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.

I think there is a tale to tell about the consumer surplus and it goes like this.

Alice loves widgets. She would pay $100 for a widget. She goes on line and finds Bob offering widgets for sale for $100. Err, that is not really what she had in mind. She imagined paying $30 for a widget, and feeling $70 better off as a consequence. She emails Bob: How about $90?

Bob feels like giving up altogether. It takes him ten hours to hand craft a widget and the minimum wage where he lives is $10 an hour. He was offering widgets for $150. $100 is the absolute minimum. Bob replies: No.

While Alice is deciding whether to pay $100 for a widget that is only worth $100 to her, Carol puts the finishing touches to her widget making machine. At the press of a button Carol can produce a widget for only $10. She activates her website, offering widgets for $40. Alice orders one at once.

How would Eve the economist like to analyse this? She would like to identify a consumer surplus of 100 - 40 = 60 dollars, and a producer surplus of 40 - 10 = 30 dollars, for a total gain from trade of 60 + 30 = 90 dollars. But before she can do this she has to telephone Alice and Carol and find out the secret numbers, $100 and... (read more)

Some of your fake numbers fall out of the common practice of shoehorning a partial order into the number line. Suppose you have some quality Foo relative to which you can compare, in a somewhat well-defined manner in at least some cases. For example, Foo = desirable: X is definitely more desirable to me than Y, or Foo = complex: software project A is definitely more complex than software project B. It's not necessarily the case that any X and Y are comparable. It's then tempting to invent a numerical notion of Foo-ness, and assign numerical values of Foo-ness to all things in such a way that your intuitive Foo-wise comparisons hold. The values turn out to be essentially arbitrary on their own, only their relative order is important.

(In mathematical terms, you have a finite (in practice) partially ordered set which you can always order-embed into the integers; if the set is dynamically growing, it's even more convenient to order-embed it into the reals so you can always find intermediate values between existing ones).

After this process, you end up with a linear order, so any X and Y are always comparable. It's easy to forget that this may not have been the case in your intuition whe... (read more)

Nitpick: utility is not just an ordering, it also has affine structure (relative intervals are preserved) because of preferences over lotteries. Software complexity is a valid example of your point, though. It's like trying to measure the "largeness" of a thing without specifying whether we mean weight, volume, surface area, or something else.

Assuming you broadly subscribe to the notion of "true numbers" and "fake numbers", how do you classify the following?

Food calories [pollid:595]

The position of an object's centre of mass [pollid:596]

The equilibrium price [pollid:597]

A population's carrying capacity [pollid:598]

The population mean [pollid:599]

Yet another is "productivity". In fact, most of software engineering consists of discussions of fake numbers. :/ This article (pdf) discusses that rather nicely.

Why is such precision required for something to count as a 'measurable quantity'? Depending on how you do the measurements, measurements of (e.g.) prices don't always agree to two decimal places, let alone three.

... (read more)Unsurprising due to range restriction - by the time you're interviewing with Google, you've gone through tons of filters (especially if you're a Stanford grad). This is the same reason that when people look at samples of elite scientists, IQ tends to not be as important a factor as one would expect - because they're

allsmart - and other things like personality factors start to correlate more.EDIT: this may be related to Spearman's law of diminishing returns

But they only hire at the top, so one would expect the subsequent performance of their hires to be little correlated with any sort of interview assessments.

Toy example: 0.8 correlation between two variables, select on one at 3 or more s.d.s above the mean, correlation within that subpopulation is around 0.2 to 0.45 (it varies a lot, even in a sample of 100000).

Let me throw in what might be a useful term: "unobservable".

Take, for example, the standard deviation of a time series. We can certainly make

estimatesof it, but the actual volatility is unobservable directly, we can only see its effects. A large chunk of statistics is, in fact, dedicated to making estimates of unobservable quantities and figuring out whether these estimates are any good.Another useful term is "well-defined". For example, look at inflation. Inflation in general (defined as "change in prices", more or less) is... (read more)

I suspect this distinction between "real" and "fake" numbers is blurrier than you are describing.

Consider voltage in classical physics. Differences in voltages are a real measurable quantity. But the "absolute" voltage at a given point is a mathematical fiction.

Or consider Kolmogorov complexity. It's only defined once you fix a specific Turing machine (which researchers rarely bother to do.) And even then, it's not decidable. Is that a real number or a fake number?

So. Entropy: real or fake number?

... even if you don't know about quantum mechanics?

To make it worse, multiplicity!

I instantly disliked your terminology of true vs fake. If I understand it correctly, you are making a distinction between widely agreed upon quantification procedures (like measuring weight) and those which are are either not well defined or contentious (cheeseburger surplus). You do not seem to stipulate that the output of these procedures be useful in any way, but maybe it is implied? Anyway, I would call these "metrics", not "numbers".

If I were to try to quantify the "trueness" of these "numbers", I would look in... (read more)

I believe the best example of 'fake numbers' may be the measurement of IQ. The problem of this sort of fake numbers is that it is not certain to tell whether IQ really represents our true intellectual being but people still use it to be judgmental or even to justify their study not knowing when to stop to regard it as a simple reference.

Fake numbers seem to prevail in our professional life as companies do quantify people's labor thanks to technology. They might be good estimates but that kind of numerical fixation affects people's mind tremendously so that the moment the numbers are revealed it now controls the people. It won't stay as a mere measurement reflecting the phenomena it gathered.

I'm sure that in AI research many programs have been written around a specific well-defined utility function. Or, by "utility" you mean utility for a human? The "complexity of value" thesis is that the latter is very hard to define / measure. I'm not sure it makes it a "bad" concept.

It is interesting that you choose mass as your prototypical "true" number. You say we can "measure" the mass of a person or car. This is true in the sense that we have a complex physical model of reality, and in one... (read more)

I don't think that dividing reality into true numbers and fake numbers is very useful.

I think it's more useful to ask yourself whether you can measure something for reasonable effort that's a good predictor of some outcome you care about. You should also ask whether it continues to be a good predictor if you optimise towards it.

Discussing whether IQ is the real measure of intelligence is irrelevant. The important question is whether it predicts performance for tasks you care about.

Apart seeking for quantities that are good predictors it also makes sense ... (read more)

Another example: financial engineers treating correlation of housing prices as if they were they were concrete and fixed.