Periodic Reporting for period 2 - REALE (Reassessing Leibniz's conception of number and the infinite)
Période du rapport: 2023-01-01 au 2023-12-31
The project aims at reconstructing Leibniz’s conception of number and his mathematical notion of infinite in light of his mereological theory (i.e. the theory of the parthood relation) of quantity. The project is articulated into three main parts: the study of Leibniz’s mereological calculus and its philosophical implications; the study of Leibniz’s theory of quantity, its metaphysical applications, and its connection with the notion of number, in particular real numbers; the role of infinity in his conception of the foundation of mathematics.
The topic is particularly relevant for the promise of reassessing Leibniz's conception of the fundamental mathematical concepts beyond those related to the infinitesimal calculus. it aims at providing a richer understating of Leibniz's role in the history of philosophy and mathematics.
The first objective is to provide a formalization of Leibniz’s mereological calculus and to study its logical properties. In particular, contemporary mereological is exploited to assess, from a logical point of view, the straighten and the weakness of Leibniz's fremework.
The second objective is to clarify Leibniz's theory of quantity and the notion of number. After reconstructing the rules and operations that regulate the notion of quantity, and after explaining Leibniz's ideas concerning the problem of estimating quantities of different natures (estimating forces, lengths, volumes, etc.), the project shows how the concept of number (more specifically, positive relation number) is derived from this theory.
The third objective is to investigate the view of the mathematical infinite in the light of the first two objectives. Particular attention is devoted to the relation between the distinction between the potential and the actual infinite and the reasons why Leibniz endorsed both of them. Moreover, it is shown how the mereological framework justifies Leibniz’s premises in his argument against the infinite number.
The topic is particularly relevant for the promise of reassessing Leibniz's conception of the fundamental mathematical concepts beyond those related to the infinitesimal calculus. it aims at providing a richer understating of Leibniz's role in the history of philosophy and mathematics.
The first objective is to provide a formalization of Leibniz’s mereological calculus and to study its logical properties. In particular, contemporary mereological is exploited to assess, from a logical point of view, the straighten and the weakness of Leibniz's fremework.
The second objective is to clarify Leibniz's theory of quantity and the notion of number. After reconstructing the rules and operations that regulate the notion of quantity, and after explaining Leibniz's ideas concerning the problem of estimating quantities of different natures (estimating forces, lengths, volumes, etc.), the project shows how the concept of number (more specifically, positive relation number) is derived from this theory.
The third objective is to investigate the view of the mathematical infinite in the light of the first two objectives. Particular attention is devoted to the relation between the distinction between the potential and the actual infinite and the reasons why Leibniz endorsed both of them. Moreover, it is shown how the mereological framework justifies Leibniz’s premises in his argument against the infinite number.
The first year has been dedicated to the first objective. The research has been focused on Leibniz’s mereology as a formal calculus. Moreover, I have also deal with the philosophical problems arising form this calculus. In particular, I worked on Leibniz’s papers on Real Addition (the Plus-Minus Calculus) where Leibniz develops his mereological calculus. The analysis of Leibniz’s mereology has also exploited the other logical papers by Leibniz (in particular the Generales Inquisitiones) and some more mathematical works where Leibniz defines a number of important concepts (such as that of similarity, congruence, equality and homogeneity) which are at the heart of his notion of part. This study has been supplemented with a continuous study of contemporary mereology, with the aim of evaluating Leibniz’s achievement with regard to contemporary mereological formal systems.
This research has produced the following papers:
1. Leibniz’s on the Empty Term Nothing;
2. Composition as identity and the logical roots of Leibniz’s nominalism;
3. Leibniz’s Mereology: a Logical Reconstruction.
The second year has been dedicated to the second objective concerning the notions of quantity and numbers. The theory of quantity, which is based on mereological notions, gives us the general background to analysis the concept of number. I worked on the notion of quantity both from a mathematical point of view (reconstructing a formal theory of quantity) and from a metaphysical point of view (where I worked I on the role of the notion of quantity in Leibniz’s philosophy more generally). I have also analyzed the way in which Leibniz’s introduce the notion of quantity in the 1680s via a procedure that is reminiscent of definitions by abstractions.
This research has produced the following papers:
4. Quantity as Limit: Leibniz on the Metaphysics of Quantity; a paper devoted to the metaphysical interpretation of the notion of quantity in Leibniz’s philosophy.
