The Mathematician’s Mind Redux, I

After a long hiatus, I have decided to re-introduce and elaborate upon what I take to be an account of what it means to think beautifully about mathematics. In an effort to do so, I will be following a path that will engage, at least in its initial stages, the work of the mathematician Jacques Hadamard exploration of the psychological elements involved in a mathematician’s effort to invent, discover, and develop further her particular field of mathematics. After looking at a summary of his thesis and at how a particular historical vignette fits into it, I will quickly diverge by framing his thesis within an account of human understanding largely developed by Bernard Lonergan. 

Written in 1945, Jacques Hadamard wrote the work The Mathematician’s Mind: The Psychology of Invention in the Mathematical Field in an effort to understand the process by which a mathematician comes to discover and invent her mathematics.[1] In the preface to the 1996 paperback edition, P.N. Johnson-Laird summarized the steps by which a mathematician goes from addressing to ultimately resolving a mathematical problem. Here they are as Laird-Johnson described them:[2]

  1. Preparation. You work on a problem, giving your conscience attention to it. 
  2. Incubation. Your conscious preparation sets going an unconscious mechanism that searches for the solution. Poincare wrote that ideas are like the hooked atoms of Epicurus: preparation sets them in motion and they continue their dance during incubation. The unconscious mechanism evaluates the resulting combinations on aesthetic criteria, but most of them are useless.
  3. Illumination. An idea that satisfies your unconscious criteria suddenly emerges into your consciousness.
  4. Verification. You carry out further conscious work in order to verify your illumination, to formulate it more precisely, and perhaps to follow up on consequences.

As Laird-Johnson notes, the articulation of these stages of the creative process can be traced back to Graham Wallace’s The Art of Thought (1926) and found in certain degrees in the writings of Helmholtz and Henri Poincare. Since we see this process as a continuous, periodic one, we will refer to it as the Hadamard cycle.  In the forgoing, we will give an account of mathematical knowledge, as centered on the mathematician qua human knower as the source of that knowledge. 

            In an effort to examine this cycle closely, let us take a look at how it occurs by repeating an account of invention and discovery given by Henri Poincare which Hadamard gives particularly close attention to.

“For fifteen days I strove to prove there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at the work table, stayed an hour or two, tried a great number of combinations and reached no result. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, making a stable combination. By next morning I had established the existence of a class of Fuchsian functions, those which come from hypergeometric series; I had only to write out the results, which took only a few hours.

Then I wanted to represent these represent these functions by the quotient of two series; this idea was perfectly conscious and deliberate, the analogy with elliptic functions guided me. I asked myself what properties these series must have if they existed, and I succeeded without difficulty in forming the series I have called theta-Fuchsian.

Just at this time I left Caen, where I was living, to go on a geological excursion under the auspices of the school of mines. The changes of travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paced the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with my conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’s sake I verified the result at my leisure. 

Then I turned my attention to the study of some arithmetic questions apparently without much success and without a suspicion of any connection with my preceding researches. Disgusted with my failure, I went to spend a few days at the seaside, and thought of something else. One morning, walking on the bluff, the idea came to me, with just the same characteristics of brevity, suddenness, and immediate certainty, that the arithmetic transformations of indeterminate ternary quadratic forms were identical with those of non-Euclidean geometry.”[3]

As Hadamard has done, we will want to dwell on how his cycle plays out regarding this concrete account of Poincare. This we will do by elaborating an account of human consciousness which fills in the details of Hadamard’s cycle in contradistinction to the analysis given by Hadamard[4].

            The course we will take will be to first identify the nature of human consciousness in light of Hadamard’s cycle and then relate that account to the role of invention and discovery specific to mathematics. In the process, we will the subject-matter of mathematics with the dynamic process of acquiring and developing mathematical knowledge. This we will do by unpacking what we mean first by the general act of knowing and of identifying knowledge and then specify what is meant by mathematical knowledge.

