Perceiving the Infinite and the Infinitesimal World: Unveiling and Optical Diagrams in Mathematics

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 objects. For example, they actually reveal the behavior of a real function not “close to” a point (as in the standard limit theory) but “in” the point. We are interested in our research in the diagrams which play an optical role –microscopes and “microscopes within microscopes”, telescopes, windows, a mirror role (to externalize rough mental models), and an unveiling role (to help create new and interesting mathematical concepts, theories, and structures). In this paper we describe some examples of optical diagrams as a particular kind of epistemic mediator able to perform the explanatory abductive task of providing a better understanding of the calculus, through a non-standard model of analysis. We also maintain they can be used in many other different epistemological and cognitive situations.     

The Infinite Turn and Speculative Explanations in Cosmology

Infinity, in various guises, has been invoked recently in order to ‘explain’ a number of important questions regarding observable phenomena in science, and in particular in cosmology. Such explanations are by their nature speculative. Here we introduce the notions of relative infinity, closure, and economy of explanation and ask: to what extent explanations involving relative or real constructed infinities can be treated as reasonable?     

<Emphasis Type="Italic">The Mathematical Intelligencer</Emphasis> Flunks the Olympics

The Mathematical Intelligencer recently published a note by Y. Sergeyev that challenges both mathematics and intelligence. We examine Sergeyev’s claims concerning his purported Infinity computer. We compare his grossone system with the classical Levi-Civita fields and with the hyperreal framework of A. Robinson, and analyze the related algorithmic issues inevitably arising in any genuine computer implementation. We show that Sergeyev’s grossone system is unnecessary and vague, and that whatever consistent subsystem could be salvaged is subsumed entirely within a stronger and clearer system (IST). Lou Kauffman, who published an article on a grossone, places it squarely outside the historical panorama of ideas dealing with infinity and infinitesimals.     

Belief Systems and Partial Spaces

One important role of belief systems is to allow us to represent information about a certain domain of inquiry. This paper presents a formal framework to accommodate such information representation. Three cognitive models to represent information are discussed: conceptual spaces (Gärdenfors in Conceptual spaces: the geometry of thought. MIT Press, Cambridge, 2000), state-spaces (van Fraassen in Quantum mechanics: an empiricist view. Clarendon Press, Oxford, 1991), and the problem spaces familiar from artificial intelligence. After indicating their weakness to deal with partial information, it is argued that an alternative, formulated in terms of partial structures (da Costa and French in Science and partial truth. Oxford University Press, New York, 2003), can be provided which not only captures the positive features of these models, but also accommodates the partiality of information ubiquitous in science and mathematics.     

结构实在论评析

曹天予提出的结构实在论知识论版本在科学哲学界有很大影响，并引发了《综合》期刊专刊讨论。他既批评反实在论的观点，也对结构实在论的知识版本与本体版本有所批判。他从结构、本体、建构这三个概念出发，论证了科学理论的客观性体现为建构性、历史性、整体性、修正性、革命性。本文对曹天予的结构实在论进行了系统的梳理与分析。笔者赞同他对“不充分决定”论旨的批评，对他批评“本体论不连续性”持中立态度，但对他的“普遍语法论证”有所存疑。     

Degenerating Families of Dendrograms

Dendrograms used in data analysis are ultrametric spaces, hence objects of nonarchimedean geometry. It is known that there exist p-adic representations of dendrograms. Completed by a point at infinity, they can be viewed as subtrees of the Bruhat-Tits tree associated to the p-adic projective line. The implications are that certain moduli spaces known in algebraic geometry are in fact p-adic parameter spaces of dendrograms, and stochastic classification can also be handled within this framework. At the end, we calculate the topology of the hidden part of a dendrogram.     

Weighted Graphs with Distances in Given Ranges

Let \( \mathcal{G} \) = (G,w) be a weighted simple finite connected graph, that is, let G be a simple finite connected graph endowed with a function w from the set of the edges of G to the set of real numbers. For any subgraph G′ of G, we define w(G′) to be the sum of the weights of the edges of G′. For any i, j vertices of G, we define D _{i,j}(\( \mathcal{G} \)) to be the minimum of the weights of the simple paths of G joining i and j. The D _{i,j}(\( \mathcal{G} \)) are called 2-weights of \( \mathcal{G} \). Weighted graphs and their reconstruction from 2-weights have applications in several disciplines, such as biology and psychology.Let \( {\left\{{m}_I\right\}}_{I\in \left(\frac{\left\{1,\dots, n\right\}}{2}\right)} \) and \( {\left\{{M}_I\right\}}_{I\in \left(\frac{\left\{1,\dots, n\right\}}{2}\right)} \) be two families of positive real numbers parametrized by the 2-subsets of {1, …, n} with m _I ≤ M _I for any I; we study when there exist a positive-weighted graph G and an n-subset {1, …, n} of the set of its vertices such that D _I (\( \mathcal{G} \)) ∈ [m _I ,M _I ] for any \( I\in \left(\frac{\left\{1,\dots, n\right\}}{2}\right) \). Then we study the analogous problem for trees, both in the case of positive weights and in the case of general weights.     

