Roy's Fortnightly Puzzle: Volume 7

A bit late - apologies. Traveling again.

Anyway, most of you will be familiar with the Yablo paradox. To refresh (and suppressing some of the more tedious details): Add a truth predicate to first-order arithmetic, and then apply the diagonal lemma to obtain a theorem of the form:

(x)(Y(x) <--> (z)(z > x <--> ~ T(Fz)))

The the Yablo paradox is the sequence Y(1), Y(2), Y(3)...

There is no 'acceptable' assignment of truth values to each of Y(1), Y(2), Y(3)... such that the usual axioms for the truth (and satisfaction) predicate(s) come out true (technically: This sequence of statements is omega-inconsistent and thus has no standard model). The question is this: Which Yabloesque variants on this construction are paradoxical and which are not?

This question has been studied (in, e.g. Cook (2004)) and continues to be studied from a graph theoretic standpoint (formalizing Yabloesque paradoxes in terms of infinitary conjunction and then studying the characteristics of the associated directed graphs). Here I would like to suggest another route: Looking at variations that can be constructed within Peano arithmetic.

To give the problem a definite form, consider omega-sequences of formulas F(1), F(2), F(3)... where, for all z:

F(z) <--> (x)((x > z & H(z, x)) --> ~ T(F(x)))

is a theorem (where H is any predicate of Peano arithmetic - not including any instances of the the truth predicate - with at most x and z free). In other words, each sentence F(z) in the Yabloesque sequence is equivalent to the claim that, for any number x greater than z such that H holds of z and x, F(x) is false.

In order to guarantee that each sentence in the list refers to at least one other sentence, we assume that:

(z)(Ex)(x > z & H(z, x))

Here are some simple results to get you started (with some notes on where they come from):

Theorem 1: If H does not contain variable z free, then the resulting Yabloesque sequence is paradoxical.

[Notes: Observed by Lavinia Picollo, Universidad de Beunos Aires].

Theorem 2: If (z)(Ew)(x)(x > w --> H(z, x)) then the resulting Yabloesque sequence is paradoxical.

[Notes: In other words, for each z, the collection of x > z such that H(z, x) is cofinite. This result is, in essence, in Yablo (2006)].

Theorem 3: Given any H(z, x) as above, if the Yabloesque sequence in question is paradoxical, then the Dual version consisting of instances of:

F(z) <--> (Ex)(x > z & H(z, x) & ~ T(F(x)))

is paradoxical as well.

[Notes: A standard sort of 'duality' result for Yabloesque constructions.]

What other conditions are necessary or sufficient for paradox?

[Notation: F is the Goedel code of F, (x) is the universal quantifier, and (Ex) is the existential quantifier.]

Cook, R. [2004], “Patterns of Paradox”, Journal of Symbolic Logic 69, [2004]: 767 – 774.

Yablo, S. [2006], “Circularity and Paradox”, in Bolander, Hendricks, & Pedersen, Self-Reference.

Mathematical reasoning and external symbolic systems

(Cross-posted at NewAPPS)

A few weeks ago I wrote a post on blind mathematicians, discussing the case of Bernard Morin and the eversion of the sphere in particular. I had been thinking about blind mathematicians then because I was working on a paper on the role of external symbolic systems (written systems such as notations in particular) for mathematical reasoning and mathematical practice. I have now completed a first, preliminary draft of the paper, and uploaded it on my academia website (it's on top of the list under 'Papers'). Should anyone be interested in taking a look, comments would be most welcome! I discuss the case of Bernard Morin all the way at the end of the paper, as well as the case of Jason Padgett, the man with acquired savant syndrome who sees shapes as fractals and can hand-draw fractals of pretty much any image you can think of. Here is the abstract:

_________________________________________

The aim of this paper is to discuss the exact status of external symbolic systems with respect to mathematical reasoning and mathematical practice. The standpoint adopted is a combination of philosophical analysis with focus on empirical studies on numerical cognition (ranging from cognitive science to developmental psychology and anthropology) and on the history of notations. Indeed, the investigation takes into account three different levels: the synchronic level of a mathematician doing mathematics at a given point; the diachronic, developmental level of how a given individual learns mathematics; and the diachronic, historical level of the development of mathematics as a discipline throughout the centuries. It will be argued that the use of external symbolic systems is constitutive of mathematical reasoning and mathematical practice in a fairly strong sense of ‘constitutive’, but not in the sense that manipulating notations is the only route to mathematical insight. Indeed, two case studies will illustrate the qualification: a man with acquired savant syndrome and a blind mathematician.

