logic

"I feel that controversies can never be finished, nor silence imposed upon the Sects, unless we give up complicated reasonings in favour of simple calculations, words of vague and uncertain meaning in favour of fixed symbols [characteres]. Thus it will appear that 'every paralogism is nothing but an error of calculation. When controversies arise, there will be no more necessity for disputation between two philosophers than between two accountants. Nothing will be needed but that they should take pen in hand, sit down with their counting-tables and (having summoned a friend, if they like) say to one another: Let us calculate.'"

"These primitive propositions … suffice to deduce all the properties of the numbers that we shall meet in the sequel. There is, however, an infinity of systems which satisfy the five primitive propositions. …All systems which satisfy the five primitive propositions are in one-to-one correspondence with the natural numbers. The natural numbers are what one obtains by abstraction from all these systems; in other words, the natural numbers are the system which has all the properties and only those properties listed in the five primitive propositions"

"I come now to the capital work of Hilbert which he communicated to the Congress of Mathematicians at Heidelberg... of which...an English translation due to Halsted appeared in The Monist. ...the author's aim is analogous to that of Russell, but on many points he diverges from his predecessor. "But," he says, "on attentive consideration we become aware that in the usual exposition of the laws of logic certain fundamental concepts of arithmetic are already employed; for example, the concept of the aggregate, in part also the concept of number. "We fall thus into a vicious circle and therefore to avoid paradoxes a partly simultaneous development of the laws of logic and arithmetic is requisite." ...what Hilbert says of the principles of logic in the usual exposition applies likewise to the logic of Russell. So for Russell logic is prior to arithmetic; for Hilbert they are 'simultaneous.' We shall find... other differences still greater... I prefer to follow step by step the development of Hubert's thought... "Let us take as the basis of our consideration first of all a thought-thing 1 (one)." Notice that in so doing we in no wise imply the notion of number, because it is understood that 1 is here only a symbol and that we do not at all seek to know its meaning. "The taking of this thing together with itself respectively two, three or more times ..." Ah! this time it is no longer the same; if we introduce the words 'two,' 'three,' and above all 'more,' 'several,' we introduce the notion of number; and then the definition of finite whole number which we shall presently find, will come too late. Our author was too circumspect not to perceive this begging of the question. So at the end of his work he tries to proceed to a truly patching-up process. Hilbert then introduces two simple objects 1 and =, and and considers all the combinations of these two objects, all the combinations of their combinations, etc. It goes without saying that we must forget the ordinary meaning of these two signs and not attribute any to them. Afterwards he separates these combinations into two classes, the class of the existent and the class of the non-existent... entirely arbitrary. Every affirmative statement tells us that a certain combination belongs to the class of the existent; every negative statement tells us that a certain combination belongs to the class of the non-existent. Note now a difference of the highest importance. For Russell any object whatsoever, which he designates by x, is an object absolutely undetermined and about which he supposes nothing; for Hilbert it is one of the combinations formed with the symbols 1 and =; he could not conceive of the introduction of anything other than combinations of objects already defined."

"It is known that geometry assumes, as things given, both the notion of space and the first principles of constructions in space. She gives definitions of them which are merely nominal, while the true determinations appear in the form of axioms. The relation of these assumptions remains consequently in darkness; we neither perceive whether and how far their connection is necessary, nor, a priori, whether it is possible. From Euclid to Legendre (to name the most famous of modern reforming geometers) this darkness was cleared up neither by mathematicians nor by such philosophers as concerned themselves with it. The reason of this is doubtless that the general notion of multiply extended magnitudes (in which space-magnitudes are included) remained entirely unworked. I have in the first place, therefore, set myself the task of constructing the notion of a multiply extended magnitude out of general notions of magnitude. It will follow from this that a multiply extended magnitude is capable of different measure-relations, and consequently that space is only a particular case of a triply extended magnitude. But hence flows as a necessary consequence that the propositions of geometry cannot be derived from general notions of magnitude, but the properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience. Thus arises the problem, to discover the simplest matters of fact from which the measure-relations of space may be determined; a problem which from the nature of the case is not completely determinate, since there may be several systems of matters of fact which suffice to determine the measure-relations of space—the most important system for our present purpose being that which Euclid has laid down as a foundation. These matters of fact are—like all matters of fact—not necessary, but only of empirical certainty; they are hypotheses. We may therefore investigate their probability, which within the limits of observation is of course very great, and inquire about the justice of their extension beyond the limits of observation, on the side both of the infinitely great and of the infinitely small."

