Monday, December 31, 2007

Knowledge is value.

As it’s the last day of the year I’ll allow myself a glimpse at the big picture. I started my second wave of philosophy with the intention of exploring the dual nature of the mad. I had seen at first hand the brilliant yet unpredictable thought processes of certified schizophrenics and manic depressives. Among the great thinkers and artists to whom we owe so much there are to be found a high density of mad men and women. Often their madness goes unmentioned in deference to their achievements. The summation is that creative minds seems to be capable of great folly and great wisdom. The rest of us look on in terror and pity.

My idea was that the mad know things that are beyond the reach of the sane. On a similar vein, the mad see the value in things that the sane overlook. The problem is that this is hard to even state, partly because the mad are seen by certain officious types as ill, their beliefs as “delusions”, or “symptoms”, their political rights as suspended and their condition one of suffering. But the real reason is that in philosophy, analytic philosophy at least, “S knows that p” is considered to be factive, so: “Einstein knows that time is relative” is only true if time is relative. Since the sane believe that a lot of what mad people believe is false, calling these things “knowledge” is against the rules, even if the mad person claims to know what he believes and we don’t claim to know that the mad person’s beliefs are false .

So I am under pressure to rephrase my initial thought as that the mad “believe” things that are beyond the reach of the sane. Now I can say that mad people believe things that are more likely to be false. This looks very promising indeed since we can explain the mad genius. If the mad believe what is more likely to be false, then they are higher risk believers. A Wall Street slogan is “the higher the risk the higher the return". When in the jungle of improbable beliefs the mad man stumbles across a true proposition, as long as he can tame it and bring back to the suburbs of probable beliefs alive, he will have found something of great value. Whereas the sane, who get their beliefs second hand from text books and TV, are scarcely likely to believe anything of any value. Of course, there is exchange value and use value. Second hand ideas can be very useful, and you can always hawk them to children or students. But beliefs that no one else have are for that reason worth a great deal. Of course this is only if they are true! But they appear to the believer just as valuable whether they are true or not, since to believe something is the same thing as believing it to be true.

But now enter the complication. Value and belief are the subject matter of decision theory. If something is improbable in decision theory, it is less valuable not more valuable. I put £36 on double six at even odds I have an improbable belief that the dice will come up two sixes. My speculation is worth £1 to anybody who knows the ways of dice. This is because out of 36 possible futures, only in one do I win. Yet if I believe it, in the sense that I know it, then my speculation would be worth £72. This is because there is only one possible future, and in that future the dice come up double six. So from my perspective I do not have an improbable belief at all. From my perspective, if I am certain the dice will come up double six, my belief is not just highly probable, it is true. So we have a pair of perspectives, from one of which my belief is improbable and from the other it is certain. Which perspective is right? The problem here is that this depends on no fact in the world!! Of course if the dice comes up snake eyes, then my belief was improbable and false. But if it comes up two sixes, those who valued my punt at £1 are still right. Their assessment of the situation is un-falsifiable in the face of experience.
The thoughts of the creative genius are clearly not of this kind. It is not as if when Einstein came up with Special Relativity it was clear to everybody else that the theory had only a 1/36 chance of being correct, and Einstein just got lucky. No one had even been able to conceive of it. A recent Lakatos award winner suggests that the pieces were all there and if Einstein hadn’t come up with it, some one else would have. It is easy to think up ideas that have already been thought up. Stephen Tiley came up with Hook’s law, I’ve come up with binomial probability distributions, “Bad News” came up with “A stairway to Heaven” by Lead Zeppelin, and a tax collector named Saul came up with Christianity, adding in his own homophobic twist.
Likewise with Darwin. The theory of evolution is, in its essence, very easy to understand. But it wasn’t a low probability hypothesis that no one but Darwin had the craziness to back. It wasn’t a hypothesis at all. Of all the people in the world, only one came up with evolution. The beliefs involved were highly improbable in this sense, they were one in a million. I am not implying that Darwin was mad incidentally, there are enough idiots who would love to go back in time and put him in a strait jacket as it is without me adding to their number.
So the decision theoretic model of improbable beliefs is not what I am after. Making wild guesses doesn’t count. One man’s wild guess is another man’s low probability hypothesis. What I am after is risky knowledge. This is not the same as betting on the thin end of the odds.
So now we have some highly philosophically interesting concepts in the pot. Knowledge, value, probability, belief contents. It seems like the value of knowledge increases in inverse proportion to its probability. But that knowledge is only knowledge if its probability is 1.
Let us look at the great mathematicians of value. They are no longer working in the abstract but are now in the stables of the super rich, hedging their wealth. Wealth does not stand still, but is alive. Leave it unattended and it will dwindle, tend to it and it will grow. The price of a stock now is an indicator of the price of the stock in the future. We can project the price into the future by seeing the range of future prices as a probability distribution. We can calculate a single value from this distribution and call that the expected value, or utility of the stock. But there is another dimension that is a little more mathematically slippery. The expected value of two stocks can be the same while the risk can be different. Black and Scholes came up with a measure of Risk in terms of variance. This can be thought of in terms of standard deviations from the mean. This again can be given a value and in derivative markets risk can be bought and sold like anything else. Here we have a mathematical equivalent of the creative genius idea. It has a probability of 1, but a high risk. However, the problem is that variance doesn’t work for probability 1. When a proposition has a probability of 1 of being true, there is no probability distribution.

Lets just try equating knowledge with value. How so? What is the value of a bet that pays £67 if p is true given that you know that p is true? Answer £67. In general the value given knowledge equals the value. Compare with probability. What is the value of an outcome with a probability of ½. Answer, ½ of the value of that outcome if it were known. Truth = probability times value. Knowledge = Value.

A contemporary question in epistemology is “what is the value of knowledge”. The answer here is that the value of knowledge is the value of the truth. Whereas the value of a partial belief is the value of the truth times the probability. What should we find out? What should be the subject of our inquiry? Which of the myriad facts that lie just beyond our reach should we step forward and grasp? The answer is, those that are of value, since the value of knowledge just is the value of the truth. It is the love of the good that should guide us through life.

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