University of Calgary

# All LogBlog Posts

## Philosophy of Math at NYU

Submitted by Richard Zach on Wed, 06/25/2008 - 7:06am

NYU Philosophy is hosting a conference on the philosophy of mathematics, October April 10-12, 2008 2009. The speakers are John Burgess, Haim Gaifman, Joel Hamkins, Kai Hauser, Peter Koellner, Stewart Shapiro, Stephen Simpson, Bill Tait, Neil Tennant, and Hugh Woodin.

## Review of Symbolic Logic 1/1

Submitted by Richard Zach on Wed, 06/25/2008 - 7:02am

## Gödel Centenary Fellowship Pics and Videos

Submitted by Richard Zach on Tue, 06/24/2008 - 6:45pm

Pictures and videos from the Gödel Centenary Fellowship are online here.

## Gödel Centenary Fellowship Pictures and Videos

Submitted by Richard Zach on Tue, 06/24/2008 - 6:45pm

Pictures and videos from the Gödel Centenary Fellowship celebration are online here.

## Towards a New Epistemology of Mathematics

Submitted by Richard Zach on Fri, 06/13/2008 - 12:38am

There's a very interesting issue of Erkenntnis just out. It's the proceedings of PhiPMSAP 1. PhiMSAP is the Network on Philosophy of Mathematics: Sociological Aspects and Mathematical Practice of the DFG, run mostly by Benedikt Löwe and Thomas Müller, the third workshop of which I just had the pleasure of attending. Contents of the issue:

Experimental Mathematics by Alan Baker
Visualizations in Mathematics Kajsa Bråting and Johanna Pejlare
A Mathematician Reflects on the Useful and Reliable Illusion of Reality in Mathematics by Keith Devlin
The Role of Axioms in Mathematics by Kenny Easwaran
What can the Philosophy of Mathematics Learn from the History of Mathematics? by Brendan Larvor
On Abstraction and the Importance of Asking the Right Research Questions: Could Jordan have Proved the Jordan-Hölder Theorem? by Dirk Schlimm
Pi on Earth, or Mathematics in the Real World by Bart Van Kerkhove and Jean Paul Van Bendegem

## First Issue of Review of Symbolic Logic Out Soon!

Submitted by Richard Zach on Wed, 06/11/2008 - 11:19am

The first issue of the Review of Symbolic Logic will be out soon (Cambridge UP page here). The Review, like the Journal and Bulletin of Symbolic Logic will be mailed to all members of the Association for Symbolic Logic. So if you're not a member (and you probably should be, if you're reading this!), join now! (Only 76$per year--38$ for students-- for three of the most important journals in logic, plus all the other benefits of membership)

A Cut-Free Simple Sequent Calculus for Modal Logic S5
FRANCESCA POGGIOLESI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
How Applied Mathematics Became Pure
PENELOPE MADDY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
The Closing of the Mind: How the Particular Quantifier Became Existentially Loaded Behind Our Backs
GRAHAM PRIEST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Recognizing Strong Random Reals
DANIEL OSHERSON AND SCOTT WEINSTEIN. . . . . . . . . . . . . . . . 56
Truth-Functionality
BENJAMIN SCHNIEDER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Probabilistic Conditionals are Almost Monotonic
MATTHEW P. JOHNSON AND ROHIT PARIKH . . . . . . . . . . . . . . . . 73
On Adopting Kripke Semantics in Set Theory
LUCA INCURVATI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
The Iterative Conception of Set
THOMAS FORSTER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
A Decision Procedure for Probability Calculus with Applications
BRANDEN FITELSON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Ultimate Truth vis-à-vis Stable Truth
P.D. WELCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

## Philosophical Logic and Mathematical Logic in the PGR

Submitted by Richard Zach on Mon, 06/09/2008 - 9:09am

Last week, Brian Leiter posted about possibly re-drawing the dividing lines between the specialty areas ranked in the Philosophical Gourmet Report

Philosophical Logic and Mathematical Logic. While there is a fair amount of divergence between the two rankings, I also see, for example, that NYU gets ranked at 4.5 in mathematical logic even though as far as I can see, almost everything the relevant people have written about logic is more naturally classified as philosophical logic than as mathematical logic. Is that a sign that the borderline between these two areas is too wide and fuzzy for the distinction to be worth making by the PGR?

