Introduction
All great mathematicians appeal to their “intuition.” One of the most famous proponents of this concept was the French mathematician Jacques Hadamard, who published a systematic inquiry into his fellow mathematicians’ practices. Hadamard reported that many major mathematical discoveries were preceded by long periods of unconscious “incubation” followed by sudden insight. He also quoted Einstein’s well-known introspection: “Words and language, whether spoken or written, do not seem to play any role in my thinking mechanisms. The mental entities that serve as elements of my thought are certain signs or images, more or less clear, that can ‘at will’ be reproduced or combined.” Henri Poincare saw intuition as the foundation ´ upon which the mathematical enterprise was based. Davis and Hersh, in their description of “the mathematical experience,” went even further (and probably too far) by viewing intuition as the single faculty that allows us to do mathematics: “In the realm of ideas, of mental objects, those ideas whose properties are reproducible are called mathematical objects, and the study of mental objects with reproducible
properties is called mathematics. Intuition is the faculty by which we can consider or examine these (internal mental) objects”.
The problem with these statements is that they fail to define what an intuition is. Indeed, the concept, like many other theoretical constructs of folk psychological origin (“attention,” “association,” “consciousness,” and others) remains suspect in present-day cognitive neuroscience because it is unclear whether it corresponds to a well-characterized mental entity or process. I would like to argue, however, that recent research in numerical cognition fleshes out the concept of intuition, at least within the small domain of elementary arithmetic. The results indicate that a sense of number is part of Homo sapiens’ core knowledge, present early on in infancy, and with a reproducible cerebral substrate. It permits a rapid evaluation of approximately how many objects are present in a scene, whether this number is more or less
than another number, and how this number is changed by simple operations of addition and subtraction. Its operation obeys three criteria that may be seen as definitional of the term “intuition”: it is fast, automatic, and inaccessible to introspection. These properties, far from implying that this intuition is inaccessible to scientific understanding, constitute a decipherable signature of intuition, and we shall see that a simple but precise quantitative mathematical model can be proposed for some of the simplest aspects of their operation.
What is mathematical intuition?
If you look at the literature on mathematics - the prefaces to math textbooks, discussion pieces by mathematicians, mathematical popularizations and biographies, philosophical works about the nature of mathematics, psychological studies of mathematical cognition, educational material on the teaching of mathematics - you will regularly find talk about intuition. This suggests that there is some role intuition plays in mathematics, specifically as a ground of belief about mathematical matters. The aim of the present chapter is to stake out some ideas about how best to understand intuition as it occurs in mathematics, i.e. about the nature of mathematical intuition [2, p. 5].
A closer look at the textbooks, discussion pieces, popularizations and biographies, philosophical works, psychological studies, and educational material reveals, however, that there are a number of distinct notions that correspond to talk about mathematical intuition. The first order of business will be to draw some distinctions between these notions and pick an appropriate focus for our present inquiry. The notion I will focus on is one according to which mathematical intuition is a kind of experience that is like sensory perception in giving its subjects non-inferential access to a world of facts, but different from sensory perception in that the facts are about abstract mathematical objects rather than concrete material objects. Let us call this the perceptualist view of intuition. It has been the dominant conception of mathematical intuition in the western philosophical tradition since Plato, and the alternatives one finds all more or less derive from it, in ways to be indicated below.
After distinguishing the perceptualist view of intuition from some others to be set aside, the plan is as follows. I will sketch some ideas about perception, by reference to which we can flesh out the analogy between mathematical intuition and perception. I explore the two main approaches to doing this in the philosophical literature - what I will call the Kantian and the Platonist views. Kantians face the problem that mathematical subject matter outstrips our sensory capacities. Platonists face the problem of accounting for how our experiences can be in contact with mathematical reality.
The question of what exactly intuition is, in general, is relevant to a variety of domains, including philosophy, mathematics, psychology, and education. Philosophers, such as Bergson and Spinoza, have contrasted intuition with reason and logic, a view that can be found in some modern conceptualizations of mathematical intuition to be discussed below. Mathematicians have traditionally regarded intuition as a way of understanding proofs and conceptualizing problems. Psychologists have examined the role of intuitive thinking in a variety of domains including clinical diagnosis, creativity, decision making, reasoning, and problem solving. The psychological study of mathematical intuition has been mainly conducted in the area of statistical reasoning. This work shows that people are susceptible to a variety of biases, such as ignoring base-rate information in making probabilistic judgments. A review of existing literature in the above areas led us to identify two primary views of
Kantian views
Kant’s view of mathematical intuition has been more influential on both the philosophical and the mathematical tradition than that of any other writer. The aim of this section is to sketch his view, relate it to the perceptualist way of thinking about intuition, and briefly discuss its influence on 12 early twentieth century developments in the foundations of mathematics. The first order of business will be to calibrate some terminology.
