Concerns the ‘problem of existence’ in mathematics: the problem of how to understand existence assertions in mathematics. The problem can best be understood by considering how Mathematical Platonists have understood such existence assertions. These philosophers have taken the existential theorems of mathematics as literally asserting the existence of mathematical objects. They have then attempted to account for the epistemological and metaphysical implications of such a position by putting forward arguments that supposedly show how humans (...) can come to know of the existence of mathematical objects. The reasoning of the most prominent philosophers in this Literalist tradition, Quine and Gödel, are critically discussed. By way of contrast, an alternative position, Heyting's Intuitionism, is examined. (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)

Many long-standing problems pertaining to contemporary philosophy of mathematics can be traced back to different approaches in determining the nature of mathematical entities which have been dominated by the debate between realists and nominalists. Through this discussion conceptualism is represented as a middle solution. However, it seems that until the 20th century there was no third position that would not necessitate any reliance on one of the two points of view. Fictionalism, on the other hand, observes mathematical entities in (...) a radically different way. This is reflected in the claim that the concepts being used in mathematics are nothing but a product of human fiction. This paper discusses the relationship between fictionalism and two traditional viewpoints within the discussion which attempts to successfully determine the ontological status of universals. One of the main points, demonstrated with concrete examples, is that fictionalism cannot be classified as a nominalist position. Since fictionalism is observed as an independent viewpoint, it is necessary to examine its range as well as the sustainability of the implications of opinions stated by their advocates. (shrink)

Over the past few decades the notion of symmetry has played a major role in physics and in the philosophy of physics. Philosophers have used symmetry to discuss the ontology and seeming objectivity of the laws of physics. We introduce several notions of symmetry in mathematics and explain how they can also be used in resolving different problems in the philosophy of mathematics. We use symmetry to discuss the objectivity of mathematics, the role of mathematical objects, the (...) unreasonable effectiveness of mathematics and the relationship of mathematics to physics. (shrink)

The solution of the problem of the future random events truth is considered in Vasil’ev’s logic. N. A. Vasil’ev graded the logic according to two levels—the level of facts, i.e. time fixed events, and the level of notions or rules, governing these facts. The mathematical construction previously suggested for imaginary Vasil’ev’s logic, extends to the early variant of his logic—a logic of notions. In the paper, we investigate the meaning of problematic and uncertain assertions introduced by Vasil’ev. As (...) a result, we developed a model of Vasil’ev’s logic of facts that resolves also the truth problem of future random events. The imaginary logic has also been extended to the level of notions, and the law of the excluded eighth is gotten in it. The correspondence between Vasil’ev’s terms “some” and “all” and modern quantifiers is discussed. (shrink)

We define the notion of Euler characteristic for definable quotients in an arbitrary o-minimal structure and prove some fundamental properties.

I build on Heras-Escribano’s ontological characterization to address issues of affordances related to mathematics education, particularly about how it can enable fruitful conceptualizations for ….

I will begin by considering how the social world appears to the scientific observer and ask the question of whether the world of scientific research, with all its categories of meaning interpretation and with all its conceptual schemes of action, is identical with the world in which the observed actor acts. Anticipating the result, I may state immediately that with the shift from one level to the other, all the conceptual schemes and all the terms of interpretation must be modified.Proceeding (...) in this direction, I encounter several problems overlapping with the problem of rational action and the “rational act unit” in Professor ParsonsParsons, Talcott’ sense. First: The conceptual scheme of rational action presupposes a more or less definite knowledge of, or orientation within, the world in which this action is performed. In this sense the term, “rationality”, is not specific to the conceptual scheme of action; it embraces the conceptual scheme of the world in general and is only one element in it. Therefore, we also must examine the problem of orientation in this world. Second: We ought to keep in mind that all interpretation of the social world has to start from the actor’s subjective point of view. Therefore, we cannot accept the term, “unit act”, as such, without trying to reduce it to its subjective meaning. Here I have to discuss in general terms why I insist on the subjective point of view. This is more than the whim of a few social scientists; social science cannot be built up except by taking cognizance of the subjective meaning the actor connects with his acting. Finally, I have to discuss the fact that there are social sciences operating on high theoretical levels that, like economics, apparently prefer to deal with statistical curves and mathematical formulae rather than with actions of human beings in the social world, yet they use the term, “rational action”, as an indispensable element of their systems. I will have to show what modification this term undergoes in these types of social sciences..). (shrink)

