S. Austin Allibone, comp. Prose Quotations from Socrates to Macaulay. 1880.
The mathematics are friends to religion, inasmuch as they charm the passions, restrain the impetuosity of imagination, and purge the mind from error and prejudice. Vice is error, confusion, and false reasoning; and all truth is more or less opposite to it. Besides, mathematical studies may serve for a pleasant entertainment for those hours which young men are apt to throw away upon their vices; the delightfulness of them being such as to make solitude not only easy but desirable.
John Arbuthnot: Usefulness of Mathematical Learning.
He that gives a portion of his time and talent to the investigation of mathematical truth will come to all other questions with a decided advantage over his opponents. He will be in argument what the ancient Romans were in the field: to them the day of battle was a day of comparative recreation, because they were ever accustomed to exercise with arms much heavier than they fought; and their reviews differed from a real battle in two respects: they encountered more fatigue, but the victory was bloodless.
The difference between the philosophy of Bacon and that of his predecessors cannot, we think, be better illustrated than by comparing his views on some important subjects with those of Plato. We select Plato, because we conceive that he did more than any other person towards giving to the minds of speculative men that bent which they retained till they received from Bacon a new impulse in a diametrically opposite direction.
It is curious to observe how differently these great men estimated the value of every kind of knowledge. Take Arithmetic for example. Plato, after speaking slightly of the convenience of being able to reckon and compute in the ordinary transactions of life, passes to what he considers as a far more important advantage. The study of the properties of numbers, he tells us, habituates the mind to the contemplation of pure truth, and raises us above the material universe. He would have his disciples apply themselves to this study, not that they may be able to buy or sell, not that they may qualify themselves to be shopkeepers or travelling merchants, but that they may learn to withdraw their minds from the ever-shifting spectacle of this visible and tangible world, and to fix them on the immutable essences of things. [Platos Republic, Book 7.] Bacon, on the other hand, valued this branch of knowledge only on account of its uses with reference to that visible and tangible world which Plato so much despised. He speaks with scorn of the mystical arithmetic of the later Platonists, and laments the propensity of mankind to employ on matters of mere curiosity powers the whole exertion of which is required for purposes of solid advantage. He advises arithmeticians to leave these trifles, and to employ themselves in framing convenient expressions which may be of use in physical researches. [De Augmentis, Lib. 3, Cap. 6.] The same reasons which led Plato to recommend the study of Arithmetic led him to recommend also the study of mathematics. 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. [Platos Republic, Book 7.] Indeed, if we are to believe Plutarch, Plato carried this 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 as unworthy of the attention of a philosopher. Archimedes in a later age imitated and surpassed Archytas. But even Archimedes was not free from the prevailing notion that geometry was degraded by being employed to produce anything useful. It was with difficulty that he was induced to stoop from speculation to practice. He was half ashamed of those inventions which were the wonder of hostile nations, and always spoke of them slightingly as mere amusements, as trifles in which a mathematician might be suffered to relax his mind after intense application to the higher parts of his science.
The opinion of Bacon on this subject was diametrically opposed to that of the ancient philosophers. He valued geometry chiefly, if not solely, on account of those uses which to Plato appeared so base. And it is remarkable that the longer Bacon lived the stronger this feeling became.
Lord Thomas Babington Macaulay: Lord Bacon, July, 1837.
As an exercise of the reasoning faculty, pure mathematics is an admirable exercise, because it consists of reasoning alone, and does not encumber the student with any exercise of judgment; and it is well always to begin with learning one thing at a time, and to defer a combination of mental exercises to a later period. But then it is important to remember that mathematics does not exercise the judgment, and, consequently, if too exclusively pursued, may leave the student very ill qualified for moral reasoning . There are probably as many steps of pure reasoning in one of the longer of Euclids demonstrations as in the whole of an argumentative treatise on some other subject, occupying perhaps a considerable volume.
Richard Whately: Annot. on Bacons Essay, Of Studies, and in Whatelys Elements of Logic.
Mathematics, in its latitude, is usually divided into pure and mixed: and though the pure do handle only abstract quantity in general, as geometry, arithmetic; yet that which is mixed doth consider the quality of some particular determinate subject: so astronomy handles the quantity of heavenly motions; music, of sounds; and mechanics, of weights and measures.