MacTutor
The Mechanism of Mental Operations among Mathematicians
The following lecture was given by Arnaud Denjoy on 25 November 1947 at the Romanian Institute of Science and Technology.
The Mechanism of Mental Operations among Mathematicians
Ladies and Gentlemen,
I must first indicate in two words the meaning of the title given to this lecture. The operations of the mind, logical reasoning, the effort of research take place in the intellect of the mathematician according to modes probably common to all active thoughts. But, the object of the mathematician's mental work being almost exclusively created by his imagination with the most negligible borrowing from suggestions of our external senses, it is not foolhardy to hope that the conditions in which our cerebral machine functions will appear with particular clarity if we observe them in the mathematician. The lightness, the simplicity of the device implemented and that the burden of external contingencies does not increase, gives the analyst's mind an ease of pace allowing him to adopt certain practices including subservience to sensitive life data that would make its use difficult and painful in other areas of knowledge or investigation.
It is all these intellectual customs of mathematicians, some common to all minds in work of invention, others special to this category of scholars, that I group under the name of the mechanism of mental operations among mathematicians.
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I will distinguish in my subject three orders of ideas: Logic, Reason, Method for invention.
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Let's start with logic. Most mathematicians obediently submit to the rules of logic, such as people reasoning; they just apply them universally. Aristotle stated its principles. "All men are mortal. Socrates is a man. So Socrates is mortal." This is the syllogism. "All of Romania is between the 42nd and 46th degrees of latitude north. Bucharest is a Romanian city. So the latitude of Bucharest is between the 42nd and the 46th degrees of latitude north". "The Romanians are lovely people. Mrs and Mr X are Romanian. Mr and Mrs X are a lovely couple." We consider a larger extended class, the class of charming people. In this class, another class, formed by the whole Romanian population. Mr and Mrs X are elements of this second class, interior to the first we defined. So Mr and Mrs X are in this one. By repeating the syllogisms we envisage a succession of classes made up of elements, each class containing the following, the first therefore containing all the subsequent ones and in particular any element of the last. This is the top-down, synthetic, didactic form of reasoning. We start by tracing the global domain where we will operate. In this one we circumscribe a part, in this one we look at a subdivision, finally we find ourselves reaching the targeted point.
We could proceed in the opposite direction. Socrates is a man. He therefore has all the characteristics of all men. In particular he is mortal. But he could belong to a class which is neither included in that of men nor an extension of that. The Greeks believed that the deities came into direct contact with certain objects belonging to the terrestrial nature, people, beasts, plants, sources. Socrates, with his demon dictating his thoughts to him, belonged to this class of souls inspired by superhuman revelations. The individuals composing this class, being the object of the interest of the gods, are exposed to the effects of the passions, of the resentments experienced by the celestial tribe. This is a character shown by Socrates without being similarly demonstrated by all men. We have included Socrates in a class, that of beings visited by the gods, a class distinct from that of men, but having elements foreign to the human species, therefore not included in this one. All the properties characterizing this class chosen by Olympus are found in Socrates, which is a particular element of this class. We have practiced the ascending syllogism, that of inventive analysis, which starts from a precise datum and which goes into the unknown of the various properties presented by this given element.
This is in essence the Aristotle's Logic. But mathematicians are not the only ones to comply with its laws. Does there exist in the logic of this school uses specific to mathematicians?
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A well-known French biologist has the habit of saying: "Mathematicians always reason on particular cases."
The first time I heard this sentence, I was surprised, I admit. I imagined, on the contrary, that my colleagues and I envisaged, with the sets of points of geometry, the numerical functions of Analysis, the abstract spaces, ideal objects of unsurpassable generality. I asked myself where did this judgment come from and I thought I understood the difference of point of view between the biologist and the mathematician.
Mathematicians indeed invoke the existence of a particular case to justify their opinion when it comes to negative reasoning, when it comes to denying the legitimacy of a general assertion.
Here is an example of such a proposition: "Before 1914 all the French citizens of the metropolis were of white race." I suppose that the biologist would have considered this allegation perfectly founded if he had found that by traversing the large cities of France, by questioning the various men of colour which he would have met, he would have learned from all that they did not enjoy rights and quality of French citizens. From his point of view as a biologist, he would have considered that the formulated law was valid.
But a mathematician would have said to him: "Your statement is false, because I know and I quote to you a sub-prefect of a district of Province who is black." This very learned biologist probably does not consider that a law proclaimed as general is ruined by the designation of a single example putting it in default.
