Some anecdotal and meandering gubbins.
I am occasionally called upon to defend what an interlocutor thinks of as a "fake number". The way I typically do this is to think of some other measure, parameter, or abstraction my interlocutor doesn't think is "fake", but has analogous characteristics to the "fake number" in question, and proceed with an argument from parallel reasoning.
(Centre of mass is actually a very good go-to candidate for this, because most people are satisfied it's a "real", "physical" thing, but have their intuitions violated when they discover their centre of mass can exist outside of their body volume. If this isn't obvious, try and visualise the centre of mass of a toroid.)
Some of the above are this sort of parallel measure I have used in the past, either with each other or with some other measures and values mentioned in comments on this post. I'm quite pleased at how divisive some of them are, though I'm surprised at the near-unanimity of food calories, which I would have expected to be more politicised.
My conclusions to date have been that the "reality" of a measure has a strong psychological component, which is strongly formed by peoples' intuitions about, and exposure to, abstract concepts with robust, well-understood or useful behaviour.
Centre of mass is actually a very good go-to candidate for this, because most people are satisfied it's a "real", "physical" thing, but have their intuitions violated when they discover their centre of mass can exist outside of their body volume.
How is the fact that the center of mass can exist outside the body volume supposed to make center of mass "fake" in any way shape or form?
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.