The manuscripts of Harriot discussed in this paper are essentially rough notes marginal to his systematic treatment of algebra, of which a small part was published posthumously. The central theme is the sign-rule for multiplication; but the incidentals open up an aspect of symbolism in mathematics entirely new for the time. A more restricted aspect of the same theme was touched on by Commandino in his Euclid, quoted by Harriot as rightly blaming ‘those that thinke that minus per minus (...) shal produce plus’; Commandino's authority was Cardano, in a similarly marginal departure from his great Ars magna, the de Aliza, which Harriot also cites in the same reference. The ostensible motivation, to preserve intact the Euclidean tradition in geometry, plays a very minor role in Harriot's notes. The basic aim, to elude the fatal link with imaginaries, they do touch on. More dimly underlying the arguments are the crucial logical facets of the problem, which Harriot faced with a greater conceptual clarity, formulating in words, however, only his own conclusion: the choice of sign-rule is the user's—a fact not fully recognised by mathematicians generally until the 19th century. (shrink)

The article explores the theme of environment in Valeriy Podoroga’s analytical anthropo­logy, offering an enactivist reading of his “The Metaphysics of Landscape”. The analyti­cal strategy of “The Metaphysics of Landscape” is contrasted with that of later works such as “Mimesis” and “Anthropograms”. Whereas in the later works the analytical tech­niques are set by a variety of optical concepts, metaphors and images, thus representing a strategy of “exclusionary observation”, in “The Metaphysics of Landscape” the analysis follows rather the movement of observer’s (...) body included in the environment and aims at the reconstruction of an integrated, sensorimotor, yet imaginary experience. Three frag­ments of Podoroga’s text, reconstructing imaginary bodies and landscapes embedded in the works of Kirkegaard, Nietzsche, and Heidegger, are interpreted as three different ways of coupling body and environment, generating three different models of bodily-environmental experience. Finding the correspondence of the fragments of the descrip­tion of this experience to some key points of enactivist epistemology and aesthetics, the author shows that the enactivist approach makes it possible to formalize those tech­niques of analysis that allow us to discover the rules of dependence between some im­plicit modus of writing and the bodily experience of reading. (shrink)

The constantly accelerating progress of contemporary natural science is indissolubly associated with the development and use of mathematics and with the processes of mathematical modeling of the phenomena of nature. The essence of this diverse and highly fertile interaction of mathematics and natural science and the dialectics of this interaction can only be disclosed through analysis of the nature of theoretical notions in general. Today, above all in the ranks of materialistically minded researchers, it is generally accepted that (...) theory possesses a value of its own. Contrary to positivist concepts of the nature of our knowledge, the meaning and significance of theory do not consist merely of the recording of experimental data, their classification and notation in abbreviated form. The meaning of theory is considerably more significant and is revealed, above all, in its explanatory and predictive functions. Contemporary theoretical knowledge is quite advanced and comprises a highly complex system, relatively self-contained and capable of internal development. Mathematics is the most interesting and distinctive phenomenon in the system of theoretical knowledge, in the system of science in general. It is in its dialectical development that the internal force and dynamics of the development of theory, in the very broadest sense of that word, find expression. Paul Lafargue tells us that Karl Marx held that "science only attains perfection when it succeeds in making use of mathematics" . In many fields of research the formulation of new ideas and concepts rests upon mathematics, its concepts and notions, and "is suggested" by the latter. "Mathematics," writes F. Dyson, "is the main source of the notions and principles out of which new theories are created" . The basic significance of mathematics to theory has been noted by outstanding thinkers over the course of the entire history of development of science. Today the expressive phrase, "the mathematization of knowledge," has come into general currency as a description of the principal direction of growth of theoretical notions in natural science. However, with respect to the growth of modern theoretical physics in general, the statement that it develops primarily by the method of the mathematical hypothesis has general validity. For example, the principal findings and discoveries of quantum theory and particle physics, starting from the corpuscularwave dualism and terminating with the omega minus hyperon and hypothetical quarks, were obtained or made "at the tip of a mathematician's pen.". (shrink)

