Fechner's conception of psychophysics included both “outer psychophysics” the relation between stimulus intensity and the response reflecting sensation strength, and “inner psychophysics” the relation between neurelectric responses and sensation strength. In his own time outer psychophysics focussed on the form of the psychophysical law, with Fechner espousing a logarithmic law, Delboeuf a variant of the logarithmic law incorporating a resting level of neural activity, and Plateau a power law. One of the issues on which the dispute was focussed concerned the (...) appearance of contrasts if the overall illumination was increased or decreased; another issue was the question of whether a sensation of a “just noticeable difference” established for one value of a sensory dimension appeared the same for a value elsewhere on the dimension. The development of “inner psychophysics” led through the works of Delboeuf, Solomons, Jastrow, and Thurstone to modern signal detection theory. A third line of research, devoted to the question of what was meant by the “measurement” of sensation strength, stemmed from the criticism of Fechner's work by von Kries (1882) and others. Although a valid body of science could be built up without the intervening variable called “sensation strength,” such a science might be a cumbersome representation of reality. When an optical contrast is set up, and its overall illumination is increased or decreased, subjective contrasts involving medium levels of lightness vary little as illumination varies (as a power law based on sensation ratios or a logarithmic law based on sensation differences predict), but subjective contrasts involving extreme levels of lightness might be subject to the effects of other variables. (shrink)

The main topic of this essay is symbolic mathematics or the method of symbolic construction, which I trace to the end of the sixteenth century when Franciscus Vieta invented the algebraic symbolism and started to use the word ‘symbolic’ in the relevant, non-ontological sense. This approach has played an important role for many of the great inventions in modern mathematics such as the introduction of the decimal place-value system of numeration, Descartes’ analytic geometry, and Leibniz’s infinitesimal calculus. It was also (...) central for the rigorization movement in mathematics in the late nineteenth century, as well as for the mathematics of modern physics in the 20th century.However, the nature of symbolic mathematics has been concealed and confused due to the strong influence of the heritage from the Euclidean and Aristotelian traditions. This essay sheds some light on what has been concealed by approaching some of the crucial issues from a historical perspective. Furthermore, I argue that the conception of modern mathematics as symbolic mathematics was essential to Wittgenstein’s approach to the foundations and nature of mathematics. This connection between Wittgenstein’s thought and symbolic mathematics provides the resources for countering the still prevalent view that he defended an uttrely idiosyncratic conception, disconnected from the progress of serious science. Instead, his project can be seen as clarifying ideas that have been crucial to the development of mathematics since early modernity. (shrink)

It is a well-known fact that mathematics plays a crucial role in physics; in fact, it is virtually impossible to imagine contemporary physics without it. But it is questionable whether mathematical concepts could ever play such a role in psychology or philosophy. In this paper, we set out to examine a rather unobvious example of the application of topology, in the form of the theory of persons proposed by Kurt Lewin in his Principles of Topological Psychology. Our aim is to (...) show that this branch of mathematics can furnish a natural conceptual system for Gestalt psychology, in that it provides effective tools for describing global qualitative aspects of the latter’s object of investigation. We distinguish three possible ways in which mathematics can contribute to this: explanation, explication and metaphor. We hold that all three of these can be usefully characterized as throwing light on their subject matter, and argue that in each case this contrasts with the role of explanations in physics. Mathematics itself, we argue, provides something different from such explanations when applied in the field of psychology, and this is nevertheless still cognitively fruitful. (shrink)

Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...) coherent (in an informal sense) and places the problem of reliability within the province of the philosophy of mathematics. (shrink)

Definitions of nature and transcendence are given, and the framework of Hindu thought is presented. The levels of reality as discovered by physics are then discussed, which leads us to revise our notions of reality and objectivity. Transcendence is defined as something beyond matter‐energy in space‐time and is explored in several contexts of modern science, as in pre‐Big‐Bang state, negative entropy, information, complexity, and others. Finally, a philosophical reflection on consciousness is presented.

