In physical science the first essential step in the direction of learning any subject is to find principles of numerical reckoning and practicable methods for measuring some quality connected with it. I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the state of Science, whatever the matter may be.
-- Lord Kelvin
If you believe that science is about describing things mathematically, you can fall into a strange sort of trap where you come up with some numerical quantity, discover interesting facts about it, use it to analyze real-world situations - but never actually get around to measuring it. I call such things "theoretical quantities" or "fake numbers", as opposed to "measurable quantities" or "true numbers".
An example of a "true number" is mass. We can measure the mass of a person or a car, and we use these values in engineering all the time. An example of a "fake number" is utility. I've never seen a concrete utility value used anywhere, though I always hear about nice mathematical laws that it must obey.
The difference is not just about units of measurement. In economics you can see fake numbers happily coexisting with true numbers using the same units. Price is a true number measured in dollars, and you see concrete values and graphs everywhere. "Consumer surplus" is also measured in dollars, but good luck calculating the consumer surplus of a single cheeseburger, never mind drawing a graph of aggregate consumer surplus for the US! If you ask five economists to calculate it, you'll get five different indirect estimates, and it's not obvious that there's a true number to be measured in the first place.
Another example of a fake number is "complexity" or "maintainability" in software engineering. Sure, people have proposed different methods of measuring it. But if they were measuring a true number, I'd expect them to agree to the 3rd decimal place, which they don't :-) The existence of multiple measuring methods that give the same result is one of the differences between a true number and a fake one. Another sign is what happens when two of these methods disagree: do people say that they're both equally valid, or do they insist that one must be wrong and try to find the error?
It's certainly possible to improve something without measuring it. You can learn to play the piano pretty well without quantifying your progress. But we should probably try harder to find measurable components of "intelligence", "rationality", "productivity" and other such things, because we'd be better at improving them if we had true numbers in our hands.
By all means, straw man away. I won't take it personally.
However, for an example — and please don't feel compelled to restrict your answer only to this — there's a variant of the good old chickens/grandmother example:
Let us say I prefer the nonextinction of chickens to their extinction (that is, I would choose not to murder all chickens, or any chickens, all else being equal). I also prefer my grandmother remaining alive to my grandmother dying. Finally, I prefer the deaths of arbitrary numbers of chickens, taking place with any probability, to any probability of my grandmother dying.
I believe this violates VNM. (Am I wrong there?) At least, I don't see how one would construe these preferences as those of an agent who is acting to maximize a quantity.
See this thread for clarification/correction.
Those preferences do not violate the VNM axioms. The willingness to kill chickens in order to eliminate an arbitrary small chance of the death of your grandmother makes the preferences a bit weird, but still VNM compliant.
A live chicken might quantum tunnel from China to your grandmother's house and eat all her heart medication. But then again, a quantum tunneling chicken could save your grandmother from tripping on fallen corn-seed. The net effect of chickens on your grandmother might be unfathomably small, but it is unlikely to ever be zero. If chick... (read more)