The philosophical problem that stems from the successful application of mathematics in the empirical sciences has recently attracted growing interest within philosophers of mathematics and philosophers of science. Nevertheless, little attention has been devoted to the converse applicability issue of how physical considerations find successful application in mathematics. In this article, focusing on some case studies, I address the latter issue and argue that some successful applications of physics to mathematics essentially depend on the use (...) of conservation principles. I conclude with a short discussion of how the successful interplay between conservation principles and mathematics can be accounted for. (shrink)

Context: the position of pure and applied mathematics in the epistemic conflict between realism and relativism. Problem: To investigate the change in the status of mathematical knowledge over historical time: specifically, the shift from a realist epistemology to a relativist epistemology. Method: Two examples are discussed: geometry and number theory. It is demonstrated how the initially realist epistemic framework – with mathematics situated in a platonic ideal reality from where it governs our physical world – became untenable, (...) with the advent of non-Euclidean geometry and the increasing abstraction of the number concept. Results: Radical constructivism offers an alternative relativist epistemology, where mathematical knowledge is constructed by the individual knower in a context of an axiomatic base and subject items chosen at her discretion, for the purpose of modelling some part of her personal experiential world. Thus it can be expedient to view the practice of mathematics as a game, played by mathematicians according to agreed-upon rules. Constructivist content: The role played by constructivism in the formulation of mathematics is discussed. This is illustrated by the historical transition from a classical (platonic) view of mathematics, as having an objective existence of its own in the “realm of ideal forms,” to the now widely accepted modern view where one has a wide freedom to construct mathematical theories to model various parts of one’s experiential world. (shrink)

The central question underlying this study was whether metacognition training could enhance the two metacognition components—knowledge and skills—and the mathematical problem-solving capacities of normal children in grade 3. We also investigated whether metacognitive training had a differential effect according to the children's mathematics level. A total of 48 participants took part in this study, divided into an experimental and a control group, each subdivided into a lower and a normal achievers group. The training programme took an interactive approach (...) in accordance with Schraw's (1998) recommendation and was carried out over five training sessions. Results indicated that children in the training group had significantly higher post-test metacognitive knowledge, metacognitive skills, and mathematical problem-solving scores. In addition, metacognitive training was particularly beneficial to the low achievers. Thus metacognitive training enabled the low achievers to make progress and solve the same number of problems on the post-test as the normal achievers solved on the pre-test. (shrink)

The terms of a mathematical problem become precise and concise if they are expressed in an appropriate notation, therefore notations are useful to mathematics. But are notations only useful, or also essential? According to prevailing view, they are not essential. Contrary to this view, this paper argues that notations are essential to mathematics, because they may play a crucial role in mathematical discovery. Specifically, since notations may consist of symbolic notations, diagrammatic notations, or a mix of symbolic (...) and diagrammatic notations, notations may play a crucial role in mathematical discovery in different ways. Some examples are given to illustrate these ways. (shrink)

Chemistry, especially its historical practice, has in the philosophy of science in recent decades attracted more and more attention, influencing the turn from the vision of science as a timeless logic-centred system of statements towards the history- and practice-centred approach. The problem of pluralism in science has become a popular topic in that context. Hasok Chang’s “active normative epistemic pluralism” manifested in his book Is water H2O? Evidence, realism and pluralism, pursuing an integrated study of history and philosophy of (...) science, has provoked quite a widespread debate. It provides a good opportunity to discuss the topical philosophical issue of the nature of disagreements. The differences among disagreements in different domains have been pointed out in the disagreement literature. It has been noticed that in mathematics and science consensus is established more clearly than in philosophy where it remains largely unachievable. However, this conclusion is derived in the context of traditional logic-centred view of science. The aim of this paper is to consider the different nature of disagreements in science and in philosophy in the context of the history- and practice-centred approach. The analysis is focused on the critique of the received view of the Chemical Revolution which played quite a central role in Chang’s becoming a pluralist about science. Unlike Chang, however, a modified Kuhnian paradigm-conception of science and scientific revolutions is defended. (shrink)

A discussion of the problem of contradiction in mechanical motion has been in progress for a long time in the pages of Filosofskie nauki. The attention given that problem is no accident. In our day, problems concerning contradictions involved in rest and motion, the continuous and the discontinuous, the finite and infinite, etc., have moved from the realm of abstract consideration to that of the concrete and current handling of the subject matter of modern physics, mathematics, and (...) other fields. It is natural that the present stage of knowledge of objective reality should occasion a revived interest on the part of logicians and philosophers in these problems, including the question of how objective contradictions may be expressed in scientific theories. (shrink)

In the paper the problem of definability and undefinability of the concept of satisfaction and truth is considered. Connections between satisfaction and truth on the one hand and consistency of certain systems of omega-logic and transfinite induction on the other are indicated.