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Here is a process which Poincaré undoubtedly made the first use of and which can allow us either to quickly discover a fault in reasoning, or to prove the emptiness of hoping to demonstrate a recklessly presumed truth.
This is Euclid's postulate, as taught from the start of elementary geometry. I remind you: A straight line is drawn or conceived. We consider a point not on the line. Through this straight line and this point passes a single plane. Euclid affirms that through the point we can draw a straight line, and only one, parallel to the first, that is to say located in the mentioned plane and such that the two straight lines, no matter how far they are extended, will not meet. This claim has never been proven logically. And precisely Poincaré has put out of doubt that we can achieve it.
A doctor, very distinguished and whose work has advanced our knowledge of epidemic diseases with great strides, having kept a certain taste for mathematical reflections from his youth, recently sent me a demonstration of Euclid's postulate. He admitted the near certainty of having committed a fault in reasoning. But neither he nor the friends he consulted discovered this one.
To find it easily, I used Poincaré's method. The words designating the objects studied are given a different meaning from that normally assigned to them. These words therefore define other objects. We examine whether the characteristic properties of the former can be transported to the latter, by also hearing them in a new sense. Once these conditions are fulfilled, everything that logical reasoning will demonstrate for the first objects will be true for the second and the converse will hold. If therefore the second category does not have a certain property, there is no point in hoping to ever be able to demonstrate that the corresponding property belongs to the first category.
And if reasoning is applied to it in order to establish this property, the chain of propositions constituting this necessarily false reasoning will at least stumble and no doubt at the moment when an incorrect proposition for the second category of objects will be introduced into the corresponding chain.
How is a line defined? It is the shortest path from one point to another. This definition is moreover very empirical. If the two points are joined by a curved line, how will the flexible metre which will have to follow the course of the curve keep a length constantly equal to itself?
Poincaré says this: I call a half-circumference a line whose diameter is located in an invariable plane and which at its two ends falls directly on this plane. We say that this half-circumference is orthogonal to the plane (that is to say the meeting at right angles). And even, instead of a plane, Poincaré considers a sphere, qualified by him as fundamental, then, relative to this sphere, arcs of a circle internal to it and limited by their two ends at this sphere, in which the arc of a circle is implanted at a peak.
To justify the analogy to the line, you have to have an unusual way of measuring distances. Poincaré uses a metre whose length examined with our eyes does not remain constant in its movements. But the closer we get to the surface of the sphere, the more the metre shrinks, so that the same length seen by us is measured with this variable meter by a larger number. Thus a length measured in centimetres gives a number a hundred times greater than if we measure it in metres. Poincaré shows that with this convention, what he calls a straight line, this arc of circle orthogonal to the sphere, is indeed the shortest path from one point to another; and also, by two points inside the sphere it passes only one of the Poincaré lines: from one point inside the sphere we can only move to one Poincaré line perpendicular to another. The Poincaré lines resting on these last two form a Poincaré plane for our eyes is a cap of a sphere, limited to the fundamental sphere, and fitting into it perpendicularly.
But on these Poincaré lines, to move away indefinitely, it consists in getting closer and closer to the fundamental sphere. Only, by a point inside the sphere and located outside a Poincaré line, we can move to, in the Poincaré plane determined by the line and the point, an infinity of lines of the same kind which, extended indefinitely, never meet the first. And that is why, on the subject of our ordinary lines, one will never be able to demonstrate the postulate of Euclid by using pure logic, because the reasoning would apply word for word, with the proper interpretation naturally, to the Poincaré lines for which the conclusion is false.
Is this conception of the great French mathematician physically absurd? Physicists tend to think that our universe is finite, endowed with a curvature; we move there indefinitely in all directions as would make perfectly flat beings on the surface of a sphere. If our universe were a sphere, its radius would be given by so many billions of billions accumulated of kilometres, that the metre of Poincaré would be of a perfectly constant length in all the part of the Universe accessible to our means of investigation, and in these limits Euclid's postulate would have the value of the most established truth of experience.
But I immediately showed my mathematical doctor the fault he was committing, stopping at the first of his successive theorems which was not true for Poincaré's lines. He admitted that an angle could move in a plane with two sides constantly perpendicular to two fixed lines, the magnitude of the angle remaining, on the other hand, constant. It was there precisely: that he was committing his petition in principle. It was immediately visible because the Poincaré lines do not offer the same possibility.
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In all of the above, the old Logic of Aristotle has been implemented alone.
