Mike Hosken's

"Hinges and Loops"

CHAPTER THREE

PARADOXES

  1. Sets
  2. Propositional Logic
  3. Incompleteness
  4. Discussion
pa'radox n. seemingly absurd though perhaps actually well-founded statement; self-contradictory or essentially absurd statement; person or thing conflicting with preconceived notions of what is reasonable or possible; paradoxical quality or character.
Concise Oxford Dictionary
If we want a really down-to-earth physical paradox we can make ourselves a one-sided piece of paper with but a single edge. All we have to do is to cut a strip of ordinary paper, bend it round tail-to-head as if we were making a link of a paper chain, but then twist the head by half a turn before gluing it to the tail. That is in my dictionary too:
Mö'bius (mer'-) n. - strip etc., (Math.) surface with only one side and edge formed by joining ends of rectangle after twisting one end through 180 degrees.
It adds that A.F. Möbius was a German mathematician who died in 1868.
If you can't believe it really does have only a single surface all you need do is to mark one surface, all the way round the loop, and then see if there is any surface left unmarked. There won't be! Similarly, colouring the whole of just one edge means that there is no edge left uncoloured. OK it's a trick, childish perhaps. At various times of childhood you probably, like me, amused yourself for a few minutes by working out verbal paradoxes. So these are probably not new to you:
  • If God is omnipotent and can do absolutely anything, can He make a rock so heavy that He can't lift it?
  • I always tell lies (or, I never tell the truth).
  • I command you not to obey me.
  • Perhaps a little more subtly, misogyny is heritable.
Now this is not intended to be just a relaxing diversion. On the contrary, similar paradoxes have been accepted in many circles as constituting philosophically significant proofs. I'm afraid we must get a bit more technical ...

§3.1 Sets

There is a subject known as set theory. A "set" can be any collection of items with a certain specification. The mathematician may be interested in such things as the set of prime numbers. But the set need not be mathematical at all: it could be the set of circular objects including wheels, balls, drawings of the sun, and so on. Even the set of lead balloons can be dealt with: rather than say there is no such set the convention copes with it as an "empty" set. Any of these ordinary sets can be combined together into sets of sets - and sets of sets of sets without limit. So the set of balloons could be made up of -
  • the set of rubber balloons
  • the set of plastic balloons
  • the set of silk balloons
  • the (empty) set of lead balloons.
There is not even any rule that a set cannot be a member of itself! So there could be a set of second-degree sets including -
  • the set of sets of balloons
  • the set of sets of aeroplanes
  • this set of sets.
In 1902 Bertrand Russell realised that this could lead to a difficulty: it was possible to have a perfectly legal set defined as -

the set of all sets which are not members of themselves.

Naming this particular set "R" for the sake of brevity, the paradox was then to decide whether or not R was itself a member of R. If it were not to be included then R would not include all sets which are not members of themselves: but if it were included it would not qualify for inclusion! Russell therefore concluded that set theory was insufficiently rigorous to cope with all contingencies and he attempted to provide a more universal and consistent formal system in its place.

§3.2 Propositional Logic

There exists another sort of hybrid land, covering an overlap between philosophy, logic and mathematics. We all know from our school days that arithmetic can be generalised into algebra:
  • 100 items @ 4 pence comes to 4 pounds
  • 100 items @ 5 pence comes to 5 pounds
  • 100 items @ 6 pence comes to 6 pounds
- and so on: in general -
  • 100 × p = £
In much the same way it is possible to express any self-consistent system as a formal set of symbols and relationships. Most common in the field of philosophy are statements of formal logic: it involves (in one form) propositions connected by relationships, especially "and", "or", "not", "if...then", and "if and only if". We don't need to go into detail but a single example may be helpful.
  • Let P = "it is raining".
  • Let Q = "water droplets are coming down, out-doors".
We can then assert the proposition that "P implies Q", or if you prefer "If P then Q", or symbolically "P → Q". In this particular case we can go further with -Q → -P which stands for "If not Q then not P". It would not, of course, be true to say that "If not P then not Q" since someone may have a sprinkler at work nearby.

It is the relationships between such propositions that make up the so-called propositional calculus of formal logic. It has its rules and its theorems: it is strict enough to be translatable into instructions for a computer.

§3.3 Incompleteness

But propositional logic is only one example of a formal language. Anyone is quite at liberty to invent a new system made up of signs, representing -
  • the subject-matter items (like the numbers in arithmetic and the variables in logic)
  • connectives (corresponding to +, −, ×, "not", → and so on)
  • punctuation or some other devices to remove ambiguities and improve clarity of expression (as with commas and brackets, etc.).

A set of starting points called axioms can be proposed, together with rules governing the way the signs are to be combined into what are known as "well-formed formulae" on a par with valid formulae in mathematics and with what we would refer to as grammatically correct sentences in ordinary language. Provided the system is built up in a self-consistent way it may well be able to extend our understanding into realms not covered by other languages. Russell, indeed, went to great lengths along one such path in his attempt to make mathematics more watertight.

