We propose a characterization of mathematical experiments in terms of a setup, a process with an outcome, and an interpretation. Using a broad notion of process, this allows us to consider arithmetic calculations and geometric constructions as components of mathematical experiments. Moreover, we argue that mathematical experiments should be considered within a broader context of an experimental research project. Finally, we present a particular case study of the genesis of a geometric construction to illustrate the experimental use (...) of hand drawings and computers in mathematical practice. (shrink)

In this article we present and compare two early attempts to establish psychology as an independent scientific discipline that had considerable influence in central Europe: the theories of Johann Friedrich Herbart (1776—1841) and Franz Brentano (1838—1917). While both of them emphasize that psychology ought to be conceived as an empirical science, their conceptions show revealing differences. Herbart starts with metaphysical principles and aims at mathematizing psychology, whereas Brentano rejects all metaphysics and bases his method on a conception of inner perception (...) (as opposed to inner observation) as a secondary consciousness, by means of which one gets to be aware of all of one’s own conscious phenomena. Brentano’s focus on inner perception brings him to deny the claim that there could be unconscious mental phenomena — a view that stands in sharp contrast to Herbart’s emphasis on unconscious, ‘repressed’ presentations as a core element of his mechanics of mind. Herbart, on the other hand, denies any role for psychological experiments, while Brentano encouraged laboratory work, thus paving the road for the more experimental work of his students like Stumpf and Meinong. By briefly tracing the fate of the schools of Herbart and Brentano, respectively, we aim to illustrate their impact on the development of psychological research, mainly in central Europe. (shrink)

Wilfred Sieg. Mechanical Procedures and Mathematical Experience.

Presents general information about meteorology, weather, and climate and includes more than thirty activities to help study these topics, including making a ...

Wilfred Sieg. Aspects of Mathematical Experience.

ome recent work by philosophers of mathematics has been aimed at showing that our knowledge of the existence of at least some mathematical objects and/or sets can be epistemically grounded by appealing to perceptual experience. The sensory capacity that they refer to in doing so is the ability to perceive numbers, mathematical properties and/or sets. The chief defense of this view as it applies to the perception of sets is found in Penelope Maddy’s Realism in Mathematics, but (...) a number of other philosophers have made similar, if more simple, appeals of this sort. For example, Jaegwon Kim (1981, 1982), John Bigelow (1988, 1990), and John Bigelow and Robert Pargetter (1990) have all defended such views. The main critical issue that will be raised here concerns the coherence of the notions of set perception and mathematical perception, and whether appeals to such perceptual faculties can really provide any justification for or explanation of belief in the existence of sets, mathematical properties and/or numbers. (shrink)

In this expository article one of the contributions of Jean Cavailles to the philosophy of mathematics is presented: the analysis of ‘mathematical experience’. The place of Cavailles on the logico-philosophical scene of the 30s and 40s is sketched. I propose a partial interpretation of Cavailles's epistemological program of so-called ‘conceptual dialectics’: mathematical holism, duality principles, the notion of formal contents, and the specific temporal structure of conceptual dynamics. The structure of mathematical abstraction is analysed in terms (...) of its complementary dimensions: paradigmatic generalization (domain extension, descriptive definitions, creative role of the symbolism...) and thematic reflexivity of concepts (promotion of operations to objects of a higher type). (shrink)

This guide is designed as a teaching tool, aimed at both teachers and student teachers. Its main purpose is to enhance the value of "The Mathematical Experience" as a textbook. It includes a sample syllabus, outlines for group work, sample exams and hints for grading essays.

Jean Dieudonné, the spokesman of the group of French mathematicians named Bourbaki, called mathematics the music of reason. This metaphor invites a phenomenological account of the affective, in contrast to the epistemic and discursive, nature of mathematics: What constitutes its charm? Mathematical reasoning is described as a perceptual experience, which in Husserl’s late philosophy would be a case of passive synthesis. Like a melody, a mathematical proof is manifest in an affective identity of a temporal object. Rather (...) than an exercise for its own sake, this account sheds a different light on both the epistemic limitation of mathematical science, and the discursive problem of social responsibility in mathematics – two issues at the heart of Husserl’s critique of science as well as of mid-20th century mathematics, for which Nicolas Bourbaki stands as a monument of rigor. (shrink)

Context: How can an enactive approach to the teaching and learning of mathematics be implemented, which fosters mathematical thinking, making intensive use of metaphorizing and taking into account ….

