In a World of Logicians and Their Ways: Russell’s Paradox Redux

This is the first in a planned series of posts on Worlds, as understood by Descriptive Psychologists. This series requires a more careful reading than most prior posts on this blog; I believe the work you put into it will be well-rewarded.

Worlds are subtle, pervasive, powerful. We are to our worlds as fish are to water: We take our world as given, existing outside of and independent of us. We live in the world; all our actions take place in and are shaped by our world; without the world we could not exist.

But fishy metaphors can only take us so far. Persons, whether we realize it or not, shape our worlds fully as much as the world shapes us. We almost certainly don’t consciously create or choose our world. But sometimes events occur that blow our worlds apart, and we are faced with the task of putting it back together, or reconstructing it in a new form. In either case we engage in active, conscious choice and creation. We recognize once and for all that we are not fish.

Case in point: In the early 1900’s Bertrand Russell was contemplating set theory as the basis of number when he happened upon a paradox that had extraordinary impact on him and his cohort of logicians. Bear with me as I lay it out for you; even if you already know it, or have concluded you just don’t get it or don’t care, you may find it worthwhile to follow along because it takes us in an unexpected direction. This post is not essentially about Russell or his paradox; it is about the world Russell lived in, and what it says about the world you live in today.

A set is simply a collection of items. Often the items have something identifiable in common, like {R}, the set of all red objects in the room, or {F}, the set of all fruits grown in California. A set can contain items that are themselves sets; you could begin with the set of all red items {R}, and the set of all blue items {B}, etc. and then construct the set of all sets of colored objects, {C}. That set would be written down as {C} = {{R}, {B}, ….} and it would contain as many items as colors you have identified. Sets can also be indefinitely long in size, like the set {N} of all real numbers. And interestingly, some sets contain themselves as a member, like {P}, the set of all sets explicitly referenced in this paragraph. The members of {P} would be {R}, {F}, {B}, {C}, {N} – and {P}!

Russell considered a set – let’s call it {S} – characterized as “The set of all sets that do not contain itself as a member.” In other words, all members of {S} are sets that do not contain themselves as a member, and {S} contains every set of that kind.

  • Russell asked: “Is {S} a member of {S}?”If you say Yes, what does that mean? {S} then is a member of {S} which means it contains itself. But all members of {S} are sets that do not contain itself – so {S} can’t be a member of {S}!
  • If you say No, then you are saying {S} does not include {S}, which makes {S} a set that does not contain itself, which by definition makes it a member of {S}!

Either answer leads to a logical contradiction.

Look – please don’t blow this off with a “Whatever…” or “I’ll take your word for it.” Work with me here. This is like a really obscure pun – you think about it and think about it and then, in a flash of insight, you get it. The sudden illumination that comes with seeing this paradox is crucial to appreciating everything that follows in this paper.

So you get it, right? A bit mindboggling? Annoying? Unsettling, perhaps? Russell and his cohort found it all that and more – in fact, they found it devastating. Their reaction tells us an enormous amount about the world they lived in, and by reflection about all the worlds we live in today.

Really, what’s the big deal? I have observed the effect of Russell’s Paradox on many people, and most people have this reaction: OK, so…?. Once they get the paradox, they find it amusing or a clever trick, maybe fun to think about a bit, or perhaps just what you would expect from people who think about things that have nothing to do with the real world. It has no consequence for them or their lives.

Some mathematicians of the time took the Paradox in stride; it did not shake their world. Henri Poincare, for instance, had long held that “logic is barren”; the Paradox just confirmed his views. David Hilbert was said to be disturbed by the Paradox, but was confident he could find a way around this.

But Russell and many other mathematician/logicians of his time had a very different reaction: OMG! That can’t be right! And if it is, I’m toast! For instance, Gottlob Frege immediately renounced his life work on “Foundations of Arithmetic” upon hearing Russell’s Paradox. To them Russell’s Paradox blew their world apart. Something happened that was impossible in the world they lived in, and it was not a small thing – it was fundamental. They were now living in a fundamentally different world.

We will now bid farewell to Russell’s Paradox because it has served its purpose in this paper, which is to prepare us to address three questions:

  1. What kind of world is it that gets blown apart by Russell’s Paradox? What held it together to begin with?
  2. What kind of person lives in that world?
  3. What other worlds are there – and what holds them together?

To address these questions we need some Descriptive Psychology.

Next: Russell’s world, what held it together – and why it blew apart.

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3 thoughts on “In a World of Logicians and Their Ways: Russell’s Paradox Redux

  1. Tony,

    I think that my first encounter with Russell’s paradox was as a student generally interested in math for its own sake. My response to it was something like: ”Gee, at this level of precision in logic, it does not take much to get into trouble!”

    When reasoning as an engineer, one who typically regards math as a tool for greatly extending the power and precision of common sense, my response is: “Well, I better somehow steer clear of this sort of thing!”

    The engineer’s position can be bolstered by that of the computer scientist, for whom the design of languages for specific purposes is a mainstream activity. From that perspective, the challenge is: “Let’s design a language for describing sets that has enough expressive power to serve us where we need it, but not so much that we encounter the Russell Paradox”. That is Zermelo-Frankel set theory (ZF), as described by Paul Halmos in his classic “Naive Set Theory”.

    So above are three worlds that are touched by the paradox, but not destroyed by it. For that to happen one would need to regard mathematical results as a foundation that supports our common sense, rather than a tool that extends our common sense. Then the paradox calls our real world into question.

    When you are choosing your mathematics (and its status in your world), Caveat emptor!

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