Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeApr 29th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexAs announced in another thread, I created [[Hilbert system]]. However, I am a bit confused about exactly how a Hilbert system formalizes mathematical practice. In particular, how does it formalize hypothetical reasoning? When I want to prove a theorem like "If $A$ and $B$ then $C$", I start out by assuming $A$ and $B$ and trying to prove $C$. I know how to formalize this in natural deduction: I start a derivation with $A$ and $B$ at the top, and when I've gotten to $C$ then I apply implies-intro, cross out the $A$ and $B$, and conclude $A\to B\to C$. And in a type theory or sequent calculus, I am trying to prove a hypothetical sequent $A,B\vdash C$, after which I apply implies-intro again to get $A\to B\to C$. But in a system where the only rules are about deducing "global" theorems, how do I formalize the hypothetical-reasoning method of proving an implication?

   As announced in another thread, I created Hilbert system.

   However, I am a bit confused about exactly how a Hilbert system formalizes mathematical practice. In particular, how does it formalize hypothetical reasoning? When I want to prove a theorem like “If $A$ and $B$ then $C$”, I start out by assuming $A$ and $B$ and trying to prove $C$. I know how to formalize this in natural deduction: I start a derivation with $A$ and $B$ at the top, and when I’ve gotten to $C$ then I apply implies-intro, cross out the $A$ and $B$, and conclude $A\to B\to C$. And in a type theory or sequent calculus, I am trying to prove a hypothetical sequent $A,B\vdash C$, after which I apply implies-intro again to get $A\to B\to C$. But in a system where the only rules are about deducing “global” theorems, how do I formalize the hypothetical-reasoning method of proving an implication?

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 30th 2016
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexDoes [this](http://www3.cs.stonybrook.edu/~cse371/chapter8.pdf) help? Example 1 shows that even $A \to A$ requires some work. Once you have the deduction theorem, section 2, things get easier.

   Does this help? Example 1 shows that even $A \to A$ requires some work. Once you have the deduction theorem, section 2, things get easier.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeApr 30th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexHmm, so the answer is that Hilbert-style proofs do not directly formalize informal proofs, but proving something Hilbert-style requires trickery that has nothing to do with how we actually write proofs?

   Hmm, so the answer is that Hilbert-style proofs do not directly formalize informal proofs, but proving something Hilbert-style requires trickery that has nothing to do with how we actually write proofs?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeApr 30th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIn particular, the deduction theorem fails for the substructural paraconsistent/relevance logics that I'm currently trying to understand: e.g. with a Hilbert meaning of $\vdash$ we have $A,B\vdash A\&B$, but we don't have $A\to (B\to A\&B)$ since $\to$ is the linear/relevant implication (a.k.a. $\multimap$). I'm trying to understand how paraconsistent/relevance logicians can "think" in these logics, given that they generally present them only as a Hilbert system.

   In particular, the deduction theorem fails for the substructural paraconsistent/relevance logics that I’m currently trying to understand: e.g. with a Hilbert meaning of $\vdash$ we have $A,B\vdash A\&B$, but we don’t have $A\to (B\to A\&B)$ since $\to$ is the linear/relevant implication (a.k.a. $\multimap$). I’m trying to understand how paraconsistent/relevance logicians can “think” in these logics, given that they generally present them only as a Hilbert system.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 30th 2016
   • (edited Apr 30th 2016)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexMy recollection of the point of Hilbert systems was that they're not for proving formally what is proved informally, but rather to prove things about a deductive system. With so few (often 1) rules of inference, it's easier to prove that something cannot be derived. I though natural deduction was set up to contrast with this approach to resemble how we reason naturally. [Wikipedia](https://en.wikipedia.org/wiki/Natural_deduction) says: >Natural deduction grew out of a context of dissatisfaction with the axiomatizations of deductive reasoning common to the systems of Hilbert, Frege, and Russell (see, e.g., Hilbert system) and Gentzen >First I wished to construct a formalism that comes as close as possible to actual reasoning. Thus arose a "calculus of natural deduction".

   My recollection of the point of Hilbert systems was that they’re not for proving formally what is proved informally, but rather to prove things about a deductive system. With so few (often 1) rules of inference, it’s easier to prove that something cannot be derived.

   I though natural deduction was set up to contrast with this approach to resemble how we reason naturally. Wikipedia says:

   Natural deduction grew out of a context of dissatisfaction with the axiomatizations of deductive reasoning common to the systems of Hilbert, Frege, and Russell (see, e.g., Hilbert system)

   and Gentzen

   First I wished to construct a formalism that comes as close as possible to actual reasoning. Thus arose a “calculus of natural deduction”.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeApr 30th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIf that's the case, then it seems backwards if Hilbert systems came first historically. What good would it be to prove that something cannot be derived in a formal system if you don't know yet that that formal system has anything to do with ordinary reasoning?

   If that’s the case, then it seems backwards if Hilbert systems came first historically. What good would it be to prove that something cannot be derived in a formal system if you don’t know yet that that formal system has anything to do with ordinary reasoning?

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 30th 2016
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexOne only needs establish the strength of a system relative to other systems that are known to able to represent ordinary reasoning. There's an intricate history to tell here, which I ought to know more about. Hilbert systems first appear in book form, I think, in > Hilbert, David and Ackermann, Wilhelm, 1928, Grundzüge der theoretischen Logik, Berlin: Springer. But perhaps had already appeared in >Hilbert, David, 1918, “Prinzipien der Mathematik”, Lecture notes by Paul Bernays. Winter-Semester 1917/18. Typescript. Bibliothek, Mathematisches Institut, Universität Göttingen. Anyway, it's certainly post-Russell and Whitehead.

   One only needs establish the strength of a system relative to other systems that are known to able to represent ordinary reasoning.

   There’s an intricate history to tell here, which I ought to know more about. Hilbert systems first appear in book form, I think, in

   Hilbert, David and Ackermann, Wilhelm, 1928, Grundzüge der theoretischen Logik, Berlin: Springer.

   But perhaps had already appeared in

   Hilbert, David, 1918, “Prinzipien der Mathematik”, Lecture notes by Paul Bernays. Winter-Semester 1917/18. Typescript. Bibliothek, Mathematisches Institut, Universität Göttingen.

   Anyway, it’s certainly post-Russell and Whitehead.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeMay 1st 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexOkay, so although natural deduction came after Hilbert systems, there might have been other systems prior to Hilbert's that also represented ordinary reasoning more directly? That still doesn't answer my real question of how people can describe a new logic with just a Hilbert system and then proceed to reason informally in it without first giving an equivalent natural deduction or sequent calculus, but it's useful to know.

   Okay, so although natural deduction came after Hilbert systems, there might have been other systems prior to Hilbert’s that also represented ordinary reasoning more directly?

   That still doesn’t answer my real question of how people can describe a new logic with just a Hilbert system and then proceed to reason informally in it without first giving an equivalent natural deduction or sequent calculus, but it’s useful to know.