If one thinks of mathematical logic as "formal logic motivated by mathematical concerns"---roughly, this is the conception according to which mathematical logic consists of model theory, set theory, recursion theory, and proof theory--then it is indeed puzzling why NYU gets ranked at the top of group 2 in the mathematical logic ranking, in addition to the top in the philosophical logic ranking. But one might also think of mathematical logic the way it's defined in the AMS Mathematics Subject Classification (03). Now how do we take "philosophical logic"? But if we take the now-standard (at least in North America) definition of "philosophical logic", then it's that part of formal logic that paradigmatically includes: "various versions of modal, temporal, epistemic, and deontic logic; constructive logics; relevance and other sub-classical logics; many-valued logics; logics of conditionals; quantum logic; decision theory, inductive logic, logics of belief change, and formal epistemology; defeasible and nonmonotonic logics; formal philosophy of language; vagueness; and theories of truth and validity" (from editorial description of the Journal of Philosophical Logic). Almost all of that is included in MSC 03Bxx! Clearly, for the purpose of the PGR at least, it would be better to define "mathematical logic" as "formal logic, but not philosophical logic".

I don't know if the evaluators for the PGR "mathematical logic" and "philosophical logic" categories get instructions on the intended scope of the category. It probably also makes a big difference--bigger than in other specialty areas--if faculty with appointments in the mathematics (or computer science) departments get included in the PGR faculty lists. As the note at the bottom of the ranking says, "much work in mathematical logic goes on in Mathematics and Computer Science departments."

Then there's also the confusion between the definition of "philosophical logic" as "formal logic motivated by philosophy" and the older (British) use of "philosophical logic" to mean "philosophy motivated by logic" (and including philosophical study of notions such as reference, necessity, truth, analyticity, etc.) Maybe a better term for that is "philosophy of logic". Leiter's proposed restructuring would have a category "philosophy of language & logic" (but no philosophical logic category).

I don't think that "philosophical logic" and "mathematical logic" should be combined in the PGR ranking. But as it currently stands--with the scope of these categories so unclear--the rankings aren't particularly informative. I'm not sure what would be more informative, but getting clarity on the definitions would be one step. Maybe it wouldn't be such a bad idea to include "philosophy of logic" in the "philosophy of language" category, and reserve "philosophical logic" for the formal work you find in the JPL or the Review of Symbolic Logic. Maybe it wouldn't even be a bad idea to merge mathematical logic into the philosophy of mathematics category. What do others think?

## Kohlenbach's Applied Proof Theory is Out!

Submitted by Richard Zach on Mon, 06/02/2008 - 5:41pm

What more do we know about a theorem if we have a proof (by restricted means) than merely that it is true? That's an old question of Kreisel's that motivated his "unwinding program": extract additional information from proofs of theorems in constructive theories, such as bounds on y in theorems of the form ?x?y A(x, y).
Ulrich Kohlenbach has worked on problems like this more than anyone else, and his book Applied Proof Theory: Proof Interpretations and Their Use in Mathematics is now out from Springer.

Ulrich Kohlenbach presents an applied form of proof theory that has led in recent years to new results in number theory, approximation theory, nonlinear analysis, geodesic geometry and ergodic theory (among others). This applied approach is based on logical transformations (so-called proof interpretations) and concerns the extraction of effective data (such as bounds) from prima facie ineffective proofs as well as new qualitative results such as independence of solutions from certain parameters, generalizations of proofs by elimination of premises.

