This month we read Godel, Escher, Bach by Douglas Hofstadter.
This is a fantastically weird and wonderful book. At one level, the book is about a parallel between three people: Johann Sebastian Bach, Kurt Godel and M.C. Escher, in particular how their work in music, math and art manages to loop back on itself and express a kind of self-referentiality. On another level, however, this book is really a hypothesis about the human mind itself, that selves and souls can come out from inanimate matter the same way that M.C. Escher’s hands draw themselves or Godel’s creates math that proves that it cannot be proven.
GÃ¶del, Escher, Bach is a wonderful exploration of fascinating ideas at the heart of cognitive science: meaning, reduction, recursion, and much more.
Here are some of the highlights from this month’s review. First off, we establish what Douglas Hofstadter is referring to when he talks about a “strange loop”; it’s perhaps easiest to refer to the visual artist M.C. Escher to explain this very sophisticated concept:
An example would be [his images] of the drawing hands where he has drawn two pictures of hands which are holding pencils and drawing themselves. Other ones Escher has includes birds which turn into other animals and shift between the foreground and background. He even has one where there is a person looking at a picture in a museum and the scene is warping so that he is part of the museum itself.
So, this is something that I think is easy to see to visually, this kind of paradox, and it’s easy to dismiss it as well as being a bit of trickery. You might not even be a fan of M.C. Escher’s art, you might consider it too obvious.
However, I think it’s a really good entry point to grasping this idea of a strange loop. Now what I think Douglas Hofstadter calls a strange loop or this tangled hierarchy is where you have layers of something where something is built on top of something else.
Next, I discuss mathematician Kurt Godel and how he showed that there were limits to logic itself:
Hilbert’s problem was that, at the time, there were no different axiomatic systems for mathematics. His idea was that if someone could come up with a proof that shows that standardized axioms exists, and was consistent and also complete, that would be a major triumph. We could feel very secure resting in our knowledge that this mechanical system would work all the time. And what Godel did in the 1930’s, was that he showed that this was actually impossible.
Godel showed that any system that you have, any set of rules that you have, is powerful enough to represent the basics of natural numbers (the basics of arithmetic) will ultimately undermine itself by creating situations that we know are true but cannot be represented.
To explain Godel’s proof in another way:
Think about how 2 + 2 = 4. You have a two, you have a plus sign, you have a two, you have an equal sign, and you have a four. What Godel is doing is saying, let’s take that 2 and put it in a number, let’s take that plus sign and put it as a number, let’s take that 2 and put it in a number, let’s take that equal sign and put it into a number and let’s take that 4 and put it into a number.
Now you have one super long number that represents 2 + 2 = 4 as not just a number but as something that involves plus signs and equals signs and has separate numbers but as just one big number.
Okay so what can you do with this? Well if you go through a little bit of work what you can end up doing with this process is you can make a statement that is the mathematical equivalent of saying “this statement is false” or more specifically what it is saying is “this statement has no proof.”
There are also mind-bending ramifications of Godel’s work on our current concept of human consciousness:
…. he’s identified a system that has reached a level of sophistication that has the ability to loop back on itself. What’s the extension here? One of them is the idea of human consciousness. He’s saying that what perhaps the self is, what we are, as not bodies but sort of an abstract quality, what makes us different from rocks or computer programs that we have today, is that our machinery for representing things is sufficiently complex and we can represent ourselves in that machinery. Moreover, we can represent ourselves representing ourselves.
We can have thoughts about our own thoughts and about the person thinking the thoughts and about their relation to the world at large.