Critical social theories look critically at the ways in which particular social arrangements hinder human flourishing, with a view to bringing about social change for the better. In this they are guided by the idea of a good society in which the identified social impediments to human flourishing would once and for all have been removed. The question of how these guiding ideas of the good life can be justified as valid across socio-cultural contexts and historical epochs is the most (...) fundamental difficulty facing critical social theories today. This problem of justification, which can be traced back to certain key shifts in the modern Western social imaginary, calls on contemporary theories to negotiate the tensions between the idea of context-transcendent validity and their own anti-authoritarian impulses. Habermas makes an important contribution towards resolving the problem, but takes a number of wrong turnings. (shrink)

It can not be our purpose to give here a complete account of the phenomenological history of mathematical doctrine. It will be enough to refer to the battle of opinions in mathematical theory which was waged within the last eighty-five years, since Riemann's inaugural lecture on Non-Euclidean Geometry. Furthermore, the revolution which Einstein's theory of general relativity created is indicative of the complete absence of any general awareness that mathematics as a science has any relation to social reality. If (...) we are going to establish such relations, we are conscious of the fact that they would be hotly contended by many mathematicians. (shrink)

This book examines the birth of the scientific understanding of motion. It investigates which logical tools and methodological principles had to be in place to give a consistent account of motion, and which mathematical notions were introduced to gain control over conceptual problems of motion. It shows how the idea of motion raised two fundamental problems in the 5th and 4th century BCE: bringing together being and non-being, and bringing together time and space. The first problem leads to the (...) exclusion of motion from the realm of rational investigation in Parmenides, the second to Zeno's paradoxes of motion. Methodological and logical developments reacting to these puzzles are shown to be present implicitly in the atomists, and explicitly in Plato who also employs mathematical structures to make motion intelligible. With Aristotle we finally see the first outline of the fundamental framework with which we conceptualise motion today. (shrink)

Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from (...) both disciplines. (shrink)

Symplectic reduction is a formal process through which degeneracy within the mathematical representations of physical systems displaying gauge symmetry can be controlled via the construction of a reduced phase space. Typically such reduced spaces provide us with a formalism for representing both instantaneous states and evolution uniquely and for this reason can be justifiably afforded the status of fun- damental dynamical arena - the otiose structure having been eliminated from the original phase space. Essential to the application of symplectic reduction (...) is the precept that the first class constraints are the relevant gauge generators. This prescription becomes highly problematic for reparameterization invariant theories within which the Hamiltonian itself is a constraint; not least because it would seem to render prima facie distinct stages of a history physically identical and observable functions changeless. Here we will consider this problem of time within non-relativistic me- chanical theory with a view to both more fully understanding the temporal struc- ture of these timeless theories and better appreciating the corresponding issues in relativistic mechanics. For the case of nonrelativistic reparameterization invariant theory application of symplectic reduction will be demonstrated to be both unnec- essary; since the degeneracy involved is benign; and inappropriate; since it leads to a trivial theory. With this anti-reductive position established we will then examine two rival methodologies for consistently representing change and observable func- tions within the original phase space before evaluating the relevant philosophical implications. We will conclude with a preview of the case against symplectic re- duction being applied to canonical general relativity. (shrink)

The text considers the impossibility of abstracting away from the sense of formal con­structions in logical and mathematical researches. The validity of the application of the “formal methodology” is allowed only after some system of conventional notations and agreements has been accepted. The context determined by such agreements is called formal. A correlation of constructions and results obtained by formal methods within sev­eral formal contexts is impossible without a consideration of the various semantic aspects of the correlated formal constructions. The (...) author calls such correlations intercontextual. The paper examines two examples of such intercontextual comparisons to demonstrate the necessity of taking into account different semantic components of the compared for­mal constructions. In the context of these conclusions, the author raises the question of the structure and origin of some senses of the “symbols” used in formal constructions and of the “sequences of symbols” constructed from them. The author identifies three main sources of the semantic load carried by formal constructions. Firstly, these are the various aspects of semiotic usage: first of all, the general cultural and general profes­sional semiotic skills of the “interpreter”. Secondly, it is the sense given to formal con­structions by verbal comments, descriptions of the construction process and the associ­ated knowledge of the “interpreter”. Thirdly, these are the senses set by the formal con­structions themselves: at the stage of defining a formal language and at the stage of con­structing a formal deductive or semantic system. The author also considers the fallacy of the assumption of the existence of some universal “global intuition” associated with the very possibility of formal methodology. (shrink)

The purpose of this paper is to provide a unitary typology for the incompatibilities of meanings at stake on different levels of Husserlian pure logic—namely, between systems of axioms and pure morphology of meanings; I show that they perfectly match by converging on the notion of Widersinn.

