If we are to obtain a priori mathematical knowledge by following proofs, then we have to be able to have a priori knowledge of the axioms. This chapter examines the major accounts of how such knowledge might be gained. It is argued that all these accounts fail.

Charles Parsons; X*—Mathematical Intuition, Proceedings of the Aristotelian Society, Volume 80, Issue 1, 1 June 1980, Pages 145–168, https://doi.org/10.1093/ari.

"Intuition" has perhaps been the least understood and the most abused term in philosophy. It is often the term used when one has no plausible explanation for the source of a given belief or opinion. According to some sceptics, it is understood only in terms of what it is not, and it is not any of the better understood means for acquiring knowledge. In mathematics the term has also unfortunately been used in this way. Thus, intuition is sometimes portrayed as (...) if it were the Third Eye, something only mathematical "mystics", like Ramanujan, possess. In mathematics the notion has also been used in a host of other senses: by "intuitive" one might mean informal, or non-rigourous, or visual, or holistic, or incomplete, or perhaps even convincing in spite of lack of proof. My aim in this book is to sweep all of this aside, to argue that there is a perfectly coherent, philosophically respectable notion of mathematical intuition according to which intuition is a condition necessary for mathemati cal knowledge. I shall argue that mathematical intuition is not any special or mysterious kind of faculty, and that it is possible to make progress in the philosophical analysis of this notion. This kind of undertaking has a precedent in the philosophy of Kant. While I shall be mostly developing ideas about intuition due to Edmund Husser! there will be a kind of Kantian argument underlying the entire book. (shrink)

The thesis is a study of the notion of intuition in the foundations of mathematics which focuses on the case of natural numbers and hereditarily finite sets. Phenomenological considerations are brought to bear on some of the main objections that have been raised to this notion. ;Suppose that a person P knows that S only if S is true, P believes that S, and P's belief that S is produced by a process that gives evidence for it. On a phenomenological (...) view the relevant evidence is provided by intuition , and this should be the case in either ordinary perceptual knowledge or in mathematical knowledge. Intuition is to be understood in terms of fulfillments of intentions. Knowledge is a product of intuition and intention. In the case of mathematical knowledge it is said that there is a construction for a mathematical statement S if and only if the intention expressed by S is fulfilled . Constructions are thus viewed as intuition processes that could actually or possibly be carried out. In elementary parts of mathematics they might be characterized in terms of certain classes of recursive functions. This view is discussed in the case where S is taken to be a singular statement about natural numbers or finite sets, and also where S is taken to be a general statement about such objects. The distinction between intuition of and intuition that is also investigated in this context. ;It is pointed out how on a phenomenological view a number of central problems about mathematical intuition can be avoided: problems about the analogousness of perceptual and mathematical intuition, about causal accounts of knowledge in mathematics, and about structuralism in mathematics. The bearing of the account on issues concerning constructivism and platonism is also discussed. (shrink)

Geometrical and physical intuition, both untutored andcultivated, is ubiquitous in the research, teaching,and development of mathematics. A number ofmathematical ``monsters'', or pathological objects, havebeen produced which – according to somemathematicians – seriously challenge the reliability ofintuition. We examine several famous geometrical,topological and set-theoretical examples of suchmonsters in order to see to what extent, if at all,intuition is undermined in its everyday roles.

The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis (...) with an often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to a philosophical frame as well as to some recent experimental studies on numerical cognition that support our claim on the cognitive origin and the constitutive role of mathematical intuition. (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)

MATHEMATICAL INTUITION IS OFTEN REGARDED AS A SPECIAL FORM\nOF PERCEPTION WHOSE OBJECTS ARE ABSTRACT ENTITIES. THE\nTHESIS OF THIS PAPER IS THAT MATHEMATICAL INTUITION IS JUST\nORDINARY PERCEPTION AND IMAGINATION OF FAMILIAR OBJECTS. IT\nIS DISTINGUISHED, HOWEVER, BY ITS MODE OF\nCONCEPTUALIZATION, WHICH UTILIZES RELATIVELY FEW PREDICATES\nAND HENCE TREATS MANY DISTINCT OBJECTS AS\nINDISTINGUISHABLE.

