Now, in regard to Peirce's definition: Mathematics is the science which draws necessary conclusions, I think two questions spring at once to the mind, and, unless these questions can be at least in a measure answered, the definition seems too vague. These questions are: first, what is meant by necessary conclusions? and, second, from what are these conclusions drawn? I take the word necessary to refer to the only legitimate (true) conclusions that can come from the premises be they true or false, that is, the conclusions must come whether the premises are true or false. This makes necessary the second questionFrom what are the conclusions drawn? In the following simple example they are undoubtedly drawn from premises that lead to a result not in accordance with our experience or observation.

Two boys, John and Will, at the same instant, start running on a straight road towards C, Will's starting point (B) being 91⁄2 miles nearer C than John's starting point (A). Will's rate is of a mile a minute. John overtakes Will at a point 1⁄2 mile from B. What is John's rate?

Hence John's rate is two miles per minute. Now here we have started with certain premises, and have arrrived at necessary conclusions. But, bringing to our aid experience and observation, we feel justified in saying that we have reached a result that is anatomically and physiologically absurd, hence we conclude that our premises are faulty. But our mathematics is all right. A conclusion arrived at mathematically will have just as much empirical truth as the premises, no more, no less. So, in a sense, the mathematician is independent of the truth or falsity of a premise. But, on the other hand, it behooves the mathematician to be very careful about his assumptions, when dealing with a problem that bears intimate relations to other

problems using common mathematical conceptions. Peirce's "necessary conclusions" must not be drawn from anything or everything, otherwise our mathematics would consist only of a series of detached propositions in logic.

Where is the starting point in a mathematical investigation? In Euclidean Geometry we might say that the starting point is to be found in certain axioms and postulates that are assumed as true. These axioms and postulates come so directly from experience and observation that Geometry has been classed as a natural science rather than a branch of mathematics. In mathematics it is a general principle that nothing must be assumed that can be proved. I cannot, and I would not if I could, enter into philosophical speculations as to the nature of premises; but it seems to me that if we are going to adopt Peirce's definition, the two questions I have asked are pertinent. It is evident that Peirce's definition emphasizes the deductive character of mathematics, but deduction does not constitute even a major part of mathematical truth. Though the hope of coming to any convincing conclusion is slight, I regret that I cannot follow up this subject further; in fact, much of what I have to say must of necessity be suggestive rather than conclusive.

Just one word more as to the nature of mathematics. It is a mistake often met with a few decades ago that mathematics is mainly reasoning, for intuition and imagination come in for a large share of the work.

"Intuitive or self-evident truths are those which are conceived in the mind immediately; that is, which are perfectly conceived by a single process of induction the moment the facts on which they depend are apprehended without the intervention of other ideas." Simple examples are the axioms of geometry, e. g. "a whole is equal to the sum of its parts." I presume that every one feels that he knows this, and would say that it is a self-evident truth. But the truth of some so-called axioms has been questioned, showing that what is apparently self-evident to one mind may not seem so to another. We need no stronger evidence than this to show us that intuition must be used with cau

'Davies: "Logic and Utility of Mathematics."

tion. It is not peculiar to Geometry, but is used tacitly, if not avowedly, in all branches of mathematics. To the student of the differential calculus, it seems intuitive that every continuous function must have a derivative, that is, that every continuous curve has tangents; but not very many years ago Weierstrass produced a continuous function without derivatives. This discovery came as a shock to the mathematical world. I give it as an example of the fatal weakness of intuition; and, yet, where would we be without intuition?

It is well known, of course, that induction is frequently employed in mathematics. Induction is at once a strong and weak weapon; very powerful in the hands of the skilful man, weak and uncertain when employed by the rash or ignorant. There are numerous instances in mathematics where important principles have been discovered by induction and later, perhaps years afterwards, proved by vigorous deduction.

As great a man as Huxley made the allegation that "Mathematics is that study which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation." The history of modern mathematical thought refutes this charge. Sylvester, in an address before the British Association in 1869, gave a powerful answer to this sweeping assertion of Mr. Huxley. In part Sylvester said: "Most, if not all, of the great ideas of modern mathematics have had their authority in observation. Lagrange, than whom no greater authority could be quoted, has expressed emphatically his belief in the importance to the mathematician of the faculty of observation; Gauss called mathematics a science of the eye. . . ; the ever to be lamented Riemann has written a thesis to show that the basis of our conception of space is purely empirical, and our knowledge of its laws the result of observation, that other kinds of space might be conceived to exist subject to laws different from those which govern the actual space in which we are immersed." So spoke Sylvester — a quarter of a century later he could have spoken with even more fervor.