Fictionalism plays a significant role in philosophy today, with defenses spanning mathematics, morality, ordinary objects, truth, modality, and more.1 Fictionalism in the philosophy of science is also gaining attention, due in particular to the revival of Hans Vaihinger’s work from the early twentieth century and to heightened interest in idealization in scientific practice.2 Vaihinger maintains that there is a ubiquity of fictions in science and, among other things, argues that Nietzsche supports the position. Yet, while contemporary commentators have focused (...) on fictionalism in Nietzsche’s moral philosophy, his view of fictions in science has remained largely unexamined.3 In this article, I begin... (shrink)

This chapter summarizes Russell’s The Problems of Philosophy, presents new biographical details about how and why Russell wrote it, and highlights its continued significance for contemporary philosophy. It also surveys Russell’s famous distinction between “knowledge by acquaintance” and “knowledge by description,” his developing views about our knowledge of physical reality, and his views about our knowledge of logic, mathematics, and other abstract objects.

The invitation of the Organizing Committee for me to speak about “Unsolved problems in mathematics” fills me as it should with considerable trepidation and a prevailing feeling of personal inadequacy. Hilbert gave a talk on this subject at the similar congress about 50 years ago and this is a very formidable precedent. He stated about a dozen unsolved problems in another widely separated areas of mathematics, and they proved to be prototypical for much of the development that followed (...) in the next decades. It would be absolutely foolish, if I tried to emulate this quite singular feat. In addition I do not know the future and the future at any rate can only be predicted ex post with any degree of reliability. I will, therefore, define what I am trying to do in a much more narrow way, hoping that in this manner I have a better chance of not failing. I will limit myself to a particular area of mathematics which I think I know and I will talk about it and about what its open ends appear to be, particularly in some directions which are not the ones that the evolution so far has mainly emphasized and which are, I think, quite important. (shrink)

The problem of God's subordination to laws of logic and mathematics in jewish philosophy and theology.

The problem of precise characterisation of traditional forms of reasoning applied in mathematics was independently investigated and successfully resolved by Jaśkowski and Gentzen in 1934. However, there are traces of earlier interests in this field exhibited by the members of the Lvov-Warsaw School. We focus on the results obtained by Jaśkowski and Leśniewski. Jaśkowski provided the first formal system of natural deduction in 1926. Leśniewski also demonstrated in some of his papers how to construct proofs in accordance with (...) intuitively correct principles. (shrink)

The conflict between Platonic realism and Constructivism marks a watershed in philosophy of mathematics. Among other things, the controversy over the Axiom of Choice is typical of the conflict. Platonists accept the Axiom of Choice, which allows a set consisting of the members resulting from infinitely many arbitrary choices, while Constructivists reject the Axiom of Choice and confine themselves to sets consisting of effectively specifiable members. Indeed there are seemingly unpleasant consequences of the Axiom of Choice. The non-constructive nature (...) of the Axiom of Choice leads to the existence of non-Lebesgue measurable sets, which in turn yields the Banach-Tarski Paradox. But the Banach-Tarski Paradox is so called in the sense that it is a counter-intuitive theorem. To corroborate my view that mathematical truths are of non-constructive nature, I shall draw upon Gödel’s Incompleteness Theorems. This also shows the limitations inherent in formal methods. Indeed the Löwenheim-Skolem Theorem and the Skolem Paradox seem to pose a threat to Platonists. In this light, Quine/Putnam’s arguments come to take on a clear meaning. According to the model-theoretic arguments, the Axiom of Choice depends for its truth-value upon the model in which it is placed. In my view, however, this is another limitation inherent in formal methods, not a defect for Platonists. To see this, we shall examine how mathematical models have been developed in the actual practice of mathematics. I argue that most mathematicians accept the Axiom of Choice because the existence of non-Lebesgue measurable sets and the Well-Ordering of reals open the possibility of more fruitful mathematics. Finally, after responding to Benacerraf’s challenge to Platonism, I conclude that in mathematics, as distinct from natural sciences, there is a close connection between essence and existence. Actual mathematical theories are the parts of the maximally logically consistent theory that describes mathematical reality. (shrink)

