It is the unanimous opinion of all who seek wisdom beyond the common lot that the best and surest way to discover and to teach truth is the method used by mathematicians in their study and exposition of the sciences, namely, that whereby conclusions are demonstrated from definitions, postulates, and axioms. And indeed rightly so. Because all sure and sound knowledge of what is unknown can be elicited and derived only from what is already known with certainty, this latter must first be built up from the ground as a solid foundation on which thereafter to construct the entire edifice of human knowledge, if that is not to collapse of its own accord or give way at the slightest blow. That the things familiar to mathematicians under the title of definitions, postulates, and axioms are of this kind cannot be doubted by anyone who has even the slightest acquaintance with that noble discipline. For definitions are merely the perspicuous explanations of the terms and names by which matters under discussion are designated, whereas postulates and axioms—that is, the common notions of the mind—are statements so clear and lucid that no one who has simply understood the words aright can possibly refuse assent.
But although this is so, you will find that with the exception of mathematics hardly any branch of learning is treated by this method. Instead, a totally different method is adopted, whereby the entire work is executed by means of definitions and logical divisions interlinked in a chain, with problems and explanations interspersed here and there. For almost all who have applied themselves to establishing and setting out the sciences have believed, and many still do believe, that the mathematical method is peculiar to mathematics and is to be rejected as inapplicable to all other branches of learning.
In consequence, nothing of what they produce is demonstrated with conclusive reasoning. They try to advance arguments that depend merely on likelihood and probability, and in this way they thrust before the public a great medley of great books in which you may look in vain for solidity and certainty. Disputes and strife abound, and what one somehow establishes with trivial arguments of no real weight is soon refuted by another, demolished and shattered with the same weapons. So where the mind, eager for unshakable truth, had thought to find for its labors a placid stretch of water that it could navigate with safety and success, thereafter attaining the haven of knowledge for which it yearned, it finds itself tossed on a stormy sea of opinion, beset on all sides with tempests of dispute, hurled about and carried away on waves of uncertainty, endlessly, with no hope of ever emerging therefrom.
Yet there have not been lacking some who have thought differently and, taking pity on the wretched plight of Philosophy, have distanced themselves from this universally adopted and habitual way of treating the sciences and have entered upon a new and indeed an arduous path bristling with difficulties, so as to leave to posterity the other parts of Philosophy, besides mathematics, demonstrated with mathematical method and with mathematical certainty. Of these, some have arranged in mathematical order and passed on to the world of letters a philosophy already accepted and customarily taught in the schools, whereas others have thus treated a new philosophy, discovered by their own exertions. For a long time, the many who undertook this task met with no success, but at last there arose that brightest star of our age, René Descartes. After bringing forth by a new method from darkness to light whatever had been inaccessible to the ancients, and in addition whatever could be wanting in his own age, he laid the unshakable foundations of philosophy on which numerous truths could be built with mathematical order and certainty, as he himself effectively proved, and as is clearer than the midday sun to all who have paid careful attention to his writings, for which no praise is too great.
Although the philosophical writings of this most noble and incomparable man exhibit the mathematical manner and order of demonstration, yet they are not composed in the style commonly used in Euclid’s Elements and other geometrical works, the style wherein Definitions, Postulates, and Axioms are first enunciated, followed by Propositions and their demonstrations. They are arranged in a very different way, which he calls the true and best way of teaching, the Analytic way. For at the end of his “Reply to Second Objections,”2 he acknowledges two modes of conclusive proof. One is by analysis, “which shows the true way by which a thing is discovered methodically and, as it were, a priori”; the other is by synthesis, “which employs a long series of definitions, postulates, axioms, theorems and problems, so that if any of the conclusions be denied, it can be shown immediately that this is involved in what has preceded, and thus the reader, however reluctant and obstinate, is forced to agree.”