"Mathematics and logic, historically speaking, have been entirely distinct studies. Mathematics has been connected with science, logic with Greek. But both have developed in modern times: logic has become more mathematical and mathematics has become more logical. The consequence is that it has now become wholly impossible to draw a line between the two; in fact, the two are one. They differ as boy and man: logic is the youth of mathematics and mathematics is the manhood of logic. This view is resented by logicians who, having spent their time in the study of classical texts, are incapable of following a piece of symbolic reasoning, and by mathematicians who have learnt a technique without troubling to inquire into its meaning or justification. Both types are now fortunately growing rarer. So much of modern mathematical work is obviously on the border-line of logic, so much of modern logic is symbolic and formal, that the very close relationship of logic and mathematics has become obvious to every instructed student. The proof of their identity is, of course, a matter of detail: starting with premises which would be universally admitted to belong to logic, and arriving by deduction at results which as obviously belong to mathematics, we find that there is no point at which a sharp line can be drawn, with logic to the left and mathematics to the right. If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the successive definitions and deductions of Principia Mathematica, they consider that logic ends and mathematics begins. It will then be obvious that any answer must be quite arbitrary."

"The philosophical tradition that goes from Descartes to Husserl, and indeed a large part of the philosophical tradition that goes back to Plato, involves a search for foundations: metaphysically certain foundations of knowledge, foundations of language and meaning, foundations of mathematics, foundations of morality, etc. […] Now, in the twentieth century, mostly under the influence of Wittgenstein and Heidegger, we have come to believe that this general search for these sorts of foundations is misguided."

"As a science mathematics has been adapted to the description of natural phenomena, and the great practitioners in this field... have never concerned themselves with the logical foundations of mathematics, but have boldly taken a pragmatic view of mathematics as an intellectual machine which works successfully. Description has been verified by further observation, still more strikingly be prediction, and sometimes, more ominously, by control of natural forces. Happily, unresolved problems... still remain as challenges."

"In conversations, some quite recent, on the present status of the foundations of mathematics, von Neumann seemed to imply that in his view, the story is far from having been told. Gödel's discovery should lead to a new approach to the understanding of the role of formalism in mathematics, rather than be considered as closing the subject."

"The abstract formulation of mathematics seems to date back to the German mathematician Moritz Pasch. At any rate, he was the first to study in detail the axioms concerning the order of points on a straight line and to state clearly the assumptions involved in the idea of "betweenness." ...But to the Italian Giuseppe Peano belongs the credit of developing this point of view systematically. His idea, which he began to elaborate about 1880, is to put the whole of mathematics on a purely formal basis, and for this purpose he invented a symbolism of his own. In 1893 he began the publication of a "Formulario di matematica," which is a synopsis of the most important propositions of the different branches of mathematical science, with their demonstrations, expressed entirely in terms of symbolic logic. ...An immense change in the point of view toward the foundations has been brought about since this abstract formulation was put forward. ...Vailati has suggested that this change is very similar to that which a nation undergoes when it changes from a monarchic or aristocratic form of government to a democracy. The point of view fifty years ago was very largely that the foundations of mathematics were axioms; and by axioms were meant self-evident truths, that is, ideas imposed upon our minds a priori, with which we must necessarily begin any rational development of the subject. So the axioms dominated over mathematical science, as it were, by the divine right of the alleged inconceivability of the opposite. And now, what is the new point of view? The self-evident truth is entirely banished. There is no such thing. What has taken the place of it? Simply a set of assumptions concerning the science which is to be developed, in the choice of which we have considerable freedom. The choice of a set of assumptions is very much like the election of men to office. There is no logical reason why we should not choose the more complex propositions; but as a matter of fact we usually choose the simpler, because it is easier to work with them. Not all propositions reach the high position of assumptions; they are elected for their fitness to serve, and their fitness is very largely determined by their simplicity, by the ease with which the other propositions may be derived from them."

