- T. Achourioti (Dept. of Philosophy, University of Amsterdam)
- Kant's theory of truth
B. Boisvert (IRIT, Université Toulouse 3),
L. Féraud (IRIT, Université Toulouse 3),
S. Soloviev (IRIT, Université Toulouse 3)
- Graph Transformations, Proofs, and Grammars
Abstract: Graph transformation systems or graph grammars are a generalization of formal grammars used
in linguistics.One of principal differences is that the structures that are transformed by transformation rules are no more linear.
In this paper we consider graph transformation systems in connection with proof theory. We develop the graph transformation framework,
where the attributes are derivable judgments of some deductive system. The computation functions are represented by partial
proofs. We consider the application of graph transformation systems to the description of Kleene-style permutation of rules, to
the derivations with “distant links”, to “parallel derivations” and “presupposed judgements”. As a “feedback” to linguistics, we
discuss applications to the linguistic derivations and their transformations, and some aspects of the notion of “presupposition”.
- P. O. R. Càrdenas (Dept. of Philosophy, University of Sheffield)
- Mathematical Structuralism, Continuity and Peirce's Diagrammatic Reasoning
Abstract: Contemporary discussions around the foundations of mathematics are traditionally stepping
into two opposite stands towards the reality of mathematical objects or structures: some of them are plainly nominalistic,
denying the existence of the objects of mathematical inquiries; the opposite stand is the extreme Platonism that defends
an account of a realm of mathematical entities where they exist. Both solutions find serious problems in accounting for
the applicability and continuity between formal theoretical inquiry and the applicability of mathematics in our best
scientific theories. In this scenario, Mathematical structuralism offers a halfway through overcoming the metaphysical
scruples of the nominalist and the extreme realism of the Platonist. At the same time enables us a criterion to find
a continuum between theory and applicability through an account of diagrammatic reasoning into the mathematical inquiries.
My aim is to show how Peirce's plea for diagrammatic reasoning fulfils the problem of our access to these structures
by an account of mathematical true continua.
- M. Colyvan (Sydney Centre for the Foundations of Science)
- From Notation to Knowledge
Abstract: There is something right about the view of mathematics as “the language of science”.
Thinking of mathematics as a language is useful in appreciating the significance of, and the difficulties encountered arriving at,
a good notational system. Good notation is far from trivial. The development of differential geometry, for example,
is intimately connected with the notation employed. But thinking of mathematics as merely language is to ignore the other roles
mathematics can play in science. I will consider the role good notation can play in prompting new ideas and new developments
in mathematics and science. Notation may even be thought to contribute to mathematical explanations.
- M. Cramer (Mathematical Institute, University of Bonn)
- Implicit dynamic function introduction and its connections to the foundations of mathematics
Abstract: We discuss a feature of the natural language of mathematics – the implicit dynamic introduction
of functions – that has, to our knowledge, not been captured in any formal system so far. If this feature is used without limitations,
it yields a paradox analogous to Russell's paradox. Hence any formalism capturing it has to impose some limitations on it. We sketch
two formalisms, both extensions of Dynamic Predicate Logic, that innovatively do capture this feature, and that differ only in
the limitations they impose onto it. One of these systems is based on a novel theory of functions that interprets ZFC, and thus
exhibits interesting connections to the foundations of mathematics.
- H. M. Dietz (Institute of Mathematics, University of Paderborn)
J. Rohde (Institute of Mathematics, University of Paderborn)
- Adventures in Reading Maths
Abstract: First semester economy students usually lack in adequate methods both of studying
and of (mathematically) working when joining courses “Mathematics for Economists”. Since 2010, we use a new concept “CAT”
in order to promote the development of adequate methods of studying and working. The virtue of this concept is investigated in the
parallel research project ECOStud. Interestingly, the ability of appropriately reading even simple mathematical expressions turned out
to be a key feature of building up a deeper understanding of their contents. We provide first insights into our ongoing work.