Suppose you come to know by intuition that circles are symmetrical about their diameters. Kant also makes a threefold distinction corresponding to the seeming, the awareness, and the whole experience that combines them, but he uses different terminology. Kant uses “intuition” for a part; I have been using “intuition” for the whole. In talking about Kant, I will use “mathematical intuition” for the whole/cognition in Kant’s sense, “intuitive awareness” for the awareness part/intuition in Kant’s sense, and “intuitive seeming” for the seeming part/that which corresponds to thought in a cognition for Kant.
Kant believed this holds for us, but not for God. The difference is that God creates the objects of his intuitive awareness, whereas we are affected by the objects of our intuitive awareness. As we’ll see, however, creation and affection are not the only options [2, p. 14].
The Kantian view of presentational phenomenology as it occurs in mathematical intuitions might be put like this: Whenever you have a mathematical intuition representing that p your mathematical intuition also makes it seem to you as if you are intuitively aware of the items in virtue of which p is true and it does so via sensory illustration of them. This is different from Felix Klein’s view because Kant thinks that mathematical intuitions do make us intuitively aware of mathematical subject matter. It is just that they always do this via sensory illustration.
Platonist view
Though in outline the view of intuition we will consider in this section has ancient and medieval adherents, Descartes put it in its modern form. For our purposes two points are crucial.
First, in contrast to Kant, Descartes argues that the natures of mathematical objects are independent of our minds.
Second, in contrast to Kant, Descartes argues that intuitive awareness is independent of our capacity for sensation—even if it sometimes involves sensory experiences [2, p. 43].
Intellect is not something that we create, nor something that affects us, nor something that must conform to forms determined by us. Rather Intellect is something that we conform to insofar as we succeed in exercising our intellectual capacities, such as the capacity for intuitive awareness of mathematical objects. So 25 the fourth way
Conclusion
Mathematical and scientific reasoning in general are not reducible to formal
conceptual structures. The history of mathematical and scientific acquisitions has been influenced by the profound tendency of individuals to produce
tacitly mental devices which enable them to believe directly in the objective
validity of their conceptions, even before a complete justification is reached. We consider that the emergence of apparently self-evident, self-consistent
cognitions - generally termed intuitions - is a fundamental condition of a
normal, fluent, productive reasoning activity. An intuition is a complex cognitive structure the role of which is to organize the available information
into apparently coherent, internally consistent, self-evident, practically meaningful representations.
But mathematics is by its very nature a formal, axiomatically organized
system of knowledge. Every statement in mathematics has to be accepted only on the grounds of an explicit, complete proof.
The dynamics of mathematical reasoning include various psychological components like beliefs and expectations, pictorial prompts, analogies and paradigms. These are not mere residuals of more primitive forms of reasoning. They are genuinely productive, active ingredients of every type of reasoning.
Mathematical and science education cannot ignore the impact of intuitive
forces on the student’s ways of reasoning. While much can be learned from history, recent studies have made an important contribution in this domain. We know now that intuitive mechanisms are organized in firm, coherent
complex structures very resistant to alterations. Much more experimental evidence is needed, but the teacher himself may very often discern such elements of resistance which conflict with the taught concepts.
In mathematics education the conflicting nature of mathematical representations has given rise to two opposite didactical strategies. On one hand, many curricula and text-book writers have tended to emphasize the intuitive, pictorial components, apparently in order to meet the child’s strong need for intuitive representations. Text-books became, then, full of beautifully colored images and diagrams. On the other hand, other authors, tried to set up programs and text-books in which the body of knowledge was presented axiomatically. In our opinion both strategies were mistaken because each of them considered only a half of the complex structure of mathematical concepts which, psychologically, are both intuitively and formally based.
A joint effort of psychologists, mathematicians and scientists, of teachers
and researchers, is necessary in order to produce new, adequate, more
efficient programs and didactical solutions in science and mathematics
education.