The Documentary Real and the Shoah

Without the support of imagination, one would not have the slightest idea of the cruel ‘real’ that has occurred in the Nazi extermination camps. Yet, in documentaries imaging the events of the Shoah, one runs the risk of missing their most basic property, namely their unimaginability. The mere idea that one is able to imagine the unimaginable comes down to a denial of the Shoah’s status as an event that defies our understanding. The unimaginable ‘real’ of the Shoah, however, is not simply located in its object, in the cruelty of what happened in the camp. The Shoah makes us at the same time facing the unimaginable ‘real’ of the modern subject—the blind spot in our own identity. If we need imagination to deal with the Shoah, it is also because of an ungraspable ‘real’ in ourselves. This is why adequate Shoah representations, acknowledging their object as being beyond representation, include the same ‘beyond’ concerning the subject of the Holocaust memory. The essay makes this clear in an elaborated comparison of Claude Lanzmann’s 1985 film, Shoah, with some conceptual works of art from the late nineties—all of this ‘fine-tuned’ in a reflection upon Ingmar Bergman’s Persona.     

The Unseen Déjà-Vu: From Erkki Huhtamo’s <Emphasis Type="Italic">Topoi</Emphasis> to Ken Jacobs’ Remakes

This commentary on Edwin Carels’ essay “Revisiting Tom Tom: Performative anamnesis and autonomous vision in Ken Jacobs’ appropriations of Tom Tom the Piper’s Son” broadens up the media-archaeological framework in which Carels places his text. Notions such as Huhtamo’s topos and Zielinski’s “deep time” are brought into the discussion in order to point out the difficulty to see what there is to see and to question the position of the viewer in front of experimental films like Tom Tom the Piper’s Son and its remakes.     

Approximation,Mad Men and the Death of JFK

In this article I take the US television series Mad Men (2007—present) as an exemplary ‘approximation’, a term I adopt to signal the way in which certain texts construct a changeable, fluid ‘truth’ resulting from collisions, exchange and dialectical argument. Approximations are layered, their formal layerings mirroring a layered, multifaceted argument. Mad Men integrates and represents real historical events within a fictional setting, and act that suggests that an event or action can never be finished, fixed and not open to reassessment. Specifically, this article examines ‘The Grown Ups’, Episode 12 of Season 3, which charts the events of 22 November 1963, the day Kennedy was assassinated. Although we might be able to bring to mind the images and conspiracy theories that have been made available since (such Abraham Zapruder’s 8 mm home movie footage of the assassination), these images were not available at the time. Mad Men as a series always strives to represent its historical milieu as authentically as possible, so the characters re-enact 22 November 1963 as authentically as possible by watching only what was on television that day (the news bulletin, Walter Kronkite’s announcement that Kennedy is dead). The contemporary backdrop to these events, including the resonances of ‘9/11’ through Mad Men, inform and collide with the authenticity on the screen.     

On the Properties of <Emphasis Type="Italic">α</Emphasis>-Unchaining Single Linkage Hierarchical Clustering

In the election of a hierarchical clustering method, theoretic properties may give some insight to determine which method is the most suitable to treat a clustering problem. Herein, we study some basic properties of two hierarchical clustering methods: α-unchaining single linkage or SL(α) and a modified version of this one, SL?(α). We compare the results with the properties satisfied by the classical linkage-based hierarchical clustering methods.     

Philosophical Issues and their Implications for the Systems Architect

Many system architects select their system methodologies without explicit consideration of the philosophical perspectives that impact their decisions. This paper describes how the concepts of ontology and epistemology apply in systems science. Ontology is how we specify terms of reference for existence, allowing us to understand the theory of existence via an ‘existence framework’. Epistemology, the theory of knowledge, allows us to explore new models and theories of knowledge acquisition so the best system-based methodologies can be deployed to solve complex system problems. After introducing these concepts, the paper presents system science issues and assesses the impact of ontological, axiological and epistemological perspectives on system methodology selection, research, system design and deployment. An ontological viewpoint such as realism, as an objective view is contrasted versus nationalism, a personal perceptional view. An epistemological viewpoint is explored comparing knowledge as a product of sensory perception or rational reflection. The paper’s significant contribution is that it helps system architects understand that their philosophical views of systems science impact their system methodology choices.     

