Thursday, June 17, 2010

Blindsight by Peter Watts

Blindsight is a 2006 science fiction book. It was mentioned in an Overcoming Bias blog and so I read it. The book is available in electronic form for free on the web. I read it for free from my local public library.

The quality of the ideas about consciousness and mind are outstanding. 5 stars, 2 thumbs up. You don't really get to the main punch line until page 325. What he has to say about consciousness and awareness and brains before then is merely excellent.

A quote: "... People aren't rational. You aren't rational. We're not thinking machines, we're --- we're feeling machines that happen to think." This is where I have been going in my own thinking about brains and consciousness. My model is Lucy, my Golden Doodle dog. In my opinion, you can't read about brains, biology, evolution, especially evolutional psychology, and not see it all laid out before you when you have your own dog. A highly pleasant way to reach your own conclusions about evolutionary psychology, get a dog. So in particular, and only sorta, you've got a lizard brain wrapped in a mammalian brain wrapped in a neocortex. Of course all mammals have a neocortex, but a dog's is a lot smaller than a human's. Whether I've got the anatomy right or not, most of what I think of as feeling is pretty similar between dogs and people.

And then you wrap it in a neocortex. With a sorta mini-dog neocortex you get a little bit of help figuring out what to be mad at, what to be hungry on, what to be horny on, what to be scared of. But mostly, if you are a dog, you are happy, sad, mad, glad, scared, excited etc., and these things dictate your actions. They also dictate your interactions. You can be a social animal without a lot of rationality. Just be loyal to what you love, angry at what seems to be frightening you and you have a pretty functional system.

Wrap it in a big-old human sized neocortex, and throw in nasty monkey emotions (have you seen chimpanzees interact?) and you get a deep need for psychiatrists, psychologists, and many other paid professionals, not to mention a significant and growing pharmacopia. Take the straight forward emotional reactions to things and graft a GIGANTIC rational model of the world, including all the people around you on to it, courtesy of your friendly local neocortex, and you have the basis of some great tragedies and comedies.

That which does not kill us, makes us stranger. -Trevor Goodchild
This is a quotation leading off a sectino of the book. I googled Trevor Goodchild, he is a character in a science fiction TV show that used to be on MTV. But I like the quote, it reminds me of Nietzsche.

I will talk more about consciousness after the "read more" link. SPOILERS about the book will be here. If you are thinking of reading the book, I recommend reading the book before reading the rest of this post.
WARNING SPOILERS! if you read the rest of this post.

Thursday, June 03, 2010

Approximations to an Exponential

Summarizing, we have the exponential function taking a complex argument
F = exp(Z)
If we write Z explicitly in terms of its real and imaginary parts,
Z = X + 1j*Y
F = exp(X) * exp(1j*Y)
But each of these pieces is simple:
FX(Z) == exp(X) is the exponential function on a real argument, very small for negative X, very large for positive X, and always positive for any X
FY(Z) == exp(1j*Y) is a rotating phasor. FY always has magnitude 1 and phase Y radians.
Now let us consider approximating exp(Z) with a finite polynomial. The page referred to above describes that a little more in detail, most of you will not need to check that to see what is going on. Let us write the approximation of exp(Z) to an Nth order polynomial as
expN(Z) = FXN(X)*FYN(Y)
Let us look at graphs of some of these functions. First the exponential.
Here we show the argument of the exponential. We see it depends only on the imaginary part of Z, not on the real part of Z at all. And we see it "rolls," as the phasor exp(1j*Y) rolls.
Now lets look at the magnitude of the exponential. The log of the magnitude of the exponential is what we show, we label it Real(log(exp(Z))) because this is equivalent to log(abs(exp(Z)). Now exp(X) gets extremely large (close to infinity) and extremely small (close to zero) so we take the log of exp(Z) to get that magnitude down. When we do that we see the magnitude depends only on Real(Z), as we would expect from the equations above.
Now what might we expect from approximations to the exponential? Lets put up the results for a 51 term polynomial approximation to exp(Z).
Here is the argument, the phase, of the approximation. We see inside a 'C' shape, it is a pretty good match for what we had with the complete exponential. Outside the 'C', it is quite different.
What defines the 'C' shape? The 'x' marks on the plot show the roots of the 51 term polynomial which we are using to approximate the exponential. There are 51 roots. They lie on the 'C' shape. Apparently, the approximation to the exponential is pretty good as long as we are inside the rough shape defined by the roots of the approximating polynomial, and are wildly bad if we are outside that shape.
We notice a similar result for the magnitude of the approximation. We have plotted the magnitude of the approximation using the same scales as we plotted the original exponential. Inside the 'C' shape defined by the roots, you can see rather good agreement: the colors match. Outside the 'C' shape, things are much worse. Indeed, Red corresponds to very large values, outside the 'C' shape the magnitude of the approximation rises to very large values, it only manages to stay very close to zero inside the 'C' shape.
We have shown the result for a 51 term polynomial. How does this result change as the order of the polynomial changes?
The figure shows an estimate of the radius of the 'C' of roots as the order of the polynomial increases from 0 to 99. The 'C' wraps around the point Z=0. The estimate above is made by measuring the distance from Z=0 to the top of the 'C'.
From this one learns that the radius of the 'C' is growing linearly with the order of the polynomia. Indeed, for larger N, the radius is essentially 0.42*N.
What this shows is how the exponential function, with no zeros (no roots), arises from increasingly accurate polynomial approximation. Even though each higher order polynomial has more roots, the roots are moving away from the origin, being pushed out towards the edge of the complex plane.
In some sense, one might say, in the limit the exponential function has an infinite number of roots but that they are all at infinity. Now many real mathematicians might hate that statement. But I suspect the intuitivists might like it. The intuitivists believe it is not the limit at infinity that tells you what is happening, but the journey to get there.

Note added: from comments a great page about this very same problem:

Wednesday, June 02, 2010

How many possible worlds exist?

One. I know this is true because I read it in a blog.

But I have an idea why we think there is more than one. Its because we are wrong about what is possible. Its because the model we carry around in our head of the world is quinzillions of times simpler than the actual real world.

So couldn't there possibly be a world where I posted this blog with a spelling error? Well within the super-duper over-simplification of the world we can think about, which has to fit inside a brain which is, again, one-quinzillionth the size of the actual universe, it sure SEEMS like that is possible. But that's just because we are overlooking like a quinzillion (give or take a trillion) facts about the actual universe, and probably a bazillion of those, if we knew them, would completely rule out a different past than the one we see.

Now of course, my mind isn't even beginning to be big enough to know for sure that there can only be one possible universe when all the facts are taken in to account. But following a nice idea from the blog I linked above, the multiple worlds hypotheses, as a class, are not the best explanation of anything. So in my economy of explanation, I'm not ever going to seriously use them. So I don't believe in them, q.e.d.