The Problem of Deduction: Turtles All The Way Down

A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: "What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise." The scientist gave a superior smile before replying, "What is the tortoise standing on?" "You're very clever, young man, very clever," said the old lady. "But it's turtles all the way down!"

     -- Stephen Hawking,  A Brief History of Time

So in our giant adventure into the epistemological underpinnings of science so far, we've discovered that induction is entirely self-referential.  Of course you know that the sun will rise tomorrow, but other than the fact that's it's always risen before, you don't have that strong an argument.  The reason most people consider it convincing is because of our faith in an orderly universe.

The other big tool in the knowledge toolbox is deduction.  As its name implies, it's kind of the opposite of induction.  With induction you take specific instances (the sun has risen every day since I can remember) and apply a general rule (the sun always rises). With deduction you have  general rule and apply it to specific instances. This is the realm of mathematics and logic. 

For example:

Premise:          A triangle has three angles that add up to 180º.

Observation:   Two angles of a certain triangle add up to 120º.

Conclusion:    The third angle is 60º.

Elementary, my dear Watson.

But this is not quite as elementary as it gets.  How do we know a triangle has 180º?  Well, in Euclidean geometry we base it on a more fundamental theorem, and the more fundamental theorems are based ultimately on axioms.  These are the really obvious things, like all right angles are equal, or the whole is greater than a part.  Euclid, the Greek Father of Geometry who lived around 300 BC, developed five postulates that have stood quite nicely for the last 2500 years, and there are plenty of proofs out there that use these axioms to prove that a triangle, indeed, has 180º.  And using these geometric or mathematical axioms and theorems developed on these axioms, we get all kinds of wonderful things like a good chunk of geometry, mathematics and physics.

But what about these axioms?  They aren't, in fact, based on anything.  They are supposed to be self-evident.  I mean, who wouldn't accept Euclid's notion that "things that are equal to the same thing are also equal to one another"?

We have no justification for them at all apart from "Well, it's obvious, isn’t it?"  So in fact we have a similar problem as we had with induction. We're left to accept that the axioms are true but unprovable. The entire edifice is constructed on the backs of these turtles that we call axioms.  But what are the turtles standing on?

Well, brilliant mathematician and philosopher Bertrand Russell made it his early life's work to sort this.  His masterpiece, co-written with Alfred North Whitehead, was a book called Principia Mathematica, a seminal work of logic on par with Aristotle's Organon.   

Page 378 of Principia Mathematica finally gets around to 1+1=2.

They investigated what the axiom turtles were standing on, and uncovered even more basic turtles.  But what were those turtles standing on?  Here's what Russell had to say of the effort later in life:

I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers wanted me to accept, were full of fallacies ... I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.

     --Portraits from Memory, 1956

A little while later another brilliant mathematician named Kurt Gödel, in a masterful piece of logic called, informally, Gödel's Incompleteness Theorems, showed that any (non-trivial) system based on axioms would be either incomplete or internally inconsistent. 

In other words, in our commonly used axiom-based systems (geometry, mathematics, logic) there will always be truths that will remain unknown.  You can go outside the system to add new axioms (drawing a larger circle around your circle, so to speak) but that system itself will be subject to the same incompleteness or inconsistency.  And so on.

Rudy Rucker has managed better than anyone else, I think, describe Gödel's Incompleteness Theorems:

The proof of Gödel's Incompleteness Theorem is so simple, and so sneaky, that it is almost embarrassing to relate. His basic procedure is as follows:

1.                 Someone introduces Gödel to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.

2.                Gödel asks for the program and the circuit design of the UTM. The program may be complicated, but it can only be finitely long. Call the program P(UTM) for Program of the Universal Truth Machine.

3.                Smiling a little, Gödel writes out the following sentence: "The machine constructed on the basis of the program P(UTM) will never say that this sentence is true." Call this sentence G for Gödel. Note that G is equivalent to: "UTM will never say G is true."

4.                Now Gödel laughs his high laugh and asks UTM whether G is true or not.

5.                If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements.

6.                 We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact a true statement. So G is true (since G = "UTM will never say G is true").

"I know a truth that UTM can never utter," Gödel says. "I know that G is true. UTM is not truly universal."

It was a particularly mind-bending piece of work.  He basically took the knife of logic and applied it to itself. I highly recommend that, if you are interested and diligent, you read the Pulitzer Prize-winning Gödel, Escher, Bach by Douglas Hofstadter, a fun book which really does justice to the staggering implications of this.

So logic itself is based on a number of unprovable axioms which we accept on faith.  Which poses a bit or a problem as wonderfully articulated by G.K. Chesterton: 

You can only find truth with logic if you've already found truth without it.

What's more, even if these axioms are true, Gödel showed that any system upon which they are based is necessary incomplete and/or internally inconsistent.  There will be truths which the system cannot determine.

Two millennia ago, Pliny the Elder, the Roman philosopher and naturalist, said that "the only certainty is that nothing is certain."  But I see no reason why that would be certain.

Interesting that these great minds throughout the ages have applied the knife of rationality with a surgeon's precision to find one certain true thing.  But logic does not to seem up to the task.  So where then to find it?