"Mathematics is a most conservative science. Its system is so rigid and all the details of geometrical demonstration are so complete, that the science was commonly regarded as a model of perfection. Thus the philosophy of mathematics remained undeveloped almost two thousand years."

"Metageometry has always proved attractive to erratic minds. Among the professional mathematicians, however, those who were averse to philosophical speculation looked upon it with deep distrust, and therefore either avoided it altogether or rewarded its labors with bitter sarcasm. Prominent mathematicians did not care to risk their reputation, and consequently many valuable thoughts remained unpublished. Even Gauss did not care to speak out boldly, but communicated his thoughts to his most intimate friends under the seal of secrecy, not unlike a religious teacher who fears the odor of heresy. He did not mean to suppress his thoughts, but he did not want to bring them before the public unless in mature shape."

"The labors of Lobatchevsky and Bolyai are significant in so far as they prove beyond the shadow of a doubt that a construction of geometries other than Euclidean is possible and that it involves us in no absurdities or contradictions. This upset the traditional trust in Euclidean geometry as absolute truth, and it opened at the same time a vista of new problems foremost among which was the question as to the mutual relation of these three different geometries. It was Cayley who proposed an answer which was further elaborated by Felix Klein. These two ingenious mathematicians succeeded in deriving by projection all three systems from one common aboriginal form called by Klein Grundgebild or the Absolute. In addition to the three geometries hitherto known to mathematicians, Klein added a fourth one which he calls elliptic. Thus we may now regard all the different geometries as three species of one and the same genus and we have at least the satisfaction of knowing that there is terra firma at the bottom of our mathematics, though it lies deeper than was formerly supposed."

"The ease with which you have assimilated my notions of geometry has been a source of genuine delight to me, especially as so few possess a natural bent for them. I am profoundly convinced that the theory of space occupies an entirely different position with regard to our knowledge a priori from that of the theory of numbers (Grössenlehre); that perfect conviction of the necessity and therefore the absolute truth which is characteristic of the latter is totally wanting to our knowledge of the former. We must confess in all humility that a number is solely a product of our mind. Space, on the other hand, possesses also a reality outside of our mind, the laws of which we cannot fully prescribe a priori."

"I shall now address you on the subject of the present situation in research in the foundations of mathematics. Since there remain open questions in this field, I am not in a position to paint a definitive picture of it for you. But it must be pointed out that the situation is not so critical as one could think from listening to those who speak of a foundational crisis. From certain points of view, this expression can be justified; but it could give rise to the opinion that mathematical science is shaken at its roots."

"As soon as I have put it into order I intend to write and if possible to publish a work on parallels. At this moment, it is not yet finished, but the way which I have followed promises me with certainty the attainment of my aim, if it is at all attainable. It is not yet attained, but I have discovered such magnificent things that I am myself astonished at the result. It would forever be a pity, if they were lost. When you see them, my father, you yourself will concede it. Now I cannot say more, only so much that from nothing I have created another wholly new world. All that I have hitherto sent you compares to it as a house of cards to a castle."

"It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning."

"Dedekind proves mathematical induction, while Peano regards it as an axiom. This gives Dedekind an apparent superiority, which must be examined. ...not because of any logical superiority, it seems simpler to begin with mathematical induction. And it should be observed that, in Peano's method, it is only when theorems are to be proved concerning any number that mathematical induction is required. The elementary Arithmetic of our childhood, which discusses only particular numbers, is wholly independent of mathematical induction; though to prove that this is so for every particular number would itself require mathematical induction. In Dedekind's method, on the other hand, propositions concerning particular numbers, like general propositions, demand the consideration of chains. Thus there is, in Peano's method, a distinct advantage of simplicity, and a clearer separation between the particular and the general propositions of Arithmetic. But from a purely logical point of view, the two methods seem equally sound; and it is to be remembered that, with the logical theory of cardinals, both Peano's and Dedekind's axioms become demonstrable."