Techne and Truth. The problem of techne in the dispute between Gorgias and Plato -/- The source of the problem matter of the book is the Plato’s dialogue „Gorgias”. One of the main subjects of the discussion carried out in this multi-aspect work is the issue of the art of rhetoric. In the dialogue the contemporary form of the art of rhetoric, represented by Gorgias, Polos and Callicles, is confronted with Plato’s proposal of rhetoric and concept of art (...) (techne). The ingenuous and dramatic structure of the dialogue composed of three acts, in each of which Socrates’ consecutive interlocutors emerge, is difficult to interpret. It seems, however, that even though the first conversation between Socrates and Gorgias constitutes a small part of the dialogue, it forms the foundations for the discussion. This part, dealing with such important issues as the subject of art, knowledge, power and relation between art and good, is the starting point for the conversations which follow. What do the differences between the Gorgias’ and Plato’s concepts of art consist in? Any attempt to answer the question requires presentation of the background for the discussion taking place in Plato’s work as well as the reconstruction of Gorgias’ vision of the art of rhetoric and art in general, defining the relation between art and knowledge. The latter may be undertaken on the basis of two paraphrases of the treatise “On Non-existence”, two epideictic speeches entitled “Helen” and “Palamedes” and some smaller fragments which have been preserved from Gorgias’ creative output. The treatise “On Non–existence”, very paradoxical in its philosophical meaning, testifies to new, sophistic thinking represented by Gorgias of Leontinoi. The treatise is the polemic and attack on Eleatism based on certain principles and methods elaborated by the Eleatics themselves. It also shows that Gorgias – using traditional Eleatic schemes – is opposed to this tradition. Even though the treatise does not discuss the problem of art directly, it anticipates the considerations from epideictic speeches. “Helen” and “Palamedes” constitute further stages in the development of Gorgias’ thought and concept of art combining typically epideictic elements with some philosophical convictions. Gorgias refers to the Eleatic theme considering the issues of truth, knowledge and belief, yet defining them in the empirical spirit. Solutions in the cognitive sphere determine the horizon of Gorgias’ rhetoric. The word is separated from the existence. Human cognition is limited and therefore the word and visual images affect the belief. This promotes the great importance of rhetoric, painting and sculpture, i.e. the areas which influence in a specific way – creating the “delusion” (ἀπάτη). Through idealization art provides pleasure, which is its objective. Therefore the task of art is to create delusion. The rhetor of Leontinoi discusses the issue of “how?” to use the word ‘rhetoric’. Gorgias is aware that rhetoric may be used to talk about what is unknown and that technical correctness of a rhetoric statement is not identical with the truthfulness of the proclaimed message. Mentioning it, Gorgias shows his respect to the truth as a value which should be respected in speech on the one hand, and on the other emphasizes that the truth in itself has no persuasive power. For this reason it is not enough to proclaim the truth, but, to convince the listeners, the orator should possess the skill of speaking. What is then the relation between truth and art emerging from epideictic speeches? Gorgias undoubtedly emphasizes the formal aspect of speech, as rhetoric is art, thus an ability based on certain principles. According to Gorgias, the truth is the value which should be respected in speeches. Rhetoric is able to create a fictive world, as a great range of reality remains outside direct human experience. Rhetoric does not oppose the truth but discussing not only what is directly given concerns the area outside the sphere of direct experience. This concept of rhetoric is the basis of the discussion about art carried out in the dialogue “Gorgias”. Even though the first of the discussions in the dialogue – the conversation between Socrates and Gorgias – is directly concerned with rhetoric itself, numerous remarks testify to its much wider character. This is best exemplified by the way of carrying out the conversation by Socrates, who, asking questions about rhetoric practised by Gorgias, throughout the whole conversation compares it with other skills. Additionally, the manner and the structure of the conversation prove that Plato considers the issue of rhetoric from a wider perspective determined by the issue of art. The specific feature of the conversation is ordering it around main issues – such notions as πρᾶγμα, ἐπιστήμη, δύναμις, each of which has its own meaning in Plato. In the first part of Gorgias Plato presents the principles previously occurring in other dialogues in a certain order, thus constructing the structure of the conversation with Gorgias. The notions are not novum in Plato but the fact that he clearly wants to systematise them and present a certain concept of art is a new and significant aspect. This tendency is of paramount importance for the discussion because Gorgias’ answers are evaluated by Plato from the perspective of this clearly set concept consisting of not only established notions but also certain principles. The discussion between Socrates and Gorgias introduces a theme which is crucial for the dialogue: the relation between art and good, i.e. the technical and ethical spheres. The issue is introduced by Gorgias, who mentions that rhetoric may be used for evil purposes. On this basis Socrates proves that if rhetoricians can use rhetoric for evil purposes, they have no knowledge concerning the subject of their art – justice, because one who knows what is just, always acts justly. Thus the relation between the technical and ethical spheres transpires to be the main point of contention. The question is whether a technician following the principles of the art he practices may act against good. This paradoxical statement based on the well-known Socrates’ principle "nemo sua sponte peccat" acquires philosophical foundations in the dialogue. In its remaining two parts Plato presents a justification, placing the principle within the framework of a certain concept of art, as a result of which not only rhetoric but also many other skills are criticised. The notions on which the conversation with Gorgias focuses are more precisely defined in the subsequent parts of the dialogue. Additionally, Plato presents twice the conditions which art should fulfill. They oblige the “technicians” to know the nature and purpose of the art and to know the causes of undertaken activities and to be able to justify them. These conditions determine the relation between the two spheres – technical and ethical. According to Plato, art may not act against good because it should be the objective of its activities. Certain concepts of good and knowledge outlined in the dialogue are the guarantees of “technicality”. Good, which should be respected by every art, is understood objectively and is contrasted with subjective pleasure. Knowledge, which should be the basis of art, opposes experience, which uses observation and recollection. Additionally, the definition of knowledge, significance of order in the world assuming the form of “geometrical equality”, significance of mathematics becoming the ideal of exactness indicate that Plato’s concept of knowledge is formed in Gorgias. These conclusions render it impossible that the results of any activity deserving the name of art should oppose good. And this is Plato’s main reproach of Gorgias’ concept, who wrote about false speeches possessing the power of conviction because they were written according to the principles proof of art or who defined tragedy as “a deceit, where he who cheats is more just than he who does not and the cheated is wiser than he who was not cheated”. Such determination of the relation between the technical and ethical sphere resulted from Gorgias’ concept of the world and fundamental philosophical principles. The whole concept of art described by Plato in the dialogue opposes the view presented by Gorgias. According to Plato, every technical activity has a defined object (ἔργον), is based on knowledge (ἐπιστήμη) and is always aimed at the good of human soul or body. The danger revealed and presented in the dialogue by Plato exists in the very concept of art presented by Gorgias, i.e. in its empirical foundations and the resulting understanding of the relation between art and good. Gorgias’ concept of art based on empirical cognition, which concerns the world of phenomena and rejects the possibility of perfect knowledge, leads, according to Plato, to a dangerous falsification of the relation between art and good. Plato proves that this weakness of Gorgias’ concept is revealed when confronted with these representatives of “the magnificent art” who are not protected against radicalism by the adherence to the moral tradition. The comparison of two, completely contrasting concepts of art based on different philosophical understanding of knowledge and good is decisive for the criticism of rhetoric in Gorgias. The concept presented by Plato criticises contemporary rhetoric and sophistry but determines the direction of research of the real art of rhetoric. (shrink)

