Useful fictions September 26, 2010Posted by Ezra Resnick in Philosophy.
Do negative numbers really exist? To someone unfamiliar with the concept, the idea of a number smaller than zero surely seems meaningless at first. I may have three apples, or five apples, but I cannot have less apples than none. Consider the problem: If Alice has five apples, how many apples must Bob give her so that she will have three apples? It’s not difficult to recall the mindset wherein such a question seems absurd. And yet, thousands of years ago people realized that for many purposes it can be very useful to extend the number line below zero — when keeping track of credit and debt, for instance. After a little practice, we get comfortable with negative numbers and they come to feel just as real as positive numbers. The same is true for imaginary numbers (originally a derogatory term): at first, the idea of a value which yields a negative result when multiplied by itself seems absurd and meaningless, but this concept turns out to be invaluable for representing certain complex physical phenomena, like electromagnetism and quantum mechanics.
Concepts like negative and imaginary numbers are part of an enormous set of mind tools that humans have devised over the ages to help us model the world around us, and these tools underlie many of the amazing achievements of our species. We must be wary, however, of extending unwarranted physical or metaphysical status to our abstractions. Here is one of Daniel Dennett’s thought experiments from his book Consciousness Explained:
Imagine that we visited another planet and found that the scientists there had a rather charming theory: Every physical thing has a soul inside it, and every soul loves every other soul. This being so, things tend to move toward each other, impelled by the love of their internal souls for each other. We can suppose, moreover, that these scientists had worked out quite accurate systems of soul-placement, so that, having determined the precise location in physical space of an item’s soul, they could answer questions about its stability (“It will fall over because its soul is so high”), about vibration (“If you put a counterbalancing object on the side of that drive wheel, with a rather large soul, it will smooth out the wobble”), and about many much more technical topics.
What we could tell them, of course, is that they have hit upon the concept of a center of gravity (or more accurately, a center of mass), and are just treating it a bit too ceremoniously. We tell them that they can go right on talking and thinking the way they were — all they have to give up is a bit of unnecessary metaphysical baggage. There is a simpler, more austere (and much more satisfying) interpretation of the very facts they use their soul-physics to understand. They ask us: Are there souls? Well, sure, we reply — only they’re abstracta, mathematical abstractions rather than nuggets of mysterious stuff. They’re exquisitely useful fictions. It is as if every object attracted every other object by concentrating all its gravitational oomph in a single point — and it’s vastly easier to calculate the behavior of systems using this principled fiction than it would be to descend to the grubby details . . .
The “useful fiction” Dennett is trying to expose here is the concept of a self — a central authority who lives in our heads and is the experiencer of our experiences and the thinker of our thoughts. There is no supreme command center in the brain where consciousness “all comes together,” but the intuition is so strong that many people believe our minds cannot be entirely a product of our brains — there must be something magical, something mysterious, something immaterial going on. But that is to mistake fiction for fact. The idea of a self is indeed a useful fiction, but we must not allow our intuitions and familiar abstractions to constrain our thinking when we set out to investigate how the world really is.