At the beginning of this century, the Englishman Russell succeeded in rejuvenating it and giving it extension by considering, instead of this mechanism of class inclusions fitting into each other, like nests of tables, somehow vertical inclusions of horizontal combinations of classes.
An element is in the sum, in the union of two classes, if it is in one or the other. It is in the logical product of the two classes, if it is in both.
Let's take a look at the class, men who are 1.70m tall and the class of men with good stomachs.
A man who is in the sum class of the previous two either will be 1.70 m tall or will have a good stomach, or he will have both.
If he is in the logical product class of the first two, he will be at least 1.70m and a good stomach.
This Russell logic greatly increased the resources of classical reasoning. It is certain that there is nothing contradictory with this one, and many people applied, since men argued, methods borrowed from Russell's logic without suspecting that they were performing a bold and rare mental action.
It is a veritable insurrection against the old ways of reasoning led by the Dutch mathematician Brouwer, claiming to reject from legitimate logic the principle of the excluded third. Here's what it is.
When the totality of beings of a certain species is divided into two categories relatively to a certain character, Brouwer denies that one can say: since such a being is foreign to the first category, then it is certainly in the second. No, you have to show, according to Brouwer, by direct evidence, how the being in question is actually in this second category. That he does not belong to the first is not enough to affirm that he is in the second, when we know it is present in the assembly of the two.
A conjurer shows you a top hat, a hat called "an opera hat", perfectly empty. You measure the depth and the exterior height of the hairstyle. There is no double bottom. The artist waves his scarf and puts it in the hat. He shakes the scarf. A rabbit springs from the hat. You pretend to be able to say: "The rabbit was in the hat. The scarf divided the inside of the hat into two zones, one outside the scarf, between the visible part of the hat and the scarf, the other included in the folds of the scarf." This is the reasoning challenged by Brouwer. The scarf divides the hat into two complementary zones. The fact that the animal was absent from one of the zones is not sufficient to prove that it was in the other.
Someone had announced that they would attend this lecture. This person is not here. Brouwer prohibits you from concluding that they are elsewhere. Elsewhere, it is the set of places distinct from it. Brouwer agrees with us that the person cannot be both here and elsewhere. But he believes that, to have the right to assert the presence of this person elsewhere, the separate location and where that person is located should necessarily be expressly designated. Logically we do not subscribe to Brouwer's requirements. Psychologically, however, we must concede to the Dutch mathematician that the logical certainty of the presence elsewhere of the person absent from here only half satisfies us. We would like to know exactly where he is located elsewhere. We lack the definite certainty.
II
And now let's go to Reason. The word "reason" has several meanings. There were the reasons for a fact, an event, namely a system of facts, conditions explaining the coming of what has been observed, recorded. There is reasoning which is a logical succession of propositions.
But we want to deal with Reason, which is sometimes called with a thought of derision: Reason with a capital R.
This Reason is the set of judgments which are imposed on us with the authority of indisputable evidence, the proposals which we do not dream of subjecting to critical examination, the postulates from which we deduce by a logical path of truths which are incontestable to us, and to which, in our ascending analyses, we hang like a ring sealed in the rock the chain of our inductions to validate the whole.
Reason is specific to each, it expresses the synthesis of our experience, of our acquired philosophy. Pascal distinguished between fine reason and geometric reason. Reason is a system of postulate. Geometric, it is based on a small number of such postulates. Pascal said: "principles", perfectly specified, specified, and from which all our knowledge can derive by the resources of Logic, the old logic of always.
Fine Reason is made up, still according to Pascal, of a multitude of "small" principles, most of them unformulated, drowned in the vagueness of thought, but appearing to our mind, at the instant of their evocation, with clarity sparkling with truth intuitively and directly perceived.
This reason is essentially individual, its value is above all subjective.
Reason will tell one that the world with the wonders whose nature gives us everywhere, at all times, the spectacle, cannot be the result of chance and that its admirable arrangement is necessarily the work of a prior thought. This one will be deist. Another will think of the colossal deployment of energy that the smallest parcels of matter continually bring into play, and will admire that the mutual neutralisation of these forces, diverging in all directions with excessive power, is such that the universe presents only results of insignificant intensity with regard to their elementary components. He will imagine that these attempts at extreme violence, desperately launched towards innumerable goals scattered in all directions are at all times filtered by chains of innumerable successive screens from which only the stable and the permanent emerge liberated. And this one will refuse to invoke the hypothesis of a creative divinity. Reason, which made the first a believer, makes the second an atheist.
