Foundations of mathematics
study of the basic mathematical concepts
Foundations of mathematics is the study of the logical and philosophical basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics.
- 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.
- Eric Temple Bell, The Development of Mathematics (1940) p. 165
- 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.
- Eric Temple Bell, The Development of Mathematics (1940) pp. 515-516
- 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.
- 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.
- 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.
- 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 axiom of reducibility, which allows a certain amount of self-reference. And by this time everyone was pretty bored.
- Jacob Bronowski, The Origins of Knowledge and Imagination (1978)
- 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.
- Jacob Bronowski, The Origins of Knowledge and Imagination (1978)
- 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.
- Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre" Tr. "Foundations of a General Theory of Aggregates" (1883) as quoted by William Bragg Ewald, From Kant to Hilbert: A Source Book in the Foundations of Mathematics (1996)
- 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.
- An article by Henri Poincaré entitled The Nature of Matematical 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.
- 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. ...Weyle 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.
- Freeman Dyson, The Scientist As Rebel (2006).
- 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.
- 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.
- That the sum of the angles cannot be smaller than 180°; this is the real difficulty, the rock on which all endeavors are wrecked. I surmise that you have not employed yourself long with this subject. I have pondered it for more than thirty years, and I do not believe that any one could have concerned himself more exhaustively with this... than I, although I have not published anything on this subject. The assumption that the sum of the three angles is smaller than 180° leads to a new geometry entirely different from our Euclidean,—a geometry which is throughout consistent with itself, and which I have elaborated in a manner entirely satisfactory to myself, so that I can solve every problem in it with the exception of the determining of a constant, which is not a priori obtainable. The larger this constant is taken, the nearer we approach the Euclidean geometry, and an infinitely large value will make the two coincident. The propositions of this geometry appear partly paradoxical and absurd to the uninitiated, but on closer and calmer consideration it will be found that they contain in them absolutely nothing that is impossible. Thus the three angles of a triangle... can be made as small as we will, provided the sides can be taken large enough; whilst the area of a triangle, however great the sides may be taken, can never exceed a definite limit, nay, can never once reach it. All my endeavors to discover contradictions or inconsistencies in this non-Euclidean geometry have been in vain, and the only thing in it that conflicts with our reason is the fact that if it were true there would necessarily exist in space a linear magnitude quite determinate in itself, yet unknown to us. But I opine that, despite the empty word-wisdom of the metaphysicians, in reality we know little or nothing of the true nature of space, so much so that we are not at liberty to characterize as absolutely impossible things that strike us as unnatural. If the non-Euclidean geometry were the true geometry, and the constant in a certain ratio to such magnitudes as lie within the reach of our measurements on the earth and in the heavens, it could be determined a posteriori. I have, therefore, in jest frequently expressed the desire that the Euclidean geometry should not be the true geometry, because in that event we should have an absolute measure a priori.
- Carl Friedrich Gauss, Letter to Franz Adolph Taurinus (Nov 8, 1824) as quoted by Paul Carus, The Foundations of Mathematics: A Contribution to the Philosophy of Geometry (1908) p.12
- 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 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.
- Leibniz believed not only that it was a metaphysical fact that all truths are reducible to primary logical truths, but also that, given an appropriate formal language, all truths should be capable of a priori proof. The means of carrying out such proofs was the subject of one of Leibniz's earliest works, his dissertation De Arte Combinatoria (On the Art of Combinations) written in 1666... In it Leibniz reveals his vision of a Characteristica Universalis, or universal characteristic, that would operate as a formal logic through which all true propositions would be demonstrable, merely through adherence to syntactical rules...
- Nicholas Griffin, The Cambridge Companion to Bertrand Russell (2003)
- Euclid could inscribe regular polygons of 3, 4, 5, 15 sides or numbers obtained by doubling these. Those of 7, 9, 11, 13, 14 sides no man ever could or ever will geometrically inscribe. When on the evening of March 30th, 1796, Gauss showed to his student friend, the Hungarian, Wolfgang Bolyai, the formula which gave the geometric inscription of the regular polygon of 17 sides, it was with the remark that this alone could be his epitaph, if it were not a pity to omit so much that went with it. Was it this break beyond Euclid's enchanted bounds that started these two young men in that re-sifting of the very foundations of geometry which led to those new conceptions of the whole subject just now, after another hundred years, beginning to be taught in America's foremost universities?
