recommended reading and/or watching:
"Ogi formula", from NISIOISIN's "Owarimonogatari" (part of his larger "Monogatari" franchise)
"The Tower of Doors" game in "Kakegurui", comprising chapters 30 "The Offering Girl", 31 "The Enthralled Girl", 32 "The Clinging Girl", and 33 "The Logical Girl"
the first kinda mirrors (the tension in) K. Devlin's 2003 "When is a proof?", the second does it with/for Grothendieck's "hammer & chisel vs. rising sea"
Student: Sir, what is a mathematical proof?
Ideal Mathematician: You don't know that? What year are you in?
Student: Third-year graduate.
I.M.: Incredible! A proof is what you've been watching me do at the board three times a week for three years! That's what a proof is.
Student: Sorry, sir, I should have explained. I'm in philosophy, not math. I've never taken your course.
I.M.: Oh! Well, in that case - you have taken some math, haven't you? You know the proof of the fundamental theorem of calculus - or the fundamental theorem of algebra?
Student: I've seen arguments in geometry and algebra and calculus that were called proofs. What I'm asking you for isn't examples of proof, it's a definition of proof. Otherwise, how can I tell what examples are correct?
I.M.: Well, this whole thing was cleared up by the logician Tarski, I guess, and some others, maybe Russell or Peano. Anyhow, what you do is, you write down the axioms of your theory in a formal language with a given list of symbols or alphabet. Then you write down the hypothesis of your theorem in the same symbolism. Then you show that you can transform the hypothesis step by step, using the rules of logic, till you get the conclusion. That's a proof.
Student: Really? That's amazing! I've taken elementary and advanced calculus, basic algebra, and topology, and I've never seen that done.
I.M.: Oh, of course no one ever really does it. It would take forever! You just show that you could do it, that's sufficient.
Student: But even that doesn't sound like what was done in my courses and textbooks. So mathematicians don't really do proofs, after all.
I.M.: Of course we do! If a theorem isn't proved, it's nothing.
Student: Then what is a proof? If it's this thing with a formal language and transforming formulas, nobody ever proves anything. Do you have to know all about formal languages and formal logic before you can do a mathematical proof?
I.M.: Of course not! The less you know, the better. That stuff is all abstract nonsense anyway.
Student: Then really what is a proof?
I.M.: Well, it's an argument that convinces someone who knows the subject.
Student: Someone who knows the subject? Then the definition of proof is subjective; it depends on particular persons. Before I can decide if something is a proof, I have to decide who the experts are. What does that have to do with proving things?
I.M.: No, no. There's nothing subjective about it! Everybody knows what a proof is. Just read some books, take courses from a competent mathematician, and you'll catch on.
Student: Are you sure?
I.M.: Well - it is possible that you won't, if you don't have any aptitude for it. That can happen, too.
Student: Then you decide what a proof is, and if I don't learn to decide in the same way, you decide I don't have any aptitude.
I.M.: If not me, then who?
Davis, P., & Hersh, R. (1981). The mathematical experience.
People have now emerged who have mathematical minds, but are computer oriented. More precisely, these sorts of people were around earlier, but, without computers, somehow something was missing. In a sense, Euler was like that, [...] And also Ramanujan, a person who didn’t even really know mathematics.
[...] don’t worry about statistics going out of business from outside competition. If you do feel the need to worry, a better subject is our own production of useful new ideas. This relates to the “inside” of the statistics profession, the side that worries about the structure of statistical inference and how it can be extended. New ideas are the coin of the realm for an intellectual discipline. Without them a field hollows out, no matter how successful it may be in terms of funding or public recognition. Too much “inside” can be deadly for a field, cutting it off from the bigger world of science, as happened to mathematics in the twentieth century.
I don't believe that all areas of mathematics are equally valuable; I'm convinced that mathematics has no intrinsic value in and of itself. Otherwise, mathematics ends up being a kind of complicated intellectual game, and we find ourselves in the realm described by Hermann Hesse ("The Glass Bead Game"), where there are no criteria at all - except for the judgment of the professional community. [...] Mathematical ideas associated with the 1990s - mirror symmetry, Gromov–Witten invariants, Seiberg–Witten invariants, quantum cohomology - all originate in string geometry.