Eulogy for Horacio Arló-Costa

Rohit Parikh wrote a beautiful eulogy for Horacio Arló-Costa, which I post here with his permission:

________________________________________

On July 11, the TARK XIII conference gave a party in Groningen for the attendees. Afterwards, Horacio and I had a small dinner in a plaza in Groningen. Only two weeks have passed and he is already gone.

But he will be honored by various events at Columbia, at CMU and most likely also in Munich and in Buenos Aires.

For me, he was a student who became an affectionate and long standing friend. We met each other frequently in New York, where he gave several talks at CUNY. But we also met in Pittsburgh, Florence (where he introduced me to Vann McGee), and of course in Groningen. He knew my family and I knew his wife Claudia.

Horacio was a person of immense scholarly integrity and an ability to produce impressive original research. He was surely a world expert in the foundations of probability and aspects of conditionals and modal logic. It was quite frequent for me to ask him a question and get an instant answer. Some have said he was a walking encyclopedia. This is a fair assessment.

He shared my interest in Borges (or perhaps I shared his), in Tagore and in Blake. There was a deep spiritual sense in him, which apparently his father also had and which he celebrated in his eulogy to his late father. I read part of the same eulogy at Horacio’s funeral in Westchester. What Horacio said about his father, in large part, applied also to Horacio himself.

I am sure his work will be carried on by others, especially his student Arthur Paul Pedersen. I look forward to being in touch with Paul, with Horacio’s colleagues at CMU and his many friends at Columbia.

Rohit Parikh

Tree Proof Generator

When I used to teach elementary logic (Logic 1), I used to recommend students that they try using the online Tree Proof Generator, which will generate tableau proofs, or provide countermodels.

For example, given the valid formula $\forall x(Rxx \rightarrow \exists y Rxy)$, it gives the following tableau proof:

1. $\neg \forall x(Rxx \rightarrow \exists y Rxy)$
2. $\neg (Raa \rightarrow \exists yRay)$
3. $Raa$
4. $\neg \exists yRay$
5. $\neg Raa$

What's Hot in Mathematical Philosophy? #2 (The Reasoner)

Each month MCMP (well, me ...) contributes a column to The Reasoner magazine, edited by Jon Williamson at the University of Kent. The most recent edition (August 2011) has just been published.
Here is the current column (the first column is available at The Reasoner website too):

--------------------------------------
What’s hot in mathematical philosophy? (#2)

Instrumentalist nominalism: nominalism is the view that there are no numbers, sets, sequences of symbols, computer programs, languages, formal systems, symmetry groups, wavefunctions, manifolds, tensor fields, Hilbert spaces, etc., and is usually motivated by a sceptical argument: in order to know that there are exactly two continuous automorphisms of the complex field to itself, the mind must be in ‘causal contact’ with these, which is impossible. On the other hand, Quine and Putnam argued that our best scientific theories, being mathematicized, are inconsistent with nominalism and that the use of mathematics in such theories is indispensable in some sense. For example, the electromagnetic field is a function from spacetime to a vector space. How can we even formulate Maxwell’s equations without appealing to the electromagnetic field, current densities, and so on?

Standard responses have been to engage in nominalization programmes: either eliminate mathematicalia or reconstrue them as nominalistically benign. The magnetic field might be eliminated and replaced by certain intrinsic spatio-temporal relations (Field) or perhaps reconstrued as a ‘possible sentence token’ (Chihara). But there are a number of difficulties with these programmes, connected to the awkwardness of the resulting reconstructions and the logical and metaphysical resources to which they appeal (for a survey, see Burgess & Rosen 1997, A Subject with No Object).

The last decade has seen the emergence of a more radical strategy responding to the indispensability arguments: instrumentalism. Contemporary instrumentalist nominalists would like to combine realism about science with anti-realism about mathematics, while insisting that there is no need to nominalize our best scientific theories. Given an explanatory and predictively successful scientific theory T, inconsistent with nominalism, the realist says,

$T$ is a good approximation to the truth,

while the instrumentalist says,

the concrete things behave as if $T$ (or, $T$ is nominalistically adequate),

while maintaining that no intrinsic description of the concrete things need be given to replace/reconstrue $T$. So, compasses, computers and constellations behave as if there is an electromagnetic field, even though there isn’t. The approach is conceptually similar to van Fraassen’s constructive empiricism: replace ‘empirically adequate’ by ‘nominalistically adequate’ to get instrumentalist nominalism.