The goal of this paper is to answer the following question: Does it make sense to speak of thought experiments not only in physics, but also in mathematics, to refer to an authentic type of activity? One may hesitate because mathematics as such is the exercise of reasoning par excellence, an activity where experience does not seem to play an important role. After reviewing some results of the research on thought experiments in the natural sciences, we turn our attention (...) to experiments and thought experiments in mathematics, especially in fundamental mathematics, and investigate what thought experiments can teach us there. If we accept the principle that mathematical practice can sometimes be best described by a pragmatist approach where the concept of experience in mathematics is considered from a new perspective, thought experiments can in some cases be a useful instrument different from both ‘mathematical experiments’ and ‘formal’ proofs: they are mathematical experiments using deviant methods. (shrink)

Longtime teacher and teacher educator Patricia Carini argues that numbers provide one of the many crucial tools that humans use to make sense of their world.4 In making this claim, Carini retells an extended "Number Story" from Alfred North Whitehead in which a squirrel moves her three children "one by one" to a new location.5 As Whitehead recounts: when the mother had placed them on a rock outside, the family group looked to her very different from its grouping within the (...) nest. She was vaguely disturbed, and ran backwards and forwards two or three times to make quite sure no young squirrel had been left behind. She was unable to count. … All she knew was that the vague multitude on the rock seemed very unlike the... (shrink)

In this chapter, the possibility of experiments in mathematics is examined. A general scheme is proposed as a tool to handle the different forms of experiments that are being used in mathematical practices: computations, “experimental mathematics” as a new research domain in mathematics and computer science, real-world experiments, and thought experiments. In a final section, extensions of the scheme are proposed that further support the conclusion that mathematical experiments are indeed facts and the future.

Some recent work by philosophers of mathematics has been aimed at showing that our knowledge of the existence of at least some mathematical objects and/or sets can be epistemically grounded by appealing to perceptual experience. The sensory capacity that they refer to in doing so is the ability to perceive numbers, mathematical properties and/or sets. The chief defense of this view as it applies to the perception of sets is found in Penelope Maddy’s Realism in Mathematics, but (...) a number of other philosophers have made similar, if more simple, appeals of this sort. For example, Jaegwon Kim (1981, 1982), John Bigelow (1988, 1990), and John Bigelow and Robert Pargetter (1990) have all defended such views. The main critical issue that will be raised here concerns the coherence of the notions of set perception and mathematical perception, and whether appeals to such perceptual faculties can really provide any justification for or explanation of belief in the existence of sets, mathematical properties and/or numbers. (shrink)

Most disciplines make use of thought experiments, but physics and philosophy lead the pack with heavy dependence upon them. Often this is for conceptual clarification, but occasionally they provide real theoretical advances. In spite of their importance, however, thought experirnents have received rather little attention as a topic in their own right until recently. The situation has improved in the past few years, but a mere generation ago the entire published literature on thought experiments could have been mastered in a (...) long weekend. Now the subject is beginning to flourish. Given the relative newness of the field, it might be useful to have several examples at one’s finger tips, so a number of great ones will be described. Attention will also be drawn outside physics and philosophy. In mathematics there is something analogous to thought experiments -- visual reasoning and picture proofs. I will look briefly at this class of thought experiments and try using them to make a case for possibly settling the continuum hypothesis. After this, I will return to thought experiments in the sciences and propose an account of how they work. Finally, I will end with a sketch of a topic I am currently working on, a kind of progress report which, I hope, will be an inducement to others. (shrink)

Mathematical apriorists sometimes hold that our non-derived mathematical beliefs are warranted by mathematical intuition. Against this, Philip Kitcher has argued that if we had the experience of encountering mathematical experts who insisted that an intuition-produced belief was mistaken, this would undermine that belief. Since this would be a case of experience undermining the warrant provided by intuition, such warrant cannot be a priori.I argue that this leaves untouched a conception of intuition as merely an (...) aspect of our ordinary ability to reason. Thus the apriorist may still hold that some mathematical beliefs are warranted by intuition. (shrink)

This paper aims to show that in ancient Greek architecture, it is possible to find a genesis of the geometric modeling of visual perception present in propositions of Euclid's Optics, considering mathematical knowledge as a human wisdom expression. Let us start by emphasizing that mathematical thinking is not exclusively rooted in mathematical disciplines, but also includes the broad spectrum of human activities, including activities that come from everyday life. Based on this, we present a socio-cultural characterization of (...) human experience as the source and sustainer of mathematical knowledge. Thereafter, on the basis of a content and contextual analysis of _Euclid's Optics_, we explain the use and development of geometry in the study of various optical effects of visual perception in the architectural art embodied in the Acropolis of Athens; and we focus our analysis on one of the most important of these effects, the Acropolis of Athens. We focus our analysis on one of the most emblematic architectural works of ancient Greek culture: the Parthenon of Athens. (shrink)