- M. De Glas (SPHERE, CNRS–Université Paris Diderot)
- Locology and localistic logic: mathematical and epistemological aspects
Abstract: The object of this paper is to present and thoroughly study a new logic, called localistic logic,
the essential features of which are as follows. First, it relies upon a rejection of the positive paradox axiom and a weakening
of the deduction theorem. Second, localistic logic provides locology with a logical framework. Third, the concepts of prelocus
and locus provide logic (and locology) with a categorical substratum.
- D. Grigoriev (CNRS - Université Lille 1, Lille)
- Complexity Frontiers in Mathematics
Abstract: Complexity bounds show the frontiers of feasible computations. Due to the lack of complexity lower bounds
related to the famous P–NP problem, one has to lean towards the existing complexity upper bounds in estimating the feasibility of a
computational problem. A survey is supposed to be given on the complexity bounds for 3 models: symbolic, numeric and blackbox
computations. We demonstrate that a big range of complexity bounds can occur even for basic computational problems,
and that the mathematical and computational difficulties do not always correlate.
- J. Hintikka (Dept. of Philosophy, University of Boston)
- IF Logic and Linguistic Theory
Abstract: Originally, modern symbolic logic was supposed to be a disambiguated and streamlined version
of the logic of natural language. It has nevertheless failed to provide a full account of several telltale semantical phenomena
of ordinary language, including Peirce's paradox, “donkey sentences”, and more generally conditionals and different kinds of anaphora.
It is shown here by reference to examples how these phenomena can be treated by means of IF logic and its semantical basis,
game-theoretical semantics. Furthermore, methodological questions like compositionality and logical form will be discussed.
- E. A. Hirsch (St. Petersburg Department of V. A. Steklov Institute of Mathematics RAS)
- Proving heuristically
Abstract: A mathematical proof is considered as something that can be verified and cannot be wrong (otherwise
it is not a proof), and mathematical theorems also do not allow errors or statements that are “approximately true”. Once proved, a theorem
cannot become false later (in theory). On the contrary, physics and similar sciences tolerate errors, solutions that have imperfect
precision, and new theories that “correct” the truth of some statements.
We consider settings where we are allowed to claim false “theorems”, if only a small amount of them emerge in practice.
This notion leads, for example, to the existence of optimal semidecision procedures (which is not known in the classical setting).
The idea of proof is to test any “novel approach” (procedure) by performing “experiments” that can “refute” this “approach”.
- L. Jerzykiewicz (Dept. of Humanities, Vanier College, Montréal)
- Mathematical realism and conceptual semantics
Abstract: The dominant approach to analyzing the meaning of natural language sentences that express
mathematical knowledge relies on a referential, formal semantics. I discuss an argument against this approach and
in favour of an internalist, conceptual, intensional alternative. The proposed shift in analytic method offers several benefits,
including a novel perspective on what is required to track mathematical content, and hence on the Benacerraf dilemma.
The new perspective also promises to facilitate discussion between philosophers of mathematics and cognitive scientists
working on topics of common interest.
- E. F. Karavaev (Dept. of Logic, St. Petersburg State University)
- Belief and Knowledge: Kant's Heritage in Philosophy and Logic Today
Abstract: In our examination we are handling the famous Kant's aphorism: “I had to do away with
knowledge in order to make room for faith”. In the “Critique of Pure Reason” (Section “About belief,
knowledge, and faith”) Kant defines three grades of our subjective confidence in trustworthiness of our
judgments: opinion, belief (or faith), and knowledge. It seems that the philosopher did not do away with
knowledge at all and even did raise it relatively to belief (and faith). Sure, we ought to give Kant's irony
due. Indeed, there is a significant mention about a ‘Socratic method’ in the same paragraph where the
aphorism is found. Though we see that to conjecture a style is an underproductive business. It is possible
to see that there is a quite positive content in the aphorism by Kant. And it concerns grades between
opinion, belief, and knowledge. We find some deviation from stated by Kant earlier an erroneous
estimation of logic which allegedly could not take even one step forward since Aristotle and was
seemingly quite completed. Namely, Kant writes that we ought not to detach use of the judgments
accepted by means of belief which are simply ‘probabilities’ from the analytical part of logic. Today, by
means of a relation of inducibility between judgments, it is possible to formalize and explicate Kant's
conception of opinion, belief, and knowledge. Furthermore, knowing about indispensability of chance in
the world, about significant events which have got very small probabilities we understand rather better
how belief (faith) and knowledge act reciprocally.