ABSTRACTJames Hansen and others have argued that climate scientists are often reluctant to speak out about extreme outcomes of anthropogenic carbonization. According to Hansen, such reticence lessens the chance of effective responses to these threats. With the collapse of the West Antarctic Ice Sheet as a case study, reasons for scientific reticence are reviewed. The challenges faced by scientists in finding the right balance between reticence and speaking out are both ethical and methodological. Scientists need a framework within which to (...) find this balance. Such a framework can be found in the long-established practices of professional ethics. (shrink)

ABSTRACT In a recent essay, Sören Stenlund tries to align Wittgenstein’s approach to the foundations and nature of mathematics with the tradition of symbolic mathematics. The characterization of symbolic mathematics made by Stenlund, according to which mathematics is logically separated from its external applications, brings it closer to the formalist position. This raises naturally the question whether Wittgenstein holds a formalist position in philosophy of mathematics. The aim of this paper is to give a negative answer to this question, defending (...) the view that Wittgenstein always thought that there is no logical separation between mathematics and its applications. I will focus on Wittgenstein’s remarks about arithmetic during his middle period, because it is in this period that a formalist reading of his writings is most tempting. I will show how his idea of autonomy of arithmetic is not to be compared with the formalist idea of autonomy, according to which a calculus is “cut off” from its applications. The autonomy of arithmetic, according to Wittgenstein, guarantees its own applicability, thus providing its own raison d’être. RESUMO Em um recente artigo, Sören Stenlund procura alinhar a abordagem de Wittgenstein em relação aos fundamentos e à natureza da matemática com a tradição da matemática simbólica. A caracterização da matemática simbólica feita por Stenlund, de acordo com a qual a matemática é logicamente separada de suas aplicações externas, a aproxima da posição formalista. Isto naturalmente levanta a questão de se Wittgenstein defende uma posição formalista em filosofia da matemática. O objetivo deste artigo é dar uma resposta negativa a esta questão, ao defender que Wittgenstein sempre pensou não haver separação lógica entre a matemática e suas aplicações. Atenção especial será dada às observações de Wittgenstein sobre a aritmética pertencentes ao seu período intermediário, pois é neste período que uma leitura formalista de seus escritos é mais sedutora. Mostrarei como sua ideia de autonomia da aritmética não deve ser comparada à ideia formalista de autonomia, segundo a qual um cálculo é “cortado” de suas aplicações. A autonomia da aritmética, para Wittgenstein, garante ela própria sua aplicabilidade, provendo, assim, sua própria raison d’être. (shrink)

We argue that the concepts of `human equality' and `inequality' play an important role in the structure of science and philosophy. When the value of `human inequality' predominates, scientific categories are formed in accordance with the principle of `hierarchical differentiation' and concepts remain closely tied to the objects they are referring to. Following Mirowski we define this as the `anthropometric stage' of human thought and development. Contrary, Mirowski's `syndetic stage' refers to societies where the value of `human equality' prevails. Here (...) concepts appear that are universally applicable. However, because of their conventional nature these concepts cannot be `grasped' any longer by human intuition. Between the `anthropometric' and `syndetic' stages, a `lineamentric stage' appears, a period of transition from `human equality' to `human inequality'. Being both a bridge and gap between the two other stages, the `lineamentric' stage contains many contradictions between an `abstract attitude' and `concrete categories'. In this paper we examine the anthropometric, lineamentric and syndetic stages and discuss several examples taken from philosophy, logic, mathematics and physics. (shrink)

Philosophy of the Social Sciences, Volume 52, Issue 4, Page 235-257, July 2022. Charles Pigden’s 1995 article “Popper Revisited, or What is Wrong with Conspiracy Theories?” stimulated what is today a fertile sub-field of philosophical enquiry into conspiracy theories. In his article, Pigden identifies Karl Popper as the originator of the philosophical argument that it is naïve to believe in any conspiracy theory. But Popper was not criticizing belief in conspiracy theories at all, as Pigden defined them or as they (...) have usually come to be understood since about the 1960s. Pigden has therefore fundamentally and anachronistically misinterpreted Popper. The object of Popper’s criticism was, rather, the inadequate approach to social science that is limited to the discovery of human intentions, including conspiracies, in particular the will of Great Men. Popper’s critique of the conspiracy theory of society was correct and should be rehabilitated. Pigden is correct only insofar as he concludes that we should not dismiss conspiracy theories without critical evaluation, a proposition with which Popper would likely have wholeheartedly agreed. (shrink)

