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# 2-5 August 2011, The Classical Model of Science II: The Axiomatic Method, the Order of Concepts and the Hierarchy of Sciences from Leibniz to Tarski, Vrije Universiteit Amsterdam, the Netherlands

Among the invited speakers are:

Hourya Benis Sinaceur (IHPST, Paris)

Patricia Blanchette (Notre Dame)

Paola Cantù (CEPERC, Université de Provence)

Paolo Mancosu (Berkeley)

Paul Rusnock (Ottawa)

Stewart Shapiro (Ohio State University/St. Andrews)

Organising committee: Arianna Betti (chair), Hein van den Berg, Lieven Decock, Wim de Jong, Iris Loeb & Stefan Roski, VU Amsterdam

#### Introduction

This conference is devoted to the development of the axiomatic method, with particular attention for the period from Leibniz to Tarski. In particular, we aim to achieve a better historical and philosophical understanding of the way the axiomatic method in the sense of an ideal of scientific knowledge as "cognitio ex principiis" has influenced the development of modern science. The overarching framework for this will be the so-called "Classical Model of Science".

The Classical Model (or Ideal) of Science consists of the following conditions for counting a system S as properly scientific (de Jong & Betti 2010: http://bit.ly/f7QKXW):

(1) All propositions and all concepts (or terms) of S concern a specific set of objects or are about a certain domain of being(s).

(2a) There are in S a number of so-called fundamental concepts (or terms).

(2b) All other concepts (or terms) occurring in S are composed of (or are definable from) these fundamental concepts (or terms).

(3a) There are in S a number of so-called fundamental propositions.

(3b) All other propositions of S follow from or are grounded in (or are provable or demonstrable from) these fundamental propositions.

(4) All propositions of S are true.

(5) All propositions of S are universal and necessary in some sense or another.

(6) All propositions of S are known to be true. A non-fundamental proposition is known to be true through its proof in S.

(7) All concepts or terms of S are adequately known. A non-fundamental concept is adequately known through its composition (or definition).

This systematization represents a general historical hypothesis insofar as it aims at capturing an ideal that many philosophers and scientists adhered to for more than two millennia, going back ultimately to Aristotle’s "Analytica Posteriora". This cluster of conditions has been set up as a rational reconstruction of particular philosophical systems, which is also meant to serve as a fruitful interpretative framework for a comparative evaluation of the way certain concepts/ideas evolved in the history of philosophy.

#### Call for papers

The focus of this conference will be the rise of the (formal) axiomatic method in the deductive sciences from Leibniz to Tarski on the basis of the so-called Classical Model (or Ideal) of Science. Although preference will be given to contributions matching this focus, *we welcome and strongly encourage submissions* discussing historical developments of the ideal of scientific knowledge as "cognitio ex principiis" as sketched above *concerning any epoch or longer period*. The historical studies should aim at a philosophical understanding of the role and development of the seven conditions listed above in the rise of modern science. Contributed papers will be programmed in parallel sessions (30-40 minute presentations, of which about half for discussion).

Topics of interest include, but are not limited to:

- Leibniz's Characteristica universalis, and the ideals of "lingua characteristica" and "calculus ratiocinator"

- Analysis and proper scientific explanation in Wolff and Kant

- Grounding and Logical Consequence from Bolzano to Tarski

- Explanation in mathematics from Leibniz to Tarski

- Epistemology and metatheory in Frege

- The relation between descriptive psychology, ontology, logic and axiomatic method in Meinong

- Knowing the principles and self-evidence in Husserl's conception of logic

- Mereology and axiomatics in 19th century mathematics

- The role of mereology as formal ontology in the system of sciences

- The notion of form in 19th and 20th century logic and mathematics

- Russell’s conception of axiomatics

- The disappearance of epistemology from 19th and 20th century geometry

- Axiomatics, truth and consequence in the Lvov-Warsaw School

- Logic as calculus, logic as language

- Type theory, range of quantifiers and domain of discourse in the early 20th century

- Interpretation, satisfaction and the history of model theory

- The axiomatisation of particular disciplines such as logic, mereology, set theory, geometry and physics but also biology, chemistry and linguistics

- Constitution systems

- The analytic-synthetic distinction

- The unity of science

- Axiomatics and model theory

- Axiomatics and extensionality constraints

Abstracts (maximum 500 words) must be sent in electronic form to axiom.erc at gmail.com. They must contain the author's name, address, institutional affiliation and e-mail address.

Deadline for submission: April 15th, 2011 Authors will be notified of the acceptance of their submission by May 1st, 2011.

*Please notice that we are currently trying to arrange conference child care for speakers. More information on this facility will follow.*

#### Additional information

The history of the methodology systematised in the model as presented above knows three milestones: Aristotle's "Analytica Posteriora", the "Logic of Port-Royal" (1662) and Bernard Bolzano's "Wissenschaftslehre" (1837). In all generality the historical influence of this model has been enormous. In particular, it dominated the philosophy of science of the Seventeenth, and Eighteenth Century (Newton, Spinoza, Descartes, Leibniz, Wolff, Kant) but its influence is still clear in Husserl, Frege and Lesniewski.

The axiomatisation of various scientific disciplines involved a strict characterisation of the 'domain' of objects and the list of primitive predicates, strict rules of composition of well-formed formulas, the determination of fundamental axioms (or axiom schemas), formal inference rules, a formalisation of the truth-concept, and a formalisation of modality. The success of the model can be seen in the formalisation of logic (Boole, Schröder, Peirce, Frege, Whitehead & Russell, Lesniewski), the axiomatisation of geometry (Hilbert, Veblen, Whitehead), the axiomatisation of set theory (Zermelo, Fraenkel, Bernays, von Neumann), the axiomatisation of physics (Vienna Circle), or in the construction of constitution systems (Carnap, Goodman). However, full and rigorous formalisation also made visible some of the intrinsic limitations of classical axiomatic methodology: problems with the determination of ontological domains (e.g. pure set theory instead of physical Ur-elements, de-interpretation and the rise of model theory), problems with the characterisation of fundamental concepts (e. g. the debate on the analytic-synthetic distinction), the separation between truth and proof, the demise of the ideal of the unity of science, etc.

The first Classical Model of Science conference took place in January 2007.

For more information on the Classical Model of Science, its formulation and its application as an interpretive tool from Proclus to Lesniewski and until today, see the papers in Betti & de Jong 2010 (http://bit.ly/hlB5yb by Arianna Betti, Paola Cantù, Wim de Jong, Tapio Korte, Sandra Lapointe and Marije Martijn) and in Betti, de Jong and Martijn forthcoming (http://bit.ly/hERked, by Hein van den Berg, Jaakko Hintikka, Anita Konzelmann-Ziv, F. A. Muller, Dirk Schlimm and Patrick Suppes).

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