the queen is naked

Still nowadays in college textbooks written under Bourbaki’s influence one finds the introduction, say of the natural numbers, given in the following way, called an “axiomatic definition”: “|N is a set satisfying the following properties” followed by a list of statements that are Peano’s axioms. This however is an explicit definition, [...]

Similarly in Bourbaki the axioms, say of group theory as above, are the definition of a class of structures, those satisfying the definition of “group” given by the conjunction of the axioms. Bourbaki’s thesis that “all mathematical theories can be considered extensions of the general theory of sets” means that every theory is presented by adding to set theory the definition of a class of structures consisting of formulae he calls “axioms”. This move, far from being an extension of set theory, is rather an extension of its language, a conservative extension of course.
 

[...] Bourbaki is missing the deep significance of the existence of univalent and polyvalent theories. Arithmetic and geometry were axiomatised with the express aim to obtain categorical theories for the two concepts that are since its historical origins the building blocks of mathematics; the other axiomatisations on the contrary sought deliberately to build theories with many models and they are a more genuine role model of the pluralistic axiomatic method. But there is a logic behind these two trends of axiomatisation: there are two logics. The axioms systems for arithmetic, the theory of real numbers and geometry are categorical only if formulated in second order logic (i.e. induction in Peano and the completeness axiom in Hilbert are written with quantifiers ranging over all subsets of the domain). If these axioms are substituted by schemata, one instance for each definable property, definable by a first order formula, it is proved in the semantic metatheory that there are always non isomorphic models, so called non standard models.

The sequence of events which brought to the solution of the apparent paradox of two contradictory theorems for a while cohabitant has been tortuous but in the end the puzzle has been worked out, and is now easily explained, of course with some knowledge of modern logic. All the ingredients were ripe and well known when Bourbaki was born. But again, the puzzle requires the acknowledgment of the presence of logic in the mathematical thought.

Structuralism is a compromise, original perhaps, but still a compromise, logically shaking, between reductionism and axiomatics. It took a foothold only because Bourbaki’s reductionism is not a real reductionism and Bourbaki’s axiomatics is not the axiomatic method.

[...] the method glorified by Bourbaki is not the axiomatic method. Let us ignore the fact that Bourbaki disregards the primary role and the study of languages (in the plural) and of their interpretations, downplaying the linguistic component of the method. The capital sin is that he forgets that set theory itself is an axiomatic theory, and as such it has infinitely many models of different cardinalities, even denumerable ones, as is known since 1922 and the theorem of Thoralf Skolem (1887–1963). It follows that the concepts Bourbaki believes to be defined in his language, for instance the structures of univalent theories, are not absolute[,] but relative to each particular model of set theory, isomorphism doesn’t cross over models. [...]

That of Bourbaki is an aged foundation with deep creases; not because of the passing of time, but because it was born old. [...] His success confirms that often in the history of thought it is not the most limpid ideas that prevail but those sustained by the strongest authority and best rhetoric.

Respected research math is dominated by men of a certain attitude. Even allowing for individual variation, there is still a tendency towards an oppressive atmosphere, which is carefully maintained and even championed by those who find it conducive to success. As any good grad student would do, I tried to fit in, mathematically. I absorbed the atmosphere and took attitudes to heart. I was miserable, and on the verge of failure. The problem was not individuals, but a system of self-preservation that, from the outside, feels like a long string of betrayals, some big, some small, perpetrated by your only support system. When I physically removed myself from the situation, I did not know where I was or what to do. First thought: FREEDOM!!!! Second thought: but what about the others like me, who don’t do math the “right way” but could still greatly contribute to the community? I combined those two thoughts and started from zero on my thesis. What resulted was a thesis written for those who do not feel that they are encouraged to be themselves. [...]

I’m unwilling to pretend that all manner of ways of thinking are equally encouraged, or that there aren’t very real issues of lack of diversity. It is not my place to make the system comfortable with itself. This may be challenging for happy mathematicians to read through; my only hope is that the challenge is accepted.