I think this is no coincidence. Mathematics has lost its general criteria because it has lost its general context. Today, far fewer people understand what is happening in science as a whole than 20 years ago, and even fewer than 40 years ago. In the absence of abstract standards, the only effective standard becomes a utilitarian one. Mathematics is only interesting to the extent that it is connected to string theory; this is a basic assumption that I don't wish to discuss right now. Relevance to physics is the only criterion we have left; and nearly all mathematics relevant to physics relates to string geometry. This thesis is well supported by the earlier observation: (almost) all interesting ideas of the past 20 years are related to string physics. [...]
Of course, one can do mathematics without understanding the broader context in which it exists; but in my view, such activity only further erodes that context, thereby worsening the blurring of standards, ignorance, and corruption, which already dominate.
From my French friends I heard that the tendency towards super-abstract generalizations is their traditional national trait. I do not entirely disagree that this might be a question of a hereditary disease, [...]
At the beginning of this century a self-destructive democratic principle was advanced in mathematics (especially by Hilbert), according to which all axiom systems have equal right to be analyzed, and the value of a mathematical achievement is determined, not by its significance and usefulness as in other sciences, but by its difficulty alone, as in mountaineering. This principle quickly led mathematicians to break from physics and to separate from all other sciences. In the eyes of all normal people, they were transformed into a sinister priestly caste of a dying religion, like Druids, parasitic on science and technology, recruiting acolytes in the mathematical schools by Zombie-like mental subjection.
Bizarre questions like Fermat's problem or problems on sums of prime numbers were elevated to supposedly central problems of mathematics. ("Why add prime numbers?" marvelled the great physicist Lev Landau. "Prime numbers are made to be multiplied, not added!") [...]
Hermann Weyl, one of the greatest mathematicians of our times (who worked, by the way, in Zurich), said, "In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain."
In the first half of the century, the devil was winning. A special "axiomatic bourbakist" method of exposition of mathematics was invented, to make it perfectly pure. [...] This formal proof [that the value of a product is unaffected by the order of the factors] is urged on students by the criminal bourbakizers and algebraizers of mathematics. [...] It is convincing only to those who have undergone a distinctive algebraic perversion of the mind [...]
A student who takes much more than five minutes to calculate the mean of sin(x)^100 with 10% accuracy has no mastery of mathematics, even if he has studied non-standard analysis, universal algebra, supermanifolds, or embedding theorems.
Set theoretic foundations have also failed to provide fully satisfying accounts of mathematical practice in certain areas, including category theory itself, and moreover have encouraged research into areas that have little or nothing to do with mathematical practice, such as large cardinals.
Axioms asserting the existence of (very) large cardinal numbers have recently given striking results. Drake and Smoryński cite this as an example of attention by logicians to the foundations. I demur, because these large cardinals seem to me to live in a never-never land.
The study of category theory for its own sake (surely one of the most sterile of all intellectual pursuits) also dates from this time ['From around 1955 to 1970']; [...]
A good many special fields of mathematics, after reaching their original goals, have continued to develop in further ways by exploiting sidelines which may or may not be dead ends. The cases best known to me lie on the fringes of mathematics. Thus category theory started out to clarify and consolidate various conceptual ideas in the mainstream of mathematics; some of its further developments turned out to involve heavy and obscure treatments of remote ideas of depressing generality. [...]
This effect of the isolation of a specialty is especially strong in mathematical logic. This field started in a study of the foundations of mathematics, but its practitioners were soon ostracized by other mathematicians. This led perforce to the isolation of mathematical logicians. Subsequently, despite splendid progress and many specific results connected with classical mathematics, the isolation has tended to continue - and at the same time mathematical logic has almost completely lost track of its original concern with foundations. Some of its practitioners are less concerned with concepts than with the demonstration that they too can solve hard problems. This they do, for example by new axions set up in the never-never land of large cardinals. Or given that one can prove the continuum hypothesis to be independent of the axioms of set theory, let us prove the independence of all sorts of combinatoric notions. Or, given that recursive functions arise in Gödel's incompleteness theorem, and that recursive functions suggest a hierarchy of degrees, let us explore all the technical difficulties in the elaborate fine structure of this hierarchy. Or, given that the axioms of set theory and the continuum hypothesis can all be satisfied in Gödel's constructible sets, let us explore the fine structure of these sets, no matter how deep the morass which they form.
The author accepts that constructive analysis is at present a minor offshoot of mathematics; he does not wish to challenge the validity of the propositions of ordinary analysis; indeed his whole aim seems to be to trace those results of ordinary analysis which can be proved by constructivist methods. It is not clear to me that this is a good thing. [...]