Context: The paper utilizes a conceptual analysis to examine the development of abstract conceptual structures in mathematical problem solving. In so doing, we address two questions: 1. How have the ideas of RC influenced our own educational theory? and 2. How has our application of the ideas of RC helped to improve our understanding of the connection between teaching practice and students’ learning processes? Problem: The paper documents how Ernst von Glasersfeld’s view of mental representation can be illustrated (...) in the context of mathematical problem solving and used to explain the development of conceptual structure in mathematical problem solving. We focus on how acts of mental re‑presentation play a vital role in the gradual internalization and interiorization of solution activity. Method: A conceptual analysis of the actions of a college student solving a set of algebra problems was conducted. We focus on the student’s problem solving actions, particularly her emerging and developing reflections about her solution activity. The interview was videotaped and written transcripts of the solver’s verbal responses were prepared. Results: The analysis of the solver’s solution activity focused on identifying and describing her cognitive actions in resolving genuinely problematic situations that she faced while solving the tasks. The results of the analysis included a description of the increasingly abstract levels of conceptual knowledge demonstrated by the solver. Implications: The results suggest a framework for an explanation of problem solving that is activity-based, and consistent with von Glasersfeld’s radical constructivist view of knowledge. The impact of von Glasersfeld’s ideas in mathematics education is discussed. (shrink)

