There is an urgent need in philosophy of mathematics for new approaches which pay closer attention to mathematical practice. This book will blaze the trail: it offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.

This paper explores the epistemology of extrapolation from model organisms to humans in molecular medicine. We take into account two common views on the issue, the homology view and the disanalogy view. In response to both interpretations, we argue that the foundational basis of extrapolations cannot simply be provided by homology and that relevant disanalogies can, thanks to the techniques of molecular biology, be experimentally controlled and exploited to allow useful and reliable extrapolations. The case of "humanised mice" in the (...) context of cancer stem cell research provides evidence of how animal models can be construed to approximate bona fide causal analogue models of human diseases. To supplement this view we show how the epistemology of model organisms needs to take into account the engineering side of molecular medicine. Model organisms are often manipulated to create analogies or remove disanalogies with the target system. We maintain that highlighting this feature is fundamental to explain what warrants extrapolation in the search for the molecular causes of disease. (shrink)

In the last two decades there has been renewed interest in visualization in logic and mathematics. Visualization is usually understood in different ways but for the purposes of this article I will take a rather broad conception of visualization to include both visualization by means of mental images as well as visualizations by means of computer generated images or images drawn on paper, e.g. diagrams etc. These different types of visualization can differ substantially but I am interested in offering a (...) characterization of visualization that is as broad as possible. The article describes and explains (1) the way in which visual thinking fell into disrepute, (2) the renaissance of visual thinking in mathematics over recent decades, (3) the ways in which visual thinking has been rehabilitated in epistemology of mathematics and logic. (shrink)

The mnemonic arts and the idea of a universal language that would capture the essence of all things were originally associated with cryptology, mysticism, and other occult practices. And it is commonly held that these enigmatic efforts were abandoned with the development of formal logic in the seventeenth century and the beginning of the modern era. In his distinguished book, Logic and the Art of Memory Italian philosopher and historian Paolo Rossi argues that this view is belied by an (...) examination of the history of the idea of a universal language. Based on comprehensive analyses of original texts, Rossi traces the development of this idea from late medieval thinkers such as Ramon Lull through Bruno, Bacon, Descartes, and finally Leibniz in the seventeenth century. The search for a symbolic mode of communication that would be intelligible to everyone was not a mere vestige of magical thinking and occult sciences, but a fundamental component of Renaissance and Enlightenment thought. Seen from this perspective, modern science and combinatorial logic represent not a break from the past but rather its full maturity. Available for the first time in English, this book (originally titled Clavis Universalis ) remains one of the most important contributions to the history of ideas ever written. In addition to his eagerly anticipated translation, Steven Clucas offers a substantial introduction that places this book in the context of other recent works on this fascinating subject. A rich history and valuable sourcebook, Logic and the Art of Memory documents an essential chapter in the development of human reason. (shrink)

Pierre de Fermat is known as the inventor of modern number theory. He invented–improved many methods useful in this discipline. Fermat often claimed to have proved his most difficult theorems thanks to a method of his own invention: the infinite descent. He wrote of numerous applications of this procedure. Unfortunately, he left only one almost complete demonstration and an outline of another demonstration. The outline concerns the theorem that every prime number of the form 4n + 1 is the sum (...) of two squares. In this paper, we analyse a recent proof of this theorem. It is interesting because: it follows all the elements of which Fermat wrote in his outline; it represents a good introduction to all logical nuances and mathematical variants concerning this method of which Fermat spoke. The assertions by Fermat will also be framed inside their historical context. Therefore, the aims of this paper are related to the history of mathematics and to the logic of proof-methods. (shrink)

After giving a brief overview of the renewal of interest in logic and the foundations of mathematics in Göttingen in the period 1914-1921, I give a detailed presentation of the approach to the foundations of mathematics found in Behmann's doctoral dissertation of 1918, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead. The dissertation was written under the guidance of David Hilbert and was primarily intended to give a clear exposition of the solution to (...) the antinomies as found in Principia Mathematica. In the process of explaining the theory of Principia, Behmann also presented an original approach to the foundations of mathematics which saw in sense perception of concrete individuals the Archimedean point for a secure foundation of mathematical knowledge. The last part of the paper points out an important numbers of connections between Behmann's work and Hilbert's foundational thought. (shrink)

At the same time, the book is a contribution to recent philosophical debates, in particular on the prospects for a successful nominalist reconstruction of .

