PhD, Postdoc with Rosalie Iemhoff

Postdoc position in Logic at Utrecht University, the Netherlands.

The postdoc is embedded in the research project “Optimal Proofs” funded by the Netherlands Organization for Scientific Research led by Dr. Rosalie Iemhoff, Department of Philosophy and Religious Studies, Utrecht University. The project in mathematical and philosophical logic is concerned with formalization in general and proof systems as a form of formalization in particular. Its mathematical aim is to develop methods to describe the possible proof systems of a given logic and establish, given various criteria of optimality, what the optimal proof systems of the logic are. Its philosophical aim is to develop general criteria for faithful formalization in logic and to thereby distinguish good formalizations from bad ones. The mathematical part of the project focusses on, but is not necessarily restricted to, the (non)classical logics that occur in computer science, mathematics, and philosophy, while the philosophical part of the project also takes into account domains where formalization in logic is less common. The postdoc is expected to contribute primarily to the mathematical part of the project. Whether the research of the postdoc also extends to the philosophical part of the project depends on his or her interests.

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Raymond Smullyan

Proof by legerdemain

Peli Grietzer shared a blog post by David Auerbach on Twitter yesterday containing the following lovely quote about Smullyan and Carnap:

I particularly delighted in playing tricks on the philosopher Rudolf Carnap; he was the perfect audience! (Most scientists and mathematicians are; they are so honest themselves ‘that they have great difficulty in seeing through the deceptions of others.) After one particular trick, Carnap said, “Nohhhh! I didn’t think that could happen in any possible world, let alone this one!”

 

In item # 249 of my book of logic puzzles titled What Is the Name of This Book?, I describe an infallible method of proving anything whatsoever. Only a magician is capable of employing the method, however. I once used it on Rudolf Carnap to prove the existence of God.

 

“Here you see a red card,” I said to Professor Carnap as I removed a card from the deck. “I place it face down in your palm. Now, you know that a false proposition implies any proposition. Therefore, if this card were black, then God would exist. Do you agree?”

 

“Oh, certainly,” replied Carnap, “if the card were black, then God would exist.”

 

“Very good,” I said as I turned over the card. “As you see, the card is black. Therefore, God exists!”

 

“Ah, yes!” replied Carnap in a philosophical tone. “Proof by legerdemain! Same as the theologians use!”

 

Raymond Smullyan, 5000 BC and Other Philosophical Fantasies. New York: St. Martin’s Press, 1983, p. 24.

See Auerbach’s post for more Carnap and Smullyan anecdotes.

Rumfitt on truth-grounds, negation, and vagueness

Philosophical Studies 175 (2018) 2079–2089

In The Boundary Stones of Thought (2015), Rumfitt defends classical logic against challenges from intuitionistic mathematics and vagueness, using a semantics of pre-topologies on possibilities, and a topological semantics on predicates, respectively. These semantics are suggestive but the characterizations of negation face difficulties that may undermine their usefulness in Rumfitt’s project.

DOI: 10.1007/s11098-018-1114-7

Preprint

Why φ?

Yesterday, @gravbeast asked on Twitter,

Does anyone know why we traditionally use Greek phi and psi for metasyntactic variables representing arbitrary logic formulas? Is it just because ‘formula’ begins with an ‘f’ sound? And chi was being used for other things?

Although Whitehead and Russell already used φ and ψ for propositional functions, the convention of using them specifically as meta-variables for formulas seems to go back to Quine’s 1940 Mathematical Logic. Quine used μ, ν as metavariables for arbitrary expressions, and reserved α, β, γ for variables, ξ, η, σ for terms, and φ, χ, ψ for statement. (ε, ι, λ had special roles.) Why φ for statements? Who knows. Perhaps simply because Whitehead and Russell used it for propositional functions in Principia? Or because “p” for “proposition” was entrenched, and in classic Greek, φ was a p sound, not f?

The most common alternative in use at the time was the use of Fraktur letters, e.g., \(\mathfrak{A}\) as a metavariable for formulas, and A as a formula variable; x as a bound variable and \(\mathfrak{x}\) as a metavariable for bound variables. This was the convention in the Hilbert school, also followed by Carnap. Kleene later used script letters for metavariables and upright roman type for the corresponding symbols of the object language. But indicating the difference by different fonts is perhaps not ideal, and Fraktur may not have been the most appealing choice anyway, both because it was the 1940s and because the type was probably not available in American print shops.