We explain the general significance of integer-based descriptors for natural shapes and show that the evolution of two such descriptors, called mechanical descriptors (the number _N_(_t_) of static balance points and the Morse–Smale graph associated with the scalar distance function measured from the center of mass) appear to capture (unlike classical geophysical shape descriptors) one of our most fundamental intuitions about natural abrasion: shapes get monotonically _simplified_ in this process. Thus mechanical descriptors help to establish a correlation between subjective and (...) objective descriptors of perceived objects. (shrink)

The collection Virtual Mathematics: the logic of difference brings together a range of new philosophical engagements with mathematics, using the work of French philosopher Gilles Deleuze as its focus. Deleuze’s engagements with mathematics rely upon the construction of alternative lineages in the history of mathematics in order to reconfigure particular philosophical problems and to develop new concepts. These alternative conceptual histories also challenge some of the self-imposed limits of the discipline of mathematics, and suggest the possibility of forging new connections (...) between philosophy and more recent developments in mathematics. This component of Deleuze’s work has, to date, been rather neglected by commentators working in the field of Deleuze studies. One of the aims of this collection is to address this critical deficit by providing a philosophical presentation of Deleuze’s relation to mathematics; one that is adequate to his project of constructing a philosophy of difference and to the exploration of some of its applications. This project developed as a result of encounters with an increasing number of researchers who have been working on the mathematical aspects of Deleuze’s work and the key role that these play in his philosophy. By bringing this work together for the first time, this collection makes an important contribution to the emerging body of work that endeavours to explore the broad range of Deleuze’s philosophy. (shrink)

The main thrust of the argument of this thesis is to show the possibility of articulating a method of construction or of synthesis--as against the most common method of analysis or division--which has always been (so we shall argue) a necessary component of scientific theorization. This method will be shown to be based on a fundamental synthetic logical relation of thought, that we shall call inversion--to be understood as a species of logical opposition, and as one of the basic monadic (...) logical operators. Thus the major objective of this thesis is to This thesis can be viewed as a response to Larry Laudan's challenge, which is based on the claim that ``the case has yet to be made that the rules governing the techniques whereby theories are invented (if any such rules there be) are the sorts of things that philosophers should claim any interest in or competence at.'' The challenge itself would be to show that the logic of discovery (if at all formulatable) performs the epistemological role of the justification of scientific theories. We propose to meet this challenge head on: a) by suggesting precisely how such a logic would be formulated; b) by demonstrating its epistemological relevance (in the context of justification) and c) by showing that a) and b) can be carried out without sacrificing the fallibilist view of scientific knowledge. OBJECTIVES: We have set three successive objectives: one general, one specific, and one sub-specific, each one related to the other in that very order. (A) The general objective is to indicate the clear possibility of renovating the traditional analytico-synthetic epistemology. By realizing this objective, we attempt to widen the scope of scientific reason or rationality, which for some time now has perniciously been dominated by pure analytic reason alone. In order to achieve this end we need to show specifically that there exists the possibility of articulating a synthetic (constructive) logic/reason, which has been considered by most mainstream thinkers either as not articulatable, or simply non-existent. (B) The second (specific) task is to respond to the challenge of Larry Laudan by demonstrating the possibility of an epistemologically significant generativism. In this context we will argue that this generativism, which is our suggested alternative, and the simplified structuralist and semantic view of scientific theories, mutually reinforce each other to form a single coherent foundation for the renovated analytico-synthetic methodological framework. (C) The third (sub-specific) objective, accordingly, is to show the possibility of articulating a synthetic logic that could guide us in understanding the process of theorization. This is realized by proposing the foundations for developing a logic of inversion, which represents the pattern of synthetic reason in the process of constructing scientific definitions. (shrink)

James Hansen and others have argued that climate scientists are often reluctant to speak out about extreme outcomes of anthropogenic carbonization. According to Hansen, such reticence lessens the chance of effective responses to these threats. With the collapse of the West Antarctic Ice Sheet as a case study, reasons for scientific reticence are reviewed. The challenges faced by scientists in finding the right balance between reticence and speaking out are both ethical and methodological. Scientists need a framework within which to (...) find this balance. Such a framework can be found in the long-established practices of professional ethics. (shrink)