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I think you're wrong on some of these points, and the author was trying to make these points but was not explicit. I remember reading an article in the intelli

I think you're wrong on some of these points, and the author was trying to make these points but was not explicit. I remember reading an article in the intelligencer by a mathematician who essentially said there are certain conjectures where if they were to be proved false rather than be thrown into a sea of uncertainty, mathematicians i would quickly move to investigate a readjustment of basic axioms rather than accept that those conjectures are incorrect. Then there are fields of mathematics around selecting different axioms. Investigating the ramifications of whether you take the undecidable "continuum hypothesis" as true or false. And then theres model theory and such. Presumably they study models of interest and not arbitrary ones. You're mostly correct that the methodology is mostly deductive but the point is that what we choose to use isn't arbitrary because there are things in math which are more important than axioms as they are things believed to be "real".

scoofy 5y ago
I don’t mean "arbitrary" in the sense that it doesn’t matter to the engineering of a building. I mean "arbitrary" in the sense that unprovableness within the framework. Axioms are by definition arbitrary in that sense. edit: I'll try to expand on this. Math is weird because it's a sort-of in-framework study. We use math when we do physics, and we use physics when we do engineering. If the physical world ever started disagreeing with our physics, then we need new physics. If the math ever started showing inconsistent results in our physics, then we would need new math. None of that is to say that Neutonian Physics, say, is "wrong in the sense that it's internally inconsistent", only that "it's wrong in the sense that it is not an accurate mapping of a framework to the world we find ourselves in." The first type of wrongness is the type of wrongness we concern ourselves with in deductive studies, the second type, the mapping-error, is inductive, and requires meta-analysis, and is prone to (Hume's) problem of induction, which means it's ultimately unknowable (i.e. Popper).
- cormacrelf 5y ago
  Here is where you're wrong: > You can't have a replication crisis inside of a deductive framework without changing the framework. You may change the framework, but the replication crisis would be that people don't notice it's changed. A proof can get so huge and complex that it would take a lifetime of study to understand it. One of the article's good points was that "you can replicate a math paper by reading it", but if you cannot read a proof, you cannot replicate it. If nobody can understand it, nobody can replicate it. There would be a big crisis if people started trusting huge papers blindly and using their results without stating them as assumptions. Mathematicians are largely not doing that, so there isn't really a crisis. But they might! And you would not notice that the framework had become inconsistent for a while. Therein would lie the crisis. It is nice to think that as soon as the framework changed, people would notice, because surely if it the ground moved, that would be obvious! But it is not true. Nothing about the deductive-ness or in-framework-ness of mathematics changes this. It is the same for psychology, the replication crisis is a crisis because people do not notice these results are wrong, and then rely on them in clinical guidelines and affect people's lives for generations. It is possible to stave off replication crises by using a machine, because mathematics is self-contained and you can use a computer to show that you proved something using only these specific assumptions and they can believe you without reading and understanding it fully. But this doesn't mean mathematics cannot ever have a replication crisis. I think your arguments here are best summarised as confusing "replication crisis" with "any published results being inconsistent with previous results". Your idea of "inconsistent" is also strange, the article considers this to mean mistakes in proofs, whereas you mean new mathematics that changes the field by discovery. You're just talking about something completely foreign from the concept of a replication crisis.
  - inglor_cz 5y ago
    This scenario was actually pretty close when Andrew Wiles published his proof of the Last Fermat Theorem. The proof was huge, IIRC over 100 pages of maths, some of it pretty obscure, and verifying it took a lot of work of several people. They even found a flaw in it, but Wiles was able to fix it after a year.
    - benibela 5y ago
      Or now the "proof" of the abc conjecture with the inter-universal Teichmüller theory
  - scoofy 5y ago
    I’m talking about an event like the incompleteness theorem that ends aspirations like those of Russell and Whitehead. Paradoxes that lead to new axioms like in the development of set theory. Deductive frameworks can’t be wrong in the way that experimental results can be statistical anomalies. The author conflates errors with perfectly reasonable experimental results later shown to be anomalous upon replication.
    - cormacrelf 5y ago
      Again you are talking about the framework as if it is exactly the same thing as the papers that attempt to describe it. It is not. Your statement that "deductive frameworks can't be wrong" is irrelevant to the question of whether a math paper can be wrong. If I prove something, it can be wrong, and it may turn out that I haven't proved it at all, but I can still call it a proof, convince other people it's correct, get it published in a journal, and have other mathematicians rely on it before the mistake is discovered. Papers can be flawed. The papers are not the framework. The reason you think they are the same thing is probably that mathematics has not had that many big mistakes recently, since the renewed effort on solid foundations last century, which means the system is working. You can't define your way out of the fact that mathematicians will make mistakes and write papers that draw conclusions that are not valid. They are just pretty good at discovering the mistakes, and finding them is culturally essential to the practice of mathematics; arguably if it were not, it would be a big game of Numberwang. Psychologists are, on the other hand, not nearly as good at discovering mistakes. We didn't have to use any properties of the self-containedness of the disciplines to get to that result. It was not necessary to prove mathematically that mathematics does not or cannot have a replication problem. You can just observe it. (Edit, all of that in brief: if it were truly impossible to be wrong in maths, then why do mathematicians spend so much time trying to verify each others' proofs? Are they wasting their time? Or is it the entire reason there isn't a replication problem?)

