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Many important concepts of the calculus are difficult to grasp, and they may appear epistemologically unjustified. For example,
how does a real function appear in “small” neighborhoods of its points? How does it appear at infinity? Diagrams allow us
to overcome the difficulty in constructing representations of mathematical critical situations and o...
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Many important concepts of the calculus are dif- ficult to grasp, and they may appear epistemologi- cally unjustified. For example, how does a real func- tion appear in "small" neighborhoods of its points? How does it appear at infinity? Diagrams allow us to overcome the difficulty in constructing repre- sentations of mathematical critical situatio...
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... It seems to me that Magnani tends to approach visual abduction by invoking geometrical diagrams. Even though does not discuss geometrical diagrams extensively, Magnani already dealt with the role of model-based and manipulative abductions in geometrical reasoning in several places ( Magnani and Dossena 2005). Since there have been many attempts to understand Peirce's philosophy of mathematics focusing on his distinction between corollarial and theorematic reasoning ( Levy 1997;Hoffmann 2000;Sternfelt 2007Sternfelt , 2011), Magnani's results in visual abduction can be easily combined with previous results in diagrammatic reasoning in geometry. ...
The book shows how eastern and western perspectives and conceptions can be used to addresses recent topics laying at the crossroad between philosophy and cognitive science. It reports on new points of view and conceptions discussed during the International Conference on Philosophy and Cognitive Science (PCS2013), held at the Sun Yat-sen University, in Guangzhou, China, and the 2013 Workshop on Abductive Visual Cognition, which took place at KAIST, in Deajeon, South Korea.
The book emphasizes an ever-growing cultural exchange between academics and intellectuals coming from different fields. It juxtaposes research works investigating new facets on key issues between philosophy and cognitive science, such as the role of models and causal representations in science; the status of theoretical concepts and quantum principles; abductive cognition, vision, and visualization in science from an eco-cognitive perspective. Further topics are: ignorance immunization in reasoning; moral cognition, violence, and epistemology; and models and biomorphism. The book, which presents a unique and timely account of the current state-of-the art on various aspects in philosophy and cognitive science, is expected to inspire philosophers, cognitive scientists and social scientists, and to generate fruitful exchanges and collaboration among them.
... A more detailed graphic representation may be found in Figure 4. 21 The key remark, due to Robinson, is that the limit in the A-approach and the standard part function in the B-approach are essentially equivalent tools. More specifically, the limit of a sequence (u n ) can be expressed, in the context of a hyperreal 21 For a recent study of optical diagrams in nonstandard analysis, see (Dossena and Magnani [31], [83]) and (Bair and Henry [8]). enlargement of the number system, as the standard part of the value u H of the natural extension of the sequence at an infinite hypernatural index n = H. ...
... The standard part function is illustrated inFigure 3. A more detailed graphic representation may be found inFigure 4. 21 The key remark, due to Robinson, is that the limit in the A-approach and the standard part function in the B-approach are essentially equivalent tools. More specifically, the limit of a sequence (u n ) can be expressed, in the context of a hyperreal 21 For a recent study of optical diagrams in nonstandard analysis, see (Dossena and Magnani [31], [83]) and (Bair and Henry [8]). enlargement of the number system, as the standard part of the value u H of the natural extension of the sequence at an infinite hypernatural index n = H. ...
We examine prevailing philosophical and historical views about the origin of
infinitesimal mathematics in light of modern infinitesimal theories, and show
the works of Fermat, Leibniz, Euler, Cauchy and other giants of infinitesimal
mathematics in a new light. We also detail several procedures of the historical
infinitesimal calculus that were only clarified and formalized with the advent
of modern infinitesimals. These procedures include Fermat's adequality;
Leibniz's law of continuity and the transcendental law of homogeneity; Euler's
principle of cancellation and infinite integers with the associated infinite
products; Cauchy's infinitesimal-based definition of continuity and "Dirac"
delta function. Such procedures were interpreted and formalized in Robinson's
framework in terms of concepts like microcontinuity (S-continuity), the
standard part principle, the transfer principle, and hyperfinite products. We
evaluate the critiques of historical and modern infinitesimals by their foes
from Berkeley and Cantor to Bishop and Connes. We analyze the issue of the
consistency, as distinct from the issue of the rigor, of historical
infinitesimals, and contrast the methodologies of Leibniz and Nieuwentijt in
this connection.
