CHAPTER XIV

MATHEMATICS AND PHILOSOPHY

=1401.= Socrates is praised by all the centuries for having called philosophy from heaven to men on earth; but if, knowing the condition of our science, he should come again and should look once more to heaven for a means of curing men, he would there find that to mathematics, rather than to the philosophy of today, had been given the crown because of its industry and its most happy and brilliant successes.--HERBART, J. F.

_Werke [Kehrbach], (Langensalza, 1890), Bd. 5, p. 95._

=1402.= It is the embarrassment of metaphysics that it is able to accomplish so little with the many things that mathematics offers her.--KANT, E.

_Metaphysische Anfangsgründe der Naturwissenschaft, Vorrede._

=1403.= Philosophers, when they have possessed a thorough knowledge of mathematics, have been among those who have enriched the science with some of its best ideas. On the other hand it must be said that, with hardly an exception, all the remarks on mathematics made by those philosophers who have possessed but a slight or hasty or late-acquired knowledge of it are entirely worthless, being either trivial or wrong.--WHITEHEAD, A. N.

_Introduction to Mathematics (New York, 1911), p. 113._

=1404.= The union of philosophical and mathematical productivity, which besides in Plato we find only in Pythagoras, Descartes and Leibnitz, has always yielded the choicest fruits to mathematics: To the first we owe scientific mathematics in general, Plato discovered the analytic method, by means of which mathematics was elevated above the view-point of the elements, Descartes created the analytical geometry, our own illustrious countryman discovered the infinitesimal calculus--and just these are the four greatest steps in the development of mathematics.

--HANKEL, HERMANN.

_Geschichte der Mathematik im Altertum und im Mittelalter (Leipzig, 1874), pp. 149-150._

=1405.= Without mathematics one cannot fathom the depths of philosophy; without philosophy one cannot fathom the depths of mathematics; without the two one cannot fathom anything.

--BORDAS-DEMOULINS.

_Quoted in A. Rebière: Mathématiques et Mathématiciens (Paris, 1898), p. 147._

=1406.= In the end mathematics is but simple philosophy, and philosophy, higher mathematics in general.--NOVALIS.

_Schriften (Berlin, 1901), Teil 2, p. 443._

=1407.= It is a safe rule to apply that, when a mathematical or philosophical author writes with a misty profundity, he is talking nonsense.--WHITEHEAD, A. N.

_Introduction to Mathematics (New York, 1911), p. 227._

=1408.= The real finisher of our education is philosophy, but it is the office of mathematics to ward off the dangers of philosophy.--HERBART, J. F.

_Pestalozzi’s Idee eines ABC der Anschauung; Werke [Kehrbach], (Langensalza, 1890), Bd. 1, p. 168._

=1409.= Since antiquity mathematics has been regarded as the most indispensable school for philosophic thought and in its highest spheres the research of the mathematician is indeed most closely related to pure speculation. Mathematics is the most perfect union between exact knowledge and theoretical thought.--CURTIUS, E.

_Berliner Monatsberichte (1873), p. 517._

=1410.= Geometry has been, throughout, of supreme importance in the history of knowledge.--RUSSELL, BERTRAND.

_Foundations of Geometry (Cambridge, 1897), p. 54._

=1411.= He is unworthy of the name of man who is ignorant of the fact that the diagonal of a square is incommensurable with its side.--PLATO.

_Quoted by Sophie Germain: Mémoire sur les surfaces élastiques._

=1412.= Mathematics, considered as a science, owes its origin to the idealistic needs of the Greek philosophers, and not as fable has it, to the practical demands of Egyptian economics.... Adam was no zoölogist when he gave names to the beasts of the field, nor were the Egyptian surveyors mathematicians.--HANKEL, H.

_Die Entwickelung der Mathematik in den letzten Jahrhunderten (Tübingen, 1884), p. 7._

=1413.= There are only two ways open to man for attaining a certain knowledge of truth: clear intuition and necessary deduction.--DESCARTES.