- George Bruce Halstead, Introduction to John Bolyai, The Science Absolute of Space: Independent of the Truth or Falsity of Euclid's Axiom XI (which Never can be Established a Priori); Followed by the Geometric Quadrature of the Circle in the Case of the Falsity of Axiom XI (June, 1891) Reprinted from Scientiæ Baccalaureus, Vol.1, No.4
- For the mathematician the important consideration is that the foundations of mathematics and a great portion of its content are Greek. The Greeks laid down the first principles, invented the methods ab initio, and fixed the terminology. Mathematics in short is a Greek science, whatever new developments modern analysis has brought or may bring.
- 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.
- 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.'
- Gottfried Wilhelm Leibniz, De Scientia Universali seu Calculo Philosophico (c. 1680) as quoted by Robert Latta, Introduction, Leibniz, The Monadology and Other Philosophical Writings (1898) Tr. Robert Latta
- 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
- Giuseppe Peano, On what became knows as the Peano axioms, in "I fondamenti dell’aritmetica nel Formulario del 1898", in Opere Scelte Vol. III (1959), edited by Ugo Cassina, as quoted in "The Mathematical Philosophy of Giuseppe Peano" by Hubert C. Kennedy, in Philosophy of Science Vol. 30, No. 3 (July 1963)
- The very possibility of the science of mathematics seems an insoluble contradiction. If this science is deductive only in appearance, whence does it derive that perfect rigor no one dreams of doubting? If, on the contrary, all the propositions it enunciates can be deduced one from another by the rules of formal logic, why is not mathematics reduced to an immense tautology? The syllogism can teach us nothing essentially new, and, if everything is to spring from the principle of identity, everything should be capable of being reduced to it. Shall we then admit that the enunciations of all those theorems which fill so many volumes are nothing but devious ways of saying A is A! ...Does the mathematical method proceed from particular to the general, and, if so, how can it be called deductive? ...If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from a syllogism.
- 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.
- Bernhard Riemann, Habilitation Lecture, "On the Hypotheses which Constitute the Bases of Geometry" Tr. William Kingdon Clifford, Nature (1873) Vol. VIII. Nos. 183, 184, pp.14-17, 36, 37
- Hardy... in vain, tried to convince him to learn classical foundations of mathematics and, in particular, the rigorous expositive method of mathematical demonstrations. Every time Hardy introduced a problem, Ramanujan considered it ex novo [new] applying unconventional reasoning which was sometimes incomprehensible to his fellow colleagues.
- Claudio Ronchi, The Tree of Knowledge: The Bright and the Dark Sides of Science (2013)
- The present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles... and will be established by strict symbolic reasoning... The demonstration of this thesis has, if I am not mistaken, all the certainty and precision of which mathematical demonstrations are capable. As the thesis is very recent among mathematicians, and is almost universally denied by philosophers, I have undertaken... to defend... against such adverse theories as appeared most widely held or most difficult to disprove. I have also endeavoured to present, in language as untechnical as possible, the more important stages in the deductions by which the thesis is established.
The other object of this work... is the explanation of the fundamental concepts which mathematics accepts as indefinable. This is a purely philosophical task, and I cannot flatter myself that I have done more than indicate a vast field of inquiry, and give a sample of the methods by which the inquiry may be conducted. The discussion of indefinables—which forms the chief part of philosophical logic—is the endeavour to see clearly... the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of a pineapple. Where, as in the present case, the indefinables are obtained primarily as the necessary residue in a process of analysis, it is often easier to know that there must be such entities than actually to perceive them...
- 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.
- 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.
- John Searle, “The Word Turned Upside Down,” The New York Review of Books, 27 October 1983
- Much has been written on the history of calculus... However, historians tend to harp on the question of logical justification and to spend a disproportionate amount of time on the way it was handled in the nineteenth century. This not only obscures the boldness and vigor of early calculus, but it is overly dogmatic about the way in which calculus should be justified. ...the sheer diversity of foundations for calculus suggests that we have not yet got to the bottom of it.
- John Stillwell, Mathematics and its History (1989)
- 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.
