Although this does not exhaust what Descartes has to say about imagination, the list of these parallels gives evidence, I would argue, for a strong conceptual proximity with the Proclean tradition and this is no big surprise since, as we saw above, it was the main vector of circulation for the concept of mathesis universalis.
It so happens that these equivalences are rational reconstructions based on very little evidence Even if Leibniz sometimes indicates connections between mathesis universalis and other disciplines such as ars combinatoria or caracteristica universalis, he never goes as far as identifying them. Moreover, when mathesis universalis is not limited to this narrow definition, it is related, as we will see, to imagination This would remain at the background of his discussions on mathesis universalis throughout his life But it also happens that Leibniz presents larger views on the subject: mea Mathesis generalis, as he puts it It will be no suprise to see that it appears on the very first page of Couturat , Book I, chapter 1, and takes also a central place in authors such as E.
Russell , p. Weyl , Philosophie der Mathematik und Naturwissenschaft, p. Initia et specimina scientiae novae generalis. In this text, Leibniz emphasizes the fact that he aimed at a new kind of Speciosa which will go beyond what Vieta and Descartes have done because it can deal not only with quantitative, but also with qualitative aspects involved in mathematics. He then proposes a list of basic mathematical relations similitudo, hypallela, congruence, coincidence, etc.
This is the reason why Couturat presented this paper as a program for a general theory of relations of logical flavor — a characterization of mathesis universalis which is again to be found in many recent commentators Other Scholars have emphasized the proximity of this theory with papers on analysis situs, in which one can also find lists of basic geometrical relations This characterization, which allows defining the object of mathematics before the distinction in quantity the usual object of mathematics and quality, is not just a convenient way of speaking.
When one looks at the text, one can see that the basic relations are indeed derived from a general logic of imagination whose first operator is the question of discernibility. For example, similar object are defined by the fact that they cannot be discerned singulatim a definition which Leibniz elaborated as early as ; hypallela are those which can be discerned neither by their shape, nor by their magnitude; congruent as the ones of which the extremities cannot be discerned, and so forth and so on, until one reaches objects which would be completely indiscernible, or discernible only numero, that is only by situating them in an external framework given by space and time.
Interestingly enough, Leibniz states that the existence of such objects overcomes the power of mathematics, which is limited to what cannot be discerned by imagination or by sensible appearances — a quite remarkable 56 Couturat , chap. VII, p. Contrary to what we found in Proclus and Descartes, Leibniz also points to a very interesting justification: the link between mathematics, imagination and space, as he presents it, has to do with the question of discernibility.
Nonetheless, it is worth noting that the reappraisal of the Proclean tradition is not here a matter of historical accuracy. Another point should be stressed here: although it is true that Leibniz put a lot of emphasis on the role of formal reasoning arguments in forma in mathematics, this did not amount to characterize its objects as formal entities — a confusion which is at the source of the picture exposed in section 1.
Quite on the contrary, mathematical objects are usually characterized 59 On which see Rabouin It thinks the circle as extended, and although the circle is free of external matter, it possesses an intelligible matter provided by the imagination itself. This thesis is very reminiscent of Proclus, especially when coming along with the idea that the mathematical realm is unified by a logic of imagination given in a universal mathematical theory. The first and second together are imaginable but the third lie beyond the imagination.
One finds here again the typical ambivalence attached to mathematical imagination which we followed throughout this paper. Indeed imagination is supposed to be at once on the side of passive sensitivity and active conceptuality. This is not a simple matter of philosophical interpretation. When looking at Cartesian Geometry, one is struck by the coexistence of two opposite features: on the one hand, the identification of curves with equations which allows de jure a purely algebraic treatment of geometry64; on the other hand, the persistent role of geometric representations, which Descartes himself emphasized when presenting his technique to Elisabeth The same is true of Leibniz.
Whereas generations of scholars have emphasized the many passages in which Leibniz defended the formal character of mathematical reasoning, recent commentators have claimed that it does not tell us much about the nature of mathematical objects. This could explain why Leibniz can at once criticize the role of imagination in reasoning and still claim that mathematics is the science of imaginable things. This paradoxical attitude is well expressed in the characterization of analysis situs which we already mentioned: to represent without figures what depends on the imagination we saw a beautiful example of this could mean in the Elementa nova matheseos universalis.
