Open-minded does not equal right: it’s wrong to be open-minded about 2+2=4.
What does open-minded mean? If it is open-minded to agree that there’s some non-zero probability that 2+2=4 then being open-minded is a good thing. If being open-minded means assigning that possibility a high likelyhood then presumably being open-minded is very bad.
As a reasonable person I probably should agree with you, but the truth is I don’t know whether to agree or disagree, or even what it means to agree or disagree with what you said. There’s a mystery and a promising research program hiding behind your words.
Assigning non-crisp probabilities to math statements is very tricky. I haven’t yet seen a satisfactory system that manages to do that. If you ever find one, tell it to Wei Dai because he’s very interested in consistently formalizing “mathematical intuition”, e.g. assigning a credence to P=NP. Just a small example of the troubles that arise: if you’re a Bayesian, assigning <1 probability to 2+2=4 (whatever that means) may make you incoherent and susceptible to Dutch books.
Part of the problem is that we do empirically make arithmetic mistakes. We’ve all misadded at some point for embarrassingly small numbers. But yes, I agree that formalizing such notions is particularly difficult with math, especially because our probability calculations themselves use arithmetic.
2 raindrops + 2 raindrops = from 0 to to unspecified droplets.
I’m honestly not sure whether this is a fair quibble or not, but if “2 + 2 = 4” is a tautology (a semi-taughtology?) which unpacks to “I’m only talking about things which are sufficiently stable to behave in a “2 + 2 =4″ish manner”, then there might be some wiggle room.
FWIW, I was talking about 2+2=4 as a statement of formal math. Interpreting it as a statement about the material world is a whole other can of worms that I’m scared to even think about, for fear of becoming confused forever. And the nature of human intuition about whole numbers of “things” (an intuition that can’t be captured by any finite system of axioms) is another completely separate mystery.
Here’s a more interesting angle than the water droplets:
There’s presumably some chance that there will be more math discovered—something with an effect like Goedel, but ever so much more so—which will make 2 + 2 = 4 not quite so certain.
For example, PA may turn out to be inconsistent. Here’s a great story about that possibility. Similar things have happened already: many people were surprised when Russell’s paradox broke naive set theory which looked as obviously consistent then as PA looks to us now.
This is another point for my claim that we need to understand how actual humans do math, not invest everything into one set of axioms.
Similar things have happened already: many people were surprised when Russell’s paradox broke naive set theory which looked as obviously consistent then as PA looks to us now.
Note that we have good reasons for believing that PA is consistent. Gentzen’s theorem is the most obvious one. I’d say we have much more reason to believe that PA is consistent than someone would have had prior to Russell believing that naive set theory was consistent. But that may involve some hindsight bias.
Note that we have good reasons for believing that PA is consistent. Gentzen’s theorem is the most obvious one.
Gentzen’s theorem assumes a much stronger claim to prove this weaker one. Yes, it grounds consistency of PA in another mathematical intuition, and in this sense could be said to strengthen the claim a bit, but formally it’s a sham.
(Of course, modern mathematics is much stronger than a century ago, so that alone counts for a good reason to believe that PA is consistent.)
Gentzen’s theorem assumes a much stronger claim to prove this weaker one.
No. The minimal axioms needs to prove Gentzen’s theorem are not stronger than PA. There are claims in PA that cannot be proven in the minimal context for Gentzen’s theorem.
There is a lot of weirdness here, once you try to reduce the question.
Imagine an AI programmed to believe in the axioms of PA and nothing else. What credence should it assign to the arithmetical statement Con(PA)? The answer is that there’s no answer. Assuming PA is “in fact” consistent, Con(PA) is formally independent from PA—which means I may add the negation of Con(PA) to PA as an extra axiom and get a consistent system. (Equivalently, that “self-hating” system has a model in ordinary set theory. Building it is a neat little exercise.) So believing or disbelieving in the “actual” consistency of PA, as we humans can, requires some notion of Platonism or semantics that we cannot yet teach to a computer.
And the question why you instinctively believe Gentzen’s proof, which uses transfinite induction up to epsilon zero, is even more mysterious. You’re comfortable relying on the consistency of a stronger system than PA because it sounds intuitive to you, right? Where do those intuitions come from? The very same place where naive set theory came from, I think.
“There’s presumably some chance that there will be more math discovered—something with an effect like Goedel, but ever so much more so—which will make 2 + 2 = 4 not quite so certain.”
right, until you learn something about non-Euclidean systems.
If you don’t understand what I say, you can’t really say I’m wrong without reasonable doubt. But some scientists don’t really listen. Just like some theists who don’t want to listen to atheistic statements.
“it’s wrong to be open-minded about 2+2=4.”—right, until you learn something about non-Euclidean systems.
It seems that you are talking about many things you don’t know much about.
Non-Euclidean is a term used to refer to certain types of geometries.
In this context, do you mean non-Archimedean? That’s very different. But then 2, +,4 and = will generally have different meanings. It seems pretty damn likely that interpreting these terms in say Peano arithmetic, that 2+2=4 is true.