            To characterize and explicate knowledge properly, it will be important, up front, to begin removing any biases as to the status of the objects of mathematics as being subjective versus objective (in other words, invented versus discovered). This will involve resolving within the human knower the tension between how things that are external are known and how knowledge is acquired internally, i.e. the tension between what is received by the senses and what is discerned within the intellect. To that end, we identify both together as cognitive acts that the human subject is intimately aware of and which is what fundamentally defines the very nature of consciousness[5]

            To parse this nature more precisely, we first identify the act of consciousness that involves sensing, perceiving, and imagining as the perceptible. This is to be distinguished with an intelligent consciousness whose acts seek to obtain the intelligible through inquiry, illumination, and formulation. Here it is worthwhile to quote Lonergan directly regarding the precise nature of this intelligent part of consciousness:

“On this level cognitional process not merely strives for and reaches the intelligible, but in doing so it exhibits its intelligence; it operates intelligently. The awareness is present but it is the awareness of intelligence, of what strives to understand, of what is satisfied by understanding, of what formulates the understood, not as a schoolboy repeating by rote a definition, but as one that defines because he grasps why that definition hits things off.”[6]

It is important at this stage to dwell more deeply on two points regarding how consciousness operates: (1) the transition from the empirical to the intelligent and (2) the precise nature of understanding.

            In the movement from the empirical to the intelligent, there is first the recognition that the mind receives within its cognition images of external objects passively received through the senses. From these, an intentional directiveness may be consciously employed to select from those images, specific ones, which will be referred to as similitudes, from which intelligence seeks to grasp them individually or together. Let us consider these similitudes as data for intelligence. But what is the relationship between these similitudes and the external objects which provide the images through the senses? To answer this question, we now need to consider the general nature of abstraction. 

            Generally defined, abstraction is an operation of the intellect that in understanding what a thing is “distinguishes one thing from another by knowing what one is without knowing anything of the other, either that it is united to it or separated from it” but are in fact “are one in reality.”[7] Before addressing the possibility of abstraction, we identify two principles that undergird it as an operation of the intellect. The first is: the aspect which is considered separate must not depend for its intelligibility on the other aspects for which it is mentally separated (Principle of Independent Intelligibility); the second being: that which is considered separate cannot be asserted to exist apart from the thing and its other aspects with which it really does exist (Principle of Dependent Existence).[8]

From these principles, we may identify what constitutes a similitude of an image received from an existing thing by the senses. Initially, we understand such a thing as composed of matter, that is, as (possibly) existing and subject to change. Further, we may take it to be an individual that possesses a concrete particularity in time and place. In what ways it possesses such features can only be identified by the senses and, as such, is structured in terms of sensible matter. Separate from these features, but integral to it as existing, a thing exists in a material form and, as such, is subject to counting, measurements, ordered and so forth which we refer generally as being ordinable. Understanding a percept strictly under these conditions alone, it is by way of the intelligence alone that such features are grasped. It is from this perspective that matter in a thing understood in its existence and as subject to quantities (broadly understood) alone is said to be constituted of intelligible matter. It is then by abstraction that the image of a thing understood in its materially intelligibility alone is what constitutes the similitude of the thing in the intelligent consciousness. By identifying intelligible matter as what can or does exist materially subject to being ordered we are identifying two different ways for something to be composed from such by understanding it as either (1) by abstraction from what actually exists, or (2) by what can possibly exist. In both cases, knowledge of a thing in being materially intelligible is obtained quantitatively either by analysis through forms of measurement or by understanding quantities per se and their properties. This is how we will initially understand what the subject-matter of mathematics is. In order to make this more precise, we will first take a closer look at Hadamard’s notion of illumination in connection with the act of understanding and then examine what is meant by quantity in the broadest sense. 

[1] Jacques Hadamard, The Mathematician’s Mind: The Psychology of Invention in the Mathematical Field, Princeton Science Library, Princeton University Press, 1996

[2] Ibid p. x

[3] H. Poincare, “Mathematical Creation,” reprinted in The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method

[4] To that end, we will be largely channeling the work of Bernard Lonergan, specifically Insight: A Study of Human Understanding(Philosophical Library New York, Darton Longman and Todd, 1957) and Understanding and Being (Collected Works of Bernard Lonergan, vol. 5, The Robert Mollot Collection, University of Toronto Press, 1990)

[5] “Consciousness is meant an awareness immanent in cognitional acts,” Lonergan, Insight, pp. 320, 322

[6] Ibid, p. 322

[7] Thomas Aquinas, The Division and Methods of the Sciences, Questions V and VI of his Commentary on De Trinitate of Boethius, translated with introduction and notes by Armand Maurer, The Pontifical Institute of Mediaeval Studies, Toronto, Canada, 1963, p. 30

[8] Thomas C. Anderson, “Intelligible matter and the Objects of Mathematics in Aquinas,” New Scholasticism