- M. van Lambalgen (Dept. of Philosophy, University of Amsterdam)
- The completeness of Kant's Table of Judgements and its consequences for philosophy of mathematics
- Yu. I. Manin (Max–Planck–Institut für Mathematik, Bonn)
- Foundations as Superstructure (Reflections of a practicing mathematician)
- G. Mints (Dept. of Philosophy, University of Stanford)
- Classical and Intuitionistic Geometric Logic
Abstract: Geometric sequents “A implies C” where all axioms A and conclusion C
are universal closures of implications of positive formulas play distinguished role in several areas including category theory and
(recently) logical analysis of Kant's theory of cognition. They are known to form a Glivenko class: existence of a classical proof
implies existence of an intuitionistic proof. Existing effective proofs of this fact involve superexponential blow-up,
but it is not known whether such increase in size is necessary. We show that any classical proof of such a sequent can be polynomially
transformed into an intuitionistic geometric proof of (classically equivalent but intuitionistically) weaker geometric sequent.
- Yu. V. Nechitaylov (Dept. of Logic, St. Petersburg State University)
- Basic Principles for the Dynamic Elimination of the Logical Omniscience
Abstract: The problem of logical omniscience was revised from the semantical perspective. A uniform principle
of logical omniscience was formulated. Basic principles of the system without logical omniscience were sketched.
- I. D. Nevvazhay (Dept. of Philosophy, Saratov State Law Academy)
- Semiotics of Mathematical Thinking Culture
Abstract: I discuss two abilities of consciousness – responsiveness and intentionality, which determine
two alternative types of culture of mathematical thinking. The first type of culture is a culture of expression, and the second one
is a culture of rules. I prove that mathematics is special way of bilingual thinking.
- A. B. Patkul (Dept. of Philosophy, St. Petersburg State University)
- The Kant's Treatment of Logic in Historical Context
Abstract: The reconstruction of Kant's classification of kinds of logic is given in this presentation.
It is shown also that the forming of contemporary understanding of so-called “formal logic” was impossible without the Kant's notion
of pure general logic, which was formed in the framework of mentioned classification and in context of Kant's critique of reason.
Such critique led to the change of treatment of logic as organon to its treatment as canon of finite cognition.
The question of status of Kant's transcendental logic and of possibility of its formalization in account of Kant's idea
of pure general logic is also put on in the presentation.
- F. J. Pelletier (Dept. of Philosophy, University of Alberta)
- The Effect of Applying Mathematical Logic to Metaphysical Concepts of Vagueness
Abstract: Vagueness is a phenomenon whose manifestation occurs most clearly in linguistic contexts. And some scholars
believe that the underlying cause of vagueness is to be traced to features of language. When the phenomenon is viewed
this way, those theorists who wish to represent vagueness in some formal manner will naturally be drawn to techniques that
are themselves embedded within language. For instance, the use of supervaluations within formal logic is such a technique.
However, when a theorist thinks that the ultimate cause of the linguistic vagueness is due to something other than language –
for instance, due to a lack of knowledge or due to the world's being itself vague – then the formal techniques can no longer
be restricted to those that look only at within-language phenomena. For example, when a theorist wonders whether the world
itself might be vague, and thinks to represent linguistic vagueness by attributing it to this vague reality, it is natural to think
of employing many-valued logics as the appropriate formal representation theory. The present talk investigates this attempt to
ground linguistic vagueness by appealing to metaphysical vagueness and using mathematically-motivated many-valued logics
as the way to explain the phenomena. By looking at some of the metamathematical properties that many-valued logics have, I
will attempt to show that there are severe formal difficulties involved in this attempt. The result is a philosophical conclusion
concerning a linguistic phenomenon that is reached by a mathematical method.
- G. Priest (Dept. of Philosophy, Universities of Melbourne -
Dept. of Philosophy, University of St. Andrews - Dept. of Philosophy, Graduate Center, CUNY)
- The Axiom of Countability
Abstract: I discuss some considerations which speak to the plausibility of the axiom that all sets are countable.