This paper covers some large subjects: as well as intuition and Wittgenstein, it also discusses modern computing. However it only traces one thread through these topics. Basically it proposes that a computational analysis of Wittgenstein's Tractatus can shed light upon processes of discovery in mathematics.

This paper provides examples in arithmetic of the account of rational intuition and evidence developed in my book After Gödel: Platonism and Rationalism in Mathematics and Logic . The paper supplements the book but can be read independently of it. It starts with some simple examples of problem-solving in arithmetic practice and proceeds to general phenomenological conditions that make such problem-solving possible. In proceeding from elementary ‘authentic’ parts of arithmetic to axiomatic formal arithmetic, the paper exhibits some elements of the (...) genetic analysis of arithmetic knowledge that is called for in Husserl’s philosophy. This issues in an elaboration on a number of Gödel’s remarks about the meaning of his incompleteness theorems for the notion of evidence in mathematics. (shrink)

The paper offers one of Parsons’ main themes in his book Mathematical Thought and Its Objects of 2008 : the role of intuition in our understanding of arithmetic. Our discussion does not cover all of the issues that have relevance for Parsons’ account of mathematical intuition, but we focus on the question: whether our knowledge that there is a model for arithmetic can reasonably be called intuitive. We focus on this question because we have some concerns about that.

Charles Parsons’ book “Mathematical Thought and Its Objects” of 2008 (Cambridge University Press, New York) is critically discussed by concentrating on one of Parsons’ main themes: the role of intuition in our understanding of arithmetic (“intuition” in the specific sense of Kant and Hilbert). Parsons argues for a version of structuralism which is restricted by the condition that some paradigmatic structure should be presented that makes clear the actual existence of structures of the necessary sort. Parsons’ paradigmatic structure is (...) the so-called ‘intuitive model’ of arithmetic realized by Hilbert’s strings of strokes. This paper argues that Hilbert’s strings, considered as given in intuition, cannot play the role Parsons assigns to them: the criteria of identity of these strings do not have the sharpness that Parsons wants to see in them, and Parsons inadvertently projects abstract structures into his ‘intuitive model’. This diagnosis is exemplified with respect to (a) Parsons’ distinction between addition and multiplication on the one hand and exponentiation on the other and (b) his analysis of arithmetical knowledge in simple cases like “7 + 5 = 12”. All in all, it is claimed that Parsons book contains many important insights with respect to, for example, different versions structuralism, the notion of “natural number” and its uniqueness, induction, predicativity and other things, for which he is rightly famous, but that his way of drawing on the notion of intuition leaves too many questions unanswered. (shrink)

The best known and most widely discussed aspect of Kurt Gödel's philosophy of mathematics is undoubtedly his robust realism or platonism about mathematical objects and mathematical knowledge. This has scandalized many philosophers but probably has done so less in recent years than earlier. Bertrand Russell's report in his autobiography of one or more encounters with Gödel is well known:Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal “not” was laid up in heaven, where (...) virtuous logicians might hope to meet it hereafter.On this Gödel commented:Concerning my “unadulterated” Platonism, it is no more unadulterated than Russell's own in 1921 when in the Introduction to Mathematical Philosophy … he said, “Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.” At that time evidently Russell had met the “not” even in this world, but later on under the infuence of Wittgenstein he chose to overlook it.One of the tasks I shall undertake here is to say something about what Gödel's platonism is and why he held it.A feature of Gödel's view is the manner in which he connects it with a strong conception of mathematical intuition, strong in the sense that it appears to be a basic epistemological factor in knowledge of highly abstract mathematics, in particular higher set theory. Other defenders of intuition in the foundations of mathematics, such as Brouwer and the traditional intuitionists, have a much more modest conception of what mathematical intuition will accomplish. (shrink)

