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The continuum fallacy is to deny the meaningfulness of discrete categories, just because they are a somewhat arbitrary partition of a continuum. More generally, it is to deny the meaningfulness of fuzzy and/or somewhat arbitrary categories, just because they cannot be precisely defined in terms of necessary and sufficient conditions.
The spectrum of visible light is a common example. There is a continuum of frequencies. Each frequency of light corresponds to a point on the spectrum. We (somewhat arbitrarily) partition the spectrum into discrete colors: red, yellow, green, blue, violet. This discretization is not random. Each color is an interval of the spectrum. It would be fallacious to claim that colors do not exist, just because they exist on a spectrum and the categories are somewhat arbitrary.
The continuum fallacy is a type of representational nihilism. It rejects a representation of reality just because it isn’t exactly the same as reality. Of course, there is room to debate whether a representation is useful for describing and explaining reality. It becomes a fallacy when a representation is rejected just because it is a representation, not reality. For example, it would be fallacious to reject a map of a city, just because the map is not the city.
The continuum fallacy is typically used as a form of tactical nihilism. It is an attempt to pull the semantic rug out from under a debate, to avoid losing the debate.
The continuum fallacy has an inverse, which is to deny the underlying reality of a continuum, just because we have discretized it. For example, it would be a fallacy to insist that all colors are really mixtures of red, yellow, green, blue and violet. This denies the underlying reality that light is a continuum of frequencies, not some number of discrete colors. It confuses a representation with reality.
Are colors real? Are red, yellow, green, blue and violet real? Yes and no. They are real in the sense that there is a real spectrum of light frequencies, which we can somewhat arbitrarily partition into discrete categories. Blue and red are clearly different. Blue and green are also different, but there is a fuzzy boundary between them. Color categories are meaningful, but we should not confuse them with the underlying reality of color. We can use discrete categories to represent color, but color is not discrete in reality.
The inverse continuum fallacy (or discretization fallacy) involves a confusion between representation and reality. Such confusions are quite common. We need to use representations to think and talk about reality. We can still be aware that certain ways of representing reality are simplified or metaphorical, and thus not entirely accurate. However, people often lose track of those limitations over time. For example, a biologist might use the metaphor of “competition” to think about the population dynamics of two species in an ecosystem, and eventually start thinking of a species as a kind of agent that competes with other agents.
There is another fallacy worth mentioning here, which we could call “the false continuum fallacy”. It involves claiming that a discrete reality is actually a continuum. The best example is the denial of the male | female distinction, just because there are rare cases of people with ambiguous sexual characteristics. This is another instance of representational nihilism.
There is also the claim that “gender” is a social construct, which is an attempt to deny the biological reality of sex. This isn’t a confusion between representation and reality. To some extent, it is simply a denial that reality exists independently of our representations of it. Normally, we think of representations as being determined by reality, not vice versa. I won’t go into the social constructivist position in detail here. It is a more complicated example of tactical nihilism.
See Theories of Knowledge and The Trans Paradox.
Races and Reality
The continuum fallacy often comes up in discussions of race realism. Race denialists reject the notion of race (in certain contexts), just because it is a somewhat arbitrary discretization of human genetic variation. This is usually a bad-faith attempt to avoid confronting the reality of race differences in mental traits.
Race idealists, on the other hand, often commit the inverse continuum fallacy by thinking of racial categories as really discrete, rather than a somewhat arbitrary discretization of human variation.
Race idealists often point to clustering (without understanding it) as proof that humanity really is composed of discrete groups. But the groups generated by a clustering algorithm are fuzzy, not crisp, and they are somewhat arbitrary. The ability of a clustering algorithm to generate clusters does not prove that humanity really consists of discrete groups.
Some divisions of humanity are pretty close to discrete. For example, the Australian aborigenes descended from a relatively small founder population, roughly 50,000 years ago. Since that time, they have probably been mostly or entirely isolated from the rest of humanity, until the recent discovery and colonization of Australia by Europeans. Since then, there has been considerable mixing.
But in most cases, racial categories are a somewhat arbitrary discretization of genetic variation. There is no crisp boundary between Europeans and East Asians, or between Europeans and North Africans, etc. Central Asians are not a mix of two discrete races, European and East Asian, any more than cyan is a mixture of two discrete colors, blue and green. They are part of a continuum of genetic variation, just as cyan is part of a continuum of color.
You could argue that “continuum of genetic variation” is also a simplifying representation, and I would agree. The reality consists of millions of individual people and their genes. The important point is that humanity doesn’t come packaged neatly into discrete races.
The race idealist thinks of a race as analogous to a body, with individuals as cells. However, the relation of an individual to a race is very different from the relation of a cell to a body. A body is a discrete entity with quite crisp boundaries. There is no continuum between my body and your body. The cells of my body descend from a single ancestral cell: a zygote. The cells of your body descend from a different ancestral zygote.
It would be very convenient for race idealists if races were crisp, discrete entities, like bodies, rather than fuzzy, somewhat arbitrary clusters of genetic variation. But that’s what they are.
The inverse continuum fallacy often occurs in biology and anthropology. For example, an anthropologist might say that the modern European genome was created by “admixture” between Western hunter-gatherers (WHG) and early European farmers (EEF). This is a simplified way of describing a more complex reality. There were not two discrete groups of humanity, WHG and EEF. There was a complex reality, consisting of genetic variation distributed in various ways, which changed over time in various ways. To talk about this complex reality, it might be very convenient to adopt discrete categories, such as WHG and EEF. But we should keep in mind that reality isn’t (and wasn’t) conveniently discretized.
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Models of reality are necessarily simpler than what they represent. A map is simpler than a city. It is a fallacy to reject a model just because it is a model, and not reality itself. However, we should not forget that a model is a model, and not reality itself.