Mathematical truth

“ Our author begins by clearly recognizing that mathematics is not so much a department of human thought as the ground-plan of all sane thought; he treats mathematical Science, not as a special set of truths, but as a map of the country in which Truth is to be found. He of course entirely repudiates the profane notion that mathematical Logic can afford any proof or disproof of religious truths ”

“ Why are mathematical certainty, and the evidence of demonstration, common phrases to express the very highest degree of assurance attainable by reason? Why are mathematics by almost all philosophers, and (by many) even those branches of natural philosophy which, through the medium of mathematics, have been converted into deductive sciences, considered to be independent of the evidence of experience and observation, and characterized as systems of Necessary Truth? ”

“ Now, sir, to cut the matter short, and not dally with your learned ideas, for fashion's sake, you have here given your opinion, without reserve, that revelation, the knowledge of God, prophetic vision, the truth of eternity, cannot be solved as a mathematical problem. The first question then is, what is a mathematical problem? and the natural answer is, a statement, proposition or question that can be solved, ascertained, unfolded or demonstrated, by knowledge, facts or figures, for 'mathematical' is an adjective derived from Mathesis (gr.) ”

“ The terms synthesis and analysis are used in mathematics in a more special sense than in logic. In ancient mathematics they had a different meaning from what they now have. The oldest definition of mathematical analysis as opposed to synthesis is that given in Euclid, XIII. 5, which in all probability was framed by Eudoxus: "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it." ”

“ In fact, one of the most attractive phases in the development of mathematics is the discovery of the relations existing between what was supposed to be unrelated. In other words, the unifying of mathematical truths is one of the chief concerns of many of the workers in this domain. ”

“ I dispute the availability, and thus the value, of that reason which is cultivated in any special form other than the abstractly logical. I dispute, in particular, the reason educed by mathematical study. The mathematics are the science of form and quantity; mathematical reasoning is merely logic applied to observation upon form and quantity. The great error lies in supposing that even the truths of what is called pure algebra are abstract or general truths. ”

“ The discovery that all mathematics follows inevitably from a small collection of fundamental laws is one which immeasurably enhances the intellectual beauty of the whole; to those who have been oppressed by the fragmentary and incomplete nature of most existing chains of deduction this discovery comes with all the overwhelming force of a revelation; like a palace emerging from the autumn mist as the traveller ascends an Italian hill-side, the stately storeys of the mathematical edifice appear in their due order and proportion, with a new perfection in every part. ”

“ The close study of pure mathematics, by directing the mind to processes of calculation rather than to phenomena, induces that sublime indifference to facts which has characterized the purely mathematical intellect of all ages. ”

“ ALL objects in the world which can be made the subjects of our contemplation are subordinate to the conditions of Space, Time, and Number; and on this account, the doctrines of pure mathematics have most numerous and extensive applications in every department of our investigations of nature. And there is a peculiarity in these Ideas, which has caused the mathematical sciences to be, in all cases, the first successful efforts of the awakening speculative powers of nations at the commencement of their intellectual progress. ”

“ No doubt this logical purpose is its application in the world of phenomena, and the mathematical concept must thus fit the world so absolutely that it can be conceived as a description of the world after abstracting not only from the will relations, as physics does, but also from the content. Mathematics would then be the phenomenalistic science of the form and order of the world. ”

“ On the contrary, he says, I think that wherever, from the side of the theory of knowledge, or in geometry, or from the theories of natural or physical science, mathematical ideas come up, the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms, that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts. ”

“ This necessary perfection of geometry, obtained essentially by the application of mathematical analysis, which it so eminently admits, is apt to produce erroneous views of the real nature of this fundamental science, which most minds at present conceive to be a purely logical science quite independent of observation. ”

“ The nature of infinity and continuity, for example, belonged in former days to philosophy, but belongs now to mathematics. Mathematical philosophy, in the strict sense, cannot, perhaps, be held to include such definite scientific results as have been obtained in this region; the philosophy of mathematics will naturally be expected to deal with questions on the frontier of knowledge, as to which comparative certainty is not yet attained. But speculation on such questions is hardly likely to be fruitful unless the more scientific parts of the principles of mathematics are known. ”

“ Thirdly, the tremendous future effect of mathematical knowledge on the lives of men, on their daily avocations, on their habitual thoughts, on the organization of society, must have been even more completely shrouded from the foresight of those early thinkers. Even now there is a very wavering grasp of the true position of mathematics as an element in the history of thought. ”

“ Every branch of physics gives rise to an application of mathematics. A prophecy may be hazarded that in the future these applications will unify themselves into a mathematical theory of a hypothetical substructure of the universe, uniform under all the diverse phenomena. ”

“ These fundamental equations being once discovered, analysis will enable us to deduce from them a multitude of consequences which it would have been previously impossible even to suspect; it will perfect the science to an immense degree, both with respect to the generality of its conceptions and to the complete co-ordination established between them. But mere mathematical analysis could never be sufficient to form the bases of any natural science, not even to demonstrate them anew when they have once been established. ”