scoofy 5y ago

I don’t mean "arbitrary" in the sense that it doesn’t matter to the engineering of a building. I mean "arbitrary" in the sense that unprovableness within the framework. Axioms are by definition arbitrary in that sense. edit: I'll try to expand on this. Math is weird because it's a sort-of in-framework study. We use math when we do physics, and we use physics when we do engineering. If the physical world ever started disagreeing with our physics, then we need new physics. If the math ever started showing inconsistent results in our physics, then we would need new math. None of that is to say that Neutonian Physics, say, is "wrong in the sense that it's internally inconsistent", only that "it's wrong in the sense that it is not an accurate mapping of a framework to the world we find ourselves in." The first type of wrongness is the type of wrongness we concern ourselves with in deductive studies, the second type, the mapping-error, is inductive, and requires meta-analysis, and is prone to (Hume's) problem of induction, which means it's ultimately unknowable (i.e. Popper).

cormacrelf 5y ago
Here is where you're wrong: > You can't have a replication crisis inside of a deductive framework without changing the framework. You may change the framework, but the replication crisis would be that people don't notice it's changed. A proof can get so huge and complex that it would take a lifetime of study to understand it. One of the article's good points was that "you can replicate a math paper by reading it", but if you cannot read a proof, you cannot replicate it. If nobody can understand it, nobody can replicate it. There would be a big crisis if people started trusting huge papers blindly and using their results without stating them as assumptions. Mathematicians are largely not doing that, so there isn't really a crisis. But they might! And you would not notice that the framework had become inconsistent for a while. Therein would lie the crisis. It is nice to think that as soon as the framework changed, people would notice, because surely if it the ground moved, that would be obvious! But it is not true. Nothing about the deductive-ness or in-framework-ness of mathematics changes this. It is the same for psychology, the replication crisis is a crisis because people do not notice these results are wrong, and then rely on them in clinical guidelines and affect people's lives for generations. It is possible to stave off replication crises by using a machine, because mathematics is self-contained and you can use a computer to show that you proved something using only these specific assumptions and they can believe you without reading and understanding it fully. But this doesn't mean mathematics cannot ever have a replication crisis. I think your arguments here are best summarised as confusing "replication crisis" with "any published results being inconsistent with previous results". Your idea of "inconsistent" is also strange, the article considers this to mean mistakes in proofs, whereas you mean new mathematics that changes the field by discovery. You're just talking about something completely foreign from the concept of a replication crisis.
- inglor_cz 5y ago
  This scenario was actually pretty close when Andrew Wiles published his proof of the Last Fermat Theorem. The proof was huge, IIRC over 100 pages of maths, some of it pretty obscure, and verifying it took a lot of work of several people. They even found a flaw in it, but Wiles was able to fix it after a year.
  - benibela 5y ago
    Or now the "proof" of the abc conjecture with the inter-universal Teichmüller theory
- scoofy 5y ago
  I’m talking about an event like the incompleteness theorem that ends aspirations like those of Russell and Whitehead. Paradoxes that lead to new axioms like in the development of set theory. Deductive frameworks can’t be wrong in the way that experimental results can be statistical anomalies. The author conflates errors with perfectly reasonable experimental results later shown to be anomalous upon replication.
  - cormacrelf 5y ago
    Again you are talking about the framework as if it is exactly the same thing as the papers that attempt to describe it. It is not. Your statement that "deductive frameworks can't be wrong" is irrelevant to the question of whether a math paper can be wrong. If I prove something, it can be wrong, and it may turn out that I haven't proved it at all, but I can still call it a proof, convince other people it's correct, get it published in a journal, and have other mathematicians rely on it before the mistake is discovered. Papers can be flawed. The papers are not the framework. The reason you think they are the same thing is probably that mathematics has not had that many big mistakes recently, since the renewed effort on solid foundations last century, which means the system is working. You can't define your way out of the fact that mathematicians will make mistakes and write papers that draw conclusions that are not valid. They are just pretty good at discovering the mistakes, and finding them is culturally essential to the practice of mathematics; arguably if it were not, it would be a big game of Numberwang. Psychologists are, on the other hand, not nearly as good at discovering mistakes. We didn't have to use any properties of the self-containedness of the disciplines to get to that result. It was not necessary to prove mathematically that mathematics does not or cannot have a replication problem. You can just observe it. (Edit, all of that in brief: if it were truly impossible to be wrong in maths, then why do mathematicians spend so much time trying to verify each others' proofs? Are they wasting their time? Or is it the entire reason there isn't a replication problem?)