And in all fields, political, moral, social, the Reason of each dictates a system of principles. Reason will say to some that the property of the rich is sacred, to others that only the work of the poor is respectable.
Reason has its gender and its age. The reason for the child is not that of the girl, herself different from that of the young woman: and that of the mature woman, that of the elderly woman, that of the adolescent, that of the young man, of the old man, so many distinct Reasons, each shaped by personal experience and lessons learned from it.
The Reason of youth is called "illusions" by old people. The Reason of these appears to juveniles as a lie opposed to the compelling suggestions of life.
Is fine Reason used in mathematics?
Yes, without a doubt. And it is guilty of mischief. Fine Reason is that which brings to the objects of geometry or analysis certain qualities not at all included in their definition, and which we nevertheless suppose inherent in the nature of concepts of such a category, because the most commonly encountered of individuals belonging to this species, precisely presented by these characters, are logically foreign to the strict conditions expressed in their definition.
Geometers consider so-called "developable" surfaces; namely generated by the movement of a mobile straight line and tangent to the same plane along this line. A cylinder, a cone of revolution, give an example. Or again a rigid, warped and forced cardboard surface. Placed on a horizontal table and given a flip, this surface will make an oscillating movement. It will not stop touching the table all along a straight line, moving both on the cardboard and on the table. This is the tangent plane enjoying its property along the same straight line generating the developable surface.
On the other hand, we define the surfaces "applicable onto a plane", that is to say such that we can spread them on a plane without tearing them or folding them back on themselves.
Ancient geometers showed the identity of the two classes. And it is true that the developable surface is applicable on a plane. But to show that, conversely, any surface applicable on a plane is developable, the geometers called upon the inspiration of their fine Reason, which assured them: first, of the existence, at any point of any surface whatsoever, of a tangent plane to this surface, secondly, from the continuous variation of this plane with its point of contact moving continuously on the surface.
However, the great French mathematician Lebesgue. then very young, barely out of the École Normale Supérieure in Paris, said, drawing from his pocket a crumpled handkerchief: "This is a surface that can be applied to a plane". Because the handkerchief spreads out exactly on a plane when it is unfolded. "But," added Lebesgue, "according to science these days, my crumpled handkerchief is a developable surface. Where are its rectilinear generators and the constant tangent plane along any generator?''.
The ancient geometers had added to the features of the definition of the developable surface, a feature perfectly specified and to which the geometric Reason should be applied, implicit features, existence, continuity of the tangent plane, of which only their fine Reason vouched for them. And since they had not expressly formulated these additional features, their reasoning was not logically correct.
But this rectification brought by Lebesgue was of such great and profound importance that it was at the origin of the ideas of this mathematician on the metrics of sets. The famous Lebesgue integral ensued. And all modern mathematics received a tremendous boost.
The fine Reason introduced into mathematics inspires the belief that the elements of geometry or analysis are subject to certain rules of taste, decency, good manners. And individuals belonging to these same elements because they meet the strict conditions of admission, but shaggy and neglected in their aspects, are repelled with horror by the delicate, subservient to their fine Reason. But it is generally among these disadvantaged and disgraced beings that the true nature of the species is revealed in the most immediate way.
Geometric Reason, that is to say the effort to attach all judgments to the least possible number of principles, hence logic alone, denying intuitive conviction, would draw all our truths, this Geometric Reason would paralyse thought in areas foreign to mathematics.
And fine Reason, invoked by the mathematician, only responds to it with opinions that generate false reasoning.
III
Now let's go to the Inventive Method, used by mathematicians. It is easy to define. It does not exist.
The mathematician, faced with a difficult question, manipulates his mind, uses a tactic, obeying only the inspirations suggested by the nature and position of the problem.
If he succeeds in conquering it, and if he goes back later in his memory to the chain of stages covered, he sometimes finds that the operations carried out in his mind can be interpreted, so as to provide the elements of a general instruction for those who want to direct their thinking during research.
But when the subject of his study changes, the same mathematician will forget by what means he had triumphed in a previous test, and again the only conditions of the case opposed to his effort will provide him with the elements of an attack, possibly show him the ways to success.
There are sciences where the method plays a primordial role: the criticism of texts, the historical sciences. Among mathematicians, the recommendation of a method is devoid of any useful effect. You know that advice should not be given, because those who follow it do not need it and those who need it do not follow it.
After this beginning contesting the existence and the usefulness of the method in mathematics, I will all the same give you examples, borrowed from the memories of a mathematician that I know, and which undoubtedly contain in themselves the principle of a method.