_Rules for the Direction of the Mind; Torrey’s The Philosophy of Descartes (New York, 1892), p. 104._

=1414.= Mathematicians have, in many cases, proved some things to be possible and others to be impossible, which, without demonstration, would not have been believed.... Mathematics afford many instances of impossibilities in the nature of things, which no man would have believed, if they had not been strictly demonstrated. Perhaps, if we were able to reason demonstratively in other subjects, to as great extent as in mathematics, we might find many things to be impossible, which we conclude, without hesitation, to be possible.--REID, THOMAS.

_Essay on the Intellectual Powers of Man, Essay 4, chap. 3._

=1415.= If philosophers understood mathematics, they would know that indefinite speech, which permits each one to think what he pleases and produces a constantly increasing difference of opinion, is utterly unable, in spite of all fine words and even in spite of the magnitude of the objects which are under contemplation, to maintain a balance against a science which instructs and advances through every word which it utters and which at the same time wins for itself endless astonishment, not through its survey of immense spaces, but through the exhibition of the most prodigious human ingenuity which surpasses all power of description.--HERBART, J. F.

_Werke Kehrbach (Langensalza, 1890), Bd. 5, p. 105._

=1416.= German intellect is an excellent thing, but when a German product is presented it must be analysed. Most probably it is a combination of intellect (I) and tobacco-smoke (T). Certainly I₃T₁, and I₂T₁, occur; but I₁T₃ is more common, and I₂T₁₅ and I₁T₂₀ occur. In many cases metaphysics (M) occurs and I hold that I_{a}T_{b}M_{c} never occurs without b + c > 2a.

N. B.--Be careful, in analysing the compounds of the three, not to confound T and M, which are strongly suspected to be isomorphic. Thus, I₁T₃M₃ may easily be confounded with I₁T₆. As far as I dare say anything, those who have placed _Hegel, Fichte_, etc., in the rank of the extenders of _Kant_ have imagined T and M to be identical.--DE MORGAN, A.

_Graves’ Life of W. R. Hamilton (New York, 1882-1889), Vol. 13, p. 446._

=1417.= The discovery [of Ceres] was made by G. Piazzi of Palermo; and it was the more interesting as its announcement occurred simultaneously with a publication by Hegel in which he severely criticized astronomers for not paying more attention to philosophy, a science, said he, which would at once have shown them that there could not possibly be more than seven planets, and a study of which would therefore have prevented an absurd waste of time in looking for what in the nature of things could never be found.--BALL, W. W. R.

_History of Mathematics (London, 1901), p. 458._

=1418.=

But who shall parcel out His intellect by geometric rules, Split like a province into round and square? --WORDSWORTH.

_The Prelude, Bk. 2._

=1419.=

And Proposition, gentle maid, Who soothly ask’d stern Demonstration’s aid, .... --COLERIDGE, S. T.

_A Mathematical Problem._

=1420.= Mathematics connect themselves on the one side with common life and physical science; on the other side with philosophy in regard to our notions of space and time, and in the questions which have arisen as to the universality and necessity of the truths of mathematics and the foundation of our knowledge of them.--CAYLEY, ARTHUR.

_British Association Address (1888); Collected Mathematical Papers, Vol. 11, p. 430._

=1421.= Mathematical teaching ... trains the mind to capacities, which ... are of the closest kin to those of the greatest metaphysician and philosopher. There is some color of truth for the opposite doctrine in the case of elementary algebra. The resolution of a common equation can be reduced to almost as mechanical a process as the working of a sum in arithmetic. The reduction of the question to an equation, however, is no mechanical operation, but one which, according to the degree of its difficulty, requires nearly every possible grade of ingenuity: not to speak of the new, and in the present state of the science insoluble, equations, which start up at every fresh step attempted in the application of mathematics to other branches of knowledge.--MILL, J. S.

_An Examination of Sir William Hamilton’s Philosophy (London, 1878), p. 615._

=1422.= The value of mathematical instruction as a preparation for those more difficult investigations, consists in the applicability not of its doctrines, but of its methods. Mathematics will ever remain the most perfect type of the Deductive Method in general; and the applications of mathematics to the simpler branches of physics, furnish the only school in which philosophers can effectually learn the most difficult and important portion of their art, the employment of the laws of the simpler phenomena for explaining and predicting those of the more complex. These grounds are quite sufficient for deeming mathematical training an indispensable basis of real scientific education, and regarding, with Plato, one who is ἀγεωμέτρητος, as wanting in one of the most essential qualifications for the successful cultivation of the higher branches of philosophy.--MILL, J. S.