However, although both kinds of demonstration afford a certainty that lies beyond any risk of doubt, not everyone finds them equally useful and convenient. There are many who, being quite unacquainted with the mathematical sciences and therefore completely ignorant of the synthetic method in which they are arranged and of the analytic method by which they were discovered, are neither able themselves to understand nor to expound to others the things that are discussed and logically demonstrated in these books. Consequently, many who, either carried away by blind enthusiasm or influenced by the authority of others, have become followers of Descartes have done no more than commit to memory his opinions and doctrines. When the subject arises in conversation, they can only prate and chatter without offering any proof, as was once and still is the case with the followers of the Peripatetic philosophy. Therefore, to provide them with some assistance, I have often wished that someone, skilled both in the analytic and synthetic arrangement and thoroughly versed in Descartes’s writings and expert in his philosophy, should set his hand to this task, and undertake to arrange in synthetic order what Descartes wrote in analytic order, demonstrating it in the way familiar to geometricians. Indeed, though fully conscious of my incompetence and unfitness for such a task, I have frequently thought of undertaking it myself and have even made a start. But other distractions, which so often claim my attention, have prevented its completion.
I was therefore delighted to hear from our Author that, while teaching Descartes’s philosophy to a certain pupil of his, he had dictated to him the whole of Part II of the Principia and some of Part III, demonstrated in that geometric style, and also the principal and more difficult questions that arise in metaphysics and remain unresolved by Descartes, and that, at the urgent entreaties and pleadings of his friends, he has permitted these to be published as a single work, corrected and amplified by himself. So I also commended this same project, at the same time gladly offering my services, if needed, to get this published. Furthermore I urged him—indeed, besought him—to set out Part I of the Principia as well in like order to precede the rest, so that the work, as thus arranged from its very beginning, might be better understood and give greater satisfaction. When he saw how reasonable was this proposal, he could not refuse the pleas of a friend and likewise the good of the reader. He further entrusted to my care the entire business both of printing and of publishing because he lives in the country far from the city and so cannot give it his personal attention.3
Such then, honest reader, are the contents of this little book, namely, Parts I and II of Descartes’s Principia Philosophiae together with a fragment of Part III, to which we have added, as an appendix, our Author’s Cogitata Metaphysica. But when we here say Part I of the Principia, and the book’s title so announces, we do not intend it to be understood that everything Descartes says there is here set forth as demonstrated in geometric order. The title derives only from its main contents, and so the chief metaphysical themes that were treated by Descartes in his Meditations are taken from that book (omitting all other matters that concern Logic and are related and reviewed only in a historical way). To do this more effectively, the Author has transposed word for word almost the entire passage at the end of the “Reply to the Second Set of Objections,” which Descartes arranged in geometric order.4 He first sets out all Descartes’s definitions and inserts Descartes’s propositions among his own, but he does not place the axioms immediately after the definitions; he brings them in only after Proposition 4, changing their order so as to make it easier to prove them, and omitting some that he did not require.
Although our Author is well aware that these axioms (as Descartes himself says in Postulate 7) can be proved as theorems and can even more neatly be classed as propositions, and although we also asked him to do this, being engaged in more important affairs he had only the space of two weeks to complete this work, and that is why he could not satisfy his own wishes and ours. He does at any rate add a brief explanation that can serve as a demonstration, postponing for another occasion a lengthier proof, complete in all respects, with view to a new edition to follow this hurried one. To augment this, we shall also try to persuade him to complete Part III in its entirety, “Concerning the Visible World” (of which we give here only a fragment, since the Author ended his instruction at this point and we did not wish to deprive the reader of it, little as it is). For this to be properly executed, some propositions concerning the nature and property of Fluids will need to be inserted at various places in Part II, and I shall then do my best to persuade the Author to do this at the time.5
It is not only in setting forth and explaining the Axioms that our Author frequently diverges from Descartes but also in proving the Propositions themselves and the other conclusions, and he employs a logical proof far different from that of Descartes. But let no one take this to mean that he intended to correct the illustrious Descartes in these matters, but that our Author’s sole purpose in so doing is to enable him the better to retain his already established order and to avoid increasing unduly the number of Axioms. For the same reason, he has also been compelled to prove many things that Descartes propounded without proof, and to add others that he completely omitted.