The part played by imagination has perhaps never been sufficiently emphasized, because it is hard to tell exactly where reasoning stops and imagination comes in. It is perhaps this subtle

5

but strong element that has made some say that mathematics is akin to literature. It was Sylvester, I believe, that said "Mathematics is poetry." Picard said "The idea of number belongs not only to logic, but to history and psychology." Certain it is that in some phases of mathematics we must look outside its pure realms both for a starting point and for material with which to carry on the investigation.

In the opinion of the writer, hair-splitting theories of philosophy have but little place in mathematics, and mathematics would lose its conservative character if they had.

Professor Schubert' in his "Essay on the Nature of Mathematical Knowledge" aptly says: "The intrinsic character of mathematical research and knowledge is based essentially on three properties: first, on its conservative attitude towards the old truths and discoveries of mathematics; secondly, on its progressive mode of development due to the incessant acquisition of new knowledge on the basis of the old; and, thirdly, on its self-sufficiency and its consequent absolute independence."

I do not claim, however, that mathematics is the exact science, though the most exact of sciences, that the average person imagines it to be. The foundations of mathematics have not always stood the test of stability, though the superstructure has never been in danger of tottering over. Later on we shall see how, during the nineteenth century, these foundations were examined into and materially strengthened. Now all of this is preliminary to the main subject of my paper; and it may be asked why has the writer gone into these questions at all, suggesting difficulties that may not have been apparent, asking questions, and yet not answering them. As one of the main features of modern mathematics is the going back to first principles, my course is perhaps vindicated. It may be that what has been said so briefly and so imperfectly touching fundamental notions will in a measure prepare our minds for some of the wonderful onslaughts of nineteenth century mathematics.

The discovery of the Calculus by Newton and Leibnitz near the

5 See translation of his St. Louis Address in Vol. XI, No. 8, of Bulletin of American Mathematical Society.

Mathematical Essays and Recreations, translated by T. J. McCormack.

end of the seventeenth century, opened up a new and vast field for mathematical research. The most powerful tool yet devised had been put into the hands of mathematical students. After its queer notation had become familiar, and the principles and details of operations were fairly well understood, calculus became popular and mathematics made great strides.

France and Switzerland took the lead in the eighteenth century in the development of mathematics. A mere mention of the great names of that period is inspiring:- the Bernouillis, Euler, Lagrange, Laplace, Legendre, Fourier, Monge. Towards the close of the eighteenth and the beginning of the nineteenth centuries, other countries came to the front, while the French and Swiss continued their magnificent work.'

Germany had her Gauss, Jacobi, Dirichlet, and more recently such men as Steiner, von Staudt, Plücher, Clebsch, Felix Klein, Weierstrass, Riemann, Fuchs, etc.; England produced DeMorgan, Boole, Hamilton, and more recently Cayley and Sylvester. Russia entered the list with Lobatchewsky; Norway with a mathematical giant, Abel; Italy, with Cremona; Hungary, with her two Bolyais, the United States with Benjamin Peirce, while France, still well to the front, produced Cauchy, Galois, Poncelet, Chasles, and others; and among those now living, Picard, Darboux and Poincaré. These nineteenth century men, as Newton said of himself, have stood on the shoulders of giants, but they themselves are not pigmies.

It has been well said that the chief characteristic of modern mathematics is its generalizing tendency. The great propositions have been general ones, some special cases of which have often been found interesting either in themselves or in an application to some practical problem.

Mathematics may be considered under three general heads:Arithmetic (number), Geometry (form), Analysis (function). It would be more scientific to treat my subject under these three heads; but I have found it impossible to adhere strictly to this formal division, and, at best, I can merely mention a few things done

In this general resumé the writer does not wish to draw invidious comparisons, and he is entirely conscious that some names may be omitted that stand higher in the mathematical world than some that are mentioned.