I view category theory as a "top down" approach to the foundations of mathematics, whereareas traditional set theory is a "bottom up" approach. The best analogy I've been able to come up with draws from psychology/cognitive science: Category theory and set theory are as models of mathematical structures what mature Skinnerian behaviorism and nuerobiology are as models of human behavior. Neurobiology and set theory construct the models from the nuts and bolts of thier domains while CT and SB ignore all that and focus only on the relations between its components as the only relevant aspects. The "nuts-and-bolts" approach has the advantage of being explicit in the composition of the objects it postulates and why they have the properties they do follows directly and clearly from this construction. For example, there's no mystery what a function is if you understand what a Cartesian product is. The problem is the constructions get extremely complicated and difficult sometimes to accomplish explicitly. The "behavioral" approach has the advantage of simplicity and large scale organization, but that many objects remain "black boxes" that can ONLY be defined in terms of relations. I  think the relectuance to fully accept category theory is the same reluctance the psychological community had with behaviorism: It seems very counterintuitive to some people to have a methodology where one does not "gets one's hands dirty" with an explicit object of study.

Everyone would like to lighten all proofs in number theory (or any mathematics) as much as they can in any way that they can. For many number theorists that would include eliminating functorial tools. All number theorists share Lenstra’s goal of solving equations while many do not yet share his amusement:

Hendrik Lenstra, in his lecture to the conference, recounted that twenty years ago he was firm in his conviction that he DID want to solve Diophantine equations, and that he DID NOT wish to represent functors — and now he is amused to discover himself representing functors in order to solve Diophantine equations! [Mazur, 1997, p. 245, emphasis in the original].

Funny things can be true. The evidence is that functors make arithmetic easier. Indeed, up to this time, they make Wiles’s proof feasible.

What isn’t category theory? 

In a well-known piece of mathematical bitching, Miles Reid described the study of category theory for its own sake as
 
surely one of the most sterile of all intellectual pursuits

(Undergraduate Algebraic Geometry, p.116). I’ve become rather fond of that quotation, though not for the reason that Reid intended. ‘Sterile’ doesn’t only mean infertile or unproductive. It’s also what you want surgical instruments to be: clean, uncontaminated, disease-free. No one wants to be operated on with a dirty scalpel. [...]

Category theory works because it’s clean, uncontaminated, sterile.

I guess I was taught that algebraic geometry was, at its heart, the study of solving polynomials. On the other hand, I never quite believed my teachers — I couldn’t believe that all that cool Grothendieck stuff (which I knew zero about at the time) could possibly be about something so unglamorous.

[...] this constitutes the best possible definition for the main category of measure theory, both in terms of conceptuality and effectiveness, just as the best way to define the category of affine schemes is to make it equal to the opposite category of the category of commutative rings. Such a viewpoint is unfortunately highly unlikely to be adopted by analysts (especially hard analysts) considering their unwillingness to study even the most elementary notions of category theory. [...]

Summing up, the deep reason for the opposition, depreciation and misunderstandings concerning logic among mathematicians lies in their inability or unwillingness to accept the binomium language-metalanguage as a mathematical tool; they don’t even seem capable of understanding its sense.

This could be due to their habit of talking in an informal quasi-natural language, where metalanguage is flattened on the language itself, or the languages are absorbed in the metalanguage, a habit legitimated and reinforced by the set-theoretical framework.

[...] set theory is a measure (not the only one, no doubt) of the degree of abstractness of mathematics, and it is at the very least a striking fact about mathematical practice, which many set theory textbooks contrive to obscure, that even before we try to reduce levels by clever use of coding, the overwhelming majority of mathematics sits comfortably inside the first couple of dozen levels of the hierarchy above the natural numbers. [...]

Some parts of mathematics are often said to be more abstract than others. Functional analysis, for instance, is more abstract than the calculus of functions of one real variable. This use of the word ‘abstract’, which is quite familiar to most mathematicians, seems to be represented quite well by the ranks of the objects referred to in this set-theoretic modelling of the parts of mathematics in question: functional analysis is more abstract than the calculus because the objects it deals with are modelled by sets of higher rank.

