Philosoraptor actually stumbled upon a deep philosophical question, with consequences affecting mathematics, our own mind and – to some – even God. Think about Pinocchio’s paradox: the way by which the contradiction arises from self-reference was what Kurt Gödel used in 1931 to prove his Incompleteness Theorem, amongst the most important scientific discoveries of the past century.
Marcus Dominus quotes the “World’s shortest explanation of Gödel’s theorem", by Raymond Smullyan, and as it’s indeed short, I reproduce it in full:
“We have some sort of machine that prints out statements in some sort of language. It needn’t be a statement-printing machine exactly; it could be some sort of technique for taking statements and deciding if they are true. But let’s think of it as a machine that prints out statements.
In particular, some of the statements that the machine might (or might not) print look like these:
— P*x (which means that the machine will print x)
— NP*x (which means that the machine will never print x)
— PR*x (which means that the machine will print xx)
— NPR*x (which means that the machine will never print xx)
For example, NPR*FOO means that the machine will never print FOOFOO. NP*FOOFOO means the same thing. So far, so good.
Now, let’s consider the statement NPR*NPR*. This statement asserts that the machine will never print NPR*NPR*.
Either the machine prints NPR*NPR*, or it never prints NPR*NPR*.
If the machine prints NPR*NPR*, it has printed a false statement. But if the machine never prints NPR*NPR*, then NPR*NPR* is a true statement that the machine never prints.
So either the machine sometimes prints false statements, or there are true statements that it never prints.
So any machine that prints only true statements must fail to print some true statements.
Or conversely, any machine that prints every possible true statement must print some false statements too.”
Perhaps the simplest and most intuitive grasp of Gödel’s Theorem is Pinocchio’s, or the even simpler “I am lying”, as the key concept is self-reference, by which both Pinocchio or that hypothetical printer can produce statements about themselves that lead to contradictions either way. Smullyan’s explanation however, though a bit longer, also illustrates how this is related to math and what Gödel accomplished.
Because, you see, the machine capable of printing statements, including about itself, is arithmetic, in a way, math.
Here’s the context. At the beginning of the 20th century, mathematicians were trying to give math an absolutely perfect, solid, sound, foundation. The most beautiful and pure of the sciences. From that basic, rock solid foundation, more complex areas of mathematics and science would then be able to rest, and be extended into, until eventually every and all statements would be clearly defined as either true or false. One of greatest works representing this ideal was Whitehead and Russell’s Principia Mathematica, where the proof that 1+1=2 is reached at page 379 of the first volume – and completed at page 86 of the second (PDF).
With all of the great minds dedicated to the pure beauty of certainty, Kurt Gödel was among them, himself trying to prove that, until he stumbled upon the proof that this ideal was simply… impossible. Through brilliant tricks involving Gödel numbers and diagonalization, he realized and proved that the logical paradoxes such as Pinocchio’s or the barber – proposed by Russell himself – could be translated into arithmetic and thus represent a problem of all systems of propositions upon which we could build the arithmetic that we know.
This showed these paradoxes were not merely fun curiosities or problems that could be circumvented – as Russell tried to – but illustrations of fundamental and insoluble problems. Either the system of propositions was consistent and incomplete – the printer that prints only truths, but not all of them –, or complete and inconsistent – the printer that prints all the truths, but some lies too.
The consequences of Gödel’s Theorems are completely mind-boggling. There are, for instance, in current mathematics several statements that can’t be proven nor refuted. They may be true, they may be false, but the proof of that can’t be produced by the mathematical machine we are using. Usually they are considered true or false based on the usefulness of considering them true or false, becoming themselves a new proposition, an axiom. Notorious examples are the axiom of choice and the continuum hypothesis.
One could say that in the most pure and rational of sciences, Gödel’s work demonstrates that there are some statements that must be taken with faith.
Many, this writer included, would not be very comfortable with the story presented in this light, but at the very least the metaphysical question must be considered as a historical curiosity. Because Gödel himself considered it metaphysically.
Surely you can picture the image: a robot is presented with a logical paradox and halts. “It does not compute!” would be its last words before the artificial brain explodes. It goes from Forbidden Planet’s Robbie, pictured above, to the latest Terminator.
Gödel took that very seriously. To him, that we are able to see beyond these paradoxes was an indication that we are not machines, but have something else.
This belief in something else was a constant throughout Gödel’s life. One of his objectives was to turn metaphysics into a hard, exact science. It’s therefore not much a surprise that one of the things he worked on for decades was nothing less than a formal logical proof of the existence of God.
In an extremely simplified way, Gödel’s Ontological argument tried to formalize previous ideas – from Anselm and Leibniz – that, in a nutshell, stated that “God is perfect, therefore he exists”. It may sound as trivial and innocuous as “I am lying”, but remember what Gödel was able to do from such a simple paradox. Did he accomplish a similar feat with God?
Well, few people, if any, hail Gödel as the man who finally proved God exists. The answer is no, his ontological argument was not a definite, solid and revolutionary proof such as his Incompleteness Theorems and other published works. Gödel himself recognized that, as he didn’t publish it and we came to know it in more detail only after his death. It was, pardon the pun, an incomplete work. Some would say, inconsistent.
Even the belief Gödel had that our ability to see beyond logical paradoxes was evidence we had something divine is not that sound. That we are not limited to hard logic is clear, what must also be clear is that we are hardly limited to any kind of consistent logic. The evidence suggests even our most solid certainties may start as random strands of thought which are developed and rationalized a posteriori, all unconsciously. A coin may also be able to decide heads and tails without any axiomatic system or connection with a higher entity.
Ironically, Kurt Gödel’s own metaphysical faith may be seen as one of these random, inconsistencies. This is not meant as something derogatory for Gödel, although he did have psychological problems which sadly lead him to death. In a way, one can see how this randomness was also part of what lead him to develop and prove some of the most revolutionary ideas in the history of ideas.
That may seem like a paradox, but then, not all things are pure and perfectly clear.
Above, Gödel with a close friend
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