At a more advanced level, the discussion of measure theory is impressive for the fact that the elementary theory of the Lebesgue integral can be done at all; but it is even more apparent than elsewhere that this is paraplegic mathematics, struggling desperately to match the achievements of its un-handicapped ideal. [...]
I am inclined to think that the author's wish to avoid the paradoxical aspects of constructivism has in fact deprived him of the only inspiration likely to come from this approach. Constructiv- ism, as presented here, merely leads to enormous technical difficulties without shedding much light on the real questions of mathematics. Of course there are many useful flashes of ingenuity which are struck by the effort of thinking the basic theory out anew. But I should like to suggest that, if there is anything of substantial value in constructivism, it is more likely to come from a rigorous formulation, within Aristotelian logic[,] of the acceptable rules of proof — requiring, perhaps, a proof of ∀xP(x) to be a proof of P(x) which is a recursive function of x — followed by a systematic analysis of the ways in which the theory differs from real mathematics. [...]
In conclusion: No doubt it is good for constructivists to learn some functional analysis. I do not think that there is yet much reason for analysts to learn constructivism.
For example, when there were difficulties in set theory, a few small changes were needed to straighten things out. There were difficulties, but not paradoxes; there was no need to write Principia Mathematica to straighten them out. It was necessary to clarify ideas -- and that is when foundational activity is of interest. Another example is category theory in logic. [...] There are a number of paradoxical constructions which can probably be straightened out by a minimal effort. There is no need to write down axioms for category theory. [...]
The axiomatic aspects of mathematical logic has encouraged clarity and precision to a dangerous degree.
One should beware of the disease called "Axiomatics", which consists of wasting time wondering wheter a, b and c imply d, where a, b, c and d are properties selected at random.
We lament on the other hand that the authors have kept a chapter on "lattices", the uselessness of which in mathematics is even more flagrant now after 35 [sic] years than it was already in 1941.
[On regrette par contre que les auteurs aient maintenu un chapitre sur les "lattices", dont l'inutilité en mathématiques est bien plus flagrante encore après 35 [sic] ans qu'elle ne l'était déjà en 1941.]
I am afraid that mathematics will perish before the end of this century if the present trend for senseless abstraction - I call it: theory of the empty set - cannot be blocked up. Let us hope that your review may be helpful.
It is entirely clear to me what circumstances have led to the inexorable decline of mathematics from a very high level, within about 100 years, to its present nadir. The evil began with the ideas of Riemann, Dedekind and Cantor, through which the well-grounded spirit of Euler, Lagrange and Gauss was slowly eroded. Next the textbooks in the style of Hasse, Schreier and van der Waerden, had a further detrimental effect upon the next generation of scholars. And finally the works of Bourbaki here provided the last fatal shove.
[Die Entartung der Mathematik begann mit den Ideen von Riemann, Dedekind und Cantor, durch die der solide Geist von Euler, Lagrange und Gauss mehrund mehr zurückgedrängt wurde. Durch Lehrbücher im Stil von Hasse, Schreier und v. d. Waerden wurde späterhin der Nachwuchs schon empfindlich geschädigt, und das Werk von Bourbaki versetzte ihm endlich den Todesstoß.]
In other words, at a great distance from its empirical source, or after much “abstract” inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up.
[...] I feel that cardinal arithmetic of the complicated sort ("regular alephs," "accessible alephs," etc.) should be kept as far from general topology as possible. This is the ordinal part of the theory of cardinal numbers, and is essentially descriptive. It is not the task of general topology to describe objects in terms of ordinal numbers.
Logic sometimes makes monsters. Since half a century we have seen arise a crowd of bizarre functions which seem to try to resemble as little as possible the honest functions which serve some purpose. No longer continuity, or perhaps continuity, but no derivatives, etc. Nay more, from the logical point of view, it is these strange functions which are the most general, those one meets without seeking no longer appear except as particular case. There remains for them only a very small corner.
Heretofore when a new function was invented, it was for some practical end; to-day they are invented expressly to put at fault the reasonings of our fathers, and one never will get from them anything more than that.
