This position clearly retains a general similarity between the two kinds of concepts because both—specific or full concepts of finite abstract things as much as complete concepts of concrete infinite individuals—must include all their predicates and can be known a priori by Him who generated them.
This view is precisely the core of the Geometrical Method! According to Leibniz, even if human beings cannot know individuals a priori but only through empirical study or by history, God does know the complete concepts of individual substances a priori which thus exist, the subject containing the predicate.
It was this theory that would lead to paragraph 13 of the Discourse of Metaphysics , according to which the complete concept of any individual was known by God and would include every single event that would ever happen to us. When God created this world, He chose those individuals who belonged to the best of all compossible worlds. It is within this view that Leibniz sharply deviates from Luther and the Protestant way of thinking in which such an intelligibility of the world to humans is bluntly denied, due to the fall.
It is this view that makes him a true optimist, being convinced of the intelligibility of the world—even if we will never exhaust it. The use of the Geometrical Method in philosophy had often been criticized, long before Kant argued against it Kant , ; 1 st Cr.
One objection was that the Geometrical Method should be restricted to geometry and could not be used in any other field. At first glance, this seems quite convincing. But the objections against the Geometrical Method were more fundamental. What the partisans of the Geometrical Method saw as its greatest advantage in contrast to any other knowledge—the necessary conclusions and thus certainty, was considered the greatest danger by its critics.
One of the reasons for such protests was obviously the theological concern about human haughtiness as it was expressed already in the accusation against Galileo. He was blamed for claiming an equality between human knowledge and that of God, at least in geometrical things Galilei , vol. The concern about human haughtiness was not restricted to the Catholic Church, it would also cause worries among Protestants, for example, for the Cambridge Platonists, very influential to Locke and Newton, who both rejected the Geometrical Method.
In Germany, it became one of the major arguments of the Lutheran theologians and philosophers against Leibniz and Christian Wolff Goldenbaum , ; But it was not the traditional method of Euclidian geometry that caused the massive criticism of the new Geometrical Method. Rather it was its close connection to the mathematization of nature and thereby the extension of geometry from a small discipline without practical relevance to reality, making it the science of the world.
Galileo had opened the new path of modern science by using the Geometrical Method for the investigation of physical phenomena, and he was deeply convinced that nature itself is structured mathematically.
In this way, he found the law of falling bodies as well as the parabola as the trajectory of thrown bodies; neither of them could have been found by mere observation or experiment. Philosophy is written in that great book which ever is before our eyes—I mean the universe—but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written.
The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth. Galileo , There shall be only one science, mathesis universalis, by which the observed natural phenomena could be explained from their inner essences and thus necessarily. The great admirer of Galileo, Thomas Hobbes, extended the Geometrical Method to politics, claiming that his political philosophy was the beginning of political science.
Spinoza even extended the Geometrical Method to ethics and delivered a theory of human affects showing the necessity by which they would occur whenever certain circumstances came together. All these thinkers extended the Geometrical Method beyond mathematics, claiming its value for the investigation of realia, of real things instead of mere geometrical figures.
Such extension of the Geometrical Method to real things was done with the goal to produce certainty of knowledge, a certainty guaranteed by the necessity of geometrical demonstrations. But if it would indeed lead to necessary demonstrations about nature, politics, and ethics, it would introduce necessitarianism into natural, social, and moral sciences, and space would not be left for miracles and, even worse, for free will.
This can be seen in the cases of Hobbes and Spinoza, who both were strict determinists. In contrast, it was precisely the recognition of this threat of determinism or necessitarianism implied in the Geometrical Method that led Henry More very early to his criticism of Descartes and since the s to his massive rejection of Cartesianism More , Besides the theological concern about human haughtiness, it was the threat of necessitarianism that was the true source of the lasting protest against the Geometrical Method throughout the 17th and 18th centuries.
What caused the most trouble about the Geometrical Method in 17th and throughout the 18th centuries was neither its ponderous way of thinking nor its lack of success.
Rather it was the turmoil about human haughtiness and the threat that its determinism would destroy free will of God as well as that of human beings. According to Leibniz, nothing can happen without a sufficient reason—and this just proves the existence of a God who—in His perfection—could not have chosen an arbitrarily functioning world.
Two things caused deep anxiety and anger regarding this method: 1 the attempt to extend the Geometrical Method to nature, to humans, and to society taking mathematization of nature for granted , providing human beings with a godlike a priori knowledge beyond mathematics, even if limited; and 2 the threat of determinism. These threats forced theologians and Christian philosophers to reject rationalism and the Geometrical Method altogether.
