If someone tells me that a complex mathematical proof comes to a certain conclusion, than often I have to make a decision about trusting their expertise or spending a few semesters studying math to get the underlying understanding to be able to follow the proof myself.
Even in the case you described, it should be possible to lay out something to support the argument, even if it takes too long to cover the argument itself. Like, instead of just saying “I know this proof seems counterintuitive right now, but trust me, once you study a bit more, it’ll all make sense,” the person would do better to say, “Well, I know it sounds absurd that you’d be able to take a single object, disassemble it, and reassemble it to form two of the same object, but in fact it has been proven to be possible given infinite divisibility and something called the Axiom of Choice. If you’re not familiar with that, I’d suggest reading a bit about set theory.” My credence in the latter case would be much higher than in the former case, and it didn’t seem to take that long just to say, “Okay, this statement has its roots in X, Y, and Z, so to understand, you’ll want to study those.” I maintain that it should be possible to give some context for the experience argument no matter what, and that if you don’t do so, you’re not trying to argue in good faith.
“Well, I know it sounds absurd that you’d be able to take a single object, disassemble it, and reassemble it to form two of the same object, but in fact it has been proven to be possible given infinite divisibility and something called the Axiom of Choice. If you’re not familiar with that, I’d suggest reading a bit about set theory.”
I actually can give you an “intuitive” justification of the Banach-Tarski theorem.
Suppose you have a rigid ball full of air. If you take half the air out and put it into another, identical ball, you now have twice the volume of air, at half the density. However, the points in a mathematical ball are infinitely dense—half of infinity is still infinity, so it turns out that if you do it just right, you can take out “half” of the points from a mathematical ball, put it inside another one, and end up with two balls that are both “completely full” and identical to the original one.
Your explanation suggests the wrong intuition for Banach-Tarski.
It’s relatively easy to show that there’s a bijection between the points contained in one ball and the points contained in two balls. (Similarly, there is a bijection between the interval [0,1] and the interval [0,2].)
The Banach-Tarski theorem proves a harder statement: you can take a unit ball, partition it into finitely many pieces (I think it can be done with five), and then rearrange those pieces, using only translations and rotations, into two unit balls.
(If there’s a canonical weird thing about the theorem, it’s that we can do this in three dimensions but not in two.)
Agreed; it’s not a real justification, it’s just something that makes it sound less absurd. (When you look at the theorem a little bit closer, the weird part becomes not that you can make two balls out of one ball, but that you can do it with just translations and rotations. And if you look really, really hard, the weird part becomes that you can’t do it with only four pieces.)
This intuitive justification likewise indicates that one should be able to do the Banach-Tarski thing with a 2-dimensional disc rather than a 3-dimensional ball. Unfortunately, that isn’t true. (Though it is if you allow area-preserving affine transformations as well as isometries.)
Not every experience has a concise summary.
If someone tells me that a complex mathematical proof comes to a certain conclusion, than often I have to make a decision about trusting their expertise or spending a few semesters studying math to get the underlying understanding to be able to follow the proof myself.
Even in the case you described, it should be possible to lay out something to support the argument, even if it takes too long to cover the argument itself. Like, instead of just saying “I know this proof seems counterintuitive right now, but trust me, once you study a bit more, it’ll all make sense,” the person would do better to say, “Well, I know it sounds absurd that you’d be able to take a single object, disassemble it, and reassemble it to form two of the same object, but in fact it has been proven to be possible given infinite divisibility and something called the Axiom of Choice. If you’re not familiar with that, I’d suggest reading a bit about set theory.” My credence in the latter case would be much higher than in the former case, and it didn’t seem to take that long just to say, “Okay, this statement has its roots in X, Y, and Z, so to understand, you’ll want to study those.” I maintain that it should be possible to give some context for the experience argument no matter what, and that if you don’t do so, you’re not trying to argue in good faith.
I actually can give you an “intuitive” justification of the Banach-Tarski theorem.
Suppose you have a rigid ball full of air. If you take half the air out and put it into another, identical ball, you now have twice the volume of air, at half the density. However, the points in a mathematical ball are infinitely dense—half of infinity is still infinity, so it turns out that if you do it just right, you can take out “half” of the points from a mathematical ball, put it inside another one, and end up with two balls that are both “completely full” and identical to the original one.
Your explanation suggests the wrong intuition for Banach-Tarski.
It’s relatively easy to show that there’s a bijection between the points contained in one ball and the points contained in two balls. (Similarly, there is a bijection between the interval [0,1] and the interval [0,2].)
The Banach-Tarski theorem proves a harder statement: you can take a unit ball, partition it into finitely many pieces (I think it can be done with five), and then rearrange those pieces, using only translations and rotations, into two unit balls.
(If there’s a canonical weird thing about the theorem, it’s that we can do this in three dimensions but not in two.)
Agreed; it’s not a real justification, it’s just something that makes it sound less absurd. (When you look at the theorem a little bit closer, the weird part becomes not that you can make two balls out of one ball, but that you can do it with just translations and rotations. And if you look really, really hard, the weird part becomes that you can’t do it with only four pieces.)
This intuitive justification likewise indicates that one should be able to do the Banach-Tarski thing with a 2-dimensional disc rather than a 3-dimensional ball. Unfortunately, that isn’t true. (Though it is if you allow area-preserving affine transformations as well as isometries.)