This paperback from Houghton-Mifflin is a little over 300 pages and sold for $9.95. I saw this book on the shelves at Borders for ages and ages and always intended to buy it, but thought, "Maybe next time." One day I noticed it was no longer on the shelves. I asked and discovered it wasn't in print any more. So I had them track down a used copy for me. Took a while to get it.
I was thoroughly disappointed with the book for the first 100 or so pages. I think my problem was that I had this idea built up in my head and the book just couldn't possibly meet my expectations. It doesn't actually make a case for mathematical reality. Instead the author just says that anybody who knows anything about math believes in mathematical reality - and then he talks about the different kinds of mathematical thinking: number, space, logic, infinity, and information. That last one, information, is really what the book is building up to - information and complexity.
I started looking at it much more favorably when the book turned to a digression on Mandelbrot. Either his explanation of the meaning of fractal dimensions is better than Mandelbrots, or maybe I've just had some time to mull it over. All I can say is it made a lot more sense when I read it this time.
But see, that's really the way this book reads - one long digression. I think my opinion of it changed favorably when I realized that digressions weren't random, that they were building to a purpose, that they were, well, laying down the tools for understanding the stuff in the final chapers.
It's interesting stuff in its own right - in fact this is stuff I've studied for ages. Very interesting - but it didn't really get down to a discussion of mathematics and reality, other than to assert as I've mentioned, that mathematics IS real. Maybe it's true that most mathematicians believe that mathematics is reality. I don't know. Why do they believe that? In what sense is it true? I've seen arguments like this before, but they don't make a lot of sense.
Don't get me wrong. I *believe* in an objective reality. Furthermore, I'm convined that people who claim to reject the existence of objective reality don't really believe what they claim to believe.
And it's clear that mathematical rules govern this reality. But the question isn't whether mathematics reflects reality, but whether mathematics IS real.
He asserts on page 156 that "Mathematics tries to replace reality..." This is something I can agree with.
Again, I don't have a problem with accepting mathematical reality as an assumption. I'm just not sure we can ever prove something like this.
I'm reminded of an argument for reality that Daniel Dennett presents in his book "Consciousness Explained." Accordingly we can't be part of some great simulation based on what we know of absolute processing speeds. If we lived in a Matrix of some kind there would be no way of hiding the slow processing from us. Of course I agree with the conclusion, but the argument is a silly one. It presupposes that whatever machine is running our universe is subject to the same physical laws as this one. This is analogous to the Intelligent Design arguments for creation, "Assume that if we see a pattern, it must be design." This kind of fallacious argument assumes the consequence.
Again and again and again, I say that I believe - I'm convinced of - everyday reality. I believe that our understanding of it is never perfect, but that reality is out there somewhere. But the proof of such a thing is almost certainly beyond the scope of human inquiry. Karl Popper made no attempt to prove it in "Objective Knowledge." Instead he promoted the idea of "commonsense realism." It may very well be that the best we can ever do is assume reality and assume that our mathematics is somehow coded into reality. Mathematics is not a property of human beings, it's a property of the universe, and all we do is discover those properties.
Aside: The assertion has been made that we all see the same things when we say, for example, "we see that the object has the color blue." When my youngest daughter was only 8, she asked me - out of the blue - "Daddy, how do I know if two people feel the same thing?" "What do you mean?" "Well, say I feel a rock and I say it's smooth and someone else feels a rock and they say it's smooth or maybe they say it's rough ... how do I know we felt the same thing."
"Baby girl, I'll tell ya. Throughout history the smartest people who ever existed asked that question and you know what - none of them gave very good answers. Maybe you'll be the one to figure it out!"
I think maybe some psychologists think they have figured it out and that they believe we do, in fact, see the same things, but I would like to relate something that happened to me. I work in distributed simulations. Imagine one of these online games like Halo or Planetside or Rogue Spear. We were doing this same kind of thing years before most people ever heard of the internet. We had tank simulators in Germany networked in with helicopter simulators back in the states and we could have this vast battles. One day I was in a tank simulator and noticed something very strange - completely by accident.
I took my right eye and looked through the eyepiece for the left eye - and to my great surprise discovered that the colors were very different. The colors were the same, but the brightness was different. I've done this experiment several times. I see slightly different things out of my individual eyes. I cannot detect this at all in normal life. I only notice it when I look into the simulated environments. Maybe I have a slightly different number of rods and cones in my two eyes. I don't know. But it's a curious thing. My two eyes see the same thing - but they see the same thing slightly differently.
I'm not sure what conclusion could be drawn from this. Probably nothing. But it's a sobering reminder for someone like myself who accepts realism.
