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I disagree with this on two points. First, oftentimes the interest in proving long-standing, difficult mathematical problems is because we hope a proof will de

by Twisol 2y ago

I disagree with this on two points. First, oftentimes the interest in proving long-standing, difficult mathematical problems is because we hope a proof will demonstrate new tools for tackling similar problems. In that sense, the exact content of a proof is quite important. Not to mention, there is value in having multiple proofs that each demonstrate quite different toolkits. Mere truth is not often the most important thing -- after all, mathematicians can (and do!) take certain propositions as premises of downstream work. If we discover a proof for one of those premises, that just means we can drop the premise from downstream work. Not having a proof doesn't block us from making use of the proposition anyway. Second, sometimes the content of the proof is relevant formally. A sibling comment gave an example in terms of paths between two points; it is often the case that you care not only that the points are merely connected, but you also have a preference for which path is taken. Or, you can do an analysis of the paths themselves, and determine their length or average furthest distance from the nearest McDonalds. A path is "just" a proof of connectivity, but the individual paths can be quite distinct when studied themselves. Less abstractly, a constructive proof will yield an algorithm that can be performed, and we know quite well that the variety of sorting algorithms (that "merely" prove that "all lists are sortable") actually vary in quite important ways, including asymptotics and stability.

lupire 2y ago
Asymptotics and stability are different theorems. An algorithm is not a proof. It is a technique for proof. Two algorithms can be different, while not being meaningfully different profs that a list is sortable. To the extent that they are different, they proof different theorems, such as "list can be sorted in O(f) time" for an f of interest.
- Twisol 2y ago
  > An algorithm is not a proof. That is an opinion that many do not share. FWIW, I framed my response as an opinion; you gave yours as a blanket statement. It is not wrong to treat algorithms as valid proofs. In a dependent type theory, propositions are represented as types; the proposition that "all lists can be sorted" could be represented represented as the type "forall (t : Type) -> (le : Ordered t) -> forall (xs : List t) -> exists (ys : List t). (Increasing le ys, PermutationOf xs ys)". A proof of this proposition is exactly a program (algorithm) with that type; the sorted list is the `ys` component of the returned existential product. Yet the inhabitants of this type are not graded by asymptotics or stability; any sorting algorithm will do. In a setting where inhabitants of the above type are distinguishable, you could then write proofs of asymptotics or stability against individual algorithms. That is, the proofs of the sorting proposition are themselves the subjects of subsequent propositions and proofs thereof.
  - js8 2y ago
    > It is not wrong to treat algorithms as valid proofs. I think that's what PP meant, i.e. if you want to differentiate between sorting algorithms in terms of efficiency, you somehow should encode this demand into the types (specification).
    - Twisol 2y ago
      You can do that, yes. My argument is that you don't have to do that -- you can prove asymptotics or stability after the fact, having previously only given an algorithm as a proof of sortability. Putting these properties in the specification assumes you already know (or suspect) that your algorithm has these properties; then you are simply verifying what you knew. If you develop your algorithm first, then want to analyze it in terms of (say) asymptotics, then not only is it far too late to change the type, you don't even know what the asymptotics are yet. You'd still like to treat the algorithm formally in order to determine those asymptotics, but since you don't know them yet, the algorithm can't inhabit a type that states those asymptotics outright.
  - yMEyUyNE1 2y ago
    >> An algorithm is not a proof. > That is an opinion that many do not share. Say, we have an algorithm to color a planar graph with 4 colors, can we say that we have a proof for 4-coloring a planar graph?
pkoird 2y ago
I don't think you have disagreed with me. You have advocated that different tools/methods are useful for different problems and may have unique properties that make them interesting in specific contexts. I completely agree and I have not stated anything against it. My opinion, admittedly abstract and stated without proof, was simply that if you have two ways of showing something to be true, they must be logically equivalent (in some sense of the word) if you are willing to dig deep enough. This does not necessarily imply that certain abstractions are not useful on their own, merely that at a certain level, they should represent the same thing. I fully understand that this is not a concrete argument and I have not stated my opinion with desirable rigor (but the author of the original article does provide a few examples in support). Maybe someone with a better grasp on abstract mathematical concept could convey my arguments better (if they think it's true).
- Twisol 2y ago
  That's a fair response; thanks for taking the time. I was primarily reacting to this part of your message... > I'd argue that two correct proofs are always the same. ...with emphasis on the "always". To my eyes, a proof is any object that witnesses the truth of a proposition. The proof can be more than a witness, but to be called a proof, it must do at least that much. To say that "two correct proofs are always the same" is, to me, to say that proofs can be no more than witnesses of the proposition; to be the same means to be indistinguishable. My argument is that two correct proofs may be distinct in useful ways. I suppose this discussion depends on what "same" means ("depends on what the meaning of the word 'is' is", heh). Do you mean something other than what I mean? Your use of "logically equivalent" is probably telling -- so, two proofs should have the same deductive power? We often speak of propositions this way, but I'm not sure how to understand that notion on proofs. Terence Tao gives some possible notions of proof equivalence in a comment on the OP [0]; you might enjoy reading them and considering which is closest to your intuitive idea of equivalence :) [0]: https://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-198 https://gowers.wordpress.com/2007/10/04/when-are-two-proofs-...
  - pkoird 2y ago
    I can attempt to semi-formalize it but I'm sure I'd butcher it along the way so feel free to point out anything that doesn't feel correct. Consider a set of premises P that are assumed to be true. Also, consider that we are trying to analyze a statement s0 assuming P. One proof could be of the form: P1: s0 -> s1 -> s2 -> ... -> T/F. Another proof could be of the form: P2: s0 -> s11 -> s12 -> ... -> T/F. Where T/F represent a single terminal symbol i.e. either T (true) or F (false) and s1... and s11... etc. could be different abstractions that have been employed to illustrate the veracity of the statement. Regardless, both of these abstractions make use of the same Logical rules at each step so you could argue that the logical chain of both P1 and P2 are equivalent in some sense. If you think about it, it does seem obvious though, because if P1 and P2 disagreed with P1 yielding T and P2 yielding F, while using the same set of logical rules, it must be the case that either the logic is not consistent or one or both of the chain has errors. So now, one could argue that all such correct logical chains (maybe of different lengths) that start with a statement s0 and terminate at a single symbol (say T) should essentially be the same. s1 -> s2 -> s3 -> ... -> sn ↑................................↓ s0 -> s11 -> s12 -> ...->T You could also argue that there must be exactly one such chain of the smallest possible complexity (in some sense) and that all other chains should be reducible to this one chain (not sure how). At the end, I still agree with you in that two correct proofs can be distinct in useful ways but since proofs, to me, are a series of application of logic under certain premise to obtain a terminal symbol, all such logically sound chains must actually correspond to the one fundamental chain that's irreducible (in some sense).