Instrumentalist nominalism was criticized by John Burgess (‘Why I am Not a Nominalist’, Notre Dame J. Formal Logic 1983) and further discussed in Burgess & Rosen 1997. Over the last decade or so, several authors have proposed similar views, occasionally called ‘fictionalism’, often incorporating ideas from the literature on the semantics and pragmatics of fictional discourse. Examples are Joseph Melia (‘Weaseling Away the Indispensability Argument’, Mind 2000), Gideon Rosen (‘Nominalism, Naturalism, Epistemic Relativism’, Nous 2001), Stephen Yablo (‘Go Figure: A Path Through Fictionalism’, Midwest Studies in Philosophy 2001), Mary Leng (‘Revolutionary Fictionalism: A Call to Arms’, Philosophia Mathematica 2005), and some recent work by Richard Pettigrew. Building on her previous work, Leng has recently published a monograph defending instrumentalism (Mathematics and Reality, OUP 2010), reviewed by Burgess in Philosophia Mathematica (Vol. 18, 2010) and by Chris Pincock in Metascience (forthcoming). Criticisms of instrumentalism include recent articles by Stathis Psillos (‘Scientific Realism: Between Platonism and Nominalism’, Philosophy of Science 2010), Mark Colyvan (‘There is No Easy Road Nominalism’, Mind 2010) and myself (‘Nominalistic Adequacy’, Proceedings of the Aristotelian Society 2011). Whether viable or not, instrumentalist nominalism has become a major topic in contemporary philosophy of mathematics.

Adding an Interpretation to a Collection of Structures

This is the third in a series of posts discussing the notion of structural representation, and related to the question of what a scientific theory is. Normally, scientists put forward conjectures, hypothesis, assumptions, etc., and then examine them in the light of evidence and other theories. One might conjecture that there is a black hole at the centre of every galaxy; or conjecture that global warming is caused by human diet; or that there was a species of dinosaur that invented algebraic geometry, etc.

The model-theoretic conception of scientific theories states that a theory is collection $\Sigma$ of mathematical structures. As I've argued below, this implies that theories are not truth bearers. And if one tries to resolve this by defining what "$\mathcal{M}$ represents the world" means, one is lead either to Newman's Objection ($\mathcal{M}$ represents the world iff the world has large enough cardinality) or back to the standard view, along with the claim that the world $\textit{is}$ a structure $\mathbb{W}$.

The underlying problem here is that mathematical structures in $\Sigma$ are $\textit{uninterpreted}$. So, we need to introduce some notion of $\textit{interpreting}$ a structure. And the previous post has proposed an analysis of what an interpretation $\mathcal{I}$ of a structure is, and a definition of what it is for a structure $\mathcal{M}$ to be correct under an interpretation.

Suppose then that one accepts the two objections described earlier - the Truth-Bearer Objection and the Newman Objection. The resolution is to introduce an interpretation, so that we can now talk of the structures being correct or incorrect (under the interpretation). So, we revise the Model-Theoretic Conception to:

The Interpreted-Model-Theoretic Conception:
A theory $T$ is a pair $(\Sigma, \mathcal{I})$ consisting of a collection $\Sigma$ of structures, all of the same signature, and an interpretation $\mathcal{I}$, which assigns an interpretation for each $\mathcal{M} \in \Sigma$. This $\mathcal{I}$ is the intended interpretation of $T$.

(Here there is a slight unclarity: the interpretation $\mathcal{I}$ must be specified for every $\mathcal{M} \in \Sigma$. Because all have the same signature, the interpretation just assigns the same referent $R^{\mathcal{I}}$, when $R$ is the distinguished relation for the structure in question. But for each structure $\mathcal{M}$, a specific denotation function, from $dom(\mathcal{M})$ to $D_{\mathcal{I}}$, must be specified too.)
Then we can define:

(D) $T$ is true iff for some $\mathcal{M} \in \Sigma$, $\mathcal{M}$ is correct under $\mathcal{I}$.

This at least provides some sort of an answer the Truth-Bearer Objection and the Newman Objection.

Interpreting a Structure

This post is related somewhat to the previous one, but has nothing to do what what a theory is. Rather it gives an analysis of what it means to say that a structure $\mathcal{M}$ is correct under some interpretation $\mathcal{I}$.