[La logique parfois engendre des monstres. Depuis un demi-siècle on a vu surgir une foule de fonctions bizarres qui semblent s’efforcer de ressembler aussi peu que possible aux honnêtes fonctions qui servent à quelque chose. Plus de continuité, ou bien de la continuité, mais pas de dérivées, etc. Bien plus, au point de vue logique, ce sont ces fonctions étranges qui sont les plus générales, celles qu’on rencontre sans les avoir cherchées n’apparaissent plus que comme un cas particulier. Il ne leur reste qu’un tout petit coin. Autrefois, quand on inventait une fonction nouvelle, c’était en vue de quelque but pratique; aujourd’hui, on les invente tout exprès pour mettre en défaut les raisonnements de nos pères, et on n’en tirera jamais que cela.]
I turn away with dread and horror from this appalling plague, continuous functions with no derivatives.
[Je me détourne avec effroi et horreur de cette plaie lamentable des fonctions continues qui n’ont pas de dérivée.]
(iii) Escaping from the cage of deterministic models of mathematical development: The adoption of strictly linear evolutionary models of progress in mathematics of the sort discussed in (i) tends to be highly attractive to many mathematicians in light of the intoxicating simplicity of such strictly linear evolutionary models, by comparison to the more complicated point of view discussed in (ii). This intoxicating simplicity also makes such strictly linear evolutionary models — together with strictly linear numerical evaluation devices such as the “number of papers published”, the “number of citations of published papers”, or other like-minded narrowly defined data formats that have been concocted for measuring progress in mathematics — highly enticing to administrators who are charged with the tasks of evaluating, hiring, or promoting mathematicians. Moreover, this state of affairs that regulates the collection of individuals who are granted the license and resources necessary to actively engage in mathematical research tends to have the effect, over the long term, of stifling efforts by young researchers to conduct long-term mathematical research in directions that substantially diverge from the strictly linear evolutionary models that have been adopted, thus making it exceedingly difficult for new “unanticipated” evolutionary branches in the development of mathematics to sprout. Put another way,
inappropriately narrowly defined strictly linear evolutionary models of progress in mathematics exhibit a strong and unfortunate tendency in the profession of mathematics as it is currently practiced to become something of a self-fulfilling prophecy — a “prophecy” that is often zealously rationalized by dubious bouts of circular reasoning.
In particular, the issue of
escaping from the cage of such narrowly defined deterministic models of mathematical development stands out as an issue of crucial strategic importance from the point of view of charting a sound, sustainable course in the future development of the field of mathematics, i.e., a course that cherishes the priviledge to foster genuinely novel and unforeseen evolutionary branches in its development.
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.
Suppose that we are interested in detecting and understanding whatever relationships we can find. Then we might wish not to be wedded to any point of view. We might, instead, try on a few hats until some interesting patterns appear where before there seemed to have been only disorder. We might find that one hat helps time and again, but we will be well-advised not to forget that we are wearing it. For if we never take it off, then we risk forever overlooking logical relationships of considerable interest. Worse, we risk coming to think of the relationships we can detect as “in the world,” “preconditions of thought,” or some such thing.
With the introduction of axiomatic set theory in the early 1900s, it appeared that mathematics was the study of consequences of ZFC. But, by now it is clear that modern mathematics is the study of models of ZFC, [...]
If any human institution is held to be exempt from the petty, self-serving, and corrupting motivations that plague us all, the result will almost inevitably be the creation of a priestly caste demanding adulation and required to answer to no one but itself.
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. [...]
I doubt that measure theory should be considered a part of analysis at all. For example, smooth manifolds were once considered part of analysis (think of multivariable calculus) but now they are not. The subject became much more clear and conceptual when it was detached from analysis. (Of course we still sometimes use analysis to prove theorems like Calabi-Yau theorem, but such proofs are not considered final and in the end of the day they will be replaced by more conceptual and geometric proofs.) Measure theory will undergo the same transition.
Cantor’s theory of the infinite had no basis in the older mathematics. You can argue about this as you like, but this was a new mathematics, a new way to think about mathematics, a new way to produce mathematics. [...]
I must explain to you how I imagine mathematics. I am an emotional Platonist (not a rational one: there are no rational arguments in favor of Platonism).
Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding.
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.
One central aim of a general theory of logics is to get some universal results that can be applied more or less directly to specific logics, this is one reason to call such a theory universal logic.