by Twisol 2y ago

lupire 2y ago

Asymptotics and stability are different theorems. An algorithm is not a proof. It is a technique for proof. Two algorithms can be different, while not being meaningfully different profs that a list is sortable. To the extent that they are different, they proof different theorems, such as "list can be sorted in O(f) time" for an f of interest.

Twisol 2y ago
> An algorithm is not a proof. That is an opinion that many do not share. FWIW, I framed my response as an opinion; you gave yours as a blanket statement. It is not wrong to treat algorithms as valid proofs. In a dependent type theory, propositions are represented as types; the proposition that "all lists can be sorted" could be represented represented as the type "forall (t : Type) -> (le : Ordered t) -> forall (xs : List t) -> exists (ys : List t). (Increasing le ys, PermutationOf xs ys)". A proof of this proposition is exactly a program (algorithm) with that type; the sorted list is the `ys` component of the returned existential product. Yet the inhabitants of this type are not graded by asymptotics or stability; any sorting algorithm will do. In a setting where inhabitants of the above type are distinguishable, you could then write proofs of asymptotics or stability against individual algorithms. That is, the proofs of the sorting proposition are themselves the subjects of subsequent propositions and proofs thereof.
- js8 2y ago
  > It is not wrong to treat algorithms as valid proofs. I think that's what PP meant, i.e. if you want to differentiate between sorting algorithms in terms of efficiency, you somehow should encode this demand into the types (specification).
  - Twisol 2y ago
    You can do that, yes. My argument is that you don't have to do that -- you can prove asymptotics or stability after the fact, having previously only given an algorithm as a proof of sortability. Putting these properties in the specification assumes you already know (or suspect) that your algorithm has these properties; then you are simply verifying what you knew. If you develop your algorithm first, then want to analyze it in terms of (say) asymptotics, then not only is it far too late to change the type, you don't even know what the asymptotics are yet. You'd still like to treat the algorithm formally in order to determine those asymptotics, but since you don't know them yet, the algorithm can't inhabit a type that states those asymptotics outright.
- yMEyUyNE1 2y ago
  >> An algorithm is not a proof. > That is an opinion that many do not share. Say, we have an algorithm to color a planar graph with 4 colors, can we say that we have a proof for 4-coloring a planar graph?

