Why do those criteria form a gold standard for a "true" number? I can invent formal, mathematical or empirical definitions for things all day long that don't correspond to anything useful or meaningful or remotely "real" outside of the most tautological sense.
Let's say the Woowah of a sample is the log of its mean minus the lowest number in the sample. If I sample the heights of fifty men aged 18-30, that sample has a mathematically well-defined Woowah. There's even an underlying true value for the Woowah of the population, but so what? When I talk about the Woowah, I'm not really talking about anything.
Meanwhile a population's carrying capacity (currently the most "fake" number in the poll) isn't something I can directly observe. I have to infer it through observation. (For all practical purposes I have to infer the Woowah through observation too, but in principle I could sample the entire population and find the true Woowah). I can't directly measure the carrying capacity because it isn't a direct property of the population. It's a parameter in a model of the population which happens to refer to a property that population would have in a specific and probably counterfactual case. It's questionable whether there is an underlying "true" population carrying capacity, but it's definitely talking about something important and meaningful.
The OP's definition of a "true" number isn't that it's useful, meaningful or corresponding to something "real". It's merely that it's objectively measurable and actually measured..
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",
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.