The book first develops the necessary logical machinery emphasizing novel forms of Goedel's famous functional (Dialectica) interpretation. It then establishes general logical metatheorems that connect these techniques with concrete mathematics. Finally, two extended case studies (one in approximation theory and one in fixed point theory) show in detail how this machinery can be applied to concrete proofs in different areas of mathematics.

1 Introduction

2 Unwinding proofs (‘Proof Mining’)
2.1 Introductory remark
2.2 Informal treatment of ineffective proofs
2.3 Herbrand’s theorem and the no-counterexample interpretation

3 Intuitionistic and classical arithmetic in all finite types
3.1 Intuitionistic and classical predicate logic
3.2 Intuitionistic (‘Heyting’) arithmetic HA and Peano arithmetic PA
3.3 Extensional intuitionistic (‘Heyting’) and classical (‘Peano’)
arithmetic in all finite types
3.4 Fragments of (W)E-HA^? and (W)E-PA^?
3.5 Fragments corresponding to the Grzegorczyk hierarchy
3.6 Models of E-PA^?

4 RepresentationofPolish metric spaces
4.1 Representation of real numbers
4.2 Representation of complete separable metric (‘Polish’) spaces
4.3 Special representation of compact metric spaces
4.4 Fragments, exercises, historical comments and suggested further

5 Modified realizability
5.1 The soundness and program extraction theorems
5.2 Remarks on fragments of E-HA^?

6 Majorizability and the fan rule
6.1 A syntactic treatment of majorization and the fan rule

7 Semi-intuitionistic systems and monotone modified realizability
7.1 The soundness and bound extraction theorems
7.2 Fragments, exercises, historical comments and suggested further

8 Gödel’s functional (‘Dialectica’) interpretation
8.1 Introduction
8.2 The soundness and program extraction theorems
8.3 Fragments, exercises, historical comments and suggested further

9 Semi-intuitionistic systems and monotone functional interpretation
9.1 The soundness and bound extraction theorems
9.2 Applications of monotone functional interpretation
9.3 Examples of axioms ? : Weak König’s lemmaWKL
9.4 WKL as a universal sentence ?
9.5 Fragments, exercises, historical comments and suggested further

10 Systems based on classical logic and functional interpretation
10.1 The negative translation
10.2 Combination of negative translation and functional interpretation
10.3 Application: Uniform weak König’s lemma UWKL
10.4 Elimination of extensionality
10.5 Fragments of (W)E-PA^?
10.6 The computational strength of full extensionality

11 Functional interpretation of full classical analysis
11.1 Functional interpretation of full comprehension
11.2 Functional interpretation of dependent choice
11.3 Functional interpretation of arithmetical comprehension
11.4 Functional interpretation of (IPP) by finite bar recursion
11.5 Models of bar recursion

12 A non-standard principle of uniform boundedness
12.1 The ?^0_1 -boundedness principle
12.2 Applications of ?^0_1 -boundedness
12.3 Remarks on the fragments E-G_nA^?

13 Elimination of monotone Skolem functions
13.1 Skolem functions of type degree 1 in fragments of finite type
arithmetic
13.2 Elimination of Skolem functions for monotone formulas
13.3 The principle of convergence for bounded monotone sequences
of real numbers (PCM)
13.4 ?^0_1 -CA and ?^0_1 -AC
13.5 The Bolzano-Weierstraß property for bounded sequences in R^d

14 The Friedman A-translation
14.1 The A-translation

15 Applications to analysis: general metatheorems I
15.1 A general metatheorem for Polish spaces
15.2 Applications to uniqueness proofs
15.3 Applications to monotone convergence theorems
15.4 Applications to proofs of contractivity
15.5 Remarks on fragments of T^?

16 Case study I: Uniqueness proofs in approximation theory
16.1 Uniqueness proofs in best approximation theory
16.2 Best Chebycheff approximation I
16.3 Best Chebycheff approximation II
16.4 Best L_1-approximation

17 Applications to analysis: general metatheorems II
17.1 Introduction
17.2 Main results in the metric and hyperbolic case
17.3 The case of normed spaces
17.4 Proofs of theorems 17.35, 17.52 and 17.69
17.5 Further variations
17.6 Treatment of several metric or normed spaces X_1 . . . , X_n
simultaneously
17.7 A generalized uniform boundedness principle ?-UB^X
17.8 Applications of ?-UB^X
17.9 Fragments of A^? [. . .]

