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.
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.
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.
Rohit Parikh wrote a beautiful eulogy for Horacio Arló-Costa, which I post here with his permission:
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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
1. $\neg \forall x(Rxx \rightarrow \exists y Rxy)$
2. $\neg (Raa \rightarrow \exists yRay)$
3. $Raa$
4. $\neg \exists yRay$
5. $\neg Raa$
$T$ 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.
The Interpreted-Model-Theoretic Conception:(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.)
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 $T$.
(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.
1. The notion of an $\mathcal{L}$-interpretation, $\mathcal{I}$.By analogy, one wants to define:
2. The notion of a formula $\phi$ being true under $\mathcal{I}$.
1. The notion of an interpretation $\mathcal{I}$ of a structure $\mathcal{M}$.The proposed definition of "interpretation of a structure" is as follows:
2. The notion of $\mathcal{M}$ being correct under $\mathcal{I}$.
(D1) Suppose a structure $\mathcal{M}$ is given. An interpretation $\mathcal{I}$ of $\mathcal{M}$ is specified by three components: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 $D$.
(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 $\mathcal{I}$.
(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:
$D_{\mathcal{I}} = \{a, b\}$.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 $\mathcal{I}$.
$f_{\mathcal{I}}(0) = a$.
$f_{\mathcal{I}}(1) = b$.
$R^{\mathcal{I}} = \{(a, a)\}$.
(SCT) Standard Conception of Theories: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.
A theory $T$ is a collection of statements, propositions, conjectures, etc.
(MCT) The Model-Theoretic Conception of Theories:This view has been advocated by Suppe, van Fraassen, French, Ladyman and others. E.g.,
A theory $T$ is a collection $\Sigma$ of structures.
Van Fraassen elaborated and generalized Beth's approach, arguing that theories and models are essentially mathematical structures ... (Ladyman & Ross 2007, p. 116).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,
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.)
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.
(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.)
(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”.)
(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.)
(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.)
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 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?
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!)
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.