Some people may have the impression that such general universal results are trivial. This impression is generally due to the fact that these people have a concrete-oriented mind, and that something which is not specified has no meaning for them, and therefore universal logic appears as logical abstract nonsense. They are like someone who understands perfectly what is Felix, his cat, but for whom the concept of cat is a meaningless abstraction. This psychological limitation is in fact a strong defect because, as we have pointed through the example of the completeness theorem, what is trivial is generally the specific part, not the universal one which requires what is the fundamental capacity of human thought: abstraction.
Without doubt the most attractive quality of mathematics is its apparent lack of subjectivity. [...] The other side of this pleasant coin is that mathematics attracts people who have a great need for certainty and encourages them to develop into rigidly dogmatic thinkers.
The charge is made against advocates of constructive mathematics — it was made against Kronecker, against Brouwer, against Bishop — that they are dogmatists who implacably advocate unreasonably extreme views. But what distinguishes them from their accusers is neither the extremity of their views nor the tenacity with which they hold them, but the mere fact that their views differ from those of their accusers. The feeling on both sides too often is, "I am not convinced by your arguments because your arguments are unconvincing; you are not convinced by my arguments because you are dogmatic."
Of course mathematicians feel that mathematics is pure reason and therefore immune to such controversy. But there are plenty of controversies in mathematics.
[...] 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.
To a large extent I was attracted first to mathematics and, subsequently, to mathematical logic by their generality, anticipating that this is the normal attitude; it seems I was mistaken. I have always felt that examples usually just confuse you (though not always), having always specific properties that are traps, as they do not hold in general. [...]
From another angle, I was amazed to find that many of my colleagues, including some of the best minds in the field of set theory, feel apologetic about their subject. Many are apologetic toward mathematicians (implying somehow that there are mathematicians and there are logicians, as if they are disjoint species) working in fields which are surely deeper, harder, more profound and meaningful, etc., and so feel that we have to justify our existence by finding applications of “logic” to “mathematics”. [...] Now, I love to prove theorems in as many areas of mathematics as I can, but I do not like this servile attitude.
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.
Gert Schubring ["The conception of pure mathematics as an instrument in the professionalization of mathematics"] has argued that in the professionalisation of mathematics in Prussia in the early 1800s, the 'meta-conception' of pure mathematics played an important role. By defining 'mathematics' as separate from externally defined objectives, the mathematicians oriented the discipline to internal values that they could control. To do this, support from the state had to be available first. Given state patronage for academic positions, the mathematicians could proceed to establish a discipline by establishing training which channelled students into the new professional orientation, reducing the number of self-taught mathematicians obtaining jobs in the field and socialising students into the meta-conception of pure mathematics. This account meshes nicely with that of Grabiner ["Is mathematical truth time-dependent?"].
This process continues today. Especially in universities, the home grounds of pure mathematics, mathematicians stake their claims to autonomy and resources on their exclusive rights, as experts, to judge research in mathematics. This is no different from the claims of many other disciplines and professions. The point is that if mathematicians emphasised application as their primary value, their claims to status and social resources would be dependent on the value of the application. The conception of 'pure' mathematics enables an exclusive claim to control over the discipline to be made. [...]
The question, 'What is the link between mathematics and social interests?', is usually answered in advance by assumptions about what mathematics really is. If mathematics is taken to be that body of mathematical knowledge which sits above or outside of human interests, then by definition social interests can only be involved in the practice of mathematics, not in mathematics itself. This Platonic-like conception sees mathematics as value-free, but is itself a value-laden conception: it serves to deflect attention from the many links between mathematics and society. [...]
The belief that mathematics is a body of truth independent of society is deeply embedded in education and research. This situation, by hiding the social role of mathematics behind a screen of objectivity, serves those groups which preferentially benefit from the present social system of mathematics. Exposing the links between mathematics and social interests should not be seen as a threat to 'mathematics' but rather as a threat to the groups that reap without scrutiny the greatest material and ideological benefits from an allegedly value-free mathematics.
The choice of axioms, the types of theorems, the style of proofs and a host of other facets of mathematics can be shaped by factors such as views about the nature of social reality.
An example here is game theory, a mathematical theory which deals with conflict situations, originally developed to model economic systems (Martin 1978). Key concepts of the theory include the 'players' in a game, each of which has a number of 'choices', followed by 'payoffs'. The mathematical theory of games is built around determining the optimal strategies for making choices. The players, choices and payoffs are usually assumed to be fixed; competition is built in; payoffs tend to be quantifiable. Hence, game theory is especially suited for applications which assume and reinforce individualism and competition.