One cannot define correctness of a structure without specifying an $\textit{interpretation}$ of the structure. Similarly, one cannot define truth of a linguistic string, say, "bajfgdhsalsbs" unless one specifies an $\textit{interpretation}$ of the language to which the string belongs. So, given a language $\mathcal{L}$, one usually defines:

1. The notion of an $\mathcal{L}$-interpretation, $\mathcal{I}$.
2. The notion of a formula $\phi$ being true under $\mathcal{I}$.

By analogy, one wants to define:

1. The notion of an interpretation $\mathcal{I}$ of a structure $\mathcal{M}$.
2. The notion of $\mathcal{M}$ being correct under $\mathcal{I}$.

The proposed definition of "interpretation of a structure" is as follows:

(D1) Suppose a structure $\mathcal{M}$ is given. An interpretation $\mathcal{I}$ of $\mathcal{M}$ is specified by three components:
(i) A domain $D_{\mathcal{I}}$.
(ii) A function $f_{\mathcal{I}} : dom(\mathcal{M}) \rightarrow D_{\mathcal{I}}$. Call this the denotation function.
(iii) For each distinguished relation $R$ in $\mathcal{M}$, a relation $R^{\mathcal{I}}$. Call this the referent of $R$ under $\mathcal{I}$.

This analogous to the standard idea of interpreting a language $\mathcal{L}$. We specify a domain $D_{\mathcal{I}}$ for the quantifiers to range over, and we specify, for each primitive predicate symbol $P$, a relation $P^{\mathcal{I}}$ on the domain $D$.

Next, we can define what it means for a structure $\mathcal{M}$ to be correct, in some sense, under any given interpretation $\mathcal{I}$.

(D2) Let $\mathcal{M}$ be a structure and let $\mathcal{I}$ be an interpretation. Then $\mathcal{M}$ is correct under $\mathcal{I}$ just if $f_{\mathcal{I}}$ is a bijection and, for each distinguished relation $R$ of $\mathcal{M}$, $f_{\mathcal{I}}(R) = R^{\mathcal{I}}$.

An example is the following. Suppose we have a very simple structure $\mathcal{M}$ such that $dom(\mathcal{M}) = \{0,1\}$, with a single distinguished relation $R = \{(0,1)\}$. This is more or less, a simple directed graph with two nodes, with one connected to the other. Let us specify an interpretation $\mathcal{I}$ as follows:

$D_{\mathcal{I}} = \{a, b\}$.
$f_{\mathcal{I}}(0) = a$.
$f_{\mathcal{I}}(1) = b$.
$R^{\mathcal{I}} = \{(a, a)\}$.

This interpretation treats $0$ as denoting $a$, treats $1$ as denoting $b$, and interprets the relation $R$ as $\{(a, a)\}$. Clearly, $\mathcal{M}$ is not correct under $\mathcal{I}$.

(This post is based on a talk I gave at Leeds University in 2003. Here is a photo, taken by Joseph Melia, of the talk:

)

On the Model-Theoretic Conception of Scientific Theories

Ordinarily, in mathematical and scientific practice, the notion of a “theory” is understood as follows:

(SCT) Standard Conception of Theories:
A theory $T$ is a collection of statements, propositions, conjectures, etc.

A theory claims that things are thus and so. The theory may be true, and may be false. A theory $T$ is true if things are as $T$ says they are, and $T$ is false if things are not as T says they are.

One can make this Aristotelian explanation more precise, as Tarski showed, in the cases where we understand how to give precise logical analyses of theories, by identifying an interpreted language $(\mathcal{L}, \mathcal{I})$ in which $T$ may be formulated. Here $\mathcal{L}$ is some formalized language and $\mathcal{I}$ is an $\mathcal{L}$-interpretation. (The language can be higher-order, infinitary, contain infinitely many predicates, or uncountably many constants, etc.) One can define the satisfaction relation $\models$, holding between $\mathcal{L}$-sentences $\phi$ and $\mathcal{L}$-interpretations, and then define the notion "$\mathcal{L}$-sentence $\phi$ is true in $(\mathcal{L}, \mathcal{I})$" as "$\mathcal{I} \models \phi$". What is essential about this is that theories are $\textit{truth bearers}$. They are bearers of semantic properties.

The standard conception of theories thus takes theories to be truth bearers. In particular, given the history of science, we need to be able to make sense of saying of a theory $T$ that it is $\textit{false}$. Furthermore, it seems clear that any account of theories according to which a theory is not a truth bearer – i.e., an account which rejects semantics – is surely not acceptable. We are obliged by the facts concerning scientific practice to provide an analysis of the notion of a theory in such a way that electromagnetic theory, special relativity, evolutionary theory, and all the various other theories that may interest us, are capable of being either true or false.