The aim of this article is to show that intuition plays no role in mathematics. That intuition plays a role in mathematics is mainly associated to the view that the method of mathematics is the axiomatic method. It is assumed that axioms are directly (Gödel) or indirectly (Hilbert) justified by intuition. This article argues that all attempts to justify axioms in terms of intuition fail. As an alternative, the article supports the view that the method of mathematics is the analytic (...) method, a method originally used by the mathematician Hippocrates of Chios and the physician Hippocrates of Cos, and first explicitly formulated by Plato. The article examines the main features of the analytic method, and argues that, while intuition plays an essential role in the axiomatic method, it plays no role in the analytic method, either in the discovery or in the justification of hypotheses. That the method of mathematics is the analytic method involves that mathematical knowledge is not absolutely certain but only plausible, but the article argues that this is the best we can achieve. (shrink)

As part of the development of an epistemology for mathematics, some Platonists have defended the view that we have (i) intuition that certain mathematical principles hold, and (ii) intuition of the properties of some mathematical objects. In this paper, I discuss some difficulties that this view faces to accommodate some salient features of mathematical practice. I then offer an alternative, agnostic nominalist proposal in which, despite the role played by mathematical intuition, these difficulties do not emerge.

Both phenomenologists and analytical philosophers of mathematics should profit from this excellent exposition and defense of Husserl's account of mathematical knowledge. Tieszen places Philosophie der Arithmetik in the context of Husserl's later phenomenological thinking and demonstrates thereby that Husserl's contributions to the philosophy of mathematics are much more important than most of us, influenced by Frege's denigrating critique of Husserl's early psychologism, had thought. He also makes a convincing case for the phenomenological approach to constructive mathematics.

Charles Parsons’ book “Mathematical Thought and Its Objects” of 2008 (Cambridge University Press, New York) is critically discussed by concentrating on one of Parsons’ main themes: the role of intuition in our understanding of arithmetic (“intuition” in the specific sense of Kant and Hilbert). Parsons argues for a version of structuralism which is restricted by the condition that some paradigmatic structure should be presented that makes clear the actual existence of structures of the necessary sort. Parsons’ paradigmatic structure is (...) the so-called ‘intuitive model’ of arithmetic realized by Hilbert’s strings of strokes. This paper argues that Hilbert’s strings, considered as given in intuition, cannot play the role Parsons assigns to them: the criteria of identity of these strings do not have the sharpness that Parsons wants to see in them, and Parsons inadvertently projects abstract structures into his ‘intuitive model’. This diagnosis is exemplified with respect to (a) Parsons’ distinction between addition and multiplication on the one hand and exponentiation on the other and (b) his analysis of arithmetical knowledge in simple cases like “7 + 5 = 12”. All in all, it is claimed that Parsons book contains many important insights with respect to, for example, different versions structuralism, the notion of “natural number” and its uniqueness, induction, predicativity and other things, for which he is rightly famous, but that his way of drawing on the notion of intuition leaves too many questions unanswered. (shrink)

Charles Parsons has argued that we have the ability to apprehend, or "intuit", certain kinds of abstract objects; that among the objects we can intuit are some which form a model for arithmetic; and that our knowledge that the axioms of arithmetic are true in this model involves our intuition of these objects. I find a problem with Parson's claim that we know this model is infinite through intuition. Unless this problem can be resolved. I question whether our knowledge that (...) the arithmetical axioms are true can reasonably be called intuitive in the way Parsons intends. (shrink)

European Journal of Philosophy, Volume 30, Issue 2, Page 578-600, June 2022.

In his first philosophy book, Science and Hypothesis, Poincare provides a picture in which the different sciences are arranged in a hierarchy. Arithmetic is the most general of all the sciences because it is presupposed by all the others. Next comes mathematical magnitude, or the analysis of the continuum, which presupposes arithmetic; and so on. Poincare's basic view was that experiment in science depends on fixing other concepts first. More generally, certain concepts must be fixed before others: hence the (...) hierarchy. This paper attempts to dissolve some potential problems regarding Poincare's hierarchy. One is an apparent epistemological circularity in the hierarchy. A more serious problem regarding the epistemology of analysis is also addressed. (shrink)

info:eu-repo/semantics/publishedVersion.