Game theory has been applied in many areas, such as international relations. What often happens in practice is that the values of the modellers are incorporated into the game theoretic formulation, which usually ensures that the game gives results which legitimate those very same values. Game theory in this situation provides a 'mystifying filter': values are built into an ostensibly value-free mathematical framework, which thus provides 'scientific' justification for the decision desired. Arguably, game theory has become popular because its mathematical framework makes it easy to use in this way. [...]
Many people who have been involved in mathematical modelling will realise the great opportunities for building the values of the modeller into the model. I have seen this process at work in a variety of areas, including mathematical ecology, game theory, stratospheric chemistry and dynamics, voting theory, wind power and econometrics.
A good example is the systems of difference equations used in the early 1970s to determine the 'limits to growth'. The choice of equations and parameters more or less ensured that global instability would result (Cole et al. 1973). When different assumptions were used by different modellers, different results - for example, that promotion of global social equality would prevent global breakdown - were obtained, nicely compatible with the values of the modellers. [...]
Mathematical models are socially significant in two principal ways: as practical applications of mathematics and as legitimations of policies or practices. Most models are closely tied to practical applications, such as in industry. The narrow specialisation involved in the modelling ensures that few other than those developing or funding the application would be interested in or capable of using the model. This sort of applied mathematics is closely linked to the social interests making the specific application. Whether the application is telecommunications satellites, anti-personnel weapons or solar house design, one may judge the mathematics by the same criteria used to judge that application. It is not adequate to say that the killer is guilty while the murder weapon is innocent, for in these sorts of applications the mathematical 'weapon' is especially tailored for its job. Certainly applied mathematicians cannot escape responsibility for their work by referring to 'neutral tools', whether this refers to their mathematical constructions or to themselves.
Models serving as legitimations are involved in a more complicated dynamic. In many cases such as limits-to-growth studies the models do no more than mathematicise a conclusion which would be obvious without the model. But the models are seen as important precisely because they are mathematical, thus drawing on the image of mathematics as objective. A mathematics-based claim also has the advantage of being the work of professionals. Anyone can make a claim, but if a scientist does so, relying on the allegedly objective tools of mathematics, that is much more influential. Although exercises in mathematical modelling are often shot through with biases, for public consumption this often is overlooked; the modellers draw on an aura of objectivity which is sustained by the more esoteric researches of pure mathematicians.
[...] the main customers of mathematics are physicists and engineers, whether we like it or not, and they have given up searching for relevant results in unreadable research articles. They will rediscover by themselves whatever mathematics they need, hoping to shove mathematicians into the dustbin of history.
8. AVOID WORD PROBLEMS
I once asked a colleague of mine why he so liked word problems, and his answer was: “I like
them because one can assign good problem sets.”
My colleague’s answer betrayed a common error of reasoning. [...] My colleague’s error consisted of believing that the more testable the material, the more teachable it is. A wider spread of performance in the problem sets and in the quizzes makes the assignment of grades “more objective.” The course is turned into a game of skill, where manipulative ability outweighs understanding.
The word problems that we find in differential equations textbooks are shameful. They are artificial, dishonest, unrealistic, contrived, repetitive, and irrelevant. I cannot see how a student can learn anything by being forced to solve snowplow problems or Rube Goldberg flows of salt water in communicating tanks. [...]
10. TEACH CONCEPTS, NOT TRICKS
What can we expect students to get out of an elementary course in differential equations? I reject the “bag of tricks” answer to this question. A course taught as a bag of tricks is devoid of educational value. One year later, the students will forget the tricks, most of which are useless anyway. The bag of tricks mentality is, in my opinion, a defeatist mentality, and the justifications I have heard of it, citing poor preparation of the students, their unwillingness to learn, and the possibility of assigning clever problem sets, are lazy ways out. [...]
Who cares whether the students become skilled at working out tricky problems? What matters is their getting a feeling for the importance of the subject, their coming out of the course with the conviction of the inevitability of differential equations, and with enhanced faith in the power of mathematics. These objectives are better achieved by stretching the students’ minds to the utmost limits of cultural breadth of which they are capable, and by pitching the material at a level that is just a little higher than they can reach.
We are kidding ourselves if we believe that the purpose of undergraduate teaching is the transmission of information. Information is an accidental feature of an elementary course in differential equations; such information can nowadays be gotten in much better ways than sitting in a classroom.