"The problem of the philosophical foundation of mathematics is closely connected with the topics of Kant's Critique of Pure Reason. It is the old quarrel between Empiricism and Transcendentalism. Hence our method of dealing with it will naturally be philosophical, not typically mathematical."

"The two great conceptual revolutions of twentieth-century science, the overturning of classical physics by Werner Heisenberg and the overturning of the foundations of mathematics by Kurt Gödel, occurred within six years of each other within the narrow boundaries of German-speaking Europe. ...A study of the historical background of German intellectual life in the 1920s reveals strong links between them. Physicists and mathematicians were exposed simultaneously to external influences that pushed them along parallel paths. ...Two people who came early and strongly under the influence of Spengler's philosophy were the mathematician Hermann Weyl and the physicist Erwin Schrödinger. ...Weyl and Schrödinger agreed with Spengler that the coming revolution would sweep away the principle of physical causality. The erstwhile revolutionaries David Hilbert and Albert Einstein found themselves in the unaccustomed role of defenders of the status quo, Hilbert defending the primacy of formal logic in the foundations of mathematics, Einstein defending the primacy of causality in physics. In the short run, Hilbert and Einstein were defeated and the Spenglerian ideology of revolution triumphed, both in physics and in mathematics. Heisenberg discovered the true limits of causality in atomic processes, and Gödel discovered the limits of formal deduction and proof in mathematics. And, as often happens in the history of intellectual revolutions, the achievement of revolutionary goals destroyed the revolutionary ideology that gave them birth. The visions of Spengler, having served their purpose, rapidly became irrelevant."

"An article by Henri Poincaré entitled The Nature of Mathematical Reasoning... appeared in 1894 as the first of a series of investigations into the foundations of the exact sciences. It was a signal for a throng of other mathematicians to inaugurate a movement for the revision of the classical concepts, a movement which culminated in the nearly complete absorption of logic into the body of mathematics."

"I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction."

"In the summer of 1914 I attended Frege's course, Logik in der Mathematik. Here he examined critically some of the customary conceptions and formulations in mathematics. He deplored the fact that mathematicians did not even seem to aim at the construction of a unified, well-founded system of mathematics, and therefore showed a lack of interest in foundations. He pointed out a certain looseness in the customary formulation of axioms, definitions, and proofs, even in the works of the more prominent mathematicians. As an example he quoted Weyerstrass's definition: "A number is a series of things of the same kind"... On this he commented with an impish smile: "According to this definition, a railroad train is also a number; this number may then travel from Berlin, pass through Jena... He criticized in particular the lack of attention to certain fundamental distinctions, e.g., ...between the symbol and the symbolized, ...between a logical concept and a mental image or act, and that between a function and the value of a function. Unfortunately, his admonitions go unheeded even today."

"Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real."

"It is in set theory that we encounter the greatest diversity of foundational opinions. This is because even the most devoted advocates of the various new axioms would not argue that these axioms are justified by any basic ‘intuition’ about sets. ... One may vary the rank of sets allowed. Conventional mathematics rarely needs to consider more than four or five iterations of the power set axiom applied to the set of integers. More iterations diminish our sense of the reality of the objects involved."

"Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic."

"I should regard it as a great misfortune if you were to allow yourself to be deterred by the 'clamors of the Bœotians' from explaining your views of geometry. From what Lambert has said and [Ferdinand Karl] Schweikart orally communicated, it has become clear to me that our geometry is incomplete and stands in need of a correction which is hypothetical and which vanishes wher. the sum of the angles of a plane triangle is equal to 180°. This would be the true geometry and the Euclidean the practical, at least for figures on the earth."

