43 research outputs found
Relative systoles of relative-essential 2-complexes
We prove a systolic inequality for the phi-relative 1-systole of a
phi-essential 2-complex, where phi is a homomorphism from the fundamental group
of the complex, to a finitely presented group G. Indeed we show that
universally for any phi-essential Riemannian 2-complex, and any G, the area of
X is bounded below by 1/8 of sys(X,phi)^2. Combining our results with a method
of Larry Guth, we obtain new quantitative results for certain 3-manifolds: in
particular for Sigma the Poincare homology sphere, we have sys(Sigma)^3 < 24
vol(Sigma).Comment: 20 pages, to appear in Algebraic and Geometric Topolog
Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow
Fermat, Leibniz, Euler, and Cauchy all used one or another form of
approximate equality, or the idea of discarding "negligible" terms, so as to
obtain a correct analytic answer. Their inferential moves find suitable proxies
in the context of modern theories of infinitesimals, and specifically the
concept of shadow. We give an application to decreasing rearrangements of real
functions.Comment: 35 pages, 2 figures, to appear in Notices of the American
Mathematical Society 61 (2014), no.
Is Leibnizian calculus embeddable in first order logic?
To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro-
cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If,
as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than
modern Weierstrassian ones
Cauchy, infinitesimals and ghosts of departed quantifiers
Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been
interpreted in both a Weierstrassian and Robinson's frameworks. The latter
provides closer proxies for the procedures of the classical masters. Thus,
Leibniz's distinction between assignable and inassignable numbers finds a proxy
in the distinction between standard and nonstandard numbers in Robinson's
framework, while Leibniz's law of homogeneity with the implied notion of
equality up to negligible terms finds a mathematical formalisation in terms of
standard part. It is hard to provide parallel formalisations in a
Weierstrassian framework but scholars since Ishiguro have engaged in a quest
for ghosts of departed quantifiers to provide a Weierstrassian account for
Leibniz's infinitesimals. Euler similarly had notions of equality up to
negligible terms, of which he distinguished two types: geometric and
arithmetic. Euler routinely used product decompositions into a specific
infinite number of factors, and used the binomial formula with an infinite
exponent. Such procedures have immediate hyperfinite analogues in Robinson's
framework, while in a Weierstrassian framework they can only be reinterpreted
by means of paraphrases departing significantly from Euler's own presentation.
Cauchy gives lucid definitions of continuity in terms of infinitesimals that
find ready formalisations in Robinson's framework but scholars working in a
Weierstrassian framework bend over backwards either to claim that Cauchy was
vague or to engage in a quest for ghosts of departed quantifiers in his work.
Cauchy's procedures in the context of his 1853 sum theorem (for series of
continuous functions) are more readily understood from the viewpoint of
Robinson's framework, where one can exploit tools such as the pointwise
definition of the concept of uniform convergence.
Keywords: historiography; infinitesimal; Latin model; butterfly modelComment: 45 pages, published in Mat. Stu