"Mathematics make the mind attentive to the objects which it considers. This they do by entertaining it with a great variety of truths, which are delightful and evident, but not obvious. Truth is the same thing to the understanding as music to the ear and beauty to the eye. The pursuit of it does really as much gratify a natural faculty implanted in us by our wise Creator as the pleasing of our senses: only in the former case, as the object and faculty are more spiritual, the delight is more pure, free from regret, turpitude, lassitude, and intemperance that commonly attend sensual pleasures." (John Arbuthnot, "An Essay on the Usefulness of Mathematical Learning", 1701)
"The mathematics are the friends to religion, inasmuch as they charm the passions, restrain the impetuosity of the 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 truth 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, "An Essay on the Usefulness of Mathematical Learning", 1701)
"[…] for the saving the long progression of the thoughts to remote and first principles in every case, the mind should provide itself several stages; that is to say, intermediate principles, which it might have recourse to in the examining those positions that come in its way. These, though they are not self-evident principles, yet, if they have been made out from them by a wary and unquestionable deduction, may be depended on as certain and infallible truths, and serve as unquestionable truths to prove other points depending upon them, by a nearer and shorter view than remote and general maxims. […] And thus mathematicians do, who do not in every new problem run it back to the first axioms through all the whole train of intermediate propositions. Certain theorems that they have settled to themselves upon sure demonstration, serve to resolve to them multitudes of propositions which depend on them, and are as firmly made out from thence as if the mind went afresh over every link of the whole chain that tie them to first self-evident principles." John Locke, "The Conduct of the Understanding", 1706)
"Nothing proves more clearly that the mind seeks truth, and nothing reflects more glory upon it, than the delight it takes, sometimes in spite of itself, in the driest and thorniest researches of algebra." (Bernard de Fontenelle, "Histoire du Renouvellement de l'Académie des Sciences", 1708)
"The ideas imprinted on the senses by the Author of Nature are called real things: and those excited in the imagination being less regular, vivid and constant, are more properly termed ideas, or images of things, which they copy and represent. But then our sensations, be they never so vivid and distinct, are nevertheless ideas, that is, they exist in the mind, or are perceived by it, as truly as the ideas of its own framing. The ideas of sense are allowed to have more reality in them, that is, to be more strong, orderly, and coherent than the creatures of the mind; but this is no argument that they exist without the mind. They are also less dependent on the spirit, or thinking substance which perceives them, in that they are excited by the will of another and more powerful spirit: yet still they are ideas, and certainly no idea, whether faint or strong, can exist otherwise than in a mind perceiving it." (George Berkeley, "Principles of Human Knowledge", 1710)
"To speak the truth, Philonous, I think there are two kinds of objects, the one perceived immediately, which are likewise called ideas, the other are real things or external objects perceived by the mediation of ideas, which are their images and representations. Now I own, ideas do not exist without the mind; but the latter sort of objects do." (George Berkeley, "Three Dialogues", 1713)
"[Mathematics] guides our minds in an orderly way, and furnishes us simple and rational principles by means of which ambiguities are clarified, disorder is converted into order, and complexities are analyzed into their component parts." (Johann B Mencken, "The Charlatanry of the Learned", 1715)
"There is nothing in Nature that does more show the piercing Force of Human Understanding, the sublimity of its Speculations and deep researchers, than true Astronomy. It raises our Minds above our Senses, and even in contradiction to them, shows us the true System of the World: the faculty of Reason by which we have made these great discoveries in the Heavens must needs be derived from Heaven, since no Earthly Principle can attain so great a Perfection." (John Keill, "An Introduction to the True Astronomy", 1721)
"By the very nature of poetry it is impossible for everyone to be at the same time a sublime poet and a sublime metaphysician, for metaphysics abstracts the mind from the senses, and the poetic faculty must submerge the whole mind in the senses. Metaphysics soars up to universals, and the poetic faculty must plunge deep into particulars." (Giambattista Vico, "The New Science", 1725)
"For chemistry is no science form’d à priori; ’tis no production of the human mind, framed by reasoning and deduction: it took its rise from a number of experiments casually made, without any expectation of what follow’d; and was only reduced into an art or system, by collecting and comparing the effects of such unpremeditated experiments, and observing the uniform tendency thereof. So far, then, as a number of experimenters agree to establish any undoubted truth; so far they may be consider'd as constituting the theory of chemistry." (Hermann Boerhaave, "A New Method of Chemistry", 1727)
"As arithmetic and algebra are sciences of great clearness, certainty, and extent, which are immediately conversant about signs, upon the skillful use whereof they entirely depend, so a little attention to them may possibly help us to judge of the progress of the mind in other sciences, which, though differing in nature, design, and object, may yet agree in the general methods of proof and inquiry." (George Berkeley, "Alciphorn: or, the Minute Philosopher", 1732)
"[...] things which do not now exist in the mind itself, can only be perceived, remembered, or imagined, by means of ideas or images of them in the mind, which are the immediate objects of perception, remembrance, and imagination. This doctrine appears evidently to be borrowed from the old system; which taught, that external things make impressions upon the mind, like the impressions of a seal upon wax; that it is by means of those impressions that we perceive, remember) or imagine them; and that those impressions must resemble the things from which they are taken. When we form our notions of the operations of the mind by analogy, this way of conceiving them seems to be very natural, and offers itself to our thoughts: for as every thing which is felt must make some impression upon the body, we are apt to think, that every thing which is understood must make some impression upon the mind." (Thomas Reid, "An Inquiry into the Human Mind", 1734)
"All the perceptions of the human mind resolve themselves into two distinct kinds, which I shall call impressions and ideas. The difference betwixt these consists in the degrees of force and liveliness, with which they strike upon the mind, and make their way into our thought or consciousness. Those perceptions, which enter with most force and violence, we may name impressions; and under this name I comprehend all our sensations, passions and emotions, as they make their first appearance in the soul. By ideas I mean the faint images of these in thinking and reasoning. I believe it will not be very necessary to employ many words in explaining this distinction." (David Hume, "A Treatise of Human Nature", 1738)
"But to form the idea of an object, and to form an idea simply is the same thing; the reference of the idea to an object being an extraneous denomination, of which in itself it bears no mark or character. Now as it is impossible to form an idea of an object, that is possessed of quantity and quality, and yet is possessed of no precise degree of either; it follows, that there is an equal impossibility of forming an idea, that is not limited and confined in both these particulars. Abstract ideas are therefore in themselves individual, however they may become general in their representation. The image in the mind is only that of a particular object, though the application of it in our reasoning be the same, as if it were universal." (David Hume, "Treatise of Human Nature", 1738)
"Mathematical knowledge adds vigour to the mind, frees it from prejudice, credulity, and superstition. This it does in two ways: 1st, by accustoming us to examine, and not to take thigs upon trust. 2nd By giving us a clear and extensive knowledge of the system of the world […]." (Dr. John Arbuthnot, "Usefulness of Mathematical Learning", 1745)
"Our mental vision or conception of ideas is nothing but a revelation made to us by our Maker. When we voluntarily turn our thoughts to any object, and raise up its image in the fancy, it is not the will which creates that idea: It is the universal Creator, who discovers it to the mind, and renders it present to us." (David Hume, "An Enquiry Concerning Human Understanding", 1748)
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