In a recent article I detailed at length the methodology employed to explore the reflective and pre-reflective contents of singular intuitive experiences in contemporary mathematics in order to propose an essential structure of intuition arousal in mathematics. In this paper I present the phenomenological assessment of the essential structure according to the three formal structures as proposed by Sokolowski's scheme and show their relevance in the description of the intuitive experience in mathematics. I also show that this essential structure acknowledges (...) the perceptualist view of intuition, as proposed by Chudnoff, and discuss the analogy that is often proposed and argued between mathematical intuition and ordinary perceptive intuition. (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 (...) class='Hi'>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)

Today it is no news to point out that Kant’s doctrine of space as a form of intuition is motivated by epistemological considerations independent of his commitment to Euclidean geometry. These considerations surface—apparently without his own recognition—in Poincaré’s, Science and Hypothesis, the very work that helped turn analytically-minded philosophers away from the Critique. I argue that we should view Poincaré as refining Kant’s doctrine of space as the form of intuition, even as we see both views as arbitrarily limited—in Kant’s (...) case, to Euclidean transformations, and, in Poincaré's, to geometries of constant curvature. Both run together the question whether space is an a priori form of intuition with the question whether there are a priori constraints on its applied geometry. At his best, Kant sees—what Poincaré does not—that they are connected by a form of cognition consisting in the intuition of homogeneous pluralities, of totalities apprehended as unities. (shrink)

The literature on mathematics suggests that intuition plays a role in it as a ground of belief. This article explores the nature of intuition as it occurs in mathematical thinking. Section 1 suggests that intuitions should be understood by analogy with perceptions. Section 2 explains what fleshing out such an analogy requires. Section 3 discusses Kantian ways of fleshing it out. Section 4 discusses Platonist ways of fleshing it out. Section 5 sketches a proposal for resolving the main (...) problem facing Platonists—the problem of explaining how our experiences make contact with mathematical reality. (shrink)

The article is devoted to philosophy of mathematics. Mathematical heuristics, being a complex of methods for solving the non-standard problems of mathematics (such problems which have no known algorithms to be solved), is the main subject of the research. As a specific mechanism for thinking, generating elements of guesswork needed as the basis of mathematical heuristics, the author considers intuition. In the work, the author uses Descartes’s, Poincaré’s, Hadamard’s and Piaget’s findings. Based on Descartes’s concept of rational intuition, (...) the author develops the concept of heuristic intuition. As a result, the author turns to the question of possibility of a complete translation of the user-derived mathematical statements in a discourse, in fact, that means a maximum depth of mathematical proof, i.e. its maximum rationalization. For this purpose, it is necessary to re-attract the intuition since it is able to transform the intuitive elements into the discourse ones. Therefore, from this point of view, the rationale is intuitively derived mathematical proof should be no more than a ‘multilayer‘ creative process. In general, the author, based on Poincaré’s research, proves that the essence of mathematical creativity is not to «sort out» and «choose». Referring to examples for illustration, the author reveals moments of «interference» of intuition, even in the process of solving school problems. Therefore, it is currently impossible to ignore the phenomenon of intuition and the results that have been historically derived a theory of knowledge in the study of creative mechanisms. (shrink)

It is alleged that the causal inertness of abstract objects and the causal conditions of certain naturalized epistemologies precludes the possibility of mathematical know- ledge. This paper rejects this alleged incompatibility, while also maintaining that the objects of mathematical beliefs are abstract objects, by incorporating a naturalistically acceptable account of ‘rational intuition.’ On this view, rational intuition consists in a non-inferential belief-forming process where the entertaining of propositions or certain contemplations results in true beliefs. This view is free (...) of any conditions incompatible with abstract objects, for the reason that it is not necessary that S stand in some causal relation to the entities in virtue of which p is true. Mathematical intuition is simply one kind of reliable process type, whose inputs are not abstract numbers, but rather, contemplations of abstract numbers. (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)

often insisted existence in mathematics means logical consistency, and formal logic is the sole guarantor of rigor. The paper joins this to his view of intuition and his own mathematics. It looks at predicativity and the infinite, Poincaré's early endorsement of the axiom of choice, and Cantor's set theory versus Zermelo's axioms. Poincaré discussed constructivism sympathetically only once, a few months before his death, and conspicuously avoided committing himself. We end with Poincaré on Couturat, Russell, and Hilbert.