I then shows that there are non-trivial paraconsistent theories containing all the theorems of ZF set theory plus this Axiom.
- O. B. Prosorov (St. Petersburg Department of V. A. Steklov Institute of Mathematics RAS)
- Topologies and Sheaves appeared as Syntax and Semantics of natural language
Abstract: We study the process of interpretation of a text written in some unspecified natural language,
say in English, considered as a means of communication. Our analysis concerns the only texts written “with good grace” and intended for
human understanding; we call them admissible. Whether a part of an admissible text is meaningful or not depends on some
accepted criterion of meaningfulness. We argue that the criterion of meaningfulness conveying an idealized reader's linguistic
competence meant as ability to grasp a communicative content gives rise to a genuine topology on an admissible text, which
we call phonocentric. We argue that the following properties of phonocentric topology are linguistic universals of topological
nature: (i) T0-separability, (ii) topological connectedness, (iii) acyclicity of corresponding Hasse diagram.
This way, we interpret linguistic notions in terms of topology and specialization order; their geometric studies is a kind of
Topological Formal Syntax. For an admissible text X, we define the Schleiermacher category Schl(X) of sheaves
of fragmentary meanings, and we define the category Context(X) of étale bundles of contextual meanings.
Their geometric studies is a kind of Sheaf-Theoretic Formal Semantics that allows us to generalize Frege's compositionality and
contextuality principles related by Frege Duality defined in categorical terms. It reveals that the acceptance of one of these principles
implies the acceptance of the other, and it gives rise to a functional representation of fragmentary meanings that allows us
to develop a kind of Dynamic Semantics describing how the interpretation of a given text proceeds by induction over the discrete time
as a successive extension of a continuous function, which represents a meaning, to the whole finite topological space naturally attached to interpreted text.
- A. Rodin (Institute of Philosophy RAS - St. Petersburg State University)
- Univalence and Constructive Identity
Abstract: The non-standard identity concept developed in the Homotopy Type theory allows for an alternative
analysis of Frege's famous Venus example, which explains how empirical evidences justify judgements about identities and accounts
for the constructive aspect of such judgements.
- G. Sandu (Dept. of Philosophy, History, Culture and Arts, University of Helsinki)
- IF logic and foundations of mathematics
- A. Slissenko (LACL, Université Paris-Est Créteil)
- Towards Analysis of Information Structure of Computations
Abstract: The paper presents considerations how one can try to analyze computations, and maybe computational
problems from the point of view of information contents and information evolution. The considerations are rather preliminary. The
goal is twofold: on the one hand, to find other vision of computations that may help to design and analyze algorithms, and
on the other hand, to understand what is realistic computation and what is real practical problem. The concepts of modern
computer science, that came from classical mathematics of pre-computer era, are overgeneralized, and for this reason are often
misleading and counter-productive from the point of view of applications. The present text discusses mainly what classical
notions of information/entropy might give for analysis of computations. The classical notions of information/entropy seem
to be insufficient. In order to better understand the problem, a philosophical discussion of the the essence and relation of
knowledge/information/uncertainty in algorithmic processes (not necessarily executed by computer) may be useful.
- V. Stepanov (Dorodnicyn Computing Centre RAS)
- The dynamic model of a language allowing the mechanism of a self-reference
Abstract: Article is aimed at giving to linguists the tool which they can use for studying of the mechanism
of references of one statements on others, including on itself. For this purpose the quantifier of the self-reference is entered and
approximation of the self-reference quantifier on sequences of statements of language is given. The language model on discrete dynamic
systems is defined. It allowed to define external operations on atomic self-reference sentences of a language. So entered of the
self-reference quantifier gives the tool for the analysis of self-reference statements, passing Güodel numbering which demands,
at least, arithmetics definition in studied language. However, for this purpose it is necessary to refuse of the first order language
and to plunge into a fragment of the second order language.