_System of Logic, Bk. 3, chap. 24, sect. 9._

=1423.= In metaphysical reasoning, the process is always short. The conclusion is but a step or two, seldom more, from the first principles or axioms on which it is grounded, and the different conclusions depend not one upon another.

It is otherwise in mathematical reasoning. Here the field has no limits. One proposition leads on to another, that to a third, and so on without end. If it should be asked, why demonstrative reasoning has so wide a field in mathematics, while, in other abstract subjects, it is confined within very narrow limits, I conceive this is chiefly owing to the nature of quantity, ... mathematical quantities being made up of parts without number, can touch in innumerable points, and be compared in innumerable different ways.--REID, THOMAS.

_Essays on the Powers of the Human Mind (Edinburgh, 1812), Vol. 2, pp. 422-423._

=1424.= The power of Reason ... is unquestionably the most important by far of those which are comprehended under the general title of Intellectual. It is on the right use of this power that our success in the pursuit of both knowledge and of happiness depends; and it is by the exclusive possession of it that man is distinguished, in the most essential respects, from the lower animals. It is, indeed, from their subserviency to its operations, that the other faculties ... derive their chief value.--STEWART, DUGALD.

_Philosophy of the Human Mind; Collected Works (Edinburgh, 1854), Vol. 8, p. 5._

=1425.= When ... I asked myself why was it then that the earliest philosophers would admit to the study of wisdom only those who had studied mathematics, as if this science was the easiest of all and the one most necessary for preparing and disciplining the mind to comprehend the more advanced, I suspected that they had knowledge of a mathematical science different from that of our time....

I believe I find some traces of these true mathematics in Pappus and Diophantus, who, although they were not of extreme antiquity, lived nevertheless in times long preceding ours. But I willingly believe that these writers themselves, by a culpable ruse, suppressed the knowledge of them; like some artisans who conceal their secret, they feared, perhaps, that the ease and simplicity of their method, if become popular, would diminish its importance, and they preferred to make themselves admired by leaving to us, as the product of their art, certain barren truths deduced with subtlety, rather than to teach us that art itself, the knowledge of which would end our admiration.--DESCARTES.

_Rules for the Direction of the Mind; Philosophy of Descartes [Torrey], (New York, 1892), pp. 70-71._

=1426.= If we rightly adhere to our rule [that is, that we should occupy ourselves only with those subjects in reference to which the mind is capable of acquiring certain and indubitable knowledge] there will remain but few things to the study of which we can devote ourselves. There exists in the sciences hardly a single question upon which men of intellectual ability have not held different opinions. But whenever two men pass contrary judgment on the same thing, it is certain that one of the two is wrong. More than that, neither of them has the truth; for if one of them had a clear and precise insight into it, he could so exhibit it to his opponent as to end the discussion by compelling his conviction.... It follows from this, if we reckon rightly, that among existing sciences there remain only geometry and arithmetic, to which the observance of our rule would bring us.

--DESCARTES.

_Rules for the Direction of the Mind; Philosophy of Descartes [Torrey], (New York, 1892), p. 62._

=1427.= The same reason which led Plato to recommend the study of arithmetic led him to recommend also the study of geometry. The vulgar crowd of geometricians, he says, will not understand him. They have practice always in view. They do not know that the real use of the science is to lead men to the knowledge of abstract, essential, eternal truth. (Plato’s Republic, Book 7). Indeed if we are to believe Plutarch, Plato carried his feeling so far that he considered geometry as degraded by being applied to any purpose of vulgar utility. Archytas, it seems, had framed machines of extraordinary power on mathematical principles. (Plutarch, Sympos., VIII., and Life of Marcellus. The machines of Archytas are also mentioned by Aulus Gellius and Diogenes Laertius). Plato remonstrated with his friend, and declared that this was to degrade a noble intellectual exercise into a low craft, fit only for carpenters and wheelwrights. The office of geometry, he said, was to discipline the mind, not to minister to the base wants of the body. His interference was successful; and from that time according to Plutarch, the science of mechanics was considered unworthy of the attention of a philosopher.