“ We have insisted particularly on this generalising tendency of mathematics, as it is the one which will play a prominent part when we consider the methodology of mathematical physics as applied to the world of reality. ”

“ This quantitative reduction is, however, but a first step; and even after it has been accomplished, nature might still defy mathematical investigation. The fact is that mathematics consists in compounding the similar with the similar; wide heterogeneities even among quantities would render an appeal to human mathematics useless. ”

“ Mathematics accounts for them logically and scientifically.33 The point I’m trying to make is that things which seem inconsistent and even absurd to the lay mind, are commonplaces to the mathematical intelligence. ”

“ Important as it is to comprehend the rigorous universality, in a logical point of view, of mathematical science, it is no less indispensable to consider now the great real limitations, which, through the feebleness of our intellect, narrow in a remarkable degree its actual domain, in proportion as phenomena, in becoming special, become complicated. ”

“ The great inventions of the seventeenth century—Analytical Geometry and the Infinitesimal Calculus—were so fruitful in new results that mathematicians had neither time nor inclination to examine their foundations. Philosophers, who should have taken up the task, had too little mathematical ability to invent the new branches of mathematics which have now been found necessary for any adequate discussion. ”

“ This is not known to the followers of those sciences who are not themselves mathematicians and who very often exalt their horns against the mathematics in consequence. They might as well be squaring the circle, for any sense they show in this particular.
A great many individuals, ever since the rise of the mathematical method, have, each for himself, attacked its direct and indirect consequences. I shall not here stop to point out how the very accuracy of exact science gives better aim than the preceding state of things could give. ”

“ If we had been embarking upon mathematical philosophy in earlier days, we should have had to start with something less abstract than the series of natural numbers, which we should reach as a stage on our backward journey. When the logical foundations of mathematics have grown more familiar, we shall be able to start further back, at what is now a late stage in our analysis. But for the moment the natural numbers seem to represent what is easiest and most familiar in mathematics. ”

“ There is a second reason why we cannot bring complicated phenomena under the dominion of mathematical analysis. Even if we could ascertain the mathematical law which governs each agent, taken by itself, the combination of so great a number of conditions would render the corresponding mathematical problem so far above our [Pg 39] feeble means, that the question would remain in most cases incapable of solution. ”

“ For, strangely enough, the logical outcome of this whole recent review of the bases of mathematical science can be expressed by saying that the modern mathematician rightly doubts every attempt to prove any proposition in his science unless, in trying to prove, you can first empirically show him, in a fashion that he can accept, the actual process of construction belonging to, or creative of, the ideal object of which your proposition undertakes to give an account. ”

“ The statement that mathematics is the science of number would be untrue in two different ways. On the one hand, there are recognised branches of mathematics which have nothing to do with number—all geometry that does not use co-ordinates or measurement, for example: projective and descriptive geometry, down to the point at which co-ordinates are introduced, does not have to do with number, or even with quantity in the sense of greater and less. ”

“ Yet in the mathematical realm it is not altogether so. Within limits, you create as you will, but the limits once found, are absolute. Unsubstantial, in one way, as fairyland, the creations of the pure mathematician’s ideality still may require of their maker as rigid, and often as baffling a search for a given kind or case of mathematical existence, as if he were an astronomer testing the existence of the fifth satellite of Jupiter, or of the variables of a telescopic star-cluster. ”

“ All traditional pure mathematics, including analytical geometry, may be regarded as consisting wholly of propositions about the natural numbers. That is to say, the terms which occur can be defined by means of the natural numbers, and the propositions can be deduced from the properties of the natural numbers—with the addition, in each case, of the ideas and propositions of pure logic. ”

“ But we will put aside all rivals, and consider the ground on which such a claim can be made for mathematics. The originality of mathematics consists in the fact that in mathematical science connections between things are exhibited which, apart from the agency of human reason, are extremely unobvious. Thus the ideas, now in the minds of contemporary mathematicians, lie very remote from any notions which can be immediately derived by perception through the senses; unless indeed it be perception stimulated and guided by antecedent mathematical knowledge. ”

“ That mathematical reasoning implies clear thinking and complies with the rules of logic has never been denied; nevertheless the assertion that all mathematical reasonings are of a purely deductive nature, and are reducible to the rules of logic, is an opinion which is by no means unanimous. ”

“ Considering the infinite diversity which must necessarily exist in this respect in the external world, we easily understand how far below the true difficulty our conceptions must frequently be found, especially if we add that as these elements of our analysis have been in the first place furnished to us by the mathematical consideration of the simplest phenomena, we have, à priori, no rational guarantee of their necessary suitableness to represent the mathematical law of every other class of phenomena. ”

“ In the course of a comprehensive historical survey the lecturer characterized, in a way I found utterly convincing, the present mathematical interpretation of nature as a transitional stage of human consciousness - a kind of knowing which is on the way from a past pre-mathematical to a future post-mathematical form of cognition. The importance of mathematics, whether as a discipline of the human spirit or as an instrument of natural science, was not for a moment undervalued. ”