Thursday, February 10, 2005

Things and Numbers

First of all, I would like to apologise for this entry that I am about to write because it will probably be at best a collection of vaguely related ideas and at worst incomprehensible. So let's jump into the stream of conciousness...

Yesterday, I sat in a seminar where a model of how we make decisions was presented. It was given by a psychologist and to make it short, I was not really convinced that this model even makes sense. In retrospect, I can trace back my uneasiness to the following: To come up with a quantitative model, the speaker was expressing all kinds of concepts in terms of numbers.

For example, given several options a,b,c..., the 'level of preference' P(a), P(b)... played a central role. I am completely fine with that: Level of preference is a map from the set of options to some level set L. Obviously, L is partially ordered ; I know that for some options I prefer them over others. For example I prefer getting a cup of coffe to getting a cup of coffee and having a hole drilled into my knee. However, I have a hard time to make up my mind whether I would like to have a cup of coffee or rather have my favourite soccer team win their next game. It's hard to compare them. But if pressed I would probably choose one over the other. So maybe, if pressed hard, L is even an ordered set.

Now the problems start, because everybody likes to be quantitative and there are obvious ordered set: Real (or rational or maybe integer) numbers. (In highschool we even had the set T of numbers that a given pocket calculator can represent). So we take L to be one of those number sets. So now the level of preference function assigns numbers to options.

This could still be fine, if we would only compare levels of different options. But with numbers you can do many other things: You can multiply them by other numbers ("My preference of option A is at least twice as my preference for option B"), you can add them ("My preference of A plus my preference of C is less than my preference for B", note that a priory this is different from "My preference of having A and C is less than having B"). That might still be fine if you have a linear model of preference (which the speaker at other places in the seminar didn't have as an answer for my question of what 'level of preference' really is, was that it is "like probability of choosing that option". Remember that probability is always between 0 and 1 and thus arbitrary multiples of probability are not probablilities).

It really gets crazy once you try to multiply two levels of preference, as can be done once you realize that both are numbers...

The upshot is, that I think typing (like in computer science) is important: You should always know what kinds of entities you are dealing with and what kinds of operations are allowed for those entities.

This remindes me of one of Feynman's anecdotes: He was on some board that approved physics and maths text books to be used in schools. In one of the books that tried to be creative he found a problem stating that there are stars of different colours and that these colours are related to the stars temperature. It gave the temperature for red, yellow, orange and blue stars. Then the student was asked to sum the temperatures of the above stars.

I am still trying to understand _why_ this problem is so stupid. Obviously, adding temperatures is not a good idea. But what exactly is the structure of/the allowed operations the set that temperature takes values in?

Let me mention some examples of such structures. Of course, there are groups, fields, vector spaces (where you can add and subtract and multiply by numbers, if you can multiply by something else it might be a ring or a module), there are affine spaces (where you can only take differences and those are valued in a vector space, connections come from such affine spaces). Furthermore, I have already mentioned several degrees of order.

So again, what kind of entitly (type) is temperature? First of all, it is ordered and there is a minimal point. Then, using perfect heat engines operating forward and backward you can map differences (or pairs) of temperature to mechanical work which comes from a one dimensional vector space and back. So you can compare the difference between T1 and T2 to the difference between T3 and T4, you can (by operating two engines) compare twice the difference between T1 and T2 to the difference between T3 and T4. You can even compare the difference between T1 and T2 plus the difference between T5 and T6 to the difference between T3 and T4. And of course the difference between T1 and T1 is zero. This is all explained in great detail in the Feyman lectures chapter 44-4 and 44-5. Still, it makes no sense operationally to add the temperatures of two stars. Why?

Before I end this this entry, let me mention one of the reasons why I like Perl, the programming language, so much: There you are not forced to identify finite sets with subsets of the integers. For example you can directly loop over the elements of a list rather than the integers between 1 and the number of elements of that list. So you can say

for $date(@girls){
&take_out_to_dinner($date);
}


rather than the much less natural


for i=1 to length(@girls){
&take_out_to_dinner(girl[i]);
}


You can even have arrays (called hashes) that are indexed by arbitrary names rather than numbers. Often this is much more natural to say $phone_number{$girl} where $girl is the name of today's date than assiging arbitrary numerical labels to the girls.