In sharp contrast to rationalism, Locke would even deny the possibility of any natural science because we could not have any real definitions beyond mathematics and morals:. This way of getting and improving our Knowledge in Substances only by Experience and History, which is all the weakness of our Faculties in this State of Mediocrity , which we are in this World, can attain to, makes me suspect, that natural Philosophy is not capable of being made a Science.
We are able, I imagine, to reach very little general Knowledge concerning the Species of Bodies, and their several Properties. Locke , ; Leibniz , ; IV, 12, Both approaches were applauded by Protestant theology Goldenbaum , Thus, the opposition between the two philosophical camps of rationalism and empiricism was not the result of different approaches to experience as is often claimed. Rather, it was their different and opposing stances toward the Geometrical Method and the mathematization of nature.
This new Geometrical Method was in no way a merely external way of presentation to rationalist philosophy. Instead, it constituted this philosophy. In contrast, empiricists and theologians are eager to deny such a possibility. Thus, it is the Geometrical Method that provides the explanation for the two schools of early modern philosophy. Ursula Goldenbaum Email: ugolden emory.
The Essential Significance of Definitions Although it was the geometrical demonstrations that guaranteed necessary truths, they were hardly under attack.
The Place of Empirical Knowledge in the Geometrical Method While God knows everything adequately and intuitively, we humans rarely get to know adequate ideas intuitively.
Geometrical Method and Logic of Containment Leibniz, embracing the Geometrical Method, was fully aware of his dangerous intellectual neighbors Hobbes and Spinoza , and worked hard to secure his metaphysics against strict determinism or necessitarianism in order to distinguish his metaphysical and epistemological project from these bad bedfellows. The Mathematization of Nature as a Challenge of Necessitarianism The use of the Geometrical Method in philosophy had often been criticized, long before Kant argued against it Kant , ; 1 st Cr.
Conclusion Two things caused deep anxiety and anger regarding this method: 1 the attempt to extend the Geometrical Method to nature, to humans, and to society taking mathematization of nature for granted , providing human beings with a godlike a priori knowledge beyond mathematics, even if limited; and 2 the threat of determinism.
In sharp contrast to rationalism, Locke would even deny the possibility of any natural science because we could not have any real definitions beyond mathematics and morals: This way of getting and improving our Knowledge in Substances only by Experience and History, which is all the weakness of our Faculties in this State of Mediocrity , which we are in this World, can attain to, makes me suspect, that natural Philosophy is not capable of being made a Science.
References and Further Reading a. Abbreviations A See Leibniz ff. AT See Descartes CSM See Descartes Loemker See Leibniz Leibniz-Clarke See Leibniz OL See Hobbes Corr, Charles A. Curley, W. Klever, F. Vrin: Paris, 11 vols.
Galilei, Galileo , Opere. Edizione Nazionale , vol. Studien zur Ausarbeitung des Erfahrungsbegriffes in der neuzeitlichen Philosophie , ed.
Goldenbaum, Ursula , Appell an das Publikum. Hobbes, Thomas , Opera Philosophica quae latine scripsit omnia , ed. Hobbes, Thomas , Leviathan, with selected variants from the Latin edition of , ed. Kant, Immanuel , Critique of Pure Reason , trans. Kant , Immanuel , Critique of the Power of Judgment , ed. Leibniz, Gottfried Wilhelm ff. Loemker, Reidel: Dordrecht.
Nidditch, Oxford University Press. More, Henry , Epistola H. Mori ad V. Jahrhundert , Olms: Hildesheim, New York. Spinoza , B. Spinoza , The Collected Works of Spinoza , ed. Roberts: London. Page 68 Simple homotopy type and categories of fractions Page 84 Some recent developments in the homology theory of groups Page Poincare duality groups of dimension two are surface groups Page Continuous solutions of linear equations - An old problem, its history, and its solution Page Mathematics: Questions and Answers Page Hurwitz-Radon matrices revisited: From effective solution of the Hurwitz matrix equations to Bott periodicity Page Birth of fibre spaces, and homotopy Page The Euler characteristic - a few highlights in its long history Page Topology, algebra, analysis - relations and missing links Page Die Zukunft der Mathematik.
Ein Ruckblick auf Hilberts programmatischen Vortrag vor Jahren Page Kolmogorov and contemporary mathematics Page Is algebraic topology a respectable field? Conference proceedings. Papers Table of contents 27 papers About About these proceedings Table of contents Page 1 Navigate to page number of 2. Front Matter Pages i-xi. Front Matter Pages Pages Open Access. Standard Groupoids of von Neumann Algebras. Pages Quantum Differential Equations and Helices.
Cruz y Cruz, Z. Gress, P. Following the Trail of the Operator Geometric Mean.
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