... See also P. Roquette [62] for infinitesimal reminiscences. A discussion of infinitesimal optics is in K. Stroyan [67], J. Keisler [44], D. Tall [68], and L. Magnani and R. Dossena [52,27]. Applications of infinitesimalenriched continua range from aid in teaching calculus [31,36,37] to the Bolzmann equation (see L. Arkeryd [3,4]), mathematical physics (see Albeverio et al. [1]); etc. Edward Nelson [56] in 1977 proposed an alternative to ZFC which is a richer (more stratified) axiomatisation for set theory, called Internal Set Theory (IST), more congenial to infinitesimals than ZFC. ...
Infinitesimals are natural products of the human imagination. Their history goes back to the Greek antiquity. Their role in the calculus and analysis has seen dramatic ups and downs. They have stimulated strong opinions and even vitriol. Edwin Hewitt developed hyperreal fields in the 1940s. Abraham Robinson's infinitesimals date from the 1960s. A noncommutative version of infinitesimals, due to Alain Connes, has been in use since the 1990s. We review some of the hyperreal concepts, and compare them with some of the concepts underlying noncommutative geometry.
... See also P. Roquette [62] for infinitesimal reminiscences. A discussion of infinitesimal optics is in K. Stroyan [67], J. Keisler [44], D. Tall [68], and L. Magnani and R. Dossena [52,27]. Applications of infinitesimalenriched continua range from aid in teaching calculus [31,36,37] to the Bolzmann equation (see L. Arkeryd [3,4]), mathematical physics (see Albeverio et al. [1]); etc. Edward Nelson [56] in 1977 proposed an alternative to ZFC which is a richer (more stratified) axiomatisation for set theory, called Internal Set Theory (IST), more congenial to infinitesimals than ZFC. ...
Infinitesimals are natural products of the human imagination. Their
history goes back to the Greek antiquity. Their role in the calculus and
analysis has seen dramatic ups and downs. They have stimulated strong
opinions and even vitriol. Edwin Hewitt developed hyperreal fields in
the 1940s. Abraham Robinson's infinitesimals date from the 1960s. A
noncommutative version of infinitesimals, due to Alain Connes, has been
in use since the 1990s. We review some of the hyperreal concepts, and
compare them with some of the concepts underlying noncommutative
geometry.
... The influence of Hilton Kramer (1928-2012) is obvious. [109,31], and Bair & Henry [8]. Applications of the B-continuum range from aid in teaching calculus [36,62,63,147,148] (see illustration in Figure 5) to the Bolzmann equation (see L. Arkeryd [5,6]); modeling of timed systems in computer science (see H. Rust [133]); Brownian motion and economics (see Anderson [3]); mathematical physics (see Albeverio et al. [1]); etc. ...
Many historians of the calculus deny significant continuity between
infinitesimal calculus of the 17th century and 20th century developments such
as Robinson's theory. Robinson's hyperreals, while providing a consistent
theory of infinitesimals, require the resources of modern logic; thus many
commentators are comfortable denying a historical continuity. A notable
exception is Robinson himself, whose identification with the Leibnizian
tradition inspired Lakatos, Laugwitz, and others to consider the history of the
infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies,
Robinson regards Berkeley's criticisms of the infinitesimal calculus as aptly
demonstrating the inconsistency of reasoning with historical infinitesimal
magnitudes. We argue that Robinson, among others, overestimates the force of
Berkeley's criticisms, by underestimating the mathematical and philosophical
resources available to Leibniz. Leibniz's infinitesimals are fictions, not
logical fictions, as Ishiguro proposed, but rather pure fictions, like
imaginaries, which are not eliminable by some syncategorematic paraphrase. We
argue that Leibniz's defense of infinitesimals is more firmly grounded than
Berkeley's criticism thereof. We show, moreover, that Leibniz's system for
differential calculus was free of logical fallacies. Our argument strengthens
the conception of modern infinitesimals as a development of Leibniz's strategy
of relating inassignable to assignable quantities by means of his
transcendental law of homogeneity.
... See also Roquette (2010) for infinitesimal reminiscences. A discussion of infinitesimal optics is in Stroyan (1972), Keisler (1986), Tall (1980), Magnani and Dossena (2005) and Dossena and Magnani (2007). Nelson (1977) in 1977 proposed an axiomatic theory parallel to Robinson's theory. ...
