Intuition and Reflection in Arithmetic: Bob Hale

Michael Potter considers several versions of the view that the truths of arithmetic are analytic and finds difficulties with all of them. There is, I think, no gainsaying his claim that arithmetic cannot be analytic in Kant's sense. However, his pessimistic assessment of the view that what is now widely called Hume's principle can serve as an analytic foundation for arithmetic seems to me unjustified. I consider and offer some answers to the objections he brings against it.     

Mathematical Intuition Vs. Mathematical Monsters*

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.     

Intuition and Reflection in Arithmetic: Michael Potter

If arithmetic is not analytic in Kant's sense, what is its subject matter? Answers to this question can be classified into four sorts according as they posit logic, experience, thought or the world as the source, but in each case we need to appeal to some further process if we are to generate a structure rich enough to represent arithmetic as standardly practised. I speculate that this further process is our reflection on the subject matter already obtained. This suggestion seems problematic, however, since it seems to rest on a confusion between the empirical and the metaphysical self.     

Mathematical Intuition and the Cognitive Roots of Mathematical Concepts

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.     

Intuition and Visualization in Mathematical Problem Solving

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.     

Intuition, reason, and metacognition

Dual Process Theories (DPT) of reasoning posit that judgments are mediated by both fast, automatic processes and more deliberate, analytic ones. A critical, but unanswered question concerns the issue of monitoring and control: When do reasoners rely on the first, intuitive output and when do they engage more effortful thinking? We hypothesised that initial, intuitive answers are accompanied by a metacognitive experience, called the Feeling of Rightness (FOR), which can signal when additional analysis is needed. In separate experiments, reasoners completed one of four tasks: conditional reasoning (N = 60), a three-term variant of conditional reasoning (N = 48), problems used to measure base rate neglect (N = 128), or a syllogistic reasoning task (N = 64). For each task, participants were instructed to provide an initial, intuitive response to the problem along with an assessment of the rightness of that answer (FOR). They were then allowed as much time as needed to reconsider their initial answer and provide a final answer. In each experiment, we observed a robust relationship between the FOR and two measures of analytic thinking: low FOR was associated with longer rethinking times and an increased probability of answer change. In turn, FOR judgments were consistently predicted by the fluency with which the initial answer was produced, providing a link to the wider literature on metamemory. These data support a model in which a metacognitive judgment about a first, initial model determines the extent of analytic engagement.     

Affect,Moral Intuition,and Risk

The natural way to deal with moral issues involving risk is to rely on our intuitive feelings. “How bad is it? Well, how bad does it feel?” We can also rely on reason to determine right and wrong but, as Jonathan Haidt has demonstrated, intuition comes first and dominates our responses unless we make an effort to critique and, if necessary, override it. We argue that our intuitive feelings are insensitive to large losses of life and thus mislead us in the face of natural disasters or human disasters associated with poverty, disease, and violence. Our intuitions seduce us into calmly turning away from these catastrophes, when we should be driven by outrage or sorrow to act. We propose ways to overcome this insensitivity through new forms of education and communication coupled with reliance on moral deliberation aimed at designing laws and institutions that compel attention to such problems.     

Intuition, Affect, and Peculiar Beliefs

Research with college students has found that intuitive thinking (e.g., using hunches to ascribe meaning to experiences) and positive affect interactively predict ideas of reference and odd/magical beliefs. We investigated whether these results would generalize to a diverse community sample of adults that included individuals with elevated levels of peculiar perceptions and beliefs. We measured positive and negative affect and intuitive thinking through questionnaires, and peculiar beliefs (i.e., ideas of reference and odd/magical beliefs) through structured clinical interviews. We found that peculiar beliefs were associated with intuitive thinking and negative affect, but not positive affect. Furthermore, in no instance did the interaction of affect and intuitive thinking predict peculiar beliefs. These results suggest that there are important differences in the factors that contribute to peculiar beliefs between college students and clinically meaningful samples.     

Intuition, cognitive style, and hemispheric processing

200 undergraduate volunteers were administered Westcott's Test of Intuitive Ability, Witkin's Group Embedded Figures Test, Paivio's Revised Individual Differences Questionnaire, and visual and aural dominance measures using a three-field tachistoscope and dichotic listening device, respectively. Field dependence was consistently related to poorer intuitive performance. No sex-related differences were found on intuition, embedded figures, Paivio's questionnaire, or dominance scores although the amount of intuition explained by cognitive style and dominance differed by hand and gender. Visual-verbal style was not associated with dominance while the relation of field dependence-independence and dominance was inconsistent. Neither dominance nor visual-verbal style was consistently related to intuition although complicated associations were indicated by hand and sex, suggesting differences in cognitive organization.     