Over the last forty years, a contrasting view has appeared:

(MCT) The Model-Theoretic Conception of Theories:
A theory $T$ is a collection $\Sigma$ of structures.

This view has been advocated by Suppe, van Fraassen, French, Ladyman and others. E.g.,

Van Fraassen elaborated and generalized Beth's approach, arguing that theories and models are essentially mathematical structures ... (Ladyman & Ross 2007, p. 116).
The semantic view encourages us to think about the relation between theories and the world in terms of mathematical and formal structures. (Ladyman & Ross 2007, p. 118.)

In the simplest case, a structure $\mathcal{M}$ is a package of the form $(D, \{R_i\}_{i \in I})$, where $D$ is some non-empty set, and the $R_i$ are relations on $D$. (This can be generalized in various ways.) The Model-Theoretic conception thus rejects the standard conception of theories described above. For it is meaningless to say of a structure $\mathcal{M}$ that it is true. Therefore,

According to the Model-Theoretic View, theories are not truth-bearers.

This consequence of MCT is a refutation of it. It is a minimal constraint on what a theory is that it be a truth bearer. If something isn’t a truth bearer, then it isn’t a theory.

Having mentioned this objection on various occasions, the reply one hears is that:

(R) A structure $\mathcal{M}$ is true iff $\mathcal{M}$ “represents the world”.

However, there is no such notion as that of a structure $\mathcal{M}$ “representing the world”! So, one is led to the question:

(Q) What does it mean to say of a structure $\mathcal{M}$ that it represents the world?

Advocates of MCT sometimes say that a structure $\mathcal{M}$ represents the world by "being isomorphic to it". However, prima facie, it doesn’t make any sense whatsoever to say of a structure $\mathcal{M}$ that it is "isomorphic to the world", because isomorphism is a relation that holds $\textit{between structures}$. Is the world a structure? (We return to this in a moment.)

The only answer I can think of, at least consistent with the intentions of advocates of this view, is the following:

(D) A structure $\mathcal{M} = (D, R_1, \dots, R_n)$ $\textit{represents the world}$ iff there is a subset $W$ of things in the world, and there are relations $S_1, \dots, S_n$ on $W$ such that $(D, R_1, \dots, R_n) \cong (W, S_1, \dots, S_n)$.

(Where "$\cong$" stands for “is isomorphic to”.)

However, according to definition (D), a claim of the form "$\mathcal{M}$ represents the world" is a Ramsey sentence. And then it is not difficult to prove the following:

(N) $(D, R_1, \dots, R_n)$ represents the world iff, for some subset $W$ of things in the world, $|D| = |W|$.

(This is a version of Newman’s Objection to structuralism. The left-to-right direction is trivial. The right-to-left direction is proved by assuming that $|D| = |W|$, and considering an bijection $f : D \rightarrow W$. Take the images $f(R_i)$ under $f$ of the relations $R_1, \dots, R_n$. The result is the structure $(W, f(R_1), \dots, f(R_n))$ isomorphic to $(D, R_1, \dots, R_n)$ by construction.)

This tells us that $\textit{any}$ structure $\mathcal{M}$ "represents the world" so long as the world has enough things in it.

This is surely unacceptable. The only way around this problem is to say something like the following:

(S) The world $\mathbf{is}$ a structure $\mathbb{W} = (W, S_1, \dots, S_n)$.

(One cannot replace "is" by "can be represented by". Go back and re-read the definition (D) again. For unless one accepts (S), then, as (N) tells us, the world "can be represented" by any structure, cardinality permitting.)

Then, if one accepts (S), one can say that a structure $\mathcal{M}$ "represents the world" iff it is isomorphic to $\mathbb{W}$.

However, if we say this, then we have returned to a (rather strong) formulation of the original, standard, conception of theories. For $\mathbb{W}$ is simply the intended interpretation of some language $\mathcal{L}$ with which we may formulate the original theory $T$ of which $\mathcal{M}$ may be a model (in the usual sense). Indeed, the correct theory $Th(\mathbb{W})$ is simply the set of truths in the structure $\mathbb{W}$, and a theory $T$ in this interpreted language is then true when it is subset of $Th(\mathbb{W})$.

