It’s rude to start refuting an idea before you’ve finished defining it.
One of these things is not like the others. There’s nothing wrong with giving us a history of constructive thinking, and providing us with reasons why outdated versions of the theory were found wanting. It’s good style to use parallel construction to build rhetorical momentum. It is terribly dishonest to do both at the same time—it creates the impression that the subjective reasons you give for dismissing point 3 have weight equal to the objective reasons history has given for dismissing points 1 and 2.
Your talk in point 3 about “map-territory confusion” is very strange. Mathematics is all in your head. It’s all map, no territory. You seem to be claiming that constructivsts are outside of the mathematical mainstream because they want to bend theory in the direction of a preferred outcome. You then claim that this is outside of the bounds of acceptable mathematical thinking, So what’s wrong with reasoning like this:
“Nobody really likes all of the consequences of the Axiom of Choice, but most people seem willing to put up with its bad behavior because some of the abstractions it enables—like the Real Numbers—are just so damn useful. I wonder how many of the useful properties of the Real Numbers I could capture by building up from (a possibly weakened version of) ZF set theory and a weakened version of the Axiom of Choice?”
I’m sorry, but I don’t think there was anything remotely “rude” or “terribly dishonest” about my previous comment. If you think I am mistaken about anything I said, just explain why. Criticizing my rhetorical style and accusing me of violating social norms is not something I find helpful.
Quite frankly, I also find criticisms of the form “you sound more confident than you should be” rather annoying. E.g:
it creates the impression that the subjective reasons you give for dismissing point 3 have weight equal to the objective reasons history has given for dismissing points 1 and 2.
That’s because for me, the reasons I gave in point 3 do indeed have similar weight to the reasons I gave in points 1 and 2. If you disagree, by all means say so. But to rise up in indignation over the very listing of my reasons—is that really necessary? Would you seriously have preferred that I just list the bullet points without explaining what I thought?
So what’s wrong with reasoning like this:
Nothing at all, except for the false claim that nobody likes the consequences of the Axiom of Choice. (Some people do like them, and why shouldn’t they?)
The target of my critique—and I thought I made this clear in my response to cousin_it—is the critique of mainstream mathematical reasoning, not the research program of exploring different axiomatic set theories. The latter could easily be done by someone fully on board with the soundness of traditional mathematics. Just as it is unnecessary to doubt the correctness of Euclid’s arguments in order to be interested in non-Euclidean geometry.
Criticizing my rhetorical style and accusing me of violating social norms is not something I find helpful.
Until very recently, I held a similar attitude. I think it’s common to be annoyed by this sort of criticism… it’s distracting and rarely relevant.
That said, it seems to me that the above “rarely” isn’t rare enough. If you’re inadvertently violating a social norm, wouldn’t you like to know? If you already know, what does it matter to have it pointed out to you? Just ignore the redundant information.
I think this principle extends to a lot of speculative or subjective criticism. The potential value of just one accurate critique taken to heart seems quite high. Does such criticism have a positive expected value? That depends on the overall cost of the associated inaccurate or redundant statements (i.e., the vast majority of them). It seems this cost can be made to approach zero by just not taking them personally and ignoring them when they’re misguided, so long as they’re sufficiently disentangled from “object-level” statements.
Regarding your three bullet points above:
It’s rude to start refuting an idea before you’ve finished defining it.
One of these things is not like the others. There’s nothing wrong with giving us a history of constructive thinking, and providing us with reasons why outdated versions of the theory were found wanting. It’s good style to use parallel construction to build rhetorical momentum. It is terribly dishonest to do both at the same time—it creates the impression that the subjective reasons you give for dismissing point 3 have weight equal to the objective reasons history has given for dismissing points 1 and 2.
Your talk in point 3 about “map-territory confusion” is very strange. Mathematics is all in your head. It’s all map, no territory. You seem to be claiming that constructivsts are outside of the mathematical mainstream because they want to bend theory in the direction of a preferred outcome. You then claim that this is outside of the bounds of acceptable mathematical thinking, So what’s wrong with reasoning like this:
“Nobody really likes all of the consequences of the Axiom of Choice, but most people seem willing to put up with its bad behavior because some of the abstractions it enables—like the Real Numbers—are just so damn useful. I wonder how many of the useful properties of the Real Numbers I could capture by building up from (a possibly weakened version of) ZF set theory and a weakened version of the Axiom of Choice?”
I’m sorry, but I don’t think there was anything remotely “rude” or “terribly dishonest” about my previous comment. If you think I am mistaken about anything I said, just explain why. Criticizing my rhetorical style and accusing me of violating social norms is not something I find helpful.
Quite frankly, I also find criticisms of the form “you sound more confident than you should be” rather annoying. E.g:
That’s because for me, the reasons I gave in point 3 do indeed have similar weight to the reasons I gave in points 1 and 2. If you disagree, by all means say so. But to rise up in indignation over the very listing of my reasons—is that really necessary? Would you seriously have preferred that I just list the bullet points without explaining what I thought?
Nothing at all, except for the false claim that nobody likes the consequences of the Axiom of Choice. (Some people do like them, and why shouldn’t they?)
The target of my critique—and I thought I made this clear in my response to cousin_it—is the critique of mainstream mathematical reasoning, not the research program of exploring different axiomatic set theories. The latter could easily be done by someone fully on board with the soundness of traditional mathematics. Just as it is unnecessary to doubt the correctness of Euclid’s arguments in order to be interested in non-Euclidean geometry.
Until very recently, I held a similar attitude. I think it’s common to be annoyed by this sort of criticism… it’s distracting and rarely relevant.
That said, it seems to me that the above “rarely” isn’t rare enough. If you’re inadvertently violating a social norm, wouldn’t you like to know? If you already know, what does it matter to have it pointed out to you? Just ignore the redundant information.
I think this principle extends to a lot of speculative or subjective criticism. The potential value of just one accurate critique taken to heart seems quite high. Does such criticism have a positive expected value? That depends on the overall cost of the associated inaccurate or redundant statements (i.e., the vast majority of them). It seems this cost can be made to approach zero by just not taking them personally and ignoring them when they’re misguided, so long as they’re sufficiently disentangled from “object-level” statements.