--MACAULAY.

_Lord Bacon; Edinburgh Review, July, 1837._

=1428.= The intellectual habits of the Mathematicians are, in some respects, the same with those [of the Metaphysicians] we have been now considering; but, in other respects, they differ widely. Both are favourable to the improvement of the power of _attention_, but not in the same manner, nor in the same degree.

Those of the metaphysician give capacity of fixing the attention on the subjects of our consciousness, without being distracted by things external; but they afford little or no exercise to that species of attention which enables us to follow long processes of reasoning, and to keep in view all the various steps of an investigation till we arrive at the conclusion. In mathematics, such processes are much longer than in any other science; and hence the study of it is peculiarly calculated to strengthen the power of steady and concatenated thinking,--a power which, in all the pursuits of life, whether speculative or active, is one of the most valuable endowments we can possess. This command of attention, however, it may be proper to add, is to be acquired, not by the practice of modern methods, but by the study of Greek geometry, more particularly, by accustoming ourselves to pursue long trains of demonstration, without availing ourselves of the aid of any sensible diagrams; the thoughts being directed solely by those ideal delineations which the powers of conception and of memory enable us to form.--STEWART, DUGALD.

_Philosophy of the Human Mind, Part 3, chap. 1, sect. 3._

=1429.= They [the Greeks] speculated and theorized under a lively persuasion that a Science of every part of nature was possible, and was a fit object for the exercise of a man’s best faculties; and they were speedily led to the conviction that such a science must clothe its conclusions in the language of mathematics. This conviction is eminently conspicuous in the writings of Plato.... Probably no succeeding step in the discovery of the Laws of Nature was of so much importance as the full adoption of this pervading conviction, that there must be Mathematical Laws of Nature, and that it is the business of Philosophy to discover these Laws. This conviction continues, through all the succeeding ages of the history of the science, to be the animating and supporting principle of scientific investigation and discovery.

--WHEWELL, W.

_History of the Inductive Sciences, Vol. 1, bk. 2, chap. 3._

=1430.= For to pass by those Ancients, the wonderful _Pythagoras_, the sagacious _Democritus_, the divine _Plato_, the most subtle and very learned _Aristotle_, Men whom every Age has hitherto acknowledged as deservedly honored, as the greatest Philosophers, the Ring-leaders of Arts; in whose Judgments how much these Studies [mathematics] were esteemed, is abundantly proclaimed in History and confirmed by their famous Monuments, which are everywhere interspersed and bespangled with Mathematical Reasonings and Examples, as with so many Stars; and consequently anyone not in some Degree conversant in these Studies will in vain expect to understand, or unlock their hidden Meanings, without the Help of a Mathematical Key: For who can play well on _Aristotle’s_ Instrument but with a Mathematical Quill; or not be altogether deaf to the Lessons of natural _Philosophy_, while ignorant of _Geometry?_ Who void of (_Geometry_ shall I say, or) _Arithmetic_ can comprehend _Plato’s_ _Socrates_ lisping with Children concerning Square Numbers; or can conceive _Plato_ himself treating not only of the Universe, but the Polity of Commonwealths regulated by the Laws of Geometry, and formed according to a Mathematical Plan?--BARROW, ISAAC.

_Mathematical Lectures (London, 1734), pp. 26-27._

=1431.=

And Reason now through number, time, and space Darts the keen lustre of her serious eye; And learns from facts compar’d the laws to trace Whose long procession leads to Deity --BEATTIE, JAMES.

_The Minstrel, Bk. 2, stanza 47._

=1432.= That Egyptian and Chaldean wisdom mathematical wherewith Moses and Daniel were furnished, ....--HOOKER, RICHARD.