5. Definitions by Abstraction and Leibniz's Notion of Quantity: a paper devoted to the abstractionist way in which quantity and other cognate notions are introduced by Leibniz in the 1680s.
6. From Measure to Numbers. Leibniz on Real Numbers. On the justification of real numbers in Leibniz via his own theory of measure.
The third year has been dedicated to the third objective of the project concerning the notion of infinite. First, I focused on the mathematical notion of infinite and I argued that his theory of measure clarifies Leibniz's position on the topic. In order for this theory to always provide an exact determination of ratios between magnitudes, when it deals with infinite sequences, these sequences must be considered actual, and not merely potential infinities (otherwise we would not have any exact measure, but only an approximation). Second, I focus on the metaphysical notion of infinite showing, once again, that the theory of measure plays a central role also in this context, especially with regard to Leibniz's rejection of the existence of a soul of the world.
The research has produced the following papers:
7.Estimating quantities. On how much of a thing there is.
8.There is no anima mundi. Leibniz's Argument against the Soul of the World.
This research has produced the following papers:
1. Leibniz’s on the Empty Term Nothing;
2. Composition as identity and the logical roots of Leibniz’s nominalism;
3. Leibniz’s Mereology: a Logical Reconstruction.
The second year has been dedicated to the second objective concerning the notions of quantity and numbers. The theory of quantity, which is based on mereological notions, gives us the general background to analysis the concept of number. I worked on the notion of quantity both from a mathematical point of view (reconstructing a formal theory of quantity) and from a metaphysical point of view (where I worked I on the role of the notion of quantity in Leibniz’s philosophy more generally). I have also analyzed the way in which Leibniz’s introduce the notion of quantity in the 1680s via a procedure that is reminiscent of definitions by abstractions.
This research has produced the following papers:
4. Quantity as Limit: Leibniz on the Metaphysics of Quantity; a paper devoted to the metaphysical interpretation of the notion of quantity in Leibniz’s philosophy.
5. Definitions by Abstraction and Leibniz's Notion of Quantity: a paper devoted to the abstractionist way in which quantity and other cognate notions are introduced by Leibniz in the 1680s.
6. From Measure to Numbers. Leibniz on Real Numbers. On the justification of real numbers in Leibniz via his own theory of measure.
The third year has been dedicated to the third objective of the project concerning the notion of infinite. First, I focused on the mathematical notion of infinite and I argued that his theory of measure clarifies Leibniz's position on the topic. In order for this theory to always provide an exact determination of ratios between magnitudes, when it deals with infinite sequences, these sequences must be considered actual, and not merely potential infinities (otherwise we would not have any exact measure, but only an approximation). Second, I focus on the metaphysical notion of infinite showing, once again, that the theory of measure plays a central role also in this context, especially with regard to Leibniz's rejection of the existence of a soul of the world.
The research has produced the following papers:
7.Estimating quantities. On how much of a thing there is.
8.There is no anima mundi. Leibniz's Argument against the Soul of the World.
The projects has provided a general reconstruction of Leibniz's mereological calculus, which is something that nobody did it before.
The detail reconstruction of Leibniz's theory of the estimation of quantity and its role for the concept of number is also something new in the literature. It is a merit of the project to stress the central role of Leibniz's theory of measure for a variety of fundamental mathematical notions.
A new argument for Leibniz'a acceptance of the actual infinite in mathematics (via his theory of measure) has been provided.
A new justification of the distinction between the infinite of the universe and the infinite of any organic body has been provided.
The expected impact is, first, to reshape the discussion between scholars working on Leibniz's conception of mathematics and metaphysics, and second, to stress the importance of Leibniz's views with regard to the current debates in philosophy of mathematics.
The detail reconstruction of Leibniz's theory of the estimation of quantity and its role for the concept of number is also something new in the literature. It is a merit of the project to stress the central role of Leibniz's theory of measure for a variety of fundamental mathematical notions.
A new argument for Leibniz'a acceptance of the actual infinite in mathematics (via his theory of measure) has been provided.
A new justification of the distinction between the infinite of the universe and the infinite of any organic body has been provided.
The expected impact is, first, to reshape the discussion between scholars working on Leibniz's conception of mathematics and metaphysics, and second, to stress the importance of Leibniz's views with regard to the current debates in philosophy of mathematics.