Improved Classification for Compositional Data Using the <Emphasis Type="Italic">α</Emphasis>-transformation

In compositional data analysis, an observation is a vector containing nonnegative values, only the relative sizes of which are considered to be of interest. Without loss of generality, a compositional vector can be taken to be a vector of proportions that sum to one. Data of this type arise in many areas including geology, archaeology, biology, economics and political science. In this paper we investigate methods for classification of compositional data. Our approach centers on the idea of using the α-transformation to transform the data and then to classify the transformed data via regularized discriminant analysis and the k-nearest neighbors algorithm. Using the α-transformation generalizes two rival approaches in compositional data analysis, one (when α=1) that treats the data as though they were Euclidean, ignoring the compositional constraint, and another (when α = 0) that employs Aitchison’s centered log-ratio transformation. A numerical study with several real datasets shows that whether using α = 1 or α = 0 gives better classification performance depends on the dataset, and moreover that using an intermediate value of α can sometimes give better performance than using either 1 or 0.     

The Abductive Loop: Tracking Irrational Sets

We argue from the Church-Turing thesis (Kleene Mathematical logic. New York: Wiley 1967) that a program can be considered as equivalent to a formal language similar to predicate calculus where predicates can be taken as functions. We can relate such a calculus to Wittgenstein’s first major work, the Tractatus, and use the Tractatus and its theses as a model of the formal classical definition of a computer program. However, Wittgenstein found flaws in his initial great work and he explored these flaws in a new thesis described in his second great work; the Philosophical Investigations. The question we address is “can computer science make the same leap?” We are proposing, because of the flaws identified by Wittgenstein, that computers will never have the possibility of natural communication with people unless they become active participants of human society. The essential difference between formal models used in computing and human communication is that formal models are based upon rational sets whereas people are not so restricted. We introduce irrational sets as a concept that requires the use of an abductive inference system. However, formal models are still considered central to our means of using hypotheses through deduction to make predictions about the world. These formal models are required to continually be updated in response to peoples’ changes in their way of seeing the world. We propose that one mechanism used to keep track of these changes is the Peircian abductive loop.     

Love and Realism

In this reply I try to show that, contrary to Milberry’s apparent assertion, the general intellect of the multitude does not have the explanatory robustness she accredits to it (following both Virno and the Hardt and Negri of the Empire trilogy). Digital network technologies are currently overwhelmingly effective in proletarianizing and disempowering the cognitariat and only an active technopolitics of deproletarianization could reverse this hegemonic situation. In my response to Verbeek, I attempt to correct his misinterpretation (shared by Milberry) of the Stieglerian approach as being dialectical in nature and show that, far from reinstating the humanist dichotomy between human beings and technologies, my analysis assumes their original, albeit fundamentally ambiguous and even ‘uncanny’ [unheimlich] interconnection. I conclude with pointing out some implications of this view for a ‘really realistic’ political theory of technology.     

Spin and Wind Directions I: Identifying Entanglement in Nature and Cognition

We present a cognitive psychology experiment where participants were asked to select pairs of spatial directions that they considered to be the best example of Two different wind directions. Data are shown to violate the CHSH version of Bell’s inequality with the same magnitude as in typical Bell-test experiments with entangled spins. Wind directions thus appear to be conceptual entities connected through meaning, in human cognition, in a similar way as spins appear to be entangled in experiments conducted in physics laboratories. This is the first part of a two-part article. In the second part (Aerts et al. in Found Sci, 2017) we present a symmetrized version of the same experiment for which we provide a quantum modeling of the collected data in Hilbert space.     

Toward a History of Mathematics Focused on Procedures

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework.     

On Living with Technology through Renunciation and Releasement

Marc Van den Bosche suggests that Heidegger’s conceptions of Gestell and Gelassenheit, taken together with his analysis of Nietzschean Nihilism (interpreted especially by Wolfgang Schirmacher), depicts our era in a way that “supplements” Andrew Feenberg and Don Ihde’s work. Weaving these sources together, he sees the possibility of our becoming (quoting Schirmacher) “technicians” that “live, in a released way, within the groundless.” Here, I raise some questions about whether the author has really fitted all these sources together and argue that his idea of becoming post-modern “technicians” appears to require that we first practice a very un-Heideggerian kind of “renunciation.”