It also appears in the fact that the introduction of a purely symbolical algorithm for differential calculus was accompanied by a diagram and that, more generally, the identification of some diagrammatic elements are an important feature for Leibnizian Calculus Moreover, this ambivalence is not only tied to some conceptual fuzziness.
It is a source of very important and fruitful developments. Schuhmann Eds. Dordrecht: Kluwer Academic Publishers. Early writings in the philosophy of logic and mathematics, D. Willard, Trans. Husserliana: Collected Works V. Philosophy of Arithmetic, D. Willard Trans. Husserliana: Collected Works X. Ierna, Trans. In The new yearbook for phenomenology and phenomenological philosophy V, p. Ierna Ed.
Ierna, C. Part 2: Philosophical and Mathematical Back- ground. In The new yearbook for phenomenology and phenomenological philosophy VI, p. Rollinger, R. Brentano and Husserl. Jacquette Ed. Cambridge: Cambridge University Press.
Smith, B. Austrian philosophy. In keeping with the two-fold character of phenomenological analysis, this distinction is based upon the results of subjectively as well as objectively directed phenomenological descriptions.
The first direction leads to the concept of "critical attitude" [FTL, 45, 46], which permits a distinction between the attitude Einstellung of the logician from that of the mathematician. The critical attitude of the logician is tantamount with an act of reflection, which is the necessary condition for encountering a judgment as judgment. The mathematician, on the other hand, remains for the most part in an objectively-directed attitude even after he has carried out the abstraction from the material determinations of the object.
In his characteristic reflective attitude, the logician directs his attention to the speaking about abstract objects, which makes it possible to isolate the structures of this speech.
Thus even when logic, in a fashion analogous to formal ontology, speaks about an object-sphere, it refers to the objects and relations in this sphere through the judgment [FTL, 54]. Objectively this distinction in attitude reveals itself to the extent that the judgment is the fundamental concept of formal logic. In the reformulation of the group-axioms in a first-order language, the axioms stand before us as judgments that are grammatically well formulated in the sense of inductive definitions.
Au introductory text-book on group-theory, for example, will normally introduce neither the syntax of formalized mathematical language nor a formal concept of proof. In other words, in contrast to formal logic, whose fundamental conceptual inventory includes "judgment" or "judgment-set", these concepts are never even issues in mathematics [cf. FTL, 24]. Mathematics in its traditional and abstract form remains in an unreflective attitude which does not in principle thematize the speaking about objects [cf.
FTL, ]. It is occasionally necessary of course, to adopt the critical attitude for the purpose of the fundamental mathematical activity of proof. For the mathematician, this "methodological exception" from the unreflective attitude is motivated by a methodical modalization of the judgment carried out in the direct attitude.
We shall return to this in the last two sections. Since the delimiting of formal logic and mathematics will play a decisive role in the understanding of mathematical incompleteness, this must be treated in somewhat more detail. With this distinction;, Husserl clears up a problem that many thinkers toward the end of the 19th century struggled to resolve.
On the other hand he speaks of formal logic as a special science, in which case it is up to the reader to distinguish on every occasion whether Husserl is referring to traditional Aristotelian formal logic or to modem mathematicized logic as discipline within the mathesis universalis. In the Ideen it is made explicit, in FTL analyzed in detail. The problem here exposed was aptly described by Kleene 24 years later in the following way: "In a mathematical theory, we study a system of mathematical objects.
How can a mathematical theory itself be an object for mathematical study? His concept of judgment not so far away from what is commonly referred to as "proposition" in the terminology of analytic philosophy. At the same time, it is important that the judgment i.
Since the judgment can only be given together with an expression in phenomenological parlance: the judgment is founded on the expression , the presumption of ' Platonic' metaphysics is here unjustified. Dealing with the constitution of judgments would call for detailed phenomenological-psychological investigations into the phenomenology of spoken and written language, analyses that cannot be carried through within the confines of this essay.