pkoird 2y ago

I don't think you have disagreed with me. You have advocated that different tools/methods are useful for different problems and may have unique properties that make them interesting in specific contexts. I completely agree and I have not stated anything against it. My opinion, admittedly abstract and stated without proof, was simply that if you have two ways of showing something to be true, they must be logically equivalent (in some sense of the word) if you are willing to dig deep enough. This does not necessarily imply that certain abstractions are not useful on their own, merely that at a certain level, they should represent the same thing. I fully understand that this is not a concrete argument and I have not stated my opinion with desirable rigor (but the author of the original article does provide a few examples in support). Maybe someone with a better grasp on abstract mathematical concept could convey my arguments better (if they think it's true).

Twisol 2y ago
That's a fair response; thanks for taking the time. I was primarily reacting to this part of your message... > I'd argue that two correct proofs are always the same. ...with emphasis on the "always". To my eyes, a proof is any object that witnesses the truth of a proposition. The proof can be more than a witness, but to be called a proof, it must do at least that much. To say that "two correct proofs are always the same" is, to me, to say that proofs can be no more than witnesses of the proposition; to be the same means to be indistinguishable. My argument is that two correct proofs may be distinct in useful ways. I suppose this discussion depends on what "same" means ("depends on what the meaning of the word 'is' is", heh). Do you mean something other than what I mean? Your use of "logically equivalent" is probably telling -- so, two proofs should have the same deductive power? We often speak of propositions this way, but I'm not sure how to understand that notion on proofs. Terence Tao gives some possible notions of proof equivalence in a comment on the OP [0]; you might enjoy reading them and considering which is closest to your intuitive idea of equivalence :) [0]: https://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-198 https://gowers.wordpress.com/2007/10/04/when-are-two-proofs-...
- pkoird 2y ago
  I can attempt to semi-formalize it but I'm sure I'd butcher it along the way so feel free to point out anything that doesn't feel correct. Consider a set of premises P that are assumed to be true. Also, consider that we are trying to analyze a statement s0 assuming P. One proof could be of the form: P1: s0 -> s1 -> s2 -> ... -> T/F. Another proof could be of the form: P2: s0 -> s11 -> s12 -> ... -> T/F. Where T/F represent a single terminal symbol i.e. either T (true) or F (false) and s1... and s11... etc. could be different abstractions that have been employed to illustrate the veracity of the statement. Regardless, both of these abstractions make use of the same Logical rules at each step so you could argue that the logical chain of both P1 and P2 are equivalent in some sense. If you think about it, it does seem obvious though, because if P1 and P2 disagreed with P1 yielding T and P2 yielding F, while using the same set of logical rules, it must be the case that either the logic is not consistent or one or both of the chain has errors. So now, one could argue that all such correct logical chains (maybe of different lengths) that start with a statement s0 and terminate at a single symbol (say T) should essentially be the same. s1 -> s2 -> s3 -> ... -> sn ↑................................↓ s0 -> s11 -> s12 -> ...->T You could also argue that there must be exactly one such chain of the smallest possible complexity (in some sense) and that all other chains should be reducible to this one chain (not sure how). At the end, I still agree with you in that two correct proofs can be distinct in useful ways but since proofs, to me, are a series of application of logic under certain premise to obtain a terminal symbol, all such logically sound chains must actually correspond to the one fundamental chain that's irreducible (in some sense).