Hi! I really appreciate this reply, and I stewed on it for a bit. I think the crux of our disagreement comes down to definitions of things, and that we mostly agree except for definitions of some words.
Knowledge—I think knowledge has to be correct to be knowledge, otherwise you just think you have knowledge. It seems like we disagree here, and you think that knowledge just means a belief that is likely to be true (and for the right reason?). It’s unclear to me how you would cash out “accurate map” for things that you can’t physically observe like math, but I think I get the gist of your definition. Also, side note, justified true belief is not a widely held view in modern philosophy, most theories of truth go for justified true belief + something else.
Real—We both agree it doesn’t matter for our day-to-day lives whether math is real or not. (It may matter for patent law, if it decides whether math is treated as an invention or a discovery!) I think that it would be nice to know whether math is real or not, and I try to understand the logical form of sentences I utter to know what fact about the world would make them true or false. So you say I “don’t have to worry about” whether numbers are real, and I agree – their reality or non-reality is not causing me any problem, I’m just curious.
I also view epistemic uniformity as pretty important, because we should have the same standards of knowledge across all fields. You seem to think that mathematical knowledge doesn’t exist, because mathematical “knowledge” is just what we have derived within a system. I can agree with that! The Benacerraf paper presents a big problem for realism, which you seem to buy—and you’re willing to put up with losing semantic uniformity for it.
I think our differences comes down to how much we want semantic uniformity in a theory of truth of math.
Interesting… My feeling is that we are not even using the same language. Probably because of something deep. It might be the definition of some words, but I doubt it.
Knowledge—I think knowledge has to be correct to be knowledge, otherwise you just think you have knowledge.
what does it mean for knowledge to be correct? To me it means that it can be used to make good predictions.
you think that knowledge just means a belief that is likely to be true (and for the right reason?)
well, that’s the same thing, a model that makes good predictions. “The right reason” is just another way to say “the model’s domain of applicability can be expanded without a significant loss of accuracy”.
It’s unclear to me how you would cash out “accurate map” for things that you can’t physically observe like math
You can “observe math”, as much as you can observe anything. How do you observe something else that is not “plainly visible”, like, say, UV radiation?
We both agree it doesn’t matter for our day-to-day lives whether math is real or not.
That is not quite what I said, I think. I meant that math is as real as, well, baseball.
You seem to think that mathematical knowledge doesn’t exist, because mathematical “knowledge” is just what we have derived within a system.
I… was saying the opposite. That mathematical knowledge exists just as much as any other knowledge, it just comes equipped with its own unique rigging, like proven theorems being “true”, or, in GEB’s language, a collection of valid strings or something. I don’t want to go deeper, since math is not my area.
In general, the concept of existence and reality, while useful, has a limited applicability and even lifetime. One can say that some models exist more than others, or are more real than others.
I also view epistemic uniformity as pretty important, because we should have the same standards of knowledge across all fields.
I agree with that, but those standards are not linguistic, the way (your review of) Benacerraf’s paper describes it, that they should have the same form (semantic uniformity). The standards are whether the models are accurate (in terms of their observational value) in the domain of their applicability, and how well they can be extended to other domains. Semantic uniformity is sometimes useful and sometimes not, and there is no reason that I can see that it should be universally valid.
Not sure if this made sense… Most people don’t naturally think in the way I described.
Thoroughgoing anti realism gives you a kind of semantic uniformity , at the expense of having the same level of anti realism about non mathematical entities. Do you want to give up believing in electrons?
Hi! I really appreciate this reply, and I stewed on it for a bit. I think the crux of our disagreement comes down to definitions of things, and that we mostly agree except for definitions of some words.
Knowledge—I think knowledge has to be correct to be knowledge, otherwise you just think you have knowledge. It seems like we disagree here, and you think that knowledge just means a belief that is likely to be true (and for the right reason?). It’s unclear to me how you would cash out “accurate map” for things that you can’t physically observe like math, but I think I get the gist of your definition. Also, side note, justified true belief is not a widely held view in modern philosophy, most theories of truth go for justified true belief + something else.
Real—We both agree it doesn’t matter for our day-to-day lives whether math is real or not. (It may matter for patent law, if it decides whether math is treated as an invention or a discovery!) I think that it would be nice to know whether math is real or not, and I try to understand the logical form of sentences I utter to know what fact about the world would make them true or false. So you say I “don’t have to worry about” whether numbers are real, and I agree – their reality or non-reality is not causing me any problem, I’m just curious.
I also view epistemic uniformity as pretty important, because we should have the same standards of knowledge across all fields. You seem to think that mathematical knowledge doesn’t exist, because mathematical “knowledge” is just what we have derived within a system. I can agree with that! The Benacerraf paper presents a big problem for realism, which you seem to buy—and you’re willing to put up with losing semantic uniformity for it.
I think our differences comes down to how much we want semantic uniformity in a theory of truth of math.
Interesting… My feeling is that we are not even using the same language. Probably because of something deep. It might be the definition of some words, but I doubt it.
what does it mean for knowledge to be correct? To me it means that it can be used to make good predictions.
well, that’s the same thing, a model that makes good predictions. “The right reason” is just another way to say “the model’s domain of applicability can be expanded without a significant loss of accuracy”.
You can “observe math”, as much as you can observe anything. How do you observe something else that is not “plainly visible”, like, say, UV radiation?
That is not quite what I said, I think. I meant that math is as real as, well, baseball.
I… was saying the opposite. That mathematical knowledge exists just as much as any other knowledge, it just comes equipped with its own unique rigging, like proven theorems being “true”, or, in GEB’s language, a collection of valid strings or something. I don’t want to go deeper, since math is not my area.
In general, the concept of existence and reality, while useful, has a limited applicability and even lifetime. One can say that some models exist more than others, or are more real than others.
I agree with that, but those standards are not linguistic, the way (your review of) Benacerraf’s paper describes it, that they should have the same form (semantic uniformity). The standards are whether the models are accurate (in terms of their observational value) in the domain of their applicability, and how well they can be extended to other domains. Semantic uniformity is sometimes useful and sometimes not, and there is no reason that I can see that it should be universally valid.
Not sure if this made sense… Most people don’t naturally think in the way I described.
Thoroughgoing anti realism gives you a kind of semantic uniformity , at the expense of having the same level of anti realism about non mathematical entities. Do you want to give up believing in electrons?