies? Set theory! (http://scienceblogs.com/evolutionblog/2012/08/13/set-theory-now/)
Thanks for the nice words!
Yes, I was aware of the set theory issue and will add my two cents on it in the near future.
Anyone who wants to explore the various categories of infinity in an imaginative way might enjoy Rudy Rucker’s novel White Light. Quoth Wikipedia:
I found it interesting, frustrating and disturbing; I can’t say I entirely enjoyed it.
Just to clarify the notation, is the function you are referring to the product (2^j)(3^k)?
If so, that is not a function of the rational number (j/k); it is a function of the ordered pair of integers (j,k).
I’m not sure I follow you. I actually did screw up in that post, in the sense that that isn’t a function (as I had intended it) from Q->Z, but from Q+ (positive rationals) -> N. Regardless, however, it isn’t necessary to say it is from NxN as you seem to indicate.
(If you have Q+ = {a/b in Q | a/b > 0}, with further assumption that a and b are relatively prime, then the function as written is from Q+ -> N)
Maybe I’m wrong. I’m clearly not thinking straight anyway.
Why Banach–Tarski paradox is called paradox, if it involves non-measurable sets?
The word paradox is used to describe things that, before detailed analysis, appear to be logically contradictory or to defy “common sense” intuition.
Thanks, aleph squared!
The “slices” in B-T aren’t something you can cut with a knife -- they’re infinitely subdivided, not-contiguous pieces. Nothing in the real world has that property, and no real-world process has the ability to cut like that.
This nicely summs my complains about calling it paradox.
(I have had disappointed by few math teachers/professors, when they called this a paradox, even “contradiction” -- with lack of explanation and even some unreasonable excitement. It seems that after it I have “childhood trauma” which still haunts me.)
// Sorry for my possibly weird English grammar.