Benacerraf convinced me that either mathematical sentences have different logical forms than non-mathematical sentences or that mathematical knowledge has a different form than non-mathematical knowledge. It sounds like your view is that mathematical sentences have different forms (they all have an implicit “within some mathematical system that is relevant, it is provable that...” before them), and also that mathematical knowledge is different (not real knowledge, just exists in a system). In other words, it sounds like you just think that epistemic uniformity and semantic uniformity are not important features of a theory of mathematical truth. That comes down to personal aesthetics and meta-beliefs about how theories should look, so I will just talk about what I think you’re saying in this comment.
I think what you’re trying to say is that mathematical statements are not true or false in an absolute sense, only true and false within a proof system. and their truth or falsehood is based entirely on whether they can be derived from within that system.
If that’s true, math is just a map, and maps are neither true nor false. If math is just a map, then there is no such thing as objective mathematical truth. So it sounds like you agree that knowledge about any mathematical object is impossible. But when you say that “Epistemic uniformity simply states that math is a useful model”, I think that’s a little different than what I intended it to mean. Epistemic uniformity says that evaluating the truth-value of a mathematical statement should be a similar process evaluating the truth-value of any other statement.
The issue here is that our non-mathematical statements aren’t only internally true or false, they are actually true or false. If you asked someone to justify sentence (1), and they handed you a proof about New York and London, consistent on a set of city-axioms, you would probably be pretty confused. Epistemic uniformity says that a theory of mathematical truth should look relatively like the rest of our theories of truth—why should math be special?
I’m going to take a slight objection to using the phrase “discoverable experimentally” to describe proving a theorem and thinking up numbers, but let’s talk about those examples. To me, it sounds like that is doing work within a system of math to determine whether a claim is consistent with axioms. There is some tension here between saying that math is just a tool and thinking that you can do experiments on it to discover facts about the world. No! It will tell us about the tool we are experimenting with. Doing math (under the intutionist paradigm) tells us whether something is provable within a mathematical system, but it has no bearing on whether it is true outside of our minds.
(Side note about intuitionism:
I think it’s important to prevent talking past each other by checking definitions, so I’d like to clarify what you mean by intuitionism. In the definition I’m aware of, intuitionism says roughly that math exists entirely in minds, and the corresponding account of mathematical truth is that a statement is true if someone has a mental construction proving it to be true. Please let me know if this is not what you meant!
My main objection with intuitionism is that it makes a lot of math time-dependent (e.x. 2+2 didn’t equal 4 until someone proved it for the first time). Under an intuitionist account of mathematical truth, you can make sentence (2) true by finding three examples that fit. But then that statement’s truthhood or falsehood is independent of whether the mathematical fact is really true or false (intuitionists usually don’t think there exist universal mathematical truth). It seems to me that math is a real thing in the universe, it was real because humans comprehended it, and it will remain real after humans are gone. That view is incompatible with intuitionism.)
(Another note—can you be a bit more specific about the contradition you think is avoided by giving up Platonism? I think that you still don’t have epistemic and semantic uniformity with an intuitionist/combinatorial theory of math)
It sounds like your view is that mathematical sentences have different forms (they all have an implicit “within some mathematical system that is relevant, it is provable that...” before them)
Yes. And your paraphrasing matches what I tried to express pretty well, except when you use the term “knowledge”.
math is just a map, and maps are neither true nor false. If math is just a map, then there is no such thing as objective mathematical truth.
Depends on your definition of “objective”. It’s a loaded term, and people vehemently disagree on its meaning.
So it sounds like you agree that knowledge about any mathematical object is impossible.
Not really, I just don’t think you and I use the term “knowledge” the same way. I reject the old definition “justified true belief”, because it has a weasel word “true” in it. Knowledge is an accurate map, nothing else.
Epistemic uniformity says that evaluating the truth-value of a mathematical statement should be a similar process evaluating the truth-value of any other statement.