"Now Gödel's proof, Russell's original paradox, all these things, all stem from one common root which is inherent in all symbolic languages, including the language we use. ...the problem which dogs all formal systems, the problem of self-reference; that is, the language can be used to refer to sentences in the language. Indeed, between 1900 and 1910 Russell tried to forbid this, to say you cannot do mathematics if you can do that, and so he invented the theory of types. Of course, no sooner had he invented it than it turned up you could not do mathematics at all if you obeyed the theory of types. So then he had to put in an , which allows a certain amount of self-reference. And by this time everyone was pretty bored."

"It is significant that we owe the first explicit formulation of the principle of recurrence to the genius of Blaise Pascal... Pascal stated the principle in a tract called The Arithmetic Triangle which appeared in 1654. Yet... the gist of the tract was contained in the correspondence between Pascal and Fermat regarding a problem in gambling, the same correspondence which is now regarded as the nucleus from which developed the theory of probabilities. It surely is a fitting subject for mystic contemplation that the principle of reasoning by recurrence, which is so basic in pure mathematics, and the theory of probabilities, which is the basis of all deductive sciences, were both conceived while devising a scheme for the division of stakes in an unfinished match of two gamblers."

"Despite the age-long tyranny exercised by the Aristotelian logic... Of all argument forms, there is one which, viewed as the figure of the way in which the mind gains certainty that a specified property belonging, but not immediately by definition, to each element of a denumerable assemblage of elements does so belong, enjoys the distinction of being at once perhaps the most fascinating, and, in its mathematical bearings, doubtless the most important single form in modern logic. This form is that variously known as reasoning by recurrence, induction by connection (De Morgan), mathematical induction, complete induction, and Fermatian induction—so called by C. S. Peirce, according to whom this mode of proof was first employed by Fermat. Whether or not such priority is thus properly ascribed, it is certain that the argument form in question is unknown to the Aristotelian system, for this system allows apodictic certainty in case of deduction only, while it is the distinguishing mark of mathematical induction that it yields such certainty by the reverse process, a movement from the particular to the general, from the finite to the infinite. Of the various designations of this mode argument, "mathematical induction" is undoubtedly the most appropriate, for though one not be able to agree with Poincaré that the mode in question is characteristic of mathematics, it is peculiar to science, being indeed, as he has called it, "mathematical reasoning par excellence.""

"One who extended the theory of equations somewhat further than Vieta was Albert Girard... Like Vieta this ingenious author applied algebra to geometry, and was the first who understood the use of negative roots in the solution of geometric problems. He spoke of imaginary quantities; inferred by induction that every equation has as many roots as there are units in the number expressing its degree; and first showed how to express the sums of their powers in terms of the coefficients."

"In his address to the Mathematical Section at the British Association Meeting of 1869 at Exeter, Professor J. J. Sylvester laid much stress upon the employment of inductive philosophy in mathematics. He said that he was aware that many who had not gone deeply into the principles of mathematical science believed that inductive philosophy, or the method of evolving new truths by induction, was reserved for the experimental sciences, and that the methods of investigation in mathematical science might all be classified as deductive. He went on to say that this opinion is not a correct one, and that many valuable results are obtained in mathematical science by induction, or reasoning from particulars to generals, which could not otherwise be obtained so easily. Although making a distinction between mathematical induction and the induction used in natural philosophy, De Morgan, in his article in the 'Penny Cyclopædia' on this subject, states that an instance of mathematical induction occurs in every equation of differences and in every recurring series. Taking the definition of induction as given by Dr. Whateley, namely, "a kind of argument which infers respecting a whole class what has been ascertained respecting one or more individuals of that class," it will be evident to any experimenter in chemical or physical science who is also acquainted with the use of induction in mathematical science, that mathematical induction is of a higher and more perfect kind than the induction used in the physical sciences, especially when it assumes the form of successive induction as De Morgan calls it, and as it is employed in recurring series. It is this high class of reasoning which is involved in the construction of series that recur according to a given law, that makes the use of recurring series so valuable in unitation."

