This is the fourth in a series of posts on my own philosophical journey; the first post is here.

At some point during my college years, I began to read Douglas Hofstadter’s Gödel, Escher, Bach: An Eternal Golden Braid. I started reading a copy by a friend, who’d gotten (I think) as required reading for a class on artificial intelligence. Eventually I got my own copy; and it took me years to actually read the whole thing. It’s a playful, whimsical, sprawling book, which I will not try to summarize here; and it’s also a work of philosophy, though I didn’t understand that at the time. The thread that binds the book together is Gödel’s Incompleteness Theorem, a very startling result.

Early in the 20th century, an effort was started (by Bertrand Russell, among others) to put all of Mathematics on a sound footing. Mathematics is the most certain of all human knowledge, but it wasn’t certain enough. Its foundations were felt to be a bit shaky, starting as they did with our normal, intuitive grasp of number. The goal, consequently, was to define the absolutely minimum number of axioms and postulates and derive all of Mathematics from that, thoroughly and systematically and for all time.

It was a bold plan. That which was true would be proven true; and that which was false would be proven false; and certainty would reign.

And into this bold project strode Kurt Gödel, who knocked it into a cocked hat.

Without going into great detail, and leaving out all sorts of nuances and caveats (not to mention the proof itself), what Gödel proved was that any sufficiently powerful system of axioms and postulates was incomplete: that there were truths expressible in the system that could not be proven within the system.

In short, you’re in a cleft stick. You can make your mathematical system complete, but it will be too simple to be of interested; or you can make it cover everything, but it won’t be complete. Bertrand Russell’s project was fundamentally flawed.

I had left my philosophy class with the conviction that trying to prove the whole range of philosophical truths from a small set of first principles was doomed to failure. It might work in mathematics, but it didn’t work when you expanded your scope to all of reality. Now I discovered that it didn’t really work in mathematics, either.

Please note: Gödel’s Incompleteness Theorem doesn’t mean that mathematics is useless, or that proceeding by means of axioms, postulates, proofs, and theorems is valueless. It’s an effective tool. But its power, even in the restricted realm of mathematics, is limited.

This was a fascinating result, and not surprisingly it confirmed me in my anti-philosophical prejudices.

Part V