95669 - Logic (2) (Lm)

Learning outcomes

At the end of the course, students are supposed to become acquainted with the semantics and the proof theory for classical logic or some extensions, such as modal logic, or alternative systems such as intuitionistic logic or other non classical logics.

Course contents

THE LOGICAL CRITICISM OF INFINITESIMAL CALCULUS: TWO CASE STUDIES

We will compare George Berkeley's and René Guenon's interpretations of infinitesimal calculus, in particular for the Leibnizian approach.

These two authors are strongly related by some fundamental philosophical conceptions and biographical experiences. In both authors the reference to physical-mathematical sciences is always constant, and addressed on the basis of their own intellectual perspectives in open contrast with the academic standards. This will lead them, in different ways, to converge in the writing of two works of critical analysis of infinitesimal calculus, respectively The Analyst (1734) and The Principles of the Infinitesimal Calculus (1946). In particular, both authors believed that the "usual" interpretation of infinitesimal calculus contained contradictions on a logical, philosophical and mathematical level. Despite a shared conceptual core, the two works came nevertheless to differ on specific issues, such as the nature of higher order infinitesimals and the value given to error compensation theories.

In our current age, the relationships that can be traced with contemporary logic are also stimulating, as regards the theory of deduction starting from contradictory premises (Berkeley), and the treatment of evanescent sequences in Model Theory and Non-Standard Analysis (Guénon).

In the course, after an introduction to Gottfried Leibniz's approach to infinitesimal calculus, we will move on to the analysis of the criticisms made by Berkeley and Guénon, both from a logical technical-formal point of view and from a philosophical one (in the awareness that these two points of view are in reality strongly connected, indeed, inseparable, already in the conception of the two authors themselves).

Readings/Bibliography

H.J.M. Bos: Differentials, Higher-Order Differentials and the Derivative in the Leibnizian Calculus. Archive for History of Exact Sciences, 14: 1-90, 1974.

G. Berkeley:The Analist, 1734 (any edition is fine).

R. Guénon: The Principles of Inifnitesimal Calculus, 1946 (any edition is fine). 

Non-attending students must also study the chapter on The Analist contained in L. Neri: George Berkeley. Filosofia e Critica dei Linguaggi Scientifici, CLUEB, 1992.

Teaching methods

Classes with the aid of a video projector and electronic blackboard. Lessons that are more focused on formal demonstrations will be recorded and made available online on the platform Virtuale.

Assessment methods

The exam will consist of an oral test in which students will be asked to demonstrate their understanding of the concepts covered in class through oral presentation but also through the written treatment of the definitions of the basic concepts and of the formal proofs analyzed during the lessons.

 

To take the exam, it is possible not only to register for the exams set by the teacher (approximately every two months) but also to agree on individual exams with the teacher.

 

Assessment criteria and thresholds of evaluation:

30 cum laude: Excellent as to knowledge, terminology and critical expression.

30: Excellent, knowledge is complete, well articulated and mostly correctly expressed, although with some slight faults.

27-29: Good, knowledge comprehensive and satisfactory, essentially correct expression.

24-26: Fairly good, knowledge present in significant points, but not complete and not always expressed with correctness.

21-23: Sufficient, knowledge is sometimes superficial, but the guiding general thread is included. Expression and articulation incomplete and often not appropriate

18-21:.Almost sufficient, but knowledge present only on the surface. The guiding principle is not included with continuity. The expression and articulation of the speech show important gaps.

<18: Not sufficient, knowledge absent or very incomplete, lack of guidance in discipline, expression seriously deficient. Exam failed.

Teaching tools

Electronic whiteboard.

Videoprojector.

Recording of some lessons with particular use of logical-mathematical formalism.

Office hours

See the website of Guido Gherardi