3 research outputs found
A topological interpretation of three Leibnizian principles within the functional extensions
Three philosophical principles are often quoted in connection with Leibniz:
"objects sharing the same properties are the same object" (Identity of
indiscernibles), "everything can possibly exist, unless it yields
contradiction" (Possibility as consistency), and "the ideal elements correctly
determine the real things" (Transfer). Here we give a precise
logico-mathematical formulation of these principles within the framework of the
Functional Extensions, mathematical structures that generalize at once
compactifications, completions, and elementary extensions of models. In this
context, the above Leibnizian principles appear as topological or algebraic
properties, namely: a property of separation, a property of compactness, and a
property of directeness, respectively. Abiding by this interpretation, we
obtain the somehow surprising conclusion that these Leibnizian principles may
be fulfilled in pairs, but not all three together.Comment: arXiv admin note: substantial text overlap with arXiv:1012.434
Procedures of Leibnizian infinitesimal calculus: An account in three modern frameworks
Recent Leibniz scholarship has sought to gauge which foundational framework
provides the most successful account of the procedures of the Leibnizian
calculus (LC). While many scholars (e.g., Ishiguro, Levey) opt for a default
Weierstrassian framework, Arthur compares LC to a non-Archimedean framework SIA
(Smooth Infinitesimal Analysis) of Lawvere-Kock-Bell. We analyze Arthur's
comparison and find it rife with equivocations and misunderstandings on issues
including the non-punctiform nature of the continuum, infinite-sided polygons,
and the fictionality of infinitesimals. Rabouin and Arthur claim that Leibniz
considers infinities as contradictory, and that Leibniz' definition of
incomparables should be understood as nominal rather than as semantic. However,
such claims hinge upon a conflation of Leibnizian notions of bounded infinity
and unbounded infinity, a distinction emphasized by early Knobloch.
The most faithful account of LC is arguably provided by Robinson's framework.
We exploit an axiomatic framework for infinitesimal analysis called SPOT
(conservative over ZF) to provide a formalisation of LC, including the
bounded/unbounded dichotomy, the assignable/inassignable dichotomy, the
generalized relation of equality up to negligible terms, and the law of
continuity.Comment: 52 pages, to appear in British Journal for the History of Mathematic
A topological interpretation of three Leibnizian principles within the functional extensions
Three philosophical principles are often quoted in connection with Leibniz:
"objects sharing the same properties are the same object" (Identity of
indiscernibles), "everything can possibly exist, unless it yields
contradiction" (Possibility as consistency), and "the ideal elements correctly
determine the real things" (Transfer). Here we give a precise
logico-mathematical formulation of these principles within the framework of the
Functional Extensions, mathematical structures that generalize at once
compactifications, completions, and elementary extensions of models. In this
context, the above Leibnizian principles appear as topological or algebraic
properties, namely: a property of separation, a property of compactness, and a
property of directeness, respectively. Abiding by this interpretation, we
obtain the somehow surprising conclusion that these Leibnizian principles may
be fulfilled in pairs, but not all three together