In summary, this post gives two major objections to the Model-Theoretic View. The first is the Truth-Bearer Objection. The second is the Newman Objection. The standard conception of theories takes theories to be truth bearers. These may be taken to be propositional or may be take to be statements in an interpreted language. In contrast, the model-theoretic conception of theories, associated with van Fraassen, French, Ladyman and others, identifies theories with collections of structures. Hence, the model-theoretic conception denies that theories are truth bearers. Unless we are prepared to accept the idea that theories are uninterpreted calculi for making predictions (i.e., radical instrumentalism about the semantics of theories), then this is unacceptable.

If one tries to remedy this problem, by defining "$\mathcal{M}$ represents the world", the definition yields a Ramsey sentence, and then one is faced with a version of the Newman objection. The only way around this problem is to $\textit{identify}$ the world itself with a structure $\mathbb{W}$. However, this merely plays the role of the intended interpretation of a language $\mathcal{L}$ which might be used to formulate the theory $T$ in question. One is then back to the standard conception, along with the the very strong metaphysical assumptions that the world is a structure $\mathbb{W}$.

Update:
I should include at least some background references for anyone curious about the literature.

A defence of the model-theoretic conception can be found in James Ladyman & Don Ross, 2007, Every Thing Must Go, Chapter 2, pp. 111-118. Here, Ladyman & Ross implicitly reject the notion that theories have semantic properties (e.g., reference, truth):

In the context of the syntactic approach, within which a theory is taken to be a set of sentences, realism amounts to the commitment to standard (correspondence) referential semantics, and to truth, for the whole theory. (Ladyman & Ross 2007, p. 117.)

Anjan Chakravartty raised the truth-bearer objection in his 2001 article, "The Semantic or Model-Theoretic Conception of Scientific Theories", Synthese (online here).

A version of the Newman objection is mentioned in footnote 10 of Jeffrey Ketland, 2004, "Empirical Adequacy and Ramsification", British Journal for the Philosophy of Science.

Roman Frigg raised a version of the Newman objection in his 2006 article, "Scientific Representation and the Semantic View of Theories", Theoria (online here), Sections 5 and 6.

In Memory of Horacio Arló-Costa

Very sad news from Choice & Inference: Horacio Arló-Costa, Professor of Philosophy at Carnegie Melon, has passed away. The post does not give any further details, but promises to provide updates in due course.

I have very fond memories of Horacio from the time I lived in New York and attended the weekly seminar of R. Parikh’s group, where Horacio was almost always present. We’ve also corresponded a few times in his capacity of editor of JPL and then RSL. But perhaps most important was our South-American complicity, actually. Brazilians and Argentinians may hate each other's guts when it comes to football, but in the outer world we team up for sure...

This is a great shock: Horacio will be dearly missed.

UPDATES: Here is the official obituary from CMU, and here is a post over at Certain Doubts with several initiatives to honor Horacio's life and work, including a CFP for an edited volume in his honor.

ANOTHER UPDATE: Here is Rohit Parikh's eulogy for Horacio.

Workshop: Formal Semantic Evidence

A few months ago I wrote a post on the exciting new directions that formal semantics as a field seems to be taking, and I mentioned in particular the workshop on formal semantic evidence that my former colleagues Katrin Schulz and Galit Weidman Sassoon are organizing within the Amsterdam Colloquium 2011. Now the CFP is out, and the organizers are keen on having many submissions dealing with very foundational, methodological/philosophical issues, as the CFP clearly indicates. Up to now, the Amsterdam Colloquium has not been a particularly prominent forum for discussions on this level, so they want to make sure that people who may not view themselves as working on 'Amsterdam Colloquium material' will nevertheless consider submitting for the workshop. As I suspect that at least some of the readers of this blog fall within this category, I paste here the CFP in full:

Formal Semantic Evidence

Workshop hosted by the

Amsterdam Colloquium

December 19-21, 2011

Invited speakers

Richard Breheny, University College London

Bart Geurts, Radboud University Nijmegen

Call for papers

Formal semantics as a field of linguists undergoes a rapid change with respect to the status of quantitative methodologies, the application of which is gradually becoming a standard in the field, replacing the good old 'armchair' methodology. In light of this development, we invite submissions reporting of high level formal semantic research benefiting from the use of a quantitative methodology, corpora-based, experimental, neurolinguistic, computational or other. Ideal presentations include informed reflections on the role of a particular methodology in formal semantics.

