Mathematics and Computation

A blog about mathematics for computers

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Posts in the category Logic

What is a formal proof?

Mike Shulman just wrote a very nice blog post on what is a formal proof. I much agree with what he says, but I would like to offer my own perspective. I started writing it as a comment to Mike’s post and then realized that it is too long, and that I would like to have it recorded independently as well. Please read Mike’s blog post first.

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I have not written a blog post in a while, so I decided to write up a short observation about truth values in intuitionistic logic which sometimes seems a bit puzzling.

Let $\Omega$ be the set of truth values (in Coq this would be the setoid whose underlying type is $\mathsf{Prop}$ and equality is equivalence $\leftrightarrow$, while in HoTT it is the h-propostions). Call a truth value $p : \Omega$ intermediate if it is neither true nor false, i.e., $p \neq \bot$ and $p \neq \top$. Such a “third” truth value $p$ is proscribed by excluded middle.

The puzzle is to explain how the following two facts fit together:

  1. “There is no intermediate truth value” is an intuitionistic theorem.
  2. There are models of intuitionistic logic with many truth values.
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Here are the slides from my Logic Coloquium 2014 talk in Vienna. This is joint work with Kazuto Yoshimura from Japan Advanced Institute for Science and Technology.

Abstract: In constructive mathematics we often consider implications between non-constructive reasoning principles. For instance, it is well known that the Limited principle of omniscience implies that equality of real numbers is decidable. Most such reductions proceed by reducing an instance of the consequent to an instance of the antecedent. We may therefore define a notion of instance reducibility, which turns out to have a very rich structure. Even better, under Kleene’s function realizability interpretation instance reducibility corresponds to Weihrauch reducibility, while Kleene’s number realizability relates it to truth-table reducibility. We may also ask about a constructive treatment of other reducibilities in computability theory. I shall discuss how one can tackle Turing reducibility constructively via Kleene’s number realizability.

Slides with talk notes:  lc2014-slides-notes.pdf

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A discussion on the homotopytypetheory mailing list prompted me to write this short note. Apparently a mistaken belief has gone viral among certain mathematicians that Univalent foundations is somehow limited to constructive mathematics. This is false. Let me be perfectly clear:

Univalent foundations subsume classical mathematics!

The next time you hear someone having doubts about this point, please refer them to this post. A more detailed explanation follows.

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Mathematicians are often confused about the meaning of variables. I hear them say “a free variable is implicitly universally quantified”, by which they mean that it is ok to equate a formula $\phi$ with a free variable $x$ with its universal closure $\forall x \,.\, \phi$. I am addressing this post to those who share this opinion.

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Substitution is pullback

I am sitting on a tutorial on categorical semantics of dependent type theory given by Peter Lumsdaine. He is talking about categories with attributes and other variants of categories that come up in the semantics of dependent type theory. He is amazingly good at fielding questions about definitional equality from the type theorists. And it looks like some people are puzzling over pullbacks showing up, which Peter is about to explain using syntactic categories. Here is a pedestrian explanation of a very important fact:

Substitution is pullback.
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With Peter LeFanu Lumsdaine.

Abstract: The Bourbaki-Witt principle states that any progressive map on a chain-complete poset has a fixed point above every point. It is provable classically, but not intuitionistically. We study this and related principles in an intuitionistic setting. Among other things, we show that Bourbaki-Witt fails exactly when the trichotomous ordinals form a set, but does not imply that fixed points can always be found by transfinite iteration. Meanwhile, on the side of models, we see that the principle fails in realisability toposes, and does not hold in the free topos, but does hold in all cocomplete toposes.

Download paper: bw.pdf
ArXiv version: arXiv:1201.0340v1 [math.CT]

This paper is an extension of my previous paper on the Bourbaki-Witt and Knaster-Tarski fixed-point theorems in the effective topos (arXiv:0911.0068v1).

