Physics Guy wrote: ↑Fri May 30, 2025 11:05 am
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.
Thanks, PG, for your reply.
I think I see your point abut suspension of disbelief, but I'm not sure that that quite corresponds to choosing to believe - especially in a religious sense.
I'm sure that many agnostics, for example, are quite able to suspend disbelief in gods, or a particular god, and use thi state to examine hypothetical questions such as those concerning what you might expect to see in the real world if god beings existed with specific properties and attributes.
But, in my opinion, that is not the same as believing in such, or choosing to believe in such.
Imaginary numbers, as you point out, have the huge advantage that they can be made (and have been made) subject to a consistent set of rules and formal operations that give reproduceable results that are useful in the real world.Literally anyone anywhere can do it, and get the same results every time. I'm no expert, but I imagine (!) that electrical and electronic engineering, for example, would look very different, and perhaps be of greatly reduced utility, without being able to apply complex numbers. That alone makes it worthwhile to believe.
By the way, can you believe that Euler's Identity is - I want to say "real"! - true? Can you choose to disbelieve it? How on earth, or in heaven, can
e,
i, and
π be related in this way? Yet, as with plain imaginery numbers, if you aply the "rules" it's impossible to deny the reality.
For the god(s) question, do we simply lack a consistent set of rules and formal operations that give universally reproduceable results that are useful in the real world? Might this then allow us to choose to believe in god(s). Or might it be the case that application of such rules and operations would cause the belief to emerge, to creep up on us, without our having to, or being able to, make the choice? We would simply believe/accept/whatever?