Continuum Arguments

A continuum argument purports to show that two things are the same by presenting a continuous sequence of intermediate states between the two. For example, if I were to argue that black and white were the same color, because there is clearly a way to morph white into black through a movie of grays, each frame imperceptibly different from the next, then I would be making a continuum argument.

As the example suggests, continuum arguments are almost always invalid. Indeed, Wikipedia files them under continuum fallacy. I think that is a little bit harsh, because there are at least two technical areas in which continuum arguments hold force:

In the mathematical field of topology, two shapes are explicitly considered to be the same if they can smoothly be changed into each other. As the standard joke goes, a topologist can't tell her donut from her coffee cup. In topology, then, continuum arguments are far from fallacious; rather, they are proofs by construction.
In the philosophical field of ontology, which studies the fundamental nature of existence. Here the standard joke is that at least in English, the entire field of ontology can be summarized in a two word question and one word answer: "What exists? Everything."
Continuum arguments are powerful in ontology because of the requirement that the fundamental entities actually be fundamental, and not themselves made up of other different stuff. For example, let's say you are a Cartesian dualist who believes that mind and matter are two totally separate and essentially different things. Then if I can show a continuum between something which clearly has a mind (Stephen Hawking, say) and something that does not (Hawking's great^(10^10) amoeba-like grandfather), you are in trouble. It isn't a completely knock-down argument: you can retreat into panpsychism, or say that the amount of mind slowly or abruptly decreases as you move along the continuum (but why would it do that if mind and matter were completely different?). But it does move you from a simple "common sense" dualism to a more complicated and much less attractive version.

Continuum arguments are sometimes confused with slippery slope arguments, but they are quite different, as a slippery slope is causal -- "if you allow A, then B will also happen, and B is bad" -- and a continuum argument isn't. Slippery slope arguments are also not out and out fallacies like continuum arguments are, but neither are they in my experience persuasive: if you are arguing about A, then the claim that A leads to bad thing B is likely to be just as contentious as A itself. Or more so, given that predicting the consequences of nearly anything is known to be hard, at least outside of physics where there really are nearly frictionless inclined planes ...

The recursively minded among you will now be wondering whether there are any fallacies intermediate between a continuum argument and slippery slope. I'm happy to say that there is. It doesn't have a Wikipedia page (yet), but a "no bright-line" argument tries to show that A is bad by placing it on a continuum with known bad thing B. For example: one shouldn't go to see Iron Man 2 for $5 per ticket, because one shouldn't go to see Iron Man 2 for $50,000 per ticket. The same fallacy is at work in the famous "Now are just haggling over the price" Winston Churchill joke. It's not a slippery slope because there is no causality, and it isn't a continuum argument because it isn't trying to establish identity, merely morality or prudence. The name comes from law, in which a bright-line rule is a clearly defined function of objective factors: you must be 21 to drink is a bright-line test, while "I know when I see it" isn't. Here, the presence of a bright line would put a kink in the continuum -- if there was a well known rule of thumb that one should never pay more than $100 for a movie ticket, then the difference between $5 and $50,000 would be even more readily apparent than it already is. But while being bright-line may be a desirable property for a rule to have, there is no law of nature saying that all goods things are separated from all bad things by such; the mistake of a "no bright-line" argument is to assume that there is.