The widespread idea that infinitesimals were "eliminated" by the "great
triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an
uninterrupted chain of work on infinitesimal-enriched number systems. The
elimination claim is an oversimplification created by triumvirate followers,
who tend to view the history of analysis as a pre-ordained march toward the
radiant future of Weierstrassian epsilontics. In the present text, we document
distortions of the history of analysis stemming from the triumvirate ideology
of ontological minimalism, which identified the continuum with a single number
system. Such anachronistic distortions characterize the received interpretation
of Stevin, Leibniz, d'Alembert, Cauchy, and others.
... A helpful " semicolon " notation for presenting an extended decimal expansion of a hyperreal was described by A. H. Lightstone [126]. A discussion of infinitesimal optics is in K. Stroyan [168], H. J. Keisler [116], D. Tall [172], and L. Magnani and R. Dossena [129, 58]. P. Ehrlich recently constructed an isomorphism of maximal surreals and hyperreals [62]. ...
We examine the classical/intuitionist divide, and how it reflects on modern
theories of infinitesimals. When leading intuitionist Heyting announced that
"the creation of non-standard analysis is a standard model of important
mathematical research", he was fully aware that he was breaking ranks with
Brouwer. Was Errett Bishop faithful to either Kronecker or Brouwer? Through a
comparative textual analysis of three of Bishop's texts, we analyze the
ideological and/or pedagogical nature of his objections to infinitesimals a la
Robinson. Bishop's famous "debasement" comment at the 1974 Boston workshop,
published as part of his Crisis lecture, in reality was never uttered in front
of an audience. We compare the realist and the anti-realist intuitionist
narratives, and analyze the views of Dummett, Pourciau, Richman, Shapiro, and
Tennant. Variational principles are important physical applications, currently
lacking a constructive framework. We examine the case of the Hawking-Penrose
singularity theorem, already analyzed by Hellman in the context of the
Quine-Putnam indispensability thesis.
... See also P. Roquette [69] for infinitesimal reminiscences. A discussion of infinitesimal optics is in K. Stroyan [81], H. J. Keisler [51], D. Tall [83], L. Magnani & R. Dossena [61, 29], and Bair & Henry [6]. Applications of the B-continuum range from aid in teaching calculus [31, 45, 46, 84, 85] to the Bolzmann equation (see L. Arkeryd [3, 4]); modeling of timed systems in computer science (see H. Rust [70]); mathematical economics (see R. Anderson [2]); mathematical physics (see Albeverio et al. [1]); etc. ...
Cauchy’s sum theorem of 1821 has been the subject of rival interpretations ever since Robinson proposed a novel reading in the 1960s. Some claim that Cauchy modified the hypothesis of his theorem in 1853 by introducing uniform convergence, whose traditional formulation requires a pair of independent variables. Meanwhile, Cauchy’s hypothesis is formulated in terms of a single variable x, rather than a pair of variables, and requires the error term rn=rn(x) to go to zero at all values of x, including the infinitesimal value generated by 1 n explicitly specified by Cauchy. If one wishes to understand Cauchy’s modification/clarification of the hypothesis of the sum theorem in 1853, one has to jettison the automatic translation-to-limits.
... P. Roquette [100] for infinitesimal reminiscences. A discussion of infinitesimal optics is in K. Stroyan [113], H. J. Keisler [71], D. Tall [114], and L. Magnani and R. Dossena [86,29]. Applications of the B-continuum range from aid in teaching calculus [34,64,65,115,116] to the Bolzmann equation, see L. Arkeryd [2,3]. ...
Cauchy's contribution to the foundations of analysis is often viewed through
the lens of developments that occurred some decades later, namely the
formalisation of analysis on the basis of the epsilon-delta doctrine in the
context of an Archimedean continuum. What does one see if one refrains from
viewing Cauchy as if he had read Weierstrass already? One sees, with Felix
Klein, a parallel thread for the development of analysis, in the context of an
infinitesimal-enriched continuum. One sees, with Emile Borel, the seeds of the
theory of rates of growth of functions as developed by Paul du Bois-Reymond.
One sees, with E. G. Bjorling, an infinitesimal definition of the criterion of
uniform convergence. Cauchy's foundational stance is hereby reconsidered.