Another mathematician, Hilbert, a quarter of a century later was attempting a total formalisation of all aspects of mathematics. The hope was that it would be possible to say, and indeed to prove, of any properly constructed mathematical assertion, whether it was true or false. But in 1931 the Austrian logician Kurt Gödel argued that such an undertaking was doomed to failure. Again it was a paradox that lay at the heart of the matter.

There would be, within Gödel's formalism, whole series of assertions, known as propositional functions. To keep track of them -

  • they could all be given names, conveniently in the form of letters, perhaps "A",
  • within which they could be assigned numbers, though when considering general cases we would use letters as in ordinary algebra, perhaps "p", "q", "r" and so on:
  • so a particular function would then be, perhaps, "Aq".
If we were talking about squaring a number we could say that the function was "times itself": we might then label that function M2 with cubing as M3 and so on. Applying M2 to the number 6 could be shown as M2(6): it would give 36. Applying M2 in general could be specified as M2(k), and using any of the M family of functions on number k as Mq(k).

But unlike ordinary arithmetical functions such as squaring, Gödel's more universal system was to cope with proofs and indeed statements about proofs - and their absence. And furthermore, such functions could apply to themselves: Mh(h) would stand for propositional function M applied to itself using h as the case in point.

Gödel realised that "There is no proof of *** within the system" would be a legitimate well-formed formula within his system when a function was put in place of the "***". But such a function could be a self-referencing one: it would be valid to have -

Pk(k) means "There is no proof of Pk(k) within the system".

This then was the paradox case. If the system could not prove Pk(k) then it was not a totally comprehensive formalism. But if it could prove it then it was proving something that was incorrect and so the formalism was in an even worse case - it was non-valid! In short, any attempt to produce a totally comprehensive and self-consistent formal mathematical language was doomed to failure. This came to be known as Gödel's incompleteness theorem. It has been accepted in many circles as setting limits to the objective certainty of knowledge: that is why we bothered with it here.

§3.4 Discussion

That ends our list of paradoxes. We are left only to assess their significance. And here I must make it clear that I am expressing my own opinions.

I do not believe that the concepts of surface and edge are made unacceptable by the existence of a Möbius strip. It is quite clear to me that the piece of paper out of which the strip was contrived had and still has two opposite surfaces and two opposite edges. Uniting head to tail by gluing does indeed destroy two edges, just as surely as the converse, cutting, creates two new edges. There are no clearly defined boundaries to the edges but that cannot be taken as denying the existence of two edges unless it is sensible also to regard a circular piece of paper as having but a single edge. Similarly of course, a solid sphere can be thought of as having only a single surface. In short, any discussion seems to be about the meaning of the words "edge" and "surface" without denying the possibility of knowing all there is to know about a Möbius strip.
  • If God is omnipotent and can do absolutely anything, can He make a rock so heavy that He can't lift it?
    Obviously in that form the enquiry is childish. But it takes on a much greater significance if it is tweaked into a different form:
    • If God is omnipotent and can do absolutely anything, did He have a free choice in deciding the strength of the gravitational constant when He designed the universe?
      Physicists argue about this one but it is outside the scope of our present discussion. Whether we consider the child's or the cosmologist's version though, I do not think this paradox is enough logically to deny the omnipotence of God as normally conceived.
  • I always tell lies (or, I never tell the truth).
  • I command you not to obey me.
    I regard these two as being purely language paradoxes. Just as any fool can ask questions that the wisest cannot answer, so any fool can mix up legitimate elements of any language (of words or of symbols) in such a way as to make varying degrees of nonsense. Every word of "I am not me" is perfectly acceptable but the combination is just plain nonsensical. "I tell lies" is OK: it may even be true. "I always tell the truth" is technically a well-formed formula: it is valid though not necessarily true. But the fact that it is possible to construct the statement "I always tell lies" does not in my view say anything new or significant about the language.
Two things may be significant about these paradoxes, though:
  • They are self-referencing -
    • The person making the statement is making that statement about his or her statements.
    • An order is being given about the carrying out of orders.
  • They are negative (assuming that telling lies is regarded as being negative).
The existence of the impossible diagram in Chapter One -

See Chapter One

- does not, in my view, set limits on the extent to which draughtsmanship can portray truth - and nonsense. Similarly, the existence of language paradoxes does not, in my view, set limits on the extent to which words can portray truth - and nonsense.

Now Russell's paradox was of exactly the same form as the last two examples. Recall that the trouble arose from -

the set of all sets which are not members of themselves.

This is clearly self-referencing and it is negative. It does not, in my view, negate the validity of the remainder of set theory nor say anything about any limits on what can be known.

Similarly with Gödel:

Pk(k) means "There is no proof of Pk(k) within the system".

Again the self-reference: again the internal negation. Again, in my view, the lack of any deep significance. Incidentally, "misogyny is heritable" is also self-referencing and negative (assuming procreation is linked to love rather than hatred).

In short, I see paradoxes as amusing entertainments. I would not make so bold as to state that they are all a waste of mental effort in all other respects, nor that they all lack any real significance. But I am confident enough to conclude this Chapter with a generalisation which seems to me to arise naturally from the discussion.

HOSKEN'S HYPOTHESIS
No self-referencing negative paradox can be significant.

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