_Ecclesiastical Polity, Bk. 3, sect. 8._

=1433.= General and certain truths are only founded in the habitudes and relations of _abstract ideas_. A sagacious and methodical application of our thoughts, for the finding out of these relations, is the only way to discover all that can be put with truth and certainty concerning them into general propositions. By what steps we are to proceed in these, is to be learned in the schools of mathematicians, who, from very plain and easy beginnings, by gentle degrees, and a continued chain of reasonings, proceed to the discovery and demonstration of truths that appear at first sight beyond human capacity. The art of finding proofs, and the admirable method they have invented for the singling out and laying in order those intermediate ideas that demonstratively show the equality or inequality of unapplicable quantities, is that which has carried them so far and produced such wonderful and unexpected discoveries; but whether something like this, in respect of other ideas, as well as those of magnitude, may not in time be found out, I will not determine. This, I think, I may say, that if other ideas that are the real as well as the nominal essences of their species, were pursued in the way familiar to mathematicians, they would carry our thoughts further, and with greater evidence and clearness than possibly we are apt to imagine.--LOCKE, JOHN.

_An Essay concerning Human Understanding, Bk. 4, chap. 12, sect. 7._

=1434.= Those long chains of reasoning, quite simple and easy, which geometers are wont to employ in the accomplishment of their most difficult demonstrations, led me to think that everything which might fall under the cognizance of the human mind might be connected together in a similar manner, and that, provided only that one should take care not to receive anything as true which was not so, and if one were always careful to preserve the order necessary for deducing one truth from another, there would be none so remote at which he might not at last arrive, nor so concealed which he might not discover.--DESCARTES.

_Discourse upon Method, part 2; The Philosophy of Descartes [Torrey], (New York, 1892), p. 47._

=1435.= If anyone wished to write in mathematical fashion in metaphysics or ethics, nothing would prevent him from so doing with vigor. Some have professed to do this, and we have a promise of mathematical demonstrations outside of mathematics; but it is very rare that they have been successful. This is, I believe, because they are disgusted with the trouble it is necessary to take for a small number of readers where they would ask as in Persius: _Quis leget haec_, and reply: _Vel duo vel nemo._

--LEIBNITZ.

_New Essay concerning Human Understanding, Langley, Bk 2, chap. 29, sect. 12._

=1436.= It is commonly asserted that mathematics and philosophy differ from one another according to their _objects_, the former treating of _quantity_, the latter of _quality_. All this is false. The difference between these sciences cannot depend on their object; for philosophy applies to everything, hence also to _quanta_, and so does mathematics in part, inasmuch as everything has magnitude. It is only the _different kind of rational knowledge or application_ of reason in mathematics and philosophy which constitutes the specific difference between these two sciences. For philosophy is _rational knowledge from mere concepts_, mathematics, on the contrary, is _rational knowledge from the construction of concepts_.

We construct concepts when we represent them in intuition _a priori_, without experience, or when we represent in intuition the object which corresponds to our concept of it.--The mathematician can never apply his reason to mere concepts, nor the philosopher to the construction of concepts.--In mathematics the reason is employed _in concreto_, however, the intuition is not empirical, but the object of contemplation is something _a priori_.

In this, as we see, mathematics has an advantage over philosophy, the knowledge in the former being intuitive, in the latter, on the contrary, only _discursive_. But the reason why in mathematics we deal more with quantity lies in this, that magnitudes can be constructed in intuition _a priori_, while qualities, on the contrary, do not permit of being represented in intuition.--KANT, E.

_Logik; Werke [Hartenstein], (Leipzig, 1868), Bd. 8, pp. 23-24._

=1437.= Kant has divided human ideas into the two categories of quantity and quality, which, if true, would destroy the universality of Mathematics; but Descartes’ fundamental conception of the relation of the concrete to the abstract in Mathematics abolishes this division, and proves that all ideas of quality are reducible to ideas of quantity. He had in view geometrical phenomena only; but his successors have included in this generalization, first, mechanical phenomena, and, more recently, those of heat. There are now no geometers who do not consider it of universal application, and admit that every phenomenon may be as logically capable of being represented by an equation as a curve or a motion, if only we were a