- George Frederick James Temple, 100 Years of Mathematics: a Personal Viewpoint (1981)
- The general truths concerning relations of space which depend upon the axioms and definitions contained in Euclid's Elements, and which involve only properties of straight lines and circles, are termed Elementary Geometry: all beyond this belongs to the Higher Geometry. To this latter province appertain... all propositions respecting the lengths of any portions of curve lines; for these cannot be obtained by means of the principles of the Elements alone. Here then we must ask to what other principles the geometer has recourse, and from what source these are drawn. Is there any origin of geometrical truth which we have not yet explored?
The Idea of a Limit supplies a new mode of establishing mathematical truths. ...a curve is not made up of straight lines, and therefore we cannot by means of any of the doctrines of elementary geometry measure the length of any curve. But we may make up a figure nearly resembling any curve by putting together many short straight lines, just as a polygonal building of very many sides may nearly resemble a circular room. And in order to approach nearer and nearer to the curve we may make the sides more and more small more and more numerous. ...by multiplying the sides we may approach more and more closely to the curve till no appreciable difference remains. The curve line is the Limit of the polygon; and in this process we proceed on the Axiom, that "What is true up to the limit is true at the limit." ... thus the relations of the elementary figures enable us to advance to the properties of the most complex cases.
A Limit is a peculiar and fundamental conception, the use of which in proving the propositions of the Higher Geometry cannot be superseded by any combination of other hypotheses and definitions. ...The ancients did not expressly introduce this conception of a Limit into their mathematical reasonings, although in the application of what is termed the Method of Exhaustions they were in fact proceeding upon an obscure apprehension of principles equivalent to those of the Method of Limits.
- From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2.
- 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.
- John Wesley Young, "Consistency, Independence, and Categoricalness of a Set of Assumptions," Lectures on Fundamental Concepts of Algebra and Geometry (1911) Lecture V, pp.52-53.
The Foundations of Mathematics (1908)Edit
- : A Contribution to the Philosophy of Geometry by Paul Carus
- 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.
- It would be wrong... to assume that the mathematicians of former ages were not conscious of the difficulty. They always felt that there was a flaw in the Euclidean foundation of geometry, but they were satisfied to supply any need of basic principles in the shape of axioms, and it has become quite customary (I might almost say orthodox) to say that mathematics is based upon axioms. In fact, people enjoyed the idea that mathematics, the most lucid of all the sciences, was at bottom as mysterious as the most mystical dogmas of religious faith.
- 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.
- A store of information may be derived from Bertrand A. W. Russell's essay on the Foundations of Geometry. He divides the history of metageometry into three periods: The synthetic, consisting of suggestions made by Legendre and Gauss; the metrical, inaugurated by Riemann and characterized by Lobatchevsky and Bolyai; and the projective, represented by Cayley and Klein, who reduce metrical properties to projection and thus show that Euclidean and non-Euclidean systems may result from "the absolute."
- Prof. B. J. Delbœuf and Prof. H. Poincaré have expressed their conceptions as to the nature of the bases of mathematics, in articles contributed to The Monist. The latter treats the subject from a purely mathematical standpoint, while Dr. Ernst Mach in his little book Space and Geometry, in the chapter "On Physiological, as Distinguished from Geometrical, Space" attacks the problem in a very original manner and takes into consideration mainly the natural growth of space conception. His exposition might be called "the physics of geometry."
- Note: citing "Are the Dimensions of the Physical World Absolute?" by Prof B. J. Delboeuf, The Monist January, 1894; "On the Foundations of Geometry," by Prof. H. Poincaré, The Monist October, 1898; also "Relations Between Experimental and Mathematical Physics," The Monist, July, 1902.
- Hermann Grassmann's Lineare Ausdehnungslehre is the best work on the philosophical foundation of mathematics from the standpoint of a mathematician. ...Victor Schlegel called attention to the similarity of Hamilton's theory of vectors to Grassmann's concept of Strecke, both being limited straight lines of definite direction. Suddenly a demand for Grassmann's book was created in the market; but alas! no copy could be had, and the publishers deemed it advisable to reprint the destroyed edition of 1844.
- 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 data of mathematics are not without their premises; they are not, as the Germans say, voraussetzungslos; and though mathematics is built up from nothing; the mathematician does not start with nothing. He uses mental implements, and it is they that give character to his science.
- 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.
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