"M. Poincaré finds the answer to these questions in the so-called 'mathematical induction' which proceeds from the particular to the more general, but at the same time does so by steps of the highest degree of certitude. In this process he sees the creative force of mathematics, which leads to real proofs and not mere sterile verifications. The illustrations used to make the thought clear are taken from the beginnings of arithmetic, where mathematical thought has remained least elaborated and uncomplicated by the difficult questions related to the notion of space. In successive instances it is shown how more general results are obtained from fundamental definitions and from previous results by means of mathematical induction. In each case the advance is made by virtue of that "power of the mind which knows that it can conceive of the indefinite repetition of the same act as soon as this act is at all possible. The mind has a direct intuition of this power and experience gives only the opportunity to use it and to become conscious of it." The conviction that the method of mathematical induction is valid our author regards as truly an à priori synthetic judgment; the mind can not tolerate nor conceive its contradictory and could not even draw any theoretic consequences from the assumption of the contradictory. No arithmetic could be built up, rejecting the axiom of mathematical induction, as the non-Euclidean geometries have been built up, rejecting the postulate of Euclid."

"A more modern attempt to explain the fruitfulness of mathematical reasoning is that of Poincaré, who finds it all due to the principle of mathematical induction. This principle of mathematical induction is undoubtedly of wide application, though there are many regions even in arithmetic where it is difficult to see its application, e.g., the science of prime numbers, a science dealing entirely with non-recurring individuals. But the important thing to observe is that this principle of mathematical induction is entirely different from the induction that prevails in the physical sciences."

"The world is totally connected. Whatever explanation we invent at any moment is a partial connection, and its richness derives from the richness of such connections as we are able to make. ...mathematics suffer from the same partiality. Gödel, Turing, and Tarski all proved this. Gödel proved that you cannot have a complete axiomatization of the whole of mathematics, that every system which you devise is partial and suffers from one great shortcoming. If it is consistent, there are theorems which are true that cannot be proved in it. And Turing showed that every machine that we can devise is like a formal system, and that therefore no machine can do all of mathematics. And Tarski put it even more boldly when he said that no universal language for all of science can exist in all cases without paradox."

"The propositions of arithmetic, the... operations, for instance, which play such a fundamental rôle even in the most simple calculations, must be demonstrated by deductive methods. What is the principle involved? Well, this principle has been variously called mathematical induction, and complete induction, and that of reasoning by recurrence. The latter is the only acceptable name, the others being misnomers. The term induction conveys an entirely erroneous idea of the method, for it does not imply systematic trials."

"It is absolutely certain that if a proposition is established by mathematical induction, it will never be disproved, i.e., if a general proposition is true of n + 1 whenever it is true of n, and also of 1, then no possible number can arise of which this proposition is not true, for the principle of mathematical induction is used in defining all finite integers. Whether, therefore, we agree with Russell and call the principle of mathematical induction a definition, or concede to Poincaré that it is a special axiom, a synthetic proposition a priori, the fact remains that reasoning from it is a purely deductive procedure."

"But, one will say, if raw experience can not legitimatize reasoning by recurrence, is it so of experiment aided by induction? We see successively that a theorem is true of the number 1, of the number 2, of the number 3 and so on; the law is evident, we say, and it has the same warranty as every physical law based on observations, whose number is very great but limited. But there is an essential difference. Induction applied to the physical sciences is always uncertain, because it rests on the belief in a general order of the universe, an order outside of us. Mathematical induction, that is, demonstration by recurrence, on the contrary, imposes itself necessarily, because it is only the affirmation of a property of the mind itself."