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.
Professor Mac Lane has for at least twenty years been saying that "set theory is obsolete," that "measurable cardinals are bizarre," and so on, and he has written one large book ("Mathematics: Form and Function") and many articles in order to present his view of mathematics.
It is the purpose of this essay to examine his stance, and to suggest that insofar as Mac Lane urges the unity of mathematics, he is to be supported, but insofar as he secretly desires the uniformity of mathematics, he is to be opposed. [...]
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.
Too often have mathematicians scolded their peers for work not to their taste. Too often have the prejudices of the world entered into what should be an oasis of intellectual freedom. [...] I am, first of all, an optimist. I believe that most mathematicians make sound decisions on what they themselves should study. I am also a pessimist. I fear that these same mathematicians can make terrible decisions about what others should study.
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.
[...] Our critics, especially those well-meaning pedagogues, should come to realize that mathematics becomes simpler only through abstraction. The mathematics that represented the conceptual limit for the minds of Newton and Leibniz is taught regularly in our high schools, because we now have a clear (i.e. abstract) notion of a function and of the real numbers.
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.
Mathematical practice, perspicuity, anthropocentrism, history, now politics - what a different world from the eternal unchanging real of Platonic entities! No wonder traditional Platonists are annoyed by the idea of mathematical practice. In defense of [Hao] Wang, we would do well to recall the Aristotelian slogan; "Of course I love Plato, but I love truth more."
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.
The study of large cardinals is not non-foundational, but anti-foundational: The goal of the study of the large cardinals is to destroy the foundations of mathematics by turning it from a Euclidean into a Popperian discipline of the sort Morris Kline mistakenly believes we already have. Less glamourously put: Gödel observed that higher axioms of infinity increased our ability to prove theorems. The proper goal of the study of such axioms is thus to use ever larger cardinals in necessary ways in ordinary mathematical practice, that is to establish pleasant mathematical results on ever shakier and shakier grounds.
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.
In any case, what is completely clear is that no notion of: set of arbitrary transfinite type, or even notions of set obtained by some definite iteration (beyond ω + ω) of the power set operation, is relevant, as of now, to mathematical practice, or even understood by mathematicians.
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.
Many of the latter [mathematicians themselves] have been unwilling for a long time to see in axiomatics anything else than futile logical hairsplitting not capable of fructifying any theory whatever. This critical attitude can probably be accounted for by a purely historical accident. [...] But the further development of the [axiomatic] method has revealed its power; and the repugnance which it still meets here and there, can only be explained by the natural difficulty of the mind to admit, in dealing with a concrete problem, that a form of intuition, which is not suggested directly by the given elements (and which often can be arrived at only by a higher and frequently difficult stage of abstraction), can turn out to be equally fruitful.
Two things which have the same logical form are analogous. The value of analogy is that a thing which has a certain logical form may be represented by another which has the same structure, i.e. which is analogous to it. [...]
Abstraction is the consideration of logical form apart from content. The reason why people distrust abstractions is simply that they do not know how to make and use them correctly, so that abstract thought leads them into error and bewilderment. Abstraction is perhaps the most powerful instrument of human understanding.
Abstracted forms are called concepts. Because nature is full of analogies, we can understand certain parts of it in the light of a few very general concepts. [...]
Logic deals with any forms whatever without reference to content. A knowledge of forms and their relations greatly facilitates any study of their possible applications. Logic is a tool of philosophical thought as mathematics is a tool of physics.
[...] 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.]
But every superfluous constraint on the urge to mathematical investigation seems to me to bring with it a much greater danger, all the more serious because in fact absolutely no justification for such constraints can be advanced from the essence of the science — for the essence of mathematics lies precisely in its freedom.
[Dagegen scheint mir aber jede überflüssige Einengung des mathematischen Forschungstriebes eine viel grössere Gefahr mit sich zu bringen und eine um so grössere, als dafür aus dem Wesen der Wissenschaft wirklich keinerlei Rechtfertigung gezogen werden kann; denn das Wesen der Mathematik liegt gerade in ihrer Freiheit.]
But then with me the horrid doubt always arises whether the convictions of man’s mind, which has been developed from the mind of the lower animals, are of any value or at all trustworthy. Would any one trust in the convictions of a monkey’s mind, if there are any convictions in such a mind?
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.]
