This paper aims to investigate certain aspects of Weyl’s account of implicit definitions. The paper takes under consideration Weyl’s approach to a certain kind of implicit definitions i.e. abstraction principles introduced by Frege.ion principles are bi-conditionals that transform certain equivalence relations into identity statements, defining thereby mathematical terms in an implicit way. The paper compares the analytic reading of implicit definitions offered by the Neo-Fregean program with Weyl’s account which has phenomenological leanings. The paper suggests that Weyl’s account should be (...) construed as putting emphasis on intentionality of human mind towards certain invariant features of the elements of initial domains of discourse that are involved in equivalence relations. Definition of terms like direction, shape, number etc. is achieved by a kind of transformation of those invariants into ideal objects that is involved in intuition. Then the paper argues that at the period of 1926 Weyl’s writings on implicit definitions, he is inclined to endorse symbolic construction as a way to explicate the objectivity of certain processes as those that are carried out in case of implicit definitions. (shrink)

In this paper we attempt to clear out the ground concerning the Aristotelian notion of density. Aristotle himself appears to confuse mathematical density with that of mathematical continuity. In order to enlighten the situation we discuss the Aristotelian notions of infinity and continuity. At the beginning, we deal with Aristotle’s views on the infinite with respect to addition as well as to division. In the sequel, we focus our attention to points and discuss their status with respect to the actuality–potentiality (...) distinction. Then we focus our attention to the nature of continuity, which Aristotle tends to consider as leading to infinite divisibility and potential density of the extended. (shrink)

In this paper we offer a critical account of Aristotelian theory of time. After a brief presentation of the main views of Aristotle on the infinite, we focus the attention to the status of points with respect to the potentiality-actuality distinction. Then we address Aristotle’s views on time on the basis of the Aristotelian notion of continuity. We construe the “nows” as potentialities awaiting to be actualized. We show that it is the intervention of an agent, who, through finitely many (...) unitary mental acts of noticing or perceiving, guarantees the actualization of particular “nows”, constructs or brings into existence time intervals and through their comparison measures them, so producing what Aristotle calls time. (shrink)

This paper addresses Harman’s approach to enumerative induction as a case of inference to the best explanation. Αfter taking under brief consideration Hume’s critique to induction, the paper argues that Harman’s proposal does not improve the situation since the same characteristics of induction and the kind of skepticism associated with it reappear in case of inference to the best explanation. Then the paper questions Armstrong’s attempt to upgrade Harman’s suggestion by regarding a necessitation relation among two universals (a natural law) (...) as the best explanans of the n observed cases of an enumerative induction procedure. It holds that Armstrong’s modification of Harman’s approach faces some difficulties which arise in the context of his views about instantiated universals. Instead, the paper investigates the possibility to assign the role of the explanans to a metaphysically stronger condition. It suggests that accounts of natural laws as metaphysically necessary facts concerning natural kinds can contribute in an attempt for a more fertile application of Harman’s approach. (shrink)

This paper addresses the issue of stipulation in three cases of implicit definitions. It argues that the alleged implicit definitions do not have a purely stipulative status. Stipulation of the vehicles of the implicit definitions in question should end up with true postulates. However, those postulates should not be taken to be true only in virtue of stipulation since they have extra commitments. Horwich’s worry emerges in all three kinds of implicit definitions under consideration, since the existence of meanings so (...) that the alleged postulates are true depends on extra requirements that should be fulfilled. Moreover, if Ramseyfication method is applied to the three kinds of implicit definition, they are split up into two components from which the first one is broadly factual while the second one is purely stipulative. The paper argues that their definitional task in each case should be assigned to their second component i.e. their Carnap-conditional. (shrink)

This paper discusses some of C. S. Peirce’s insights about continuity in his attempt to grasp the concept of the mathematical continuum. After a discussion of his earlier notions which he called ‘Kanticity’ and ‘Aristotelicity’ we arrive at his later belief that a continuum is rather a system of potential points. In his mature views, Peirce grasps a continuum as “a whole range of possibilities” without points at all. In the sequel, we turn to take into account some of Leibniz’s (...) attempts to deal with continuity and the continuum and we compare Peirce and Leibniz’s approaches detecting certain impressive similarities and differences. (shrink)

In this paper I address G. C. Rota’s account of mathematical identity and I attempt to relate it with aspects of Frege as well as Husserl’s views on the issue. After a brief presentation of Rota’s distinction among mathematical facts and mathematical proofs, I highlight the phenomenological background of Rota’s claim that mathematical objects retain their identity through different kinds of axiomatization. In particular, I deal with Rota’s interpretation of the ontological status of mathematical objects in terms of ideality. Then (...) I detect certain similarities among Rota’s views and Frege’s account of the constitution of arithmetical identity on the grounds of 1–1 correspondence. I point out an epistemic as well as an ontological aspect of this issue. In the sequel, I attempt to deal with the problem of “mixed identities” in mathematics stated by Benacerraf by taking in account Rota’s use of the phenomenological notion “Fundierung”. (shrink)