"Many years ago I published in the Formulaire de Mathématique of Professor Peano an account of the first discovery of mathematical induction as due to the Italian Maurolycus. But this paper seems to have had only a small diffusion. ...the most original of his works is the treatise on arithmetic "Arithmeticorum libri duo" written in the year 1557 and printed in Venice in the year 1575 in the collection "D. Francisci Maurolyci Opuscula mathematica." In the Prolegomena to this work he points out that neither in Euclid nor in any other Greek or Latin writer (among them he enumerates Iamblichus, Nicomachus, Boetius) is there, to his knowledge, a treatment of the polygonal and polyhedral numbers, and he reproaches Jordanus for having been content with a useless repetition of what was written by Euclid. "Nos igitur [he says] conabimur ea, quae super hisce numerariis formis nobis occurrunt, exponere: multa interim faciliori via demonstrantes, et ab aliis authoribus aut neglecta, aut non animadversa supplentes." This new and easy way is nothing else than the principle of mathematical induction. This principle is used at the beginning of the work only in the demonstration of very simple propositions, but in the course of the treatise is applied to the more complicated theorems in a systematic way. ...Was Pascal unaware of the book of Maurolycus? In his Traité du triangle arithmétique printed perhaps in the year 1657, he never mentions Maurolycus, notwithstanding that, in my opinion, this treatise is only an application of the method discovered by Maurolycus. But Pascal, shortly after, being engaged in the polemic concerning the cycloid, in the well known letter, "Lettre de Dettonville à Carcavi" had to demonstrate a proposition concerning the triangular and pyramidal numbers. He says then:"Cela Est Aisé Par Maurolic."It is strange to point out that not even the name of Maurolycus has been included in the Table analytique of the old edition of the works of Pascal, and more strange that the editors of the new edition of the "Oeuvres" of Pascal in a very incomplete historical note before the reimpression of the Traité du triangle arithmétique never mention the name of one of the greatest European mathematicians of the sixteenth century."

"We can not... escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of contradiction. ...Neither can this rule come to us from experience... This rule, inaccessible to analytic demonstration and to experience, is the veritable type of the synthetic a priori judgment. On the other hand, we can not think of seeing in it a convention, as in some of the postulates of geometry. ...it is only the affirmation of the power of the mind which knows itself capable of conceiving the indefinite repetition of the same act when once this act is possible. The mind has a direct intuition of this power, and experience can only give occasion for using it and thereby becoming conscious of it."

"This procedure is the demonstration by recurrence. We first establish a theorem for n = 1; then we show that if it is true of n - 1, it is true of n, and thence conclude that it is true for all the whole numbers. ..Here then we have the mathematical reasoning par excellence, and we must examine it more closely. ...The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms. ...to arrive at the smallest theorem [we] can not dispense with the aid of reasoning by recurrence, for this is an instrument which enables us to pass from the finite to the infinite. This instrument is always useful, for, allowing us to overleap at a bound as many stages as we wish, it spares us verifications, long, irksome and monotonous, which would quickly become impracticable. But it becomes indispensable as soon as we aim at the general theorem... In this domain of arithmetic,.. the mathematical infinite already plays a preponderant rôle, and without it there would be no science, because there would be nothing general."

"We could call it "proof from n to n + 1" or still simpler "passage to the next integer." Unfortunately, the accepted technical term is "mathematical induction." This name results from a random circumstance. ...Now, in many cases... the assertion is found experimentally, and so the proof appears as a mathematical complement to induction; this explains the name."

"The use of mathematical induction in demonstrations was, in the past, something of a mystery. There seemed no reasonable doubt that it was a valid method of proof, but no one quite knew why it was valid. Some believed it to be really a case of induction, in the sense in which that word is used in logic. Poincaré considered it to be a principle of the utmost importance, by means of which an infinite number of syllogisms could be condensed into one argument. We now know that all such views are mistaken, and that mathematical induction is a definition, not a principle."

"Mathematical induction, which is purely ordinal... may be stated as follows: A series generated by a one-one relation, and having a first term, is such that any property, belonging to the first term and to the successor of any possessor of the property, belongs to every term of the series."