As in science and philosophy, thought experiments in mathematics link a problem to new epistemic resources that are unavailable in a given practice, e.g., Euclidean geometry. Thought experiments invite us to perform an imaginary scenario involving counterfactual, deductive and sensory elements. This chapter aims to pinpoint the beneficial peculiarities of thought experiments in mathematics in comparison with inferences, diagrams and calculative procedures. Reflection about thought experiments assists us to realize both the limits and opportunities in mathematical thinking. Henceforth, the (...) analysis of examples suggests a broader understanding of mathematical, thought and experiment. (shrink)

This paper explores the connections between K. Wynn's well-known experiments in cognitive psychology and later Wittgenstein's views on the philosophy of mathematics.

Like many of their contemporaries Bernard Nieuwentijt and Pieter van Musschenbroek were baffled by the heterodox conclusions which Baruch Spinoza drew in the Ethics. As the full title of the Ethics—Ethica ordine geometrico demonstrata—indicates, these conclusions were purportedly demonstrated in a geometrical order, i.e. by means of pure mathematics. First, I highlight how Nieuwentijt tried to immunize Spinoza’s worrisome conclusions by insisting on the distinction between pure and mixed mathematics. Next, I argue that the anti-Spinozist underpinnings of Nieuwentijt’s distinction between (...) pure and mixed mathematics resurfaced in the work of van Musschenbroek. By insisting on the distinction between pure and mixed mathematics, Nieuwentijt and van Musschenbroek argued that Spinoza abused mathematics by making claims about things that exist in rerum natura by relying on a pure mathematical approach. In addition, by insisting that mixed mathematics should be painstakingly based on mathematical ideas that correspond to nature, van Musschenbroek argued that René Descartes’ natural-philosophical project abused mathematics by introducing hypotheses, i.e. ideas, that do not correspond to nature. (shrink)

According to a very common view, the main tenet of empiricism is the conviction that all human knowledge derives from sensory experience. But classic philosophers representing empiricism hold that mathematical knowledge is a priori. Mill intended to demonstrate that the laws of arithmetic and geometry have inductive origins. But Frege and others authors showed that Mill’s arguments were wrong. Benacerraf held that, since mathematical objects are abstract entities, they could not have any causal relationship with human beings, (...) so they cannot be known by us. On the other hand, biology and psychology show that in animals and human creatures we can find innate behaviours, in accordance with the theory on natural selection. Experiments performed by Wynn and by other psychologists strongly support that very young babies can determine the results of simple arithmetical operations without any previous learning. We conclude that there are convincing reasons to accept the rationalist thesis about the a priori character of mathematical knowledge. (shrink)

The question of whether mathematics depends on experience, including experience of the external world, is problematic because, while it is clear that natural sciences depend on experience, it is not clear that mathematics depends on experience. Indeed, several mathematicians and philosophers think that mathematics does not depend on experience, and this is also the view of mainstream philosophy of mathematics. However, this view has had a deleterious effect on the philosophy of mathematics. This article argues (...) that, in fact, the view is not valid. Mathematics depends on experience because experience influences the making of mathematics, indeed much mathematics arises from experience and is evaluated on the basis of experience. (shrink)

In the ten years following the publication of Galileo Galilei's Discorsi e dimostrazioni matematiche intorno a due nuove scienze , the new science of motion was intensely debated in Italy, France and northern Europe. Although Galileo's theories were interpreted and reworked in a variety of ways, it is possible to identify some crucial issues on which the attention of natural philosophers converged, namely the possibility of complementing Galileo's theory of natural acceleration with a physical explanation of gravity; the legitimacy of (...) Galileo's methods of proof and of his theory of the composition of continuous magnitudes; and the adequacy of the experimental evidence in favor of his theory.Through his published works and his correspondence, Marin Mersenne contributed in a significant way to each of these discussions. In the following, I shall provide a sketch of Mersenne's multiple engagement with the new science of motion, while trying to account for his growing skepticism concerning Galileo's law of free fall. Whereas in the 1630s, Mersenne still firmly believed in the validity of this law and tried to devise experimental and mathematical proofs in its favor, in the 1640s he developed serious doubts regarding the possibility of constructing an exact science of motion. As we shall see, Mersenne's evolving attitude sheds an interesting light on his views concerning the relation between physics and mathematics.---------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ----------------------------------------------------. (shrink)