One of the trends we can trace in the development of mathematics, especially during the 20th century, is a move towards greater abstractness in the sense just defined. Nevertheless, the overwhelming majority of 20th century mathematics is straightforwardly representable by sets of fairly low infinite ranks, certainly less than ω + 20.

In particular, the existence of many possible models of mathematics is difficult to accept upon first encounter, so that a possible reaction may very well be that somehow axiomatic set theory does not correspond to an intuitive picture of the mathematical universe, and that these results are not really part of normal mathematics. [...] I can assure that, in my own work, one of the most difficult parts of proving independence results was to overcome the psychological fear of thinking about the existence of various models of set theory as being natural objects in mathematics about which one could use natural mathematical intuition.

For example, the correct notion of a derivative and thus of the slope of a tangent line is somewhat complicated. But whatever definition is chosen, the slope of a horizontal line (and hence the derivative of a constant function) must be zero. If the definition of a derivative does not yield that a horizontal line has zero slope, it is the definition that must be viewed as wrong, not the intuition behind the example.

For another example, consider the definition of the curvature of a plane curve, [...] The formulas are somewhat ungainly. But whatever the definitions, they must yield that a straight line has zero curvature, that at every point of a circle the curvature is the same and that the curvature of a circle with small radius must be greater than the curvature of a circle with a larger radius (reflecting the fact that it is easier to balance on the earth than on a basketball). If a definition of curvature does not do this, we would reject the definitions, not the examples.

Riemann spells out what every student experiences in the very first lectures [on analysis]:

"This makes the proofs of the fundamental theorems of infinitesimal calculus... somewhat more involved."

("Es werden die Beweise der Fundamentalsätze der Infinitesimalrechnung dadurch... etwas umständlicher")

Of course, when it comes to obtaining new insights, epsilontics is largely irrelevant; it is an algorithm for securing previously found insights, canonically prescribed by habitual consensus. Other innovations, pursued purposively by Riemann, are far more important for the progress of mathematics.

Mathematicians have developed habits of communication that are often dysfunctional. Organizers of colloquium talks everywhere exhort speakers to explain things in elementary terms. Nonetheless, most of the audience at an average colloquium talk gets little of value from it. Perhaps they are lost within the first 5 minutes, yet sit silently through the remaining 55 minutes. Or perhaps they quickly lose interest because the speaker plunges into technical details without presenting any reason to investigate them. [...]

This pattern is similar to what often holds in classrooms, where we go through the motions of saying for the record what we think the students “ought” to learn, while the students are trying to grapple with the more fundamental issues of learning our language and guessing at our mental models. Books compensate by giving samples of how to solve every type of homework problem. Professors compensate by giving homework and tests that are much easier than the material “covered” in the course, and then grading the homework and tests on a scale that requires little understanding. We assume that the problem is with the students rather than with communication: that the students either just don’t have what it takes, or else just don’t care.

Outsiders are amazed at this phenomenon, but within the mathematical community, we dismiss it with shrugs.

In the mid-60's, my former colleague Holbrook MacNeille, who worked for the Atomic Energy Commission before becoming the first Executive Director of the American Mathematical Society, remarked often that whereas laboratory scientists were mutually supportive in evaluating research proposals, mathematicians were seldom loath to dump on each other. [...]

In the United States, instead of trying to nurture and sustain our mathematical community, we seem to turn our backs as a small but influential group wreaks havoc. I call them B.A.D.: Bigoted And Destructive. They have always been with us; what has increased in recent years is their ability to be destructive. [...]

Academics usually have great difficulty admitting, even to themselves, that they act in their own self-interest, so the mathematical bigots have little trouble in rationalizing their selfish or dishonest acts as the maintenance of high standards.

The prestige of a field changes with time, sometimes for good reason, but often as a result of power struggles which have an impact on granting agencies and the composition of editorial boards. This puts those not on the faculty of elite institutions in the position of playing against loaded dice. A small number of nasty referee's reports or evasive letters from editors are often enough to push "outsiders" out of research.