"Dedekind proves mathematical induction, while Peano regards it as an axiom. This gives Dedekind an apparent superiority, which must be examined. ...not because of any logical superiority, it seems simpler to begin with mathematical induction. And it should be observed that, in Peano's method, it is only when theorems are to be proved concerning any number that mathematical induction is required. The elementary Arithmetic of our childhood, which discusses only particular numbers, is wholly independent of mathematical induction; though to prove that this is so for every particular number would itself require mathematical induction. In Dedekind's method, on the other hand, propositions concerning particular numbers, like general propositions, demand the consideration of chains. Thus there is, in Peano's method, a distinct advantage of simplicity, and a clearer separation between the particular and the general propositions of Arithmetic. But from a purely logical point of view, the two methods seem equally sound; and it is to be remembered that, with the logical theory of cardinals, both Peano's and Dedekind's axioms become demonstrable."

"When we examine the classical set-theoretic foundations of mathematics, we see that the only sets that play a role are sets of restricted type; at the risk of understatement, only sets of rank < ω + ω. Further examination reveals four fundamental principles about sets used: the existence of an infinite set; the existence of the power set of any set; every property determines a subset of any set; and the axiom of choice."

"Fourier's analytical theory of heat (final form, 1822), devised in the Galileo-Newton tradition of controlled observation plus mathematics, is the ultimate source of much modern work in the theory of functions of a real variable and in the critical examination of the foundation of mathematics."

"I have also in my leisure hours frequently reflected upon another problem, now of nearly forty years' standing. I refer to the foundations of geometry. I do not know whether I have ever mentioned to you my views on this matter. My meditations here also have taken more definite shape, and my conviction that we cannot thoroughly demonstrate geometry a priori is, if possible, more strongly confirmed than ever. But it will take a long time for me to bring myself to the point of working out and making public my very extensive investigations on this subject, and possibly this will not be done during my life, inasmuch as I stand in dread of the clamors of the Bœotians, which would be certain to arise, if I should ever give full expression to my views. It is curious that in addition to the celebrated flaw in Euclid's Geometry, which mathematicians have hitherto endeavored in vain to patch and never will succeed, there is still another blotch in its fabric to which, so far as I know, attention has never yet been called and which it will by no means be easy, if at all possible, to remove. This is the definition of a plane as a surface in which a straight line joining any two points lies wholly in that plane. This definition contains more than is requisite to the determination of a surface, and tacitly involves a theorem which is in need of prior proof."

"The truth is that the mathematical sciences are growing in complete security and harmony. The ideas of Dedekind, Poincare, and Hilbert have been systematically developed with great success, without any conflict in the results. It is only from the philosophical point of view that objections have been raised. They bear on certain ways of reasoning peculiar to analysis and set theory. These modes of reasoning were first systematically applied in giving a rigorous form to the methods of the calculus. [According to them,] the objects of a theory are viewed as elements of a totality such that one can reason as follows: For each property expressible using the notions of the theory, it is [an] objectively determinate [fact] whether there is or there is not an element of the totality which possesses this property. Similarly, it follows from this point of view that either all the elements of a set possess a given property, or there is at least one element which does not possess it."

"These doubts did not halt mathematical creation. Technicians working on the superstructure did not drop their tools and scurry down to the basement because some of their underpinning needed reinforcing. Continuing their own highly specialized labors, they left the necessary task to experts who understood what they were about. The building had not collapsed as late as 1940; and while those engaged in elaborating the superstructure but seldom concerned themselves with what the consolidators of the foundations were doing, they had at least come to tolerate their presence in the building. The misconceptions and recriminations of the 1900's gave way in the 1930's to a first crude approximation to harmony. It was as if the American Federation of Labor and the Committee for Industrial Organization had at last decided to bury the hatchet elsewhere than in either's skull, and get on with the job."

"At the bottom of the difficulty there lurks the old problem of apriority, proposed by Kant and decided by him in a way which promised to give to mathematics a solid foundation in the realm of transcendental thought. And yet the transcendental method finally sent geometry away from home in search of a new domicile in the wide domain of empiricism."