I’d restrict the notion of “truth” to proved theorems. Not just provable, but actually proved. Which also means that different people have different mathematical truths. If I don’t know what the eighth decimal digit of pi is, the statement that it is equal six is neither true nor false for me, not without additional evidence. In that sense, a set of mathematical axioms carves out a piece of territory that is in the model space. There is nothing particularly contradictory about it, we are all embedded agents, and any map is also a territory, in the minds of the agents. I agree that math is not very special, except insofar as it has a specific structure, a set of axioms that can be combined to prove theorems, and those theorems can sometimes serve as useful maps of the territory outside the math itself.
I am not sure what your objection is to the statement that mathematical truths can be discovered experimentally. Seems like we are saying the same thing?
Doing math (under the intutionist paradigm) tells us whether something is provable within a mathematical system, but it has no bearing on whether it is true outside of our minds.
It’s worse than that. “Truth” is not a coherent concept outside of the parts of our minds that do math.
My main objection with intuitionism is that it makes a lot of math time-dependent (e.x. 2+2 didn’t equal 4 until someone proved it for the first time).
A better way to state this is that the theorem 2+2=4 was not a part of whatever passed for math back then. We are in the process of continuous model building, some models work out and persist for a time, some don’t and fade away quickly. Some models propagate through multiple human minds and take over as “truths”, and others remain niche, even if they are accurate and useful. That depends on the memetic power of the model, not just on how accurate it is. Religions, for example, have a lot of memetic power, even if their predictions are wildly inaccurate.
It seems to me that math is a real thing in the universe, it was real because humans comprehended it, and it will remain real after humans are gone. That view is incompatible with intuitionism.
Again, “real” does all the work here. Math is useful to humans. The model that “[math] will remain after humans are gone” is content-free unless you specify how it can be tested. And that requires a lot of assumptions, such as “what if another civilization arose, would it construct mathematics the way humans do?”—and we have no way to test that, given that we know of no other civilizations.
can you be a bit more specific about the contradition you think is avoided by giving up Platonism? I think that you still don’t have epistemic and semantic uniformity with an intuitionist/combinatorial theory of math
If you give up Platonism as some independent idea-realm, you don’t have to worry about meaningless questions like “are numbers real?” but only about “are numbers useful?” Semantic uniformity disappears except as a model that is sometimes useful and sometimes not. In the examples given it is not useful. Epistemic uniformity is trivially true, since all mathematical “knowledge” is internal to the mathematical system in question.
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?
Benacerraf convinced me that either mathematical sentences have different logical forms than non-mathematical sentences or that mathematical knowledge has a different form than non-mathematical knowledge. It sounds like your view is that mathematical sentences have different forms (they all have an implicit “within some mathematical system that is relevant, it is provable that...” before them), and also that mathematical knowledge is different (not real knowledge, just exists in a system). In other words, it sounds like you just think that epistemic uniformity and semantic uniformity are not important features of a theory of mathematical truth. That comes down to personal aesthetics and meta-beliefs about how theories should look, so I will just talk about what I think you’re saying in this comment.
I think what you’re trying to say is that mathematical statements are not true or false in an absolute sense, only true and false within a proof system. and their truth or falsehood is based entirely on whether they can be derived from within that system.
If that’s true, math is just a map, and maps are neither true nor false. If math is just a map, then there is no such thing as objective mathematical truth. So it sounds like you agree that knowledge about any mathematical object is impossible. But when you say that “Epistemic uniformity simply states that math is a useful model”, I think that’s a little different than what I intended it to mean. Epistemic uniformity says that evaluating the truth-value of a mathematical statement should be a similar process evaluating the truth-value of any other statement.
The issue here is that our non-mathematical statements aren’t only internally true or false, they are actually true or false. If you asked someone to justify sentence (1), and they handed you a proof about New York and London, consistent on a set of city-axioms, you would probably be pretty confused. Epistemic uniformity says that a theory of mathematical truth should look relatively like the rest of our theories of truth—why should math be special?