Is it desirable to press mathematicians all to think in the same way? I say not: if you take someone who wishes to become a set theorist and force him to do (say) algebraic topology, what you get is not a topologist but a neurotic. Uniformity is not desirable, and an attempt to attain it, by (say) manipulating the funding agencies, will have unhealthy consequences. 

It is now taken for granted that every analysis student study measure theory, but the subject, formally inaugurated in 1902 by Lebesgue's thesis, remained somewhat suspect as late as the thirties.

The mathematical community is no more eager than other communities to accept what it considers alien ideas. When Saks lectured in Cambridge (Massachusetts) on what is now called the Vitali-Hahn-Saks theorem there was professional grumbling at what was considered the extremely abstract nature of the topic.

Coping with radical novelty requires an orthogonal method. One must consider one's own past, the experiences collected, and the habits formed in it as an unfortunate accident of history, and one has to approach the radical novelty with a blank mind, consciously refusing to try to link it with what is already familiar, because the familiar is hopelessly inadequate. One has, with initially a kind of split personality, to come to grips with a radical novelty as a dissociated topic in its own right. Coming to grips with a radical novelty amounts to creating and learning a new foreign language that can not be translated into one's mother tongue. [...] Needless to say, adjusting to radical novelties is not a very popular activity, for it requires hard work. For the same reason, the radical novelties themselves are unwelcome.

By now, you may well ask why I have paid so much attention to and have spent so much eloquence on such a simple and obvious notion as the radical novelty. My reason is very simple: radical novelties are so disturbing that they tend to be suppressed or ignored, to the extent that even the possibility of their existence in general is more often denied than admitted.

Many mathematicians of my generation were confused by (I might go so far as to say suspicious of) the Gödel revolution - the reasoning of logic was something we respected, but preferred to respect it from a distance. It looked like a strange cousin several times removed from our immediate mathematical family - similar but at the same time ineffably different. [...] the logical, recursive, axiom-watching point of view [...] is not the point of view of most working mathematicians, the kind who know that the axiom of choice is true and use it several times every morning before breakfast without even being aware that they are using it.

In recent years we have become much more preoccupied with streamlining and organizing our subject than with maintaining its overall vitality. If we are not careful, a great adventure of the mind will become yet another profession. Please, do not misunderstand me. The ancient professions of medicine, engineering or law are in no way inferior as disciplines to mathematics, physics or astronomy. But they are different, and the differences have been, since time immemorial, emphasized by vastly different educational approaches. The main purpose of professional education is development of skills; the main purpose of education in subjects like mathematics, physics or philosophy is development of attitudes. When I speak of professionalism in mathematics it is mainly this distinction that I have in mind. Our graduate schools are turning out ("producing" is the horrible word which keeps creeping up more and more) specialists in topology, algebraic geometry, differential equations, probability theory or what have you, and these in turn go on turning out more specialists. And there is something disconcerting and more than slightly depressing in the whole process. For if I may borrow an analogy, we no longer climb a mountain because it is there but because we were trained to climb mountains and because mountain climbing happens to be our profession.

Mathematics is not a deductive science - that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.

Amongst university lecturers in mathematics I have been able to distinguish two basic types of tension-reduction techniques.

The first of these belongs to the formalist, definition-theorem-proof school of lecturing which has been for some time now the dominant style of teaching mathematics at the post-secondary level. The subject is seen as a fixed body of knowledge which needs to be consumed by the student. The lecturer perceives his primary task to be a logical, coherent presentation of the material.

This teaching strategy minimizes stress on the teacher while maximizing that on the student. Pressure on the teacher is reduced by means of the illusion of objectivity which is created. This illusion is based on a confusion between naming or defining a concept on the one hand and understanding it on the other. [...]

It is the feeling of objectivity which allows the teacher to keep the student and his problems at arm's length. If the latter does not grasp the material it must be due to his lack of ability - the teacher need not feel any sense of responsibility.

One of the characteristics of Bourbaki mathematics is its extraordinary unity: there is hardly any idea in one theory that does not have notable repercussions in several others, and it would therefore be absurd, and contrary to the very spirit of our science, to attempt to compartmentalize it with rigid boundaries, in the manner of the traditional division into algebra, analysis, geometry, etc. now completely obsolete.

