Well done. I've noticed this before (but not named it) when it comes to speciation. As a population evolves (and leaves traces in the fossil record), we arbitrarily label the individual fossils with different species names. We define species, however, by the ability to produce viable offspring. (We can't test that, of course, but no matter: it's certain that any creature would be unable to interbreed with a distant enough ancestor.)
The problem is that if every individual left a fossil, there would always come a point where a taxonomist would have to change species names between a parent and a child, but by any reasonable definition of species they have to be the same species.
(I could give examples from particle physics, too, but they're more abstruse.)
Our notation often forces us into your fallacy (sorry, to name a thing is to own it). The ignorant then proceed to read more significance into the names (and their attendant problems) than into the ideas.
There is a tendency on the crevo threads to argue about words rather than ideas. We see a lot of "logic" being applied on the crevo threads, as if finding a an example of verbal imprecision has some magic power to destroy the ideas behind the words.
I would like to point out that there are uses for such a fallacy. It's a method for Amplification of Randomness.
One may observe a chunk of radioactive material and track whether there are an even or odd number of counts in an interval (I know that there's a bias towards even numbers but it's really small for large time intervals.) The underlying process may be continuous but the "rounding" to just parity gives a jumpy appearance. Other sources can be used, cosmic rays, thermal noise in circuits, etc. Such methods can (and are) used to generate keys for cryptographic systems. Schrödinger's Cat is a famous example.
One could drive a control system (switching a motor on and off, for example) with a randomly generated signal and thus produce a system where a small amount of uncertainty in input has a large amount of uncertainty in output. This sort of thing happens naturaly too; piles of sand (or rocks) may become unstable but exactly how they fall depends on very small inputs (the proverbial came's last straw.)
Where there is a continuum – like earnings and temperature in your example – the quantizing is indeed arbitrary and the meaning of your term “Quantizing the Continuum” is quite apparent.
Also, as Physicist suggests where one sees speciation as a continuum in the geology record, the term expresses the ambiguity of naming species per se.
And in Doctor Stochastic’s example one might quantize a continuum for the purpose of amplifying randomness in a system. That is a useful concept for understanding how a natural system might seem to operate like a finite state machine (Rocha).
Indeed, your new term “Quantizing the Continuum” certainly has many applications, but I do not believe it can apply to the definition of biological life as you originally proposed it at post 633.
Biological life is not like earnings or temperature – or incomplete records over time – or amplified randomness in a system.
I assert the distinction between that which is alive and that which is not alive is clear to anyone who has visited a morgue, been with a dying person or animal, or looked at a dead cell under a microscope.
Dead biological organisms – or dead individual molecular machines (liver, heart etc.) within the dying organism – are quite distinctive from living ones. Dead cells are quite distinctive from living cells.
A living cell communicates (Shannon information: the reduction of uncertainty in a receiver or molecular machine in going from a before state to an after state) and is asymmetrical. A dead cell does not communicate and is symmetrical. The DNA and chemical composition does not change at the instance of death.
Also, if “Quantizing the Continuum” precludes a clear definition of biological life then it also diminishes all abiogenesis theory – because there could not be a point at which life begins. Evolutionists then could neither successfully exclude abiogenesis nor defend against the assertion of the very same term to argue against abiogenesis by definition. If the term prevails in this debate with reference to biological life, I for one will advocate the argument exactly that way.