I’m going to take a slight objection to using the phrase “discoverable experimentally” to describe proving a theorem and thinking up numbers, but let’s talk about those examples. To me, it sounds like that is doing work within a system of math to determine whether a claim is consistent with axioms. There is some tension here between saying that math is just a tool and thinking that you can do experiments on it to discover facts about the world. No! It will tell us about the tool we are experimenting with. Doing math (under the intutionist paradigm) tells us whether something is provable within a mathematical system, but it has no bearing on whether it is true outside of our minds.
(Side note about intuitionism:
I think it’s important to prevent talking past each other by checking definitions, so I’d like to clarify what you mean by intuitionism. In the definition I’m aware of, intuitionism says roughly that math exists entirely in minds, and the corresponding account of mathematical truth is that a statement is true if someone has a mental construction proving it to be true. Please let me know if this is not what you meant!
My main objection with intuitionism is that it makes a lot of math time-dependent (e.x. 2+2 didn’t equal 4 until someone proved it for the first time). Under an intuitionist account of mathematical truth, you can make sentence (2) true by finding three examples that fit. But then that statement’s truthhood or falsehood is independent of whether the mathematical fact is really true or false (intuitionists usually don’t think there exist universal mathematical truth). It seems to me that math is a real thing in the universe, it was real because humans comprehended it, and it will remain real after humans are gone. That view is incompatible with intuitionism.)
(Another note—can you be a bit more specific about the contradition you think is avoided by giving up Platonism? I think that you still don’t have epistemic and semantic uniformity with an intuitionist/combinatorial theory of math)
First, I appreciate your thoughtful reply!
Yes. And your paraphrasing matches what I tried to express pretty well, except when you use the term “knowledge”.
Depends on your definition of “objective”. It’s a loaded term, and people vehemently disagree on its meaning.
Not really, I just don’t think you and I use the term “knowledge” the same way. I reject the old definition “justified true belief”, because it has a weasel word “true” in it. Knowledge is an accurate map, nothing else.
I’d restrict the notion of “truth” to proved theorems. Not just provable, but actually proved. Which also means that different people have different mathematical truths. If I don’t know what the eighth decimal digit of pi is, the statement that it is equal six is neither true nor false for me, not without additional evidence. In that sense, a set of mathematical axioms carves out a piece of territory that is in the model space. There is nothing particularly contradictory about it, we are all embedded agents, and any map is also a territory, in the minds of the agents. I agree that math is not very special, except insofar as it has a specific structure, a set of axioms that can be combined to prove theorems, and those theorems can sometimes serve as useful maps of the territory outside the math itself.
I am not sure what your objection is to the statement that mathematical truths can be discovered experimentally. Seems like we are saying the same thing?
It’s worse than that. “Truth” is not a coherent concept outside of the parts of our minds that do math.
A better way to state this is that the theorem 2+2=4 was not a part of whatever passed for math back then. We are in the process of continuous model building, some models work out and persist for a time, some don’t and fade away quickly. Some models propagate through multiple human minds and take over as “truths”, and others remain niche, even if they are accurate and useful. That depends on the memetic power of the model, not just on how accurate it is. Religions, for example, have a lot of memetic power, even if their predictions are wildly inaccurate.
Again, “real” does all the work here. Math is useful to humans. The model that “[math] will remain after humans are gone” is content-free unless you specify how it can be tested. And that requires a lot of assumptions, such as “what if another civilization arose, would it construct mathematics the way humans do?”—and we have no way to test that, given that we know of no other civilizations.
If you give up Platonism as some independent idea-realm, you don’t have to worry about meaningless questions like “are numbers real?” but only about “are numbers useful?” Semantic uniformity disappears except as a model that is sometimes useful and sometimes not. In the examples given it is not useful. Epistemic uniformity is trivially true, since all mathematical “knowledge” is internal to the mathematical system in question.
We might be talking past each other though.
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?