malkie wrote: ↑Fri May 30, 2025 1:42 am
Neither of us could get a handle on what it means to
choose to believe something, especially in a religious sense, though apparently some people can do so.
I think it depends on exactly what one means by "believe". Belief is a somewhat fuzzy concept, and I think that there are real and important things which can reasonably count as forms of belief, and which one can or even must choose to do.
For instance, I believe that
i times
i is minus 1. To me this is a mathematical fact that by now is almost as familiar as two plus two being four. When I first learned to accept the concept of imaginary numbers, though, I think I must have found it difficult. I had no intuitive picture of what it meant for something times itself to be a negative number, the way I could picture two sets of two things merging into one set of four. I reckon I must have spent some time just saying, "Well, let's just say that it's so." Gradually it stopped taking effort. I began to take for granted that
i squared was negative one, to consider that it was a fact.
Does this example really count as choosing to believe? It's a subtle point, but I think in the end that it does.
It's subtle because whether or not imaginary numbers exist isn't exactly an empirical question that has to be either true or false as a matter of fact. It's a game rule that we're allowed to invent; it extends the concept of multiplication, and even the concept of number. Accepting a weird rule as a rule in a game isn't the same as believing it's true.
On the other hand, though, when I really think hard about it, it's not so clear to me that two plus two being four is quite exactly an empirical fact. If I look closely, there also seems to be a subtle bit of rule invention involved just in deciding that two-ness and four-ness are general qualities that are independent of any particular sets of real items. I can't help concluding that basic arithmetic is also a game whose rules we have invented, or perhaps discovered. We keep playing it because it matches common patterns we see in the world. So when we say that we believe that two-plus-two equals four, we really aren't just saying that this is a rule in our game. We're saying the rule fits the world.
And the square root of minus one is the same kind of thing, a rule in a game that fits a lot of real things. So when I say that
i times
i is minus 1, I'm also really not just saying that we can play a game in which this is one of the rules. I'm also saying that it's a good game to play, because it fits the real world, albeit in less obvious ways than the way that two-plus-two fits the world. It's a game I play all the time, now.
There is no way that I could have gotten this comfortable with using imaginary numbers if I hadn't spent a long while just suspending disbelief, by deliberate choice, and thinking as though all their weird rules were true. The utility of imaginary numbers doesn't really show up until you see things like de Moivre's theorem about the exponential of an imaginary number, or how you can get real-number roots of a cubic equation by using imaginary numbers in your arithmetic. You need to choose to entertain the weird rules, and think with them, before you can see how they make sense.
I was a teenager before it was cool.