The virtue of formal texts is that their manipulations, in order to be legitimate, need to satisfy only a few simple rules; they are, when you come to think of it, an amazingly effective tool for ruling out all sorts of nonsense that, when we use our native tongues, are almost impossible to avoid.

Instead of regarding the obligation to use formal symbols as a burden, we should regard the convenience of using them as a privilege: thanks to them, school children can learn to do what in earlier days only genius could achieve. (This was evidently not understood by the author that wrote — in 1977 — in the preface of a technical report that "even the standard symbols used for logical connectives have been avoided for the sake of clarity". The occurrence of that sentence suggests that the author's misunderstanding is not confined to him alone.) When all is said and told, the "naturalness" with which we use our native tongues boils down to the ease with which we can use them for making statements the nonsense of which is not obvious.

In a similar way, one can support the intuition that there is something wrong with A1', in spite of its not being straightforwardly question-begging, by showing that if A1' supports MPP, an exactly analogous argument would support a deductively invalid rule, say:

MM (modus morons);
From: A → B and B
to infer: A.
Thus:
A4
Supposing that 'A → B' is true and 'B' is true, 'A → B' is true → 'B' is true.
Now, by the truth-table for '→', if 'A' is true, then, if 'A → B' is true, 'B' is true. Therefore, 'A' is true.

This argument, like A1, has the very form which it is supposed to justify. For it goes:

A4'
Suppose D (if 'A → B' is true, 'B' is true).
If C, then D (if 'A' is true, then, if 'A → B' is true, 'B' is true).
So, C ('A' is true)

It is no good to protest that A4' does not justify modus morons because it uses an invalid rule of inference, whereas A4' does justify modus ponens, because it uses a valid rule of inference - for to justify our conviction that MPP is valid and MM is not is precisely what is at issue. [...]

Nor will it do to argue that MPP is, whereas MM is not, justified 'in virtue of the meaning of "→" '. For how is the meaning of '→' given? There are three kinds of answer commonly given: that the meaning of the connectives is given by the rules of inference/axioms of the system in which they occur; that the meaning is given by the interpretation, or, specifically, the truth-table, provided; that the meaning is given by the English readings of the connectives. Well, if '→' is supposed to be at least partially defined by the rules of inference governing sentences containing it, then MPP and MM would be exactly on a par. In a system containing MPP the meaning of '→' is partially defined by the rule, from 'A → B' and 'A', to infer 'B'. In a system containing MM the meaning of '→' is partially defined by the rule, from 'A → B' and 'B', to infer 'A'. In either case the rule in question would be justified in virtue of the meaning of '→', finally, since the meaning of '→' would be given by the rule. If, on the other hand, we thought of '→' as partially defined by its truth-table, we are in the difficulty discussed earlier that arguments from the truth-table to the justification of a rule of inference are liable to employ the rule in question. Nor would it do to appeal to the usual reading of '→' as 'if... then...', not just because the propriety of that reading has been doubted, but also because the question, why 'B' follows from 'if A then B' and 'A' but not 'A' from 'if A then B' and 'B', is precisely analogous to the question at issue.

At a series of recent lectures on non-Platonic mathematics, a typical comment was "Well presented, but irrelevant. Let's get back to our (Platonic) drawing boards". Undoubtedly in 1971, one can earn a living with Platonic mathematics, and if mathematician A spouts some Platonism to mathematician B and the latter responds in kind, then there is at least human significance in the act. The emperor may be walking around in his underwear, but if the court is also, they can make a life together.

To put it facetiously and anachronistically, if a Sumerian mathematician had been asked for his opinion of Euclid he might have replied that he was interested in real Mathematics and not in useless generalizations and abstractions.

[...]

The fundamental importance of the advent of non-Euclidean geometry is that by contradicting the axiom of parallels it denied the uniqueness of geometrical concepts and hence, their reality.

The dangers of Platonistic modes of speech.