You extract the list of all labels of a hash so you can loop over it but the order in that list might change. So, especially you it makes no sense to say $phone_number{10*$girl+5}.

Of course, sometimes (like in numerical programs) it is natural to index an array by numbers to make arithmetic on the labels, so there are ordinary arrays as well.

Oh, this brings me to another example of an affine space: The space of pointers in C: There are of course the integers which are a ring and there are pointers: Allowed operations are adding numbers to a pointer (giving a pointer) and subtract pointers giving an integer. Furthermore you can compare pointers by comparing their difference to 0 (in Z). But it makes no sense to add two pointers.

Oh, still I forgot one thing: Back to social scientists aiming to be quantitative by mapping everything to numbers: We all know, that for small variations, everything looks linear. Refering to this, it is common to define a utility function for the various options of a decision (thing of it as "value in US$ to have that"). And at first order this is linear: Two coins are twice as valuable as one coin. However, this is only an approximation: Having 101 apples is not as much better than having 100 apples than one vs none. So, if you are aware of this you say your utility function is convex or sublinear. But even that is not true. Having one shoe is not half as good as having a pair of shoes. Or having a can without a can opener is not really useful. But you get pretty far with your linear (or sublinear) model.

Still there is one area where it is nearly infinitely bad: That is when you try to evaluate anything that has to do with information: If you tell me one fact twice it should not be better than telling it me once. Evenmore: You still know it if you tell me. You can still tell it someone else and there is no point for me to tell you "back".

I think this is part of the reason why people brought up with the linear model of economics have such a hard time to understand the workings of copyright (the attempt to linearise the value of information) and copyleft and open source culture.

Update: In the January 2005 issue of physics today I found a picture that fits in here:

5 comments:

John Baez said...

I've always enjoyed that Feynman story too.

The reason it was completely idiotic to add the temperatures of those stars is that (I bet) they were measured in Celsius or Fahrenheit. In these systems, the "zero" of temperature is a highly arbitrary thing: the melting point of water in Celsius, and the coldest possible liquid salt-water mixture in the case of Fahrenheit.

So, addition of temperatures in these systems has the curious property of depending on complicated facts of chemistry.

It makes more sense to add temperatures in Kelvin, because the zero is not such an arbitrary thing.

As you note, temperature starts out being a mere ordered set, defined by the property that T < T' iff a body
of temperature T' will transfer heat energy to a body of temperature T.

However, more sophisticated laws of physics make it sensible to subtract
temperatures, and ultimately to make
temperatures into an affine space.

The laws of physics concerning absolute zero give this torsor a preferred origin, making it into a 1-dimensional vector space.

It takes even more work to make this 1-dimensional vector space into a copy of the real numbers: for that we need a preferred "1" of temperature, e.g. the Planck temperature.

It would be fun to work out all the details. Some philosopher of physics should have done it. I don't know if any has, but there has been a lot of careful work on the foundations of thermodynamics.

Higher Pi said...

Perhaps you would add temperatures of stars if you wanted to calculate a mean average for some statistical reason.

I don't see why having temperatures in Kelvin allows adding temperatures to make any more sense - temperature still doesn't have an additive property as far as I know (it might have, I suppose).

Regarding your psychologist - what he is describing is Utility Theory and is a core of economic research. Labelling preferences with numbers - the basis of this theory - requires, as you say, some sort of order. Economists use mathematical terms such as transitivity to describe this. (As I understand, Utility Theorists worry about the transitivity of human preference... if a person would pick an apple over a banana for lunch, and a banana over an orange, would they necessarily pick an apple over an orange?!)
I think mathematicians have a lot to add to the field of utility theory, and could work more closely with them to find uses for the theory, good models of utility, and the potential pitfalls of utility theory, which you begin to describe.

I just made a post today relating to this idea - titled Gambler's Ruin. I think you would find it interesting!

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