18 Case study II: Applications to the fixed point theory of nonexpansive
mappings
18.1 General facts
18.2 Applications of the metatheorems from chapter 17
18.3 Logical analysis of the proof of the Borwein-Reich-Shafrir
theorem
18.4 Asymptotically nonexpansive mappings
18.5 Applications of proof mining in ergodic theory

## What Sorts of People Should There Be?

Submitted by Richard Zach on Mon, 05/19/2008 - 10:07am

From my colleague up north, Rob Wilson:

The What Sorts of People blog is now up and running: check it out. This is the blog for the What Sorts of People Should There Be? network, a collaborative blog with regular contributions from around 10 team members. Short, recent posts are available on double-amputee Oscar Pistorius's bid to compete Olympically, and on a so-recent-it's-still-forthcoming piece by Steve Pinker in The New Republic on the concept of dignity and its use in a recent President's Council on Bioethics report. Biella Coleman, who was a Killam Postdoc at Alberta last year and now teaches at NYU, has just posted a tempered rant on the blog on medical genetics and eugenics.

## Open Access Logic Textbooks

Submitted by Richard Zach on Sat, 05/17/2008 - 11:44am

Submitted by Richard Zach on Wed, 05/14/2008 - 10:07am

## Finite Axiomatizability of Theories in the Predicate Calculus Using Additional Predicate Symbols (Classic Logic Papers, Pt. 4)

Submitted by Richard Zach on Wed, 05/14/2008 - 8:09am

You probably all know the result that Peano Arithmetic is not finitely axiomatizable (a result due to Ryll-Nardzewski), and a similar result for ZFC (due to Richard Montague, I believe). The standard axiom system for PA is not finite since the axiom scheme of induction stands for infinitely many sentences. Ryll-Nardzewski showed that there is no finite set S of sentences in the language of PA so that S ? ? iff PA ? ?. I don't know how widely known this is,* but by contrast it is possible to finitely axiomatize (almost) any theory T, if you allow new predicate symbols in the axiomatization. This is a result due to Kleene:

Stephen Cole Kleene, Finite axiomatizability of theories in the predicate calculus using additional predicate symbols, in: Two Papers on the Predicate Calculus. Memoirs of the American Mathematical Society, no. 10, Providence, 1952, 27-68.

The "almost" above, as you might have guessed, applies to the theory itself: it has to be recursively enumerable and, if the language includes identity, must have only infinite models (Kleene's original paper does not deal with identity). The basic idea is that you use the new predicate symbols to finitely axiomatize a weak arithmetic in which you can formalize semantics and do primitive recursion. The first part gives you a truth predicate for the language of T, the second part gives you a predicate that's satisfied of exactly the Gödel numbers of sentences of T. Then you say that all the axioms of the (r.e.) theory T are true. Kleene's result was later strengthened by Craig and Vaught in:

William Craig and Robert L. Vaught, Finite axiomatizability using additional predicates. The Journal of Symbolic Logic 23 (1958) 289-308.

A sketch of Craig and Vaught's proof is given in Michael Makkai's review.

*According to Google Scholar, Kleene's monograph is cited a lot, but as far as I can tell overwhelmingly for the other paper, which is another classic logic paper: "Permutation of inferences in Gentzen’s calculi LK and LJ."

## Gödel Colloquium Tomorrow (Live Stream)

Submitted by Richard Zach on Sat, 04/26/2008 - 2:17pm

So the colloquium honoring the recipients of the Gödel Research Fellowships is tomorrow. There should be a live feed. It starts at 9:00 CDT (that's midnight tonight on the West Coast, and 3 am Eastern). I'll try to find out if the lectures will be archived.