[...] let me insist once more that phrases like ‘our explicit knowledge of the continuum is very restricted’, used by Professor Bernays, are apt to mislead their users into believing that somewhere (perhaps in some Platonic heaven) there lies spread out the ideal continuum to be known and beheld by all people with sufficient uncluttered intuition and with sufficient readiness and training to use their mental experience. The requirement of mathematical objectivity must, and can, be met without recourse to such doubtful metaphors, to put it mildly.

As to Mostowski’s comment, let me insist once again that I fail to find in myself something I would be tempted to call a conception of ‘the intuitive set theory’ and that I have grave doubts as to what other people mean exactly when they use this phrase, in particular with the definite article. While Mostowski finds it disquieting that he and many other mathematicians do not know where to look for the new axioms that would finally codify the theory, I for one rather find Mostowski’s uneasiness disquieting.

Meta-mathematics - like Russellian logic - has its origin in the criticism of intuition; now meta-mathematicians - as did the logicists - ask us to accept their intuition as the 'ultimate' test: thereby both fall back on the same subjectivist psychologism which they once attacked. But why on earth have 'ultimate' tests, or 'final' authorities? Why foundations, if they are admittedly subjective? Why not honestly admit mathematical fallibility, and try to defend the dignity of fallible knowledge from cynical scepticism, rather than delude ourselves that we shall be able to mend invisibly the latest tear in the fabric of our 'ultimate' intuitions?

True, the axiom of choice was explicitly formulated only in the twentieth century and apparently was not implicitly used earlier than two decades before. But, at that, every mathematical principle was once expressed for the first time, mostly long after it had been used implicitly and unconsciously. The development of mathematics through the centuries has been achieved in two directions: by drawing new conclusions from previously admitted premises; as well as, in a less conspicuous way by adding new premises or principles to those admitted before, in accordance with the needs of science.

In fact one has arrived at the axiom of choice just as at other mathematical principles, viz. by a posteriori examining and logically analysing concepts, methods, and proofs actually found in mathematics whose original development in an intuitive manner rests on psychological rather than on logical foundations. This way of analysing then yielded the principle in question, and a reference to the intuitive or logical evidence of the principle was at best secondary. [...]

In particular in logic many principles are chiefly justified by the evidence of their consequences [...]

The metatheory belongs to intuitive and informal mathematics (unless the metatheory is itself formalized from a metametatheory, which here we leave out of account). The metatheory will be expressed in ordinary language, with mathematical symbols, such as metamathematical variables, introduced according to need. The assertions of the metatheory must be understood. The deductions must carry conviction. They must proceed by intuitive inferences, and not, as the deductions in the formal theory, by applications of stated rules. Rules have been stated to formalize the object theory, but now we must understand without rules how those rules work. An intuitive mathematics is necessary even to define the formal mathematics. [...]

The ultimate test whether a method is admissible in metamathematics must of course be whether it is intuitively convincing.

[...] the gradual getting down to the bed-rock of thought is characteristic of modern advance. [...] in mathematics, the Theory of Sets, the acknowledgment and singling out of which for study may be said to form the characteristic great and notable progress in mathematics in this first quarter of a century. [...]

At the turn of the century Mengenlehre was a term of which the meaning was all but unknown. The ideas were already budding out here and there, and by Georg Cantor it was perhaps already conceived as a perfect whole. But the mathematical world at large seems to have been unprepared for it. [...]

It is remarkable, too, that, though the Théorie des Ensembles had been recognized comparatively early in France, and some of her most distinguished mathematicians owe their advance to its exploitation, even Lebesgue himself, in a published criticism, when "The Theory of Sets of Points" by my wife and myself appeared in 1906, more than suggested that a work on this subject was not called for, and that its treatment should merely constitute a chapter, one of the introductory chapters, of a course of Analysis.

The reluctance with which the Theory of Sets was allowed to take its individual place in the scheme of mathematics was but a symptom of the same spirit which, in the domain of scientific studies generally, is dictating the demand to relegate mathematics to the rank of the handmaid of the other sciences. [...]