• We welcome submissions focusing on wide spread, yet not unproblematic or elusive semantic-pragmatic concepts such as 'context', 'accommodation', 'question under discussion', 'ordering source', and so forth; what methodologies can shed new light on such notions and the way they might be systematically studied and decoded?
• What kind of experimental evidence (if any) can bear on fundamental issues such as the nature of the semantic lexicon on the one hand, and compositionality and projection on the other? What is the nature of semantic infelicity or markedness? What kind of experimental evidence (if any) can support or refute hypotheses concerning the nature of the logical form or its very existence? Concerning empty categories?
• Finally, can formal semantic tools contribute to our understanding of experimental results and theoretical issues within cognitive psychology as pertaining to natural language semantics?

Instructions for authors

Authors can submit anonymous abstracts of at most two pages via the website of the Amsterdam Colloquium.

Important dates

Submission deadline: September 1, 2011

Notification of acceptance: October 15, 2011

Deadline for pre-proceedings: December 1, 2011

Deadline for registration: December 1, 2011

Conference: December 19 - 21, 2011

Organizers

Katrin Schulz

Galit Weidman Sassoon

Roy's Fortnightly Puzzle: Volume 6

This one is easy to state, but hard, I think, to answer.

As we all know, the axiom of foundation (loosely speaking, that there do not exist sets with infinitely descending membership chains in their transitive closure, but usually formulated in a manner apparently custom-designed to confuse one's students) is a standard part of the machinery of ZFC. The question is: Why?

The answer, of course, depends on the role that we think ZFC should, and must, play. Here are some options:

Option 1: Foundation blocks the paradoxes, by ruling out the existence of non-well-founded sets such as the Russell set.

Worry: This is just insane. Full ZFC is, in some intuitive sense, more likely to be inconsistent for including foundation. In other words, if full ZFC (including foundation) is consistent, then ZFC-minus-foundation is consistent. (Another way to see the point: Adding foundation to Frege's Basic Law V, or any other formulation of set theory, doesn't make the resulting theory consistent - in fact, it might make contradictions easier to prove!)

Option 2: We are attempting to formulate the one true description of the unique set-theoretic universe. This universe is given to us informally by the iterative conception (as described by Boolos and others). Foundation follows on this conception.

Worry: If this is right, then one wonders why we also include the axiom of replacement (which, it seems to me, and seemed to Boolos, does not follow on the iterative conception of set). Also, one might wonder why the iterative conception is privileged in this way. There are other conceptions of set, including but not limited to the limitation-of-size conception, that don't support foundation.

Option 3: We are looking for the 'safest' reasonable looking set theory that will support the reconstruction of current mathematics.

Worry: Setting aside what 'safe' and 'reconstruction' amount to, we need some evidence that foundation plays any ineliminable role in such a reconstruction. After all, we could have a set theory that admitted various sorts of non-well-founded sets (either Aczel style or Forster style), and then just restrict our attention to the hereditary sets (those that can be built up from the empty set, loosely - see Boolos' stuff for exact details) and proceed as before. If there are no reasons for thinking we need foundation for our reconstructions, then any reasonable notion of safe will surely rank ZFC-minus-foundation as safer than ZFC.

Option 4: We are looking for the most powerful consistent set theory.

Worry: Well, this depends on what one means by 'most powerful'. Aczel's non-well-founded set theory is (if I am remembering correctly) equiconsistent with standard ZFC, but it admits far more sets. So why isn't this more powerful? At any rate, this just means we want one of either ZFC or ZFC-minus-foundation-plus-not-foundation (since either of these is more powerful than ZFC-minu-foundation), but doesn't seem to select between them.

Thoughts?

The maths of the cube

From the MIT News Office:

Erik Demaine, an associate professor of computer science and engineering at MIT; his father, Martin Demaine, a visiting scientist at MIT’s Computer Science and Artificial Intelligence Laboratory; graduate student Sarah Eisenstat; Anna Lubiw, who was Demaine’s PhD thesis adviser at the University of Waterloo; and Tufts graduate student Andrew Winslow showed that the maximum number of moves required to solve a Rubik’s cube with N squares per row is proportional to N²/log N. "That that’s the answer, and not N², is a surprising thing," Demaine says.
(Read more.)

(I owe the pointer to Wild About Math.)

What is it like to be a blind mathematician?

(Cross-posted at NewAPPS, and in the spirit of the "Count me in" campaign.)

I am now working on a paper on the role of manipulating notations in mathematical reasoning, which reminded me of an issue that interested me a few years ago: how do blind mathematicians do their work, given that the usual manipulation of notations (which obviously appeals crucially to vision) is not an option to them?