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I am not sure whether to call this one a constructive gem or stone. I suppose it is a matter of personal taste. I think it is a gem, albeit a very unusual one: there is a topos in which $\mathbb{N}^\mathbb{N}$ can be embedded into $\mathbb{N}$.

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As a preparation for my part of a joint tutorial Programs from proofs at MFPS 27 at the end of this month with Ulrich Berger, Monika Seisenberger, and Paulo Oliva, I’ve developed in Agda some things we’ve been doing together.

Using

for giving a proof term for classical countable choice, we prove the classical infinite pigeonhole principle in Agda: every infinite boolean sequence has a constant infinite subsequence, where the existential quantification is classical (double negated).

As a corollary, we get the finite pigeonhole principle, using Friedman’s trick to make the existential quantifiers intuitionistic.

This we can run, and it runs fast enough. The point is to illustrate in Agda how we can get witnesses from classical proofs that use countable choice. The finite pigeonhole principle has a simple constructive proof, of course, and hence this is really for illustration only.

The main Agda files are

These are Agda files converted to html so that you can navigate them by clicking at words to go to their definitions. A zip file with all Agda files is available. Not much more information is available here.

The three little modules that implement the Berardi-Bezem-Coquand, Berger-Oliva and Escardo-Oliva functionals disable the termination checker, but no other module does. The type of these functionals in Agda is the J-shift principle, which generalizes the double-negation shift.

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I think I am getting addicted to Coq, or more generally to doing mathematics, including the proofs, with computers. I spent last week finalizing a formalization of Gödel’s functional interpretation of logic, also known as the Dialectica interpretation. There does not seem to be one available already, which is a good opportunity for a blog post.

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I am discovering that mathematicians cannot tell the difference between “proof by contradiction” and “proof of negation”. This is so for good reasons, but conflation of different kinds of proofs is bad mental hygiene which leads to bad teaching practice and confusion. For reference, here is a short explanation of the difference between proof of negation and proof by contradiction.

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Two versions of this talk were given at Computability and complexity in analysis 2008 and at Mathematics, Algorithms and Proofs 2008.

Joint work with Paul Taylor.

Abstract: Cauchy’s construction of reals as sequences of rational approximations is the theoretical basis for a number of implementations of exact real numbers, while Dedekind’s construction of reals as cuts has inspired fewer useful computational ideas. Nevertheless, we can see the computational content of Dedekind reals by constructing them within Abstract Stone Duality (ASD), a computationally meaningful calculus for topology. This provides the theoretical background for a novel way of computing with real numbers in the style of logic programming. Real numbers are defined in terms of (lower and upper) Dedekind cuts, while programs are expressed as statements about real numbers in the language of ASD. By adapting Newton’s method to interval arithmetic we can make the computations as efficient as those based on Cauchy reals.

Slides: slides-map2008.pdf (obsolete version: slides-cca2008.pdf)
Extended abstract: abstract-cca2098.pdf

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The hydra game

Today I lectured about the Hydra game by Laurence Kirby and Jeff Paris (Accessible Independence Results for Peano Arithmetic, Kirby and Paris, Bull. London Math. Soc. 1982; 14: 285-293). For the occasion I implemented the game in Java. I am publishing the code for anyone who wants to play, or use it for teaching.

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Recently there has been a discussion (here, here, here, and here) on the Foundations of Mathematics mailing list about completeness of Peano arithmetic (PA) with respect to “small” sentences. Harvey Friedman made several conjectures of the following kind: “All true small sentences of PA are provable.” He proposed measures of smallness, such as counting the number of distinct variables or restricting the depth of terms. Here are some statistics concerning such statements.

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Proof hacking

A neat example of propositions-as-types using recursion. → continue reading (2 comments)

How many is two?

In constructive mathematics even very small sets can be quite a bit more interesting than in classical mathematics. Since you will not believe me that sets with at most one element are very interesting, let us look at the set of truth values, which has “two” elements.

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