However, I should like it to be particularly noted that in all these writings, in Parts I and II and the fragment of Part III of the Principia and also in the Cogitata Metaphysica, our Author has simply given Descartes’s opinions and their demonstrations just as they are found in his writings, or such as should validly be deduced from the foundations laid by him. For having undertaken to teach his pupil Descartes’s philosophy, his scruples forbade him to depart in the slightest degree from Descartes’s views or to dictate anything that did not correspond with, or was contrary to, his doctrines. Therefore no one should conclude that he here teaches either his own views or only those of which he approves. For although he holds some of the doctrines to be true, and admits that some are his own additions, there are many he rejects as false, holding a very different opinion.6
Of this sort, to single out one of many, are statements concerning the Will in the Scholium to Proposition 15 of Part I of the Principia and in Chapter 12, Part II of the Appendix, although they appear to be laboriously and meticulously proved. For he does not consider the Will to be distinct from the Intellect, far less endowed with freedom of that kind. Indeed, in making these assertions, as is clear from Part 4 of the Discourse on Method, the “Second Meditation,” and other passages, Descartes merely assumes, and does not prove, that the human mind is an absolutely thinking substance. Although our Author does indeed admit that there is in Nature a thinking substance, he denies that this constitutes the essence of the human mind.7 He maintains that, just as Extension is not determined by any limits, so Thought, too, is not determined by any limits. And therefore, just as the human body is not Extension absolutely, but only as determined in a particular way in accordance with the laws of extended Nature through motion and rest, so too the human mind or soul is not Thought absolutely, but only as determined in a particular way in accordance with the laws of thinking Nature through ideas, and one concludes that this must come into existence when the human body begins to exist. From this definition, he thinks it is not difficult to prove that Will is not distinct from Intellect, far less is it endowed with the freedom that Descartes ascribes to it.8 Indeed, he holds that a faculty of affirming and denying is quite fictitious, that affirming and denying are nothing but ideas, and that other faculties such as Intellect, Desire, etc., must be accounted as figments, or at least among those notions that men have formed through conceiving things in an abstract way, such as humanity, stoniness, and other things of that kind.
Here, too, we must not omit to mention that assertions found in some passages, that this or that surpasses human understanding, must be taken in the same sense (i.e., as giving only Descartes’s opinion). This must not be regarded as expressing our Author’s own view. All such things, he holds, and many others even more sublime and subtle, can not only be conceived by us clearly and distinctly but can also be explained quite satisfactorily, provided that the human intellect can be guided to the search for truth and the knowledge of things along a path different from that which was opened up and leveled by Descartes. And so he holds that the foundations of the sciences laid by Descartes and the superstructure that he built thereon do not suffice to elucidate and resolve all the most difficult problems that arise in metaphysics. Other foundations are required if we seek to raise our intellect to that pinnacle of knowledge.
Finally, to bring my preface to a close, we should like our readers to realize that all that is here treated is given to the public for the sole purpose of searching out and disseminating truth and to urge men to the pursuit of a true and genuine philosophy. And so in order that all may reap therefrom as rich a profit as we sincerely desire for them, before they begin reading we earnestly beg them to insert omitted passages in their proper place and carefully to correct printing errors that have crept in. Some of these are such as may be an obstacle in the way of perceiving the force of the demonstration and the Author’s meaning, as anyone will readily gather from looking at them.
Notes without brackets are Spinoza’s. Bracketed notes are those of Steven Barbone and Lee Rice (main annotators for this work), translator Samuel Shirley, Spinoza’s friend Pieter Balling, and Michael L. Morgan.
1 [The frontispiece announces only Parts I and II of the PPC; Part III is not mentioned here.—S.B./L.R.]
2 [See AT7, 155–156; cf. the slight variation in the French version at AT9, 121–122.]
3 [It appears from Ep12, however, that Spinoza was able to make corrections to the page proofs.]
4 [AT7, 160–170.]
5 [For evidence that Spinoza was developing his own theory of fluids, see Ep6, 78–81.]
6 [Meyer notes three main differences: the substantiality of the human soul, the distinction between the will and intellect, and the freedom to suspend judgment. Spinoza notes his differences with Descartes; see Ep2, 62–63; Ep21, 154–158.]
7 [Cf. E2P11.]
8 [Cf. E2P48; E2P49 Cor and Schol.]