The problem posed is of the following kind: it is a question of proving that such a class of mathematical objects, class A has a certain property P. Class A is generally defined by a certain number of rules. Here is the truth which must be penetrated.
If the objects of class A do indeed possess the property P, this is generally, or almost never, the consequence of the totality of the rules defining A. The property P almost always results from a simple part of the rules of A and sometimes only one of these rules. And then, the idea enclosing in it a method of research, is this: All the other rules of A, all those which do not contribute in any way necessary to the truth of the property P, far from facilitating the research by the multiplicity of the knowledge they give us of A only dangerously obscure the question posed.
They mislead the thought, occupied in wondering if these rules, superfluous in reality, do not contain in them one of the causes of the observed or conjectured property. A man has an internal condition. Would a doctor be advanced to discover the cause and the nature of the evil, to accumulate information on the size, the colour of the hair, the cephalic type of the patient?
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Poincaré, in his studies of mechanics, had studied the movements of a point subjected to forces independent of time and subject to move on the surface of a torus, that is to say of a circular ring, with perfectly round section. And the question was to know what would become of the aspect of the movement of any material point placed on the surface of the torus, when time did not stop growing indefinitely, and if no movement described a closed cycle. There were two a priori possibilities. One is that, during its unlimited movement, the moving point will return indefinitely often, indefinitely close to any geometric point which it has already passed. It will be the fulfilment of the eternal return: the thought dear to Nietzsche. Scientists say that the trajectory of each point is stable. Or on the contrary, and at least for certain moving points, any position once occupied will remain away from the rest of the indefinite trajectory: it will be further and further away from it as time increases. The movement of these points is described as unstable.
Believing that, the more the conditions of the phenomenon would appear in harmonious form, the easier it would be to highlight the response sought, Poincaré had assumed that the forces at play were of the noblest and richest nature possible, what we call of an analytical nature. But this basket of donations added nothing to the mere effectiveness of the forces. The successful method was to assume on the forces, strictly what was necessary for the problem to have a meaning, essentially the continuity of the field of forces. In this case the problem was solved simply and in the sense, predicted by Poincaré, of possible instability.
Then we added the hypothesis of first order differential conditions, analogous for a curve to the existence of a tangent at each of its points. The answer was still obtained and it was in the same sense as in the previous case. Instability was admissible. We then moved on to conditions relating to second-order differentials.
This time the answer contradicted the presumption of Poincaré and Birkhoff. Stability was the rule. These famous mathematicians had assumed that the forces had differentials of all orders, and even more. All these graciously offered properties only served to obscure the subject.
We knew too much about the data. It was necessary to have much less information to see clearly in the question.
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In another problem, it was a question of finding what we call the second primitive of a generalised second derivative. The technical meaning of the words doesn't matter. This research was necessary to obtain the rule for calculating the coefficients of the trigonometric series. For these, we had a first intermediate derivative of a very convenient nature: this derivative was "summable". In the general case, on the contrary, the first derivative, when it existed, presented much less simple properties and it was not summable. Only when the problem had been solved, we realised that in both cases, the core of the difficulty was the same, much less voluminous in the first case (that of the trigonometric series), but all also condensed and no less inseparable; broader and much more convenient to observe in the second case. It was by attacking the most general problem that we discovered the solution. The two problems could be compared as follows:
A mountain range separates us from reaching a land. You have to find the pass where the passage is possible. The massif is cut by faults, bordered by vertical walls, connected by a more or less chaotic soil. A second chain rises at a great height from the imaginary and translucent walls extending towards the zenith those of the first chain. This is, in planar projection, identical in design to the second. The first, worn by a long erosion, is of a much less marked relief than the second, of which it is the degenerate witness.
With each attempt to cross the real chain, one engages in one of the countless trenches dug by nature in the flanks of the original solid mass, and one invariably leads, and in spite of any contrary presumption, to the bottom of a dead end formed by a low wall, but impossible to climb.
When the lesson of these failures is sufficiently learned, we decide to observe with hindsight the second chain, and then, in the profile of this one, we see the defect of the crest and the breach, the pass through which the path will pass.
Again, too much detail on the data of the problem was harmful. The difficulty was overcome, because it was replaced by a much more significant aspect, but in reality easier to tackle, because the number of its attack faces had been reduced to a minimum.
This is the lesson in method that mathematicians give to others, without knowing how to always profit from it themselves.
Last Updated July 2020