Intuition,intelligence, data compression

The main goal of my paper is to argue that data compression is a necessary condition for intelligence. One key motivation for this proposal stems from a paradox about intuition and intelligence. For the purposes of this paper, it will be useful to consider playing board games—such as chess and Go—as a paradigm of problem solving and cognition, and computer programs as a model of human cognition. I first describe the basic components of computer programs that play board games, namely value functions and search functions. I then argue that value functions both play the same role as intuition in humans and work in essentially the same way. However, as will become apparent, using an ordinary value function is just a simpler and less accurate form of relying on a database or lookup table. This raises our paradox, since reliance on intuition is usually considered to manifest intelligence, whereas usage of a lookup table is not. I therefore introduce another condition for intelligence that is related to data compression. This proposal allows that even reliance on a perfectly accurate lookup table can be nonintelligent, while retaining the claim that reliance on intuition can be highly intelligent. My account is not just theoretically plausible, but it also captures a crucial empirical constraint. This is because all systems with limited resources that solve complex problems—and hence, all cognitive systems—need to compress data.

     

Intuition

At least since the Enlightenment Western thought has been dominated by a skeptical attitude towards intuition, although it is vital for mental and creative functions. In contrast, in recent times, a re-evaluation has emerged. Intuition is now considered by many to be of significant assistance not only in everyday life but also in making complex decisions in business, people management and psychotherapy. This article describes different dimensions of intuition ranging from unconscious intelligence in everyday life through intuitive processes with their connection to the collective unconscious up to its spiritual dimension. The question of how intuition works is extensively discussed. A distinction is made between intuitive work, resonance phenomena and gut feelings. The author concludes that it is not a question as to whether we want to use intuition as a source of knowledge for ourselves but if we recognize and appreciate that we do it constantly.     

Intuition as an Influence on Creative Problem-Solving: The Effects of Intuition,Positive Affect,and Training

Intuition may be a critical component of creative thought. To test this hypothesis, a measure of individual differences in intuition was developed. After completing this measure, 320 undergraduates were asked to work on a domain-relevant creative problem-solving task under conditions where positive and neutral affect were induced and they were exposed to 1 of 3 different types of training. It was found that intuitive people produced more creative problem solutions, but that positive affect and training offset the advantage intuitive people showed in creative problem-solving. The implications of these findings for understanding the nature of intuition, and its role in creative problem-solving, are discussed.     

Intuition, Inference, and Rational Disagreement in Ethics

This paper defends a moderate intuitionism by extending a version of that view previously put forward and responding to some significant objections to it that have been posed in recent years. The notion of intuition is clarified, and various kinds of intuition are distinguished and interconnected. These include doxastic intuitions and intuitive seemings. The concept of inference is also clarified. In that light, the possibility of non-inferential intuitive justification is explained in relation to both singular moral judgments, which intuitionists do not take to be self-evident, and basic moral principles, which they typically do take to be self-evident in a sense explicated in the paper. This explanation is accomplished in part by drawing some analogies between moral and perceptual judgments in the light of a developmental conception of knowledge. The final section of the paper presents a partial account of rational disagreement and indicates how the kind of intuitionist view defended can allow for rational disagreement between apparent epistemic peers.

Function, Intuition and Ends in Aristotle's Ethics

This essay attempts to show why deliberation is not of ends for Aristotle, not only because deliberation is concerned with means, but because ends are grasped by wish. Such wishing, I argue, is a form of rational intuition that is non-discursive and analogous to seeing and therefore not at all like the discursive thought involved in deliberation. Such a reading also helps shed light on the nature of contemplation and therefore on happiness in Aristotle.     

Creativity and Intuition Revisited

This study used the recently revised Myers Briggs Type Indicator (MBTI) and a variety of validated creativity measures to replicate a classic finding — that sensing‐intuition predicts creativity — and to evaluate whether the other MBTI scales add to the prediction of creativity. 94 undergraduates completed the MBTI (form M self‐scorable) and criterion measures including the Creative Personality Scale, an abbreviated Creative Behavior Inventory (self‐reported past accomplishments) and the Test of Creative Thinking‐Drawing Production (creative product ratings using consensual assessment). Results showed that intuition predicted all three criteria but that, contrary to expectation from past research, judging‐perceiving did not add to the prediction model. Thinking‐feeling emerged as a possible suppressor variable. We suggest that judging‐perceiving predicts creativity as an artifact of its relation with intuition and that the combination of intuition and feeling best characterizes high scorers on a composite creativity measure.