## Reck on Dedekind

Submitted by Richard Zach on Tue, 04/22/2008 - 7:55am

Erich Reck's entry on Richard Dedekind in the SEP is now online. I'm particularly happy about this one: It's time Dedekind gets some of the attention for his philosophy of math that Frege's been getting for his, and Erich's entry as well as his other work, I hope, will help bring that about.

While many of Dedekind's contributions to mathematics and its foundations are thus common knowledge, they are seldom discussed together. In particular, his mathematical writings are often treated separately from his foundational ones. This entry provides a broad survey of his contributions. The main focus will be on his foundational writings, but they will be related to his other mathematical work. Indeed, it will be argued that foundational concerns are at play throughout Dedekind's work, so that any attempt to distinguish sharply between his “mathematical” and “foundational” writings is artificial and misleading. Another goal of the entry is to establish the continuing relevance of his contributions to the philosophy of mathematics. Indeed, their full significance has only started to be recognized, as should become evident along the way. This is especially so with respect to methodological and epistemological aspects of Dedekind's work, which shape and ground the logical and metaphysical views that emerge in his writings.

I also recommend: "Dedekind's Structuralism: An Interpretation and Partial Defense", Synthese 137:3, 2003, pp. 369-419 (paper)

## More SEP Entries

Submitted by Richard Zach on Thu, 04/17/2008 - 12:18am

Two more new entries in the Stanford Encyclopedia of interest to logicians:

From Jan's entry I see that his translation of the interesting third chapter of Genzen's thesis--wherein Gentzen proved normalization of natural deduction derivations--is out in the next issue of the Bulletin.

## Double-Blind Review at Journals

Submitted by Richard Zach on Mon, 04/14/2008 - 3:09am

## Linear Logic is Pseudoscience?

Submitted by Richard Zach on Sun, 04/13/2008 - 7:45pm

There's a very laudable enterprise: Blogging on Pseudoscience (at BPSDB.org) aggregates blog posts debunking or pointing out pseudo-scientific nonsense such as Intelligent Design. Lots of good stuff, PZ Meyers is part of it, etc. But, look at the logo they use:

Yes, that's a sequent calculus for linear logic. I know Girard has an idiosyncratic style, and that ludics paper may look a bit hokey. Probably only Girard can get a logic paper published in a respectable journal which contains quotes like this:

Jurassic logic keeps celebrating the wedding of Semantics and Syntax through the intercession of the Holy Meta, and, of course, every ritual deserves its own mockery, the Cross upside down, the Gospel in reverse order… In paralogics, there is no syntax, no semantics (or, better, the semantics is called syntax and vice versa). These mockeries, religious or logical, betray a paradoxical respect for a tradition that the protagonists never understood.

as well as pictures of skunks:
I mean, those two appendices are weird. But the proof theory is ok, isn't it? And there's nothing wrong with linear logic, is there?

## Gupta on Definitions

Submitted by Richard Zach on Fri, 04/11/2008 - 2:10am

New SEP entry on Definitions by Anil Gupta.

## Mancosu on Explanation in Mathematics

Submitted by Richard Zach on Sun, 04/06/2008 - 4:36pm

Paolo's Stanford Encyclopedia entry on explanation in mathematics is online.

## Kurt Gödel Centenary Research Fellowship Winners

Submitted by Richard Zach on Thu, 04/03/2008 - 4:56pm

Ok, that was faster than I expected: The winners of the Kurt Gödel Centenary Research Fellowships have been announced. They are:

## Pre-doctoral category

• David Fernández Duque (Stanford): Non-deterministic semantics for dynamic topological logic
• Pavel Hrubeš (Czech Academy of Sciences): On lengths of proofs in non-classical logics

## Senior category

All great logicians, with great projects. Congratulations!

All applicants submitted papers on their research projects. Papers by the winners (and the finalists) will shortly appear in a special issue of the Annals. There will be a colloquium honoring the winners on April 27 (the eve of Kurtele's 101st birthday)