Benzi Name, Genova, , pp. Alcan, Paris, Google Scholar L. Alcan, Paris, Google Scholar G. Crapulli, Mathesis universalis. Casanave College Publications, London, , pp. Hill, J. College Publications, London, , pp. Dasypodius, Euclidis quindecim elementorum geometriae primum C. Mylius, Strasburg, a Google Scholar C. Dasypodius, Euclidis quindecim elementorum geometriae secondum C. Mylius, Strasburg, b Google Scholar C. Dasypodius, Euclidis elementorum liber primus C.
Mylius, Strasburg, Google Scholar C. Descartes, [AT. X] Oeuvres, vol. X, ed.Philosophie der Arithmetik Psychologische und Logische Untersuchungen. Leibniz, Die Philosophische Schriften, ed. Henceforth cited with page references to the original German edition, which are included in the margins of both the Husserliana edition and the translation. Arndt , Methodo scientifica pertractatum, Berlin, W. The core of their doctrine was expressed in Ficino's statement that the perfect divine order of the universe gets mirrored in human mind due to mind's mathematical insights; thus mathematics proves capable of the role of an universal key to the knowledge; hence the denomination mathesis universalis.
However, not only does Husserl argue that proper and improper presentations authentic and inauthentic representations are logically equivalent, but also that we are not even always aware of using symbolic presenta- tions, because their use is so spontaneous and easy Husserl b, p.
Willard, Trans. Moreover, when the principles of a mathematical logic are applied to any object whatever, it becomes clear, given the identity of the judgment as posited and the judgments as supposed, that mathematical logic can also be understood as formal ontology. Text Nr.
With this distinction;, Husserl clears up a problem that many thinkers toward the end of the 19th century struggled to resolve. Genesi di un'idea nel XVI secolo, Roma as his focus is on the sixteenth century, Crapulli treats neither Descartes nor Leibniz, but only their predecessors. One could rather say that totalities are a special case of Inbegriffe, namely completed and circumscribed ones, which contain only objects of a specific kind or have a clear internal articulation. The phenomenological analysis of the sense-directed attitude leads Husserl to the following conclusions: there is a region of sense wherein a judgement is meaningful irrespective of whether or not it is exact.
Strohmeyer Ed. It also appears in the fact that the introduction of a purely symbolical algorithm for differential calculus was accompanied by a diagram and that, more generally, the identification of some diagrammatic elements are an important feature for Leibnizian Calculus
Erster Band: Prolegomena zur reinen Logik. Husserl, Philosophie der Arithmetik.
Texte aus dem Nachlass — [Studies on arithmetic and geometry. After three semesters in Rostock, and one in Freiburg, Martius transferred to the University of Munich.
For more than any other science, says Husserl, arithmetic manifests the finite and imperfect constitution of human cognition. At the end of the book, Husserl, having clarified the psychological and logical foundations of arithmetic, points ahead to the development of a mathesis universalis. Philosophical Essays in Honor of Thomas M. Leibniz, [a] Non inelegans specimen demonstrandi in abstractis, Akademieausgabe IV. Erste Lieferung Caspar Widtmann, Prague, Within four weeks, she was awarded her doctorate summa cum laude.
Au introductory text-book on group-theory, for example, will normally introduce neither the syntax of formalized mathematical language nor a formal concept of proof. Initia et specimina scientiae novae generalis. David Carr. IV, Ch. Leibniz, [a] Quid sit Idea? Gerhardt, Halle, , reed.
Husserliana: Collected Works V. The same is true of Leibniz. Genesi di un'idea nel XVI secolo, Roma as his focus is on the sixteenth century, Crapulli treats neither Descartes nor Leibniz, but only their predecessors. Leibniz distinguished between a narrower and a broader sense of mathesis universalis. A Akademie Verlag, Berlin, a , pp. Only thereby would it achieve the formality allowing it to serve as the theory-form for any science, whatever the material region to which that science is directed.
He was accompanied by Nicholas of Cusa , Leonardo da Vinci , also by Nicolaus Copernicus Au introductory text-book on group-theory, for example, will normally introduce neither the syntax of formalized mathematical language nor a formal concept of proof. On this see also the remarks on the roll of Stumpf, Brentano, Meinong, Twardowski, or Reinach in the development of the concept of Sachverhalt. Mylius, Strasburg, Google Scholar C. This could explain why Leibniz can at once criticize the role of imagination in reasoning and still claim that mathematics is the science of imaginable things.