The Benacerraf Problem of Mathematical Truth and Knowledge Before philosophical theorizing, people tend to believe that most of the claims generally accepted in mathematics—claims like “2+3=5” and “there are infinitely many prime numbers”—are true, and that people know many of them. Even after philosophical theorizing, most people remain committed to mathematical truth and mathematical knowledge. … Continue reading The Benacerraf Problem of Mathematical Truth and Knowledge →.

Given an infinite Boolean algebra B, we find a natural class of equation image-definable equivalence relations equation image such that every imaginary element from Beq is interdefinable with an element from a sort determined by some equivalence relation from equation image. It follows that B together with the family of sorts determined by equation image admits elimination of imaginaries in a suitable multisorted language. The paper generalizes author's earlier results concerning definable equivalence relations and weak elimination of imaginaries for (...) Boolean algebras, obtained in 10. (shrink)

No one will dispute, looking at the history of mathematics, that there are plenty of moments where mathematics is “in trouble”, when paradoxes and inconsistencies crop up and anomalies multiply. This need not lead, however, to the view that mathematics is intrinsically inconsistent, as it is compatible with the view that these are just transient moments. Once the problems are resolved, consistency (in some sense or other) is restored. Even when one accepts this view, what remains is (...) the question what mathematicians do during such a transient moment? This requires some method or other to reason with inconsistencies. But there is more: what if one accepts the view that mathematics is always in a phase of transience? In short, that mathematics is basically inconsistent? Do we then not need a mathematics of inconsistency? This paper wants to explore these issues, using classic examples such as infinitesimals, complex numbers, and infinity. (shrink)

The vision of integrating artificial intelligence in education is part of an ongoing push for harnessing digital solutions to improve teaching and learning. Drawing from Jasanoff and Hasse, this paper deliberates on how sociotechnical imaginaries are interrelated to the implications of new technologies, such as AI, in education. Complicating Hasses’s call for the development of Socratic ignorance to consider our predispositions about new technologies and open new prospects of thought, this paper revisits postphenomenology and Feenberg’s critical constructivist theories. While embracing (...) the notion of Socratic ignorance, this paper stresses the importance of developing a nuanced understanding of technology that realizes its lack of neutrality and supports the creation of a deeper understanding of how knowledge is produced, deployed, and interpreted in the digital age. Thus, this paper argues that an amalgam of Hasse’s call for advancing Socratic ignorance combined with postphenomenology and critical constructivism can support students in developing a critical understanding of technology and opening new landscapes of imaginaries. (shrink)

In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in mathematical practice. Then, I (...) will give an example of mathematical reasoning with a figure, and show that both visualization and intuition are involved. I claim that mathematical intuition depends on background knowledge and expertise, and that it allows to see the generality of the conclusions obtained by means of visualization. (shrink)

The paper gives an analysis of the theoretical-methodological principles of the philosophy of J. Pato?ka not only as a historian of philosophy, but also as a historian of science, especially of its revolutionary periods. The aim of the paper is to show that following the general context of his works here also Pato?ka consistently deals with the central issue of his philosophy, namely the life-world . In Pato?ka's view it was already the rise of ancient philosophy, and especially of the (...) mathematical natural science that destroyed the original uniform picture of the universe, splitting man's world into the natural world on one hand and the world construed by the mathematical natural science on the other hand. Further, the paper shows Pato?ka's view of phenomenology, which he has seen as a means, with the help of which this dichotomy could be resolved, as well as Pato?ka's criticism of the original Husserlian concept of phenomenology, which failed to avoid its subjectivist and solipsist tendencies. The problem of lifeworld led Pato?ka not only to the principles of non-subjective phenomenolog, but it also became the basis of his philosophy of the history of philosophy. (shrink)