This book offers a phenomenological conception of experiential justification that seeks to clarify why certain experiences are a source of immediate justification and what role experiences play in gaining (scientific) knowledge. Based on the author's account of experiential justification, this book exemplifies how a phenomenological experience-first epistemology can epistemically ground the individual sciences. More precisely, it delivers a comprehensive picture of how we get from epistemology to the foundations of mathematics and physics. The book is unique as it utilizes (...) methods and insights from the phenomenological tradition in order to make progress in current analytic epistemology. It serves as a starting point for re-evaluating the relevance of Husserlian phenomenology to current analytic epistemology and making an important step towards paving the way for future mutually beneficial discussions. This is achieved by exemplifying how current debates can benefit from ideas, insights, and methods we find in the phenomenological tradition. (shrink)

In §69.b of BT Heidegger attempts an existential genetic analysis of science, i.e. a phenomenology of the conceptual process of the constitution of the logical view of science (science seen as theory) starting from the Dasein. It attempts to do so by examining the special intentional-existential modification of (human) being-in-the-world, which is called the "mathematical projection of nature"; that is, by examining that special modification of our being, which places us in the state of experience that presents the (...) world to us as taught by, e.g., Galileo and Newton. The idea he puts forward is that modern mathematical physics and the corresponding experience of the real is the result of the mathematical projection of nature. Modern science, Heidegger says, stands out because it is mathematical. But we cannot get the answer to what this mathematicality consists in from mathematics, because mathematics is only a special manifestation of mathematicality. Here is what Heidegger means by "mathematicality" and, accordingly by "mathematical projection of nature." If one were to let two bodies of different weights fall simultaneously from the top of the tower of Pisa, Heidegger notes, both Galileo and an Aristotelian would see that the two bodies would reach the ground with as little time difference as possible, and not at exactly the same moment. Galileo, however, does something decisive: he claims that under conditions that would be impossible to ensure (if we removed absolutely all resistance to the motion of the bodies), the two bodies would reach the ground absolutely simultaneously. This is not something that he or the Aristotelian sees happening empirically. How, then, does Galileo insist so firmly on his position? How exactly does it happen that he has before him such a view of things? Heidegger's answer is that Galileo is mathematically projecting nature. In his Dialogues Galileo says "I conceive with my mind" (mente concipio) a body thrown to move on an infinitely extending horizontal plane, having eliminated from its path every possible obstacle. Finally, I argue that from the detailed interpretation of Heidegger's points of the mathematical and of the notion of mathematical projection of nature, we can arrive at the conclusions I had arrived at in my doctoral dissertation (2000) as regards the meaning and function of thought experiments in physics as seen from a Husserlian phenomenological point of view as intentional acts. In a nutshell, in thought experiments physics constitutes the special metaphysics (to use the Kantian term) of the primordial ontological region it investigates as well as attempts to make natural sense of the mathematical formalism in cases it proves necessary. (shrink)

Theory of linear estimation; General structure of analysis of designs; Standard designs; Applications of galois fields and finite geometry in the construction of designs; Some selected topics in design of experiments.

This chapter is devoted to discussions about the nature of experiments in mathematics during the formative period of experimental mathematics and to a quantitative and qualitative analysis of the journal Experimental Mathematics founded in 1992. Particular attention is paid to interpret the roles of experiment and observation in the mathematical practices surrounding Experimental Mathematics.

The present work is an attempt to bring attention to the application of several key ideas of Husserl ‘s Krisis in the construction of certain mathematical theories that claim to be altemative nonstandard versions of the standard Zermelo-Fraenkel set theory. In general, these theories refute, at least semantically, the platonistic context of the Cantorian system and to one or the other degree are motivated by the notions of the lifeworld as the pregiven holistic field of experience and that (...) of horizon as the boundary of human perceptions and the de facto constraint in reaching limit-idealizations. Moreover, 1 try to give convincing reasons for the existence of an ultimate constitutional ‘vacuum’ of a subjective origin that is formally refiected in the application of a notion of actual infinity in dealing generally with the mathematical infinite. (shrink)