- I. Stojanovic (Institut Jean-Nicod, ENS-EHESS-CNRS)
O. Kutz (SFB/TR8 Spatial Cognition, University of Bremen)
- Generalized Quantifiers and Ontological Commitments
Abstract: In Generalized Quantifier Semantics for Natural Language, as developed by Montague, Barwise,
Cooper and others, singular terms, such as names like “Tom Smith”, or demonstrative phrases like ‘this man’, are treated as
(generalized) quantifiers. Thus a statement like “Tom Smith is clever” does not attribute cleverness to that person, Tom,
notwithstanding intuition. Rather, it is to cleverness that such a statement will attribute a higher-order property,
namely that of belonging among Tom's properties. In our paper, we raise the following problem for such semantic approaches,
a problem that arises from ontological considerations. Singular terms such as names and demonstratives should only commit us
to the existence of individuals, hence, given that generalized quantifiers commit us to the existence not only of properties
of individuals (or sets, if you prefer), but also to the existence of properties of properties (or classes of sets),
such accounts are ontologically questionable. In discussing possible replies, we also raise and address the more general issue
of the relationship between semantics and ontology.
- J. Tellings (Dept. of Linguistics, University of California, Los Angeles)
- On the relation between language and mathematics
Abstract: This essay addresses the question of how mathematics and language are related.
I argue that there exist various relationships at different levels. Looking at mathematics from the point of view of linguistics
allows one to identify different aspects of mathematics that are more clearly discernible in linguistics: the human language ability,
the use of language and the field of science that studies language all have their counterparts in the domain of mathematics.
Most of the research in the area where mathematics and linguistics meet, involves the relation between linguistics and mathematics
as fields of science: the use of mathematical tools in linguistic theory, or vice versa. This is just one level of relationship
that I distinguish between the two fields. Considering mathematics from the side of linguistics reveals several interesting areas of research
that have received very little attention, and should deserve more consideration.
- V. L. Vasyukov (Chair of History and Philosophy of Science, Institute of Philosophy RAS)
- Categorical Ontology of Non-Classical Mathematics
Abstract: Some general technique for many logics is considered which allows to obtain
categorical ontology for non-classical mathematics based on those logics. The main idea lies in exploiting
preorder categories with an additional structure as a categorical counterpart of the respective abstract
algebraic models of logical systems. Then there are two possibilities of further moves: to construct
model in toposes using the version of Yoneda's lemma or produce non-classical toposes starting from
the structure of respective modified preorder categories. In first case one still obtains usual toposes as
an ontology of non-classical mathematics with underlying logical systems while in second case the
situation is more exotic: one obtains non-classical categorical constructions in which the respective
non-classical mathematics would be developed.
- M. A. Werning (Dept. of Philosophy, Ruhr University of Bochum)
- Making Quotation Transparent
Abstract: Quotation is traditionally regarded as an opaque context. This claim is often illustrated by the observation that,
within quotation marks, two synonymous expressions cannot be substituted for each other without changing the semantic value
of the quotation and its embedding context. Since the substitutability of synonyms salva significatione is logically equivalent to
the principle of compositionality, the view that quotation is opaque is tightly linked to the claim that it constitutes an irrevocable
exception to the principle of compositionality. This principle demands that the semantic value of a syntactically complex
expression be a syntax-dependent function of the semantic values of its parts. However, aside from the general postulate that
any semantic analysis of natural language constructions should abide by the principle of compositionality, there are a number
of linguistic phenomena that challenge the view that quotation is non-compositional. We will see that any compositionalityfriendly
analysis of quotation faces a dilemma: In order to be compositional, any analysis of quotation, on the one hand, must
avoid that the quoted expression be a syntactic part of the quotation. Only thus one can escape the substitutability objection.
On the other hand, in order for the semantic value of the quoted expression to contribute to the semantic value of the greater
linguistic context, the quoted expression must be a syntactic part of the greater linguistic context. In the paper a solution to this
dilemma and a fully compositional analysis of quotation will be developed. The solution covers both the obviously opaque and
the apparently transparent aspects of quotation. The analysis stays fully in the realm of semantics and does neither appeal to
any pragmatic use-mention shifts nor to extra-linguistic context parameters.