## Tuesday, November 1, 2016

### Friedman's effective field theory

In preparing my response to George Blackford here, I have re-read Friedman's paper [pdf] once again, and I'm surprised by how good this is from a physicist's perspective. Is Friedman anticipating effective field theory (which doesn't happen until the late 70s and early 80s) in 1953? I only had to make one correction make these paragraphs presentable in a physics class:
We may start with a simple physical example, the law of falling bodies. It is an accepted hypothesis that the acceleration of a body dropped in a vacuum is a constant [near the surface of the Earth] - g, or approximately 32 feet per second per second on the earth - and is independent of the shape of the body, the manner of dropping it, etc. This implies that the distance traveled by a falling body in any specified time is given by the formula s = ½ gt² , where s is the distance traveled in feet and t is time in seconds. The application of this formula to a compact ball dropped from the roof of a building is equivalent to saying that a ball so dropped behaves as if it were falling in a vacuum. Testing this hypothesis by its assumptions presumably means measuring the actual air pressure and deciding whether it is close enough to zero. At sea level the air pressure is about 15 pounds per square inch. Is 15 sufficiently close to zero for the difference to be judged insignificant? Apparently it is, since the actual time taken by a compact ball to fall from the roof of a building to the ground is very close to the time given by the formula. Suppose, however, that a feather is dropped instead of a compact ball. The formula then gives wildly inaccurate results. Apparently, 15 pounds per square inch is significantly different from zero for a feather but not for a ball. Or, again, suppose the formula is applied to a ball dropped from an airplane at an altitude of 30,000 feet. The air pressure at this altitude is decidedly less than 15 pounds per square inch. Yet, the actual time of fall from 30,000 feet to 20,000 feet, at which point the air pressure is still much less than at sea level, will differ noticeably from the time predicted by the formula - much more noticeably than the time taken by a compact ball to fall from the roof of a building to the ground. According to the formula, the velocity of the ball should be gt and should therefore increase steadily. In fact, a ball dropped at 30,000 feet will reach its top velocity well before it hits the ground. And similarly with other implications of the formula.
The initial question whether 15 is sufficiently close to zero for the difference to be judged insignificant is clearly a foolish question by itself. Fifteen pounds per square inch is 2,160 pounds per square foot, or 0.0075 ton per square inch. There is no possible basis for calling these numbers “small” or “large” without some external standard of comparison [n.b. scale of the theory]. And the only relevant standard of comparison is the air pressure for which the formula does or does not work under a given set of circumstances. But this raises the same problem at a second level. What is the meaning of “does or does not work”? Even if we could eliminate errors of measurement, the measured time of fall would seldom if ever be precisely equal to the computed time of fall. How large must the difference between the two be to justify saying that the theory “does not work”? Here there are two important external standards of comparison. One is the accuracy achievable by an alternative theory with which this theory is being compared and which is equally acceptable on all other grounds. The other arises when there exists a theory that is known to yield better predictions but only at a greater cost. The gains from greater accuracy, which depend on the purpose in mind, must then be balanced against the costs of achieving it.
This would only get better if Friedman had said we should measure an effective value of g for each object or for different altitudes (or add in terms v0 t + s0).

I am not defending the implications Friedman makes from this analogy about economics -- although I will say that this would be an excellent argument for defending the use of rational agents if rational agents lead to empirically successful theories. The problem is that this analogy doesn't defend against theories that fail to match the data which is a more serious issues in economics than a particular mathematical approach.

1. "I only had to make one correction make these paragraphs presentable in a physics class:"

What correction was that? Was that the bit is square brackets?

"[n.b. scale of the theory]"

or this bit:

"This would only get better if Friedman had said we should measure an effective value of g for each object or for different altitudes (or add in terms v0 t + s0)."

1. It can always be worse:
https://muircheart.wordpress.com/2016/10/22/welcome-to-the-bear-pit-when-public-engagement-goes-to-pot/

2. "..although I will say that this would be an excellent argument for defending the use of rational agents if rational agents lead to empirically successful theories."

Such breathtaking clarity of thought and devastating logic.

"The problem is that this analogy doesn't defend against theories that fail to match the data which is a more serious issues in economics than a particular mathematical approach."