But in fact self-evidence is never more than a part of the reason for accepting an axiom, and is never indispensable. The reason for accepting an axiom, as for accepting any other proposition, is always largely inductive, namely that many propositions which are nearly indubitable can be deduced from it, and that no equally plausible way is known by which these propositions could be true if the axiom were false, and nothing which is probably false can be deduced from it. If the axiom is apparently self-evident, that only means, practically, that it is nearly indubitable; for things have been thought to be self-evident and have yet turned out to be false. And if the axiom itself is nearly indubitable, that merely adds to the inductive evidence derived from the fact that its consequences are nearly indubitable: it does not provide new evidence of a radically different kind. Infallibility is never attainable, and therefore some element of doubt should always attach to every axiom and to all its consequences. In formal logic, the element of doubt is less than in most sciences, but it is not absent, as appears from the fact that the paradoxes followed from premises which were not previously known to require limitations.

Consequently, there are two conceptions of mathematics, two mentalities, in evidence. After all that has been said up to this point, I do not see any reason for changing mine. I do not mean to impose it. At the most, I shall note in its favor the arguments that I stated in the Revue generale des Sciences (30 March 1905), to wit:

1. I believe that in essence the debate [over the axiom of choice] is the same as the one which arose between Riemann and his predecessors over the notion of function. The rule that Lebesgue demands appears to me to closely resemble the analytic expression on which Riemann's adversaries insisted so strongly. (I believe it necessary to reiterate this point, which, if I were to express myself fully, appears to form the essence of the debate. From the invention of the infinitesimal calculus to the present, it seems to me, the essential progress in mathematics has resulted from successively annexing notions which, for the Greeks or the Renaissance geometers or the predecessors of Riemann, were "outside mathematics" because it was impossible to describe them.) [...]

[Ce sont donc deux conceptions des Mathématiques, deux mentalités qui sont en présence. Je ne vois, dans tout ce qui a été dit jusqu'ici, aucun motif de changer la mienne. Je ne prétends pas l'imposer. Tout au plus ferai-je valoir en sa faveur les arguments que j'ai indiqués dans la Revue générale des Sciences (3o mars 1905), savoir: 1º Je crois que le débat est au fond le même qui s'est élevé entre Riemann et ses prédécesseurs, sur la notion même de fonction. La loi qu'exige Lebesgue me paraît ressembler fort à l'expression analytique que réclamaient à toute force les adversaires de Riemann. (Je crois devoir insister un peu sur ce point de vue qui, s'il faut dire toute ma pensée, me paraît former le fond môme du débat. Il me semble que le progrès véritablement essentiel des Mathématiques, à partir de l'invention môme du Calcul infinitésimal, a consisté dans l'annexion de notions successives qui, les unes pour les Grecs, les autres pour les géomètres de la Renaissance ou les prédécesseurs de Riemann, étaient «en dehors des Mathématiques», parce qu'il était impossible de les décrire.)]

It is well that there are logicians and that there are intuitives; who would dare say whether he preferred that Weierstrass had never written or that there never had been a Riemann? We must therefore resign ourselves to the diversity of minds, or better we must rejoice in it.

[Il est bon qu’il y ait des logiciens et qu’il y ait des intuitifs; qui oserait dire s’il aimerait mieux que Weierstrass n’eût jamais écrit, ou qu’il n’y eût pas eu de Riemann. Il faut donc nous résigner à la diversité des esprits, ou mieux, il faut nous en réjouir.]

Even if there were such a theory, based on calculation, it still would not be of the highest degree of perfection, in my opinion. It is preferable, as in the modern theory of functions, to seek proofs based immediately on fundamental characteristics, rather than on calculation, and indeed to construct the theory in such a way that it is able to predict the results of calculation [...]

[Mais, lors même qu'il n'en serait pas ainsi, une telle théorie, fondée sur le calcul, n'offrirait pas encore, ce me semble, le plus haut degré de perfection; il est préférable, comme dans la théorie moderne des fonctions, de chercher à tirer les démonstrations, non plus du calcul, mais immédiatement des concepts fondamentaux caractéristiques, et d'édifier la théorie de manière qu'elle soit, au contraire, en état de prédire les résultats du calcul [...] ]

One should always generalize.

[Man muss immer generalisieren.]