Open-minded does not equal right: it’s wrong to be open-minded about 2+2=4.
I don’t understand what problem you have with scientists. It sounds like they’re right and you’re wrong, plain and simple.
What does open-minded mean? If it is open-minded to agree that there’s some non-zero probability that 2+2=4 then being open-minded is a good thing. If being open-minded means assigning that possibility a high likelyhood then presumably being open-minded is very bad.
As a reasonable person I probably should agree with you, but the truth is I don’t know whether to agree or disagree, or even what it means to agree or disagree with what you said. There’s a mystery and a promising research program hiding behind your words.
Assigning non-crisp probabilities to math statements is very tricky. I haven’t yet seen a satisfactory system that manages to do that. If you ever find one, tell it to Wei Dai because he’s very interested in consistently formalizing “mathematical intuition”, e.g. assigning a credence to P=NP. Just a small example of the troubles that arise: if you’re a Bayesian, assigning <1 probability to 2+2=4 (whatever that means) may make you incoherent and susceptible to Dutch books.
Part of the problem is that we do empirically make arithmetic mistakes. We’ve all misadded at some point for embarrassingly small numbers. But yes, I agree that formalizing such notions is particularly difficult with math, especially because our probability calculations themselves use arithmetic.
Is “2 + 2 = 4” about things or statements?
2 raindrops + 2 raindrops = from 0 to to unspecified droplets.
I’m honestly not sure whether this is a fair quibble or not, but if “2 + 2 = 4” is a tautology (a semi-taughtology?) which unpacks to “I’m only talking about things which are sufficiently stable to behave in a “2 + 2 =4″ish manner”, then there might be some wiggle room.
FWIW, I was talking about 2+2=4 as a statement of formal math. Interpreting it as a statement about the material world is a whole other can of worms that I’m scared to even think about, for fear of becoming confused forever. And the nature of human intuition about whole numbers of “things” (an intuition that can’t be captured by any finite system of axioms) is another completely separate mystery.
Here’s a more interesting angle than the water droplets:
There’s presumably some chance that there will be more math discovered—something with an effect like Goedel, but ever so much more so—which will make 2 + 2 = 4 not quite so certain.
For example, PA may turn out to be inconsistent. Here’s a great story about that possibility. Similar things have happened already: many people were surprised when Russell’s paradox broke naive set theory which looked as obviously consistent then as PA looks to us now.
This is another point for my claim that we need to understand how actual humans do math, not invest everything into one set of axioms.
Note that we have good reasons for believing that PA is consistent. Gentzen’s theorem is the most obvious one. I’d say we have much more reason to believe that PA is consistent than someone would have had prior to Russell believing that naive set theory was consistent. But that may involve some hindsight bias.
Gentzen’s theorem assumes a much stronger claim to prove this weaker one. Yes, it grounds consistency of PA in another mathematical intuition, and in this sense could be said to strengthen the claim a bit, but formally it’s a sham.
(Of course, modern mathematics is much stronger than a century ago, so that alone counts for a good reason to believe that PA is consistent.)
No. The minimal axioms needs to prove Gentzen’s theorem are not stronger than PA. There are claims in PA that cannot be proven in the minimal context for Gentzen’s theorem.
There is a lot of weirdness here, once you try to reduce the question.
Imagine an AI programmed to believe in the axioms of PA and nothing else. What credence should it assign to the arithmetical statement Con(PA)? The answer is that there’s no answer. Assuming PA is “in fact” consistent, Con(PA) is formally independent from PA—which means I may add the negation of Con(PA) to PA as an extra axiom and get a consistent system. (Equivalently, that “self-hating” system has a model in ordinary set theory. Building it is a neat little exercise.) So believing or disbelieving in the “actual” consistency of PA, as we humans can, requires some notion of Platonism or semantics that we cannot yet teach to a computer.
And the question why you instinctively believe Gentzen’s proof, which uses transfinite induction up to epsilon zero, is even more mysterious. You’re comfortable relying on the consistency of a stronger system than PA because it sounds intuitive to you, right? Where do those intuitions come from? The very same place where naive set theory came from, I think.
“There’s presumably some chance that there will be more math discovered—something with an effect like Goedel, but ever so much more so—which will make 2 + 2 = 4 not quite so certain.”
Uh, no.
“it’s wrong to be open-minded about 2+2=4.”
right, until you learn something about non-Euclidean systems.
If you don’t understand what I say, you can’t really say I’m wrong without reasonable doubt. But some scientists don’t really listen. Just like some theists who don’t want to listen to atheistic statements.
I wanna learn! What’s a non-Euclidean system and what is 2+2 in it?
It seems that you are talking about many things you don’t know much about.
Non-Euclidean is a term used to refer to certain types of geometries.
In this context, do you mean non-Archimedean? That’s very different. But then 2, +,4 and = will generally have different meanings. It seems pretty damn likely that interpreting these terms in say Peano arithmetic, that 2+2=4 is true.