I then sent a query on this topic to the FOM list, and got some very interesting replies. In particular, many people mentioned this fascinating notice published by the American Mathematical Association on blind mathematicians, which I highly recommend to anyone interested in how mathematics 'comes about'. An excerpt:

A sighted mathematician generally works by sitting around scribbling on paper: According to one legend, the maid of a famous mathematician, when asked what her employer did all day, reported that he wrote on pieces of paper, crumpled them up, and threw them into the wastebasket. So how do blind mathematicians work?

The notice focuses in particular on Bernard Morin, the mathematician who formulated the first visualization of the eversion of the sphere (also known as Smale's paradox) by construing clay models of the process. (Here is an educational video on the topic.) Quoting from this site:

To evert a sphere is to turn it inside-out by means of a continuous deformation, which allows the surface to pass through itself, but forbids puncturing, ripping, creasing, or pinching the surface. An abstract theorem proved by Smale in the late 1950s implied that sphere eversions were possible, but it remained a challenge for many years to exhibit an explicit eversion.

The bit that interests me most in Morin's achievement is that he himself claims that being blind was an asset he had over other mathematicians on this specific problem. I quote from the AMS notice:

One thing that is difficult about visualizing geometric objects is that one tends to see only the outside of the objects, not the inside, which might be very complicated. By thinking carefully about two things at once, Morin has developed the ability to pass from outside to inside, or from one “room” to another. This kind of spatial imagination seems to be less dependent on visual experiences than on tactile ones.

To me, Morin's eversion of the sphere represents a case study of the impact of perceptual interaction with the environment (including bits of notations) in mathematical reasoning. On account of being blind, he seemed to have a different perception of objects, including their 'insides'. Thus, we seem to have a case of a difference in perceptual conditions actively making a significant difference for the kind of mathematical knowledge produced.

The usual story goes that mathematical reasoning is 'purely abstract' and completely divorced from sensorimotor processing, but in recent years people like Rafael Nunez and David Landy, among others working within an embodied, extended perspective, have emphasized the role of perceptual grounding for mathematical reasoning. There are many ways in which this can be cashed out, and my own focus has been on the kind of engagement we seem to have with notations when 'doing math'; are we merely reporting independent, prior cognitive processes by means of notations, or is the manipulation of the notation itself part of these cognitive processes? Unsurprisingly, I side with the second alternative, and in that case the perceptual properties of the notation will have a significant impact. Now, this obviously does not mean that people who do not manipulate notations in this manner, e.g. a blind mathematician such as Morin, cannot do mathematics; rather, the point is that they apparently do mathematics in a different way, at times 'seeing' things that the rest of us cannot see.

(When I have the time, I will elaborate further on these ideas with a post on Jason Padgett, a man with acquired savant syndrome who 'sees' everything geometrically as fractals. So stay tuned!)

Visiting Scholarships for NIP in Aberdeen

The Northern Institute of Philosophy (NIP) in Aberdeen is announcing visiting scholarships. My guess is that many readers of this blog may be interested in the opportunity, in particular those working on formal epistemology, truth and paradoxes. I've never been to Aberdeen myself, but the NIP is clearly a very vibrant research environment, so this looks like a wonderful opportunity.

I quote from the announcement:

The Northern Institute of Philosophy (NIP) at The University of Aberdeen is pleased to announce the availability of NIP Scholarships for visits to NIP within the periods 10 October to 16 December 2011 and 21 May to 6 July 2011 .

Successful applicants will have well-developed research interests in one of the five projects currently running at NIP (Basic Knowledge, Relativism and Tolerance, Truth and Paradox, Self Knowledge, Pluralism). The Fellowships will provide an opportunity to pursue and present research in a supportive collaborative environment. Successful applicants will be expected to participate fully in the regular NIP activities and various workshops and conferences scheduled to take place during the NIP session.

The Fellowships are given in the form of the provision of travel and accommodation costs for the visit, up to a maximum of £3000.

Applications should include: a cover letter (stating any preference for an Autumn or Spring visit), a research statement and a writing sample (max 5000 words) and a letter of recommendation by a supervisor (from graduate applicants) or a CV (from faculty applicants). They should be sent to Sharon Coull on s.coull@abdn.ac.uk by Sunday 14th August 2011. Although applications for shorter visits will be considered, priority will be given to candidates proposing visits for the full period who are engaged in research on topics within the remit of one or more of the five current Institute projects.

Change in perspective

"The founding principle of the Journal of Universal Rejection (JofUR) is rejection. Universal rejection. That is to say, all submissions, regardless of quality, will be rejected. Despite that apparent drawback, here are a number of reasons you may choose to submit to the JofUR..."