Learning to perceive is faced with a classical paradox: if understanding is required for perception, how can we learn to perceive something new, something we do not yet understand? According to the sensorimotor approach, perception involves mastery of regular sensorimotor co-variations that depend on the agent and the environment, also known as the “laws” of sensorimotor contingencies (SMCs). In this sense, perception involves enacting relevant sensorimotor skills in each situation. It is important for this proposal that such skills can be (...) learned and refined with experience and yet up to this date, the sensorimotor approach has had no explicit theory of perceptual learning. The situation is made more complex if we acknowledge the open-ended nature of human learning. In this paper we propose Piaget’s theory of equilibration as a potential candidate to fulfill this role. This theory highlights the importance of intrinsic sensorimotor norms, in terms of the closure of sensorimotor schemes. It also explains how the equilibration of a sensorimotor organization faced with novelty or breakdowns proceeds by re-shaping pre-existing structures in coupling with dynamical regularities of the world. This way learning to perceive is guided by the equilibration of emerging forms of skillful coping with the world. We demonstrate the compatibility between Piaget’s theory and the sensorimotor approach by providing a dynamical formalization of equilibration to give an explicit micro-genetic account of sensorimotor learning and, by extension, of how we learn to perceive. This allows us to draw important lessons in the form of general principles for open-ended sensorimotor learning, including the need for an intrinsic normative evaluation by the agent itself. We also explore implications of our micro-genetic account at the personal level. (shrink)

A recent article of Chernikov, Hrushovski, Kruckman, Krupinski, Moconja, Pillay, and Ramsey finds the first examples of simple structures with formulas which do not fork over the empty set but are universally measure zero. In this article we give the first known simple $\omega $ -categorical counterexamples. These happen to be various $\omega $ -categorical Hrushovski constructions. Using a probabilistic independence theorem from Jahel and Tsankov, we show how simple $\omega $ -categorical structures where a formula forks over $\emptyset $ (...) if and only if it is universally measure zero must satisfy a stronger version of the independence theorem. (shrink)

We will discuss a mathematical proof found in Wittgenstein’s Nachlass, a constructive version of Euler’s proof of the infinity of prime numbers. Although it does not amount to much, this proof allows us to see that Wittgenstein had at least some mathematical skills. At the very last, the proof shows that Wittgenstein was concerned with mathematical practice and it also gives further evidence in support of the claim that, after all, he held a constructivist stance, at least during the transitional (...) period of his thought (1929-33). (shrink)

Can we build endogenous expectations about the locus of occurrence of a target without being able to describe them? Participants performed cue–target detection tasks with different proportions of valid and invalid trials, without being informed of these proportions, and demonstrated typical endogenous effects. About half were subsequently able to correctly describe the cue–target relationships . However, even non-verbalizer participants showed endogenous orienting with peripheral cues , not depending solely on practice . Explicit instructions did not bring about dramatic advantages in (...) performance .With central symbolic cues, only verbalizers showed reliable endogenous effects . We concluded that endogenous orienting with peripheral cues can occur independently of participants developing explicit hypotheses about the cue–target relationships. (shrink)

No chance of seeing her for another fortnight and it is 11 days since I saw her. Went solitary walk felt miserable but to some extent staved it off by reflecting on |$\langle$|Continuum Problem|$\rangle$|1The occasion for this review article on the life and accomplishments of Frank Ramsey is the publication in the last eight years of three important books: a biography of Frank Ramsey by his sister, Margaret Paul, a book by Steven Methven on aspects of Ramsey’s philosophy, and the (...) recent biography of Ramsey written by Cheryl Misak. In a companion piece [Mancosu, 2020] I discuss Ramsey’s contacts with logicians in continental Europe and publish previously unpublished letters from Ramsey to Abraham Fraenkel and to Heinrich Behmann.1. INTRODUCTIONFrank Plumpton Ramsey (1903–1930) will be well known to the readers of this journal. His contributions to philosophy, mathematics, and economics have left a lasting legacy. In mathematical logic and philosophy of mathematics, Ramsey’s solution to the paradoxes by means of a simple theory of types [1926a] and Ramsey’s theorem in combinatorics (both finite and infinite [1930]) are among Ramsey’s most celebrated contributions. Ramsey is a key presence in accounts of Wittgenstein’s, Russell’s, and Moore’s lives and work. It is well known that it was Ramsey who translated the Tractatus into English and wrote what is still considered a very insightful review [1923] of the Tractatus. Wittgenstein mentioned the importance of his conversations with Ramsey in 1929 in Cambridge as having influenced his transition to the views presented in the Philosophical Investigations. Ramsey also contributed, in 1924, to the second edition of Principia Mathematica (see [Linsky, 2011]). While Ramsey’s friendship with Moore does not have an analogous concrete precipitate as the work on the Tractatus or Principia Mathematica, it is evident from the correspondence and other documents that Ramsey’s work in philosophy (truth, propositions, universals, etc.) influenced Moore. Of course, the direction of influence goes also the other way around. Wittgenstein, Russell, and Moore, among others, were essential in shaping Ramsey’s approach to philosophy. (shrink)