From a purely formalist viewpoint on the philosophy of mathematics, experiments cannot (and should not) play a role in warranting mathematical statements but must be confined to heuristics. Yet, due to the incorporation of new mathematical methods such as computer-assisted experimentation in mathematical practice, experiments are now conducted and used in a much broader range of epistemic practices such as concept formation, validation, and communication. In this article, we combine corpus studies and qualitative analyses to assess and (...) categorize the epistemic roles experiments are seen—by mathematicians—to have in actual mathematical practice. We do so by text-mining a corpus of reviews from the _Mathematical Reviews_, which include the indicator word “experiment”. Our qualitative, grounded classification of samples from this corpus allows us to explore the various roles played by experiments. We thus identify instances where experiments function as references to established knowledge, as tools for heuristics or exploration, as epistemic warrants, as communication or pedagogy, and instances simply proposing experiments. Focusing on the role of experiments as epistemic warrants, we show through additional sampling that in some fields of mathematics, experiments can warrant theorems as well as methods. We also show that the expressed lack of experiments by reviewers suggests concordant views that experiments could have provided epistemic warrants. Thus, our combination of corpus studies and qualitative analyses has added a typology of roles of experiments in mathematical practice and shown that experiments can and do play roles as epistemic warrants depending on the mathematical field. (shrink)

Several promising approaches have been developed to represent conscious experience in terms of mathematical spaces and structures. What is missing, however, is an explicit definition of what a ‘mathematical structure of conscious experience’ is. Here, we propose such a definition. This definition provides a link between the abstract formal entities of mathematics and the concreta of conscious experience; it complements recent approaches that study quality spaces, qualia spaces, or phenomenal spaces; and it provides a general (...) method to identify and investigate structures of conscious experience. We hope that ultimately this work provides a basis for developing a common formal language to study consciousness. (shrink)

This paper shows why Kant's critique of empirical psychology should not be read as a scathing criticism of quantitative scientific psychology, but has valuable lessons to teach in support of it. By analysing Kant's alleged objections in the light of his critical theory of cognition, it provides a fresh look at the problem of quantifying first-person experiences, such as emotions and sense-perceptions. An in-depth discussion of applying the mathematical principles, which are defined in the Critique of Pure Reason as (...) the constitutive conditions for mathematical-numerical experience in general, to inner sense will demonstrate why it is in principle possible to justify a quantitative structure of psychological judgments on the grounds of Kant's critical thinking. In conclusion, it will propose how Kant's critique could be used in a constructive way to develop first steps towards a transcendental foundation of psychological knowledge. (shrink)

Kant's Mathematical World aims to transform our understanding of Kant's philosophy of mathematics and his account of the mathematical character of the world. Daniel Sutherland reconstructs Kant's project of explaining both mathematical cognition and our cognition of the world in terms of our most basic cognitive capacities. He situates Kant in a long mathematical tradition with roots in Euclid's Elements, and thereby recovers the very different way of thinking about mathematics which existed prior to its 'arithmetization' (...) in the nineteenth century. He shows that Kant thought of mathematics as a science of magnitudes and their measurement, and all objects of experience as extensive magnitudes whose real properties have intensive magnitudes, thus tying mathematics directly to the world. His book will appeal to anyone interested in Kant's critical philosophy -- either his account of the world of experience, or his philosophy of mathematics, or how the two inform each other. (shrink)

Leaving aside here the question of the author of the Essai de logique, I show that, if Mariotte insisted on the specificity of physics, he also sought a certain inspiration in mathematics as to the way in which to lay out the propositions in a proof. To do so, I start off from the ontological distinction made in the Essai among three types of possibles; next we will show that the three types of propositions correspond to three types of knowledge, (...) and, correlatively, that the main problem of physics is that it is impossible to establish in a certain fashion universal sensory propositions; I next characterize the notion of principles of experience; finally, I examine a bit systematically the relationship between mathematics and physics. (shrink)

In our contribution to this special issue on thought experiments and mathematics, we aim to insert theology into the conversation. There is a very long tradition of substantial inquiries into the relationship between theology and mathematics. Platonism has been provoking a consolidation of that tradition to some extent in recent decades. Accordingly, in this paper we look at James R. Brown’s Platonic account of thought experiments. Ultimately, we offer an analysis of some of the merits and perils inherent in framing (...) the use of thought experiments in mathematics and theology in terms of Platonism. (shrink)

This study applied a method of assisted introspection to investigate the phenomenology of mathematical intuition arousal. The aim was to propose an essential structure for the intuitive experience of mathematics. To achieve an intersubjective comparison of different experiences, several contemporary mathematicians were interviewed in accordance with the elicitation interview method in order to collect pinpoint experiential descriptions. Data collection and analysis was then performed using steps similar to those outlined in the descriptive phenomenological method that led to a (...) generic structure that accounts for the intuition surge in the experience of mathematics which was found to have four irreducible structural moments. The interdependence of these moments shows that a perceptualist view of intuition in mathematics, as defended by Chudnoff, is relevant to the characterization of mathematical intuition. The philosophical consequences of this generic structure and its essential features are discussed in accordance with Husserl’s philosophy of ideal objects and theory of intuition. (shrink)