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…