La struttura a livelli modella il tutto e le sue parti con grande generalità. Lo dimostrano le argomentazioni teoriche (4.3), (4.4) e (4.5), e lo comprova la casistica che è stata aggiunta. Una tale struttura non presenta affatto i problemi a), b) e c) che sono tipici della struttura ad albero. Essa corregge inoltre deviazioni concettuali che sono diffuse nel settore informatico. L’immediatezza deviazioni concettuali che sono diffuse nel settore informatico. L’immediatezza visiva, infine, ne facilita la diffusione sul piano professionale (...) e tecnico.Per concludere, ritengo che la struttura a livelli possa fornire un contributo anche alle ricerche di ontologia e in particolare possa favorire i tentativi di unificazione condotti sia sul piano teorico, sia su quello applicativo [Uschold 1996]. Le difficoltà ed i limiti che la teoria degli insiemi oppone ad un formalismo generale (ad esempio quando riduce la casistica della relazione concetto di livello possiede una valenza estremamente ampia e ciò lo predispone per gli auspicati sviluppi ontologici. (shrink)

In this study some typical aspects of human behavior are reconsidered in a new evolutionary perspective. Firstly, a theoretical paradigm is introduced, according to which animals show a natural propensity to maintain behavior far beyond the time when the triggering motivation has been removed;, these drifting conducts are defined as Vestigial Drifting Drives or. Such a paradigm is then applied to those human attitudes that once conferred an adaptive advantage on our species but have now become dysfunctional within the framework (...) of the current ecological crisis. In this context, human beings are considered the repository of VDDs that may play out as evolutionary threats. This thesis is supported by many examples and references that, hopefully, could stimulate scientific debate and further research in evolutionary anthropology, psychology, and related disciplines. (shrink)

The two theories of pancreatic enzyme secretion, those of exocytosis and transmembrane flow, are described. Data thought to support the theory of transmembrane flow of single molecules from pancreatic acinar cells are first reviewed, and the conditions which could allow these data to be explained by the theory of exocytosis of enzyme quanta, i.e. secretory granules, are then discussed.The evidence suggesting short‐term modulation of the composition of pancreatic juice is also considered, and its possible explanations at the organ and cellular (...) level are discussed, in the light of suggested theories on enzyme secretion pathways. (shrink)

ArgumentThe purpose of this paper is to provide an analysis of Giordano Bruno’s conception of mathematics. Specifically, it intends to highlight two aspects of this conception that have been neglected in previous studies. First, Bruno’s conception of mathematics changed over time and in parallel with another concept that was central to his thought: the concept of infinity. Specifically, Bruno undertook a reform of mathematics in order to accommodate the concept of the infinitely small or “minimum,” which was introduced at a (...) later stage. Second, contrary to what Héléne Védrine claimed, Bruno believed that mathematical objects were mind-dependent. To chart the parallel development of the conceptions of mathematics and infinity, a seven-year time span is considered, from the publication of Bruno’s first Italian dialogue (La cena de le ceneri,1584) to the publication of one of his last Latin works (De minimo, 1591). (shrink)

L’articolo si pone il problema del perché Vico avesse pensato alla sua impresa come ad una scienza e si fosse occupato di confrontarla con la scienza della natura. Si passano in rassegna l’uso di concetti come degnità, assiomi, postulati, tenendo fermo che Vico non fu né un cartesiano, né un galileiano, e come non si lasci racchiudere nelle gabbie dell’apriorismo e del platonismo: egli ha solo fatto sue alcune idee o tesi presenti entro quelle tradizioni. Vico si guarda bene dall’effettuare (...) una scelta precisa e definitiva tra quelle realtà alle quali attribuiamo oggi il nome di razionalismo e empirismo. Si richiama contemporaneamente e negli stessi testi sia alla tradizione del geometrismo cartesiano sia a quella di Bacone e delpensiero dei baconiani del tardo Seicento, facendone un uso alquanto libero. L’articolo contiene inoltre osservazioni su Vico lettore di Cogitata et visa e di De dignitate et augmentis scientarum di Bacone. (shrink)