Logic Colloquium, Udine

The European Summer Meeting of the Association of Symbolic Logic will be in Udine, just north of Venice, July 23-28. Abstracts for contributed talks are due on April 27. Student members of the ASL are eligible for travel grants!

lc18.uniud.it

 

The Significance of Philosophy to Mathematics

If you wanted to explain how philosophy has been important to mathematics, and why it can and should continue to be, it would be hard to do it better than Jeremy Avigad. In this beautiful plea for a mathematically relevant philosophy of mathematics disguised as a book review he writes:

Throughout the centuries, there has been considerable interaction between philosophy and mathematics, with no sharp line dividing the two. René Descartes encouraged a fundamental mathematization of the sciences and laid the philosophical groundwork to support it, thereby launching modern science and modern philosophy in one fell swoop. In his time, Leibniz was best known for metaphysical views that he derived from his unpublished work in logic. Seventeenth-century scientists were known as natural philosophers; Newton’s theory of gravitation, positing action at a distance, upended Boyle’s mechanical philosophy; and early modern philosophy, and philosophy ever since, has had to deal with the problem of how, and to what extent, mathematical models can explain physical phenomena. Statistics emerged as a response to skeptical concerns raised by the philosopher David Hume as to how we draw reliable conclusions from regularities that we observe. Laplace’s Essai philosophique sur la probabilités, a philosophical exploration of the nature of probability, served as an introduction to his monumental mathematical work, Théorie analytique des probabilités.

 

In these examples, the influence runs in both directions, with mathematical and scientific advances informing philosophical work, and the converse. Riemann’s revolutionary Habilitation lecture of 1854, Über die Hypothesen welche der Geometrie zu Grunde liegen (“On the hypotheses that lie at the foundations of geometry”), was influenced by his reading of the neo-Kantian philosopher Herbart. Gottlob Frege, the founder of analytic philosophy, was a professor of mathematics in Jena who wrote his doctoral dissertation on the representation of ideal elements in projective geometry. Late nineteenth-century mathematical developments, which came to a head in the early twentieth-century crisis of foundations, provoked strong reactions from all the leading figures in mathematics: Dedekind, Kronecker, Cantor, Hilbert, Poincaré, Hadamard, Borel, Lebesgue, Brouwer, Weyl, and von Neumann all weighed in on the sweeping changes that were taking place, drawing on fundamentally philosophical positions to support their views. Bertrand Russell and G. H. Hardy exchanged letters on logic, set theory, and the foundations of mathematics. F. P. Ramsey’s contributions to combinatorics, probability, and economics played a part in his philosophical theories of knowledge, rationality, and the foundations of mathematics. Alan Turing was an active participant in Wittgenstein’s 1939 lectures on the foundations of mathematics and brought his theory of computability to bear on problems in the philosophy of mind and the foundations of mathematics.

Go and read the whole thing, please. And feel free to suggest other examples!

The book reviewed is Proof and Other Dilemmas: Mathematics and Philosophy, Bonnie Gold and Roger A. Simons, eds., Mathematical Association of America, 2008

[Photo: Bertrand Russell and G. H. Hardy as portrayed by Jeremy Northam and Jeremy Irons in The Man Who Knew Infinity, via MovieStillsDB]

Ptolemaic Astronomy

Working on the chapters on counterfactual conditionals for the Open Logic Project, I needed some illustrations for David Lewis’s sphere models, which he jokingly called “Ptolemaic astronomy.” Since Franz Berto joked that this should just require \usepackage{ptolemaicastronomy}, I wrote some LaTeX macros to make this easier using TikZ. You can download ptolemaicastronomy.sty (it should work independently of OLP); examples are in the OLP chapter on minimal change semantics (PDF, source).

(This will probably interest a total of two people other than me so I didn’t spend much time documenting it, but if you want to use it and need help just comment here.)

Update: it’s now in its own github repository and properly documented.