Abstract
For the entire span of his philosophical career, Husserl struggled with the epistemological problem posed by imaginary elements, that is, improper or objectless representations that are nonetheless treated as if denoting something. How can it be explained that we can obtain knowledge (symbolic knowledge), as is paradigmatically the case of mathematics, by operating with symbols according to rules – even when these symbols do not represent anything? This problem presented itself very early in Husserl’s philosophical life and was a dominant factor in the development of his thought. From the first to the last work he published the task of clarifying the sense and delimitating the scope of symbolization and formalization in science and mathematics was one of Husserl’s major concerns. In this paper I want to show how Husserl dealt with the problem of symbolic knowledge in mathematics, and the central role it played in his philosophical development.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Cognitio caeca is one of the terms Leibniz—the man who brought this issue into philosophy—used for symbolic knowledge.
- 2.
We are talking, basically, of Husserl’s philosophical development during, roughly, the last decade of the nineteenth century. The relevant textual sources are the Philosophy of Arithmetic (PA, 1891), the Logical Investigations (LI, 1900–01) and minor texts of that period published in Husserliana 12, 21 and 22. In this paper I will concentrate on those where Husserl’s treatment of the many (basically three) versions of the problem of symbolic knowledge (concerning, respectively, symbolisms with and without interpretation, the role of imaginary elements in symbolisms with interpretation) comes out more clearly; namely, PA, LI, “Semiotic” (Hua 12, 340–373), the review of Schröder’s book Lectures on the Algebra of Logic (Hua 22, 3–43), and the draft for the Göttingen lectures (“The Imaginary in Mathematics,” Hua 12, 430–451). Other texts, such as “The Concept of General Arithmetic” (Hua 12, 375–379), “Arithmetic as an A Priori Science” (Hua 12, 380–384), a letter to Carl Stumpf (Hua 21, 244–251), “On Set Theory” (Hua 12, 385–407), and “Formal and Contentual Arithmetic” (Hua 21, 21–23), to mention a few, are either much shorter or not directly concerned with the problems discussed here; they will be taken mostly as subsidiary sources of information, not as loci classici of Husserl’s treatment of the problem of symbolic knowledge.
- 3.
Published originally in Göttinger Gelehrte Anzeigen, 1891, no 7, 243–278; republished in Hua 22, 3–43.
- 4.
Boole, whose logical calculus could be interpreted either as a calculus of classes or a calculus of propositions, was aware of the way winds were blowing in mathematics: “They, who are acquainted with the present state of the theory of Symbolical Algebra, are aware that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a question on the property of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics of optics” (Boole, in Bochenski 1970, §38.17). Although Brentano was an influence to be reckoned with, I believe Husserl’s realization that purely symbolic reasoning has an essential role in knowledge is mainly due to his mathematical training and the awareness of the logical relevance of symbolization that came from reading Boole and Schröder, among other formal logicians of the time—it is clear, in Husserl’s long review of Schröder’s book, that he knew well Boole’s calculus, maybe as “beautifully explained by Venn” (Schröder’s review, p. 40). Of course, it was Leibniz (who was also well aware of the fact that some symbolic systems admit different interpretations, and that in this resides their main interest) the first to bring to philosophical attention the fact (and the problem) of symbolic knowledge; Boole, Schröder and Frege were Leibniz’s natural heirs. I believe, however (hence the title of this paper) that Husserl took this problem a step further by considering purely formal, non-interpreted systems; not only interpreted ones, as many of his antecessors, including Leibniz, did.
- 5.
It is worth noticing that the notion of isomorphism, as we have it today, had not yet by then come out clearly and distinctively in mathematics, although it was already operational, as the work of Dedekind, for instance, testifies.
- 6.
His critique of technization in Crisis spares symbolic mathematics for, as André de Muralt says: “[a]lthough mathematics is a symbolic knowledge, it can nevertheless be applied and is therefore not technized on its own account. Its original sense is therefore a logical sense” (1974, 183).
- 7.
Even though, let us keep this in mind, the insights we obtain by inquiring this concept cannot distinguish between numbers proper and anything that just look like numbers, that is, any domain equiform with (isomorphic to) the domain of numbers.
- 8.
How can we know anything by simply “playing” with symbols according to prefixed rules?
- 9.
It is already clearly discernible in this passage the tasks that Husserl will in later works impose upon formal logic, to elaborate a logical grammar so as to guarantee that the formulas of a logical language be meaningful—and consequently denote sates-of-affairs—and a theory of deduction so as to guarantee that formal derivations preserve truth. In fact, according to D. Willard (Husserl 1994, XIV) “Semiotic” was “apparently Husserl’s first systematic effort towards a ‘logic,’ in his special sense, for symbolic calculation.”
- 10.
Husserl observes correctly (Hua 22, 42–43) that Schröder’s calculus has little value interpreted as a logic of deduction, for the domain of deduction it formalizes is very restrict, whereas, as a calculus of classes, it can have many applications in mathematics.
- 11.
“A geometry is still geometry if after having defined square circles it uses them in deductions?” (Hua 22, 31) A conflict is an incongruity between a symbolic system and its intended objectual domain.
- 12.
I will take this opportunity to say that one of my main criticisms of Husserl’s philosophy of mathematics is that it does not take into consideration the fact that a formal system, even when built on the intuitive apprehension of truths about a determinate objective domain, or the concept of which this domain is the extension, is never only a theory of this domain, even if it is categorical (for categoricity guarantees only the uniqueness of the structure of the domain, not its material content).
- 13.
- 14.
He reasons thus: numerical equalities and inequalities are decidable on the basis of the axioms; algebraic equalities and inequalities and general assertions are decidable because their numerical instances are decidable (needless to remark that what Husserl understands by “decidable” has nothing to do with our notion of syntactic—or even semantic—decidability).
- 15.
Another example may be this: the famous problems of ancient Greek geometry, the squaring of the circle, the trisection of the angle and the duplication of the cube could only be adequately dealt with—if not solved, at least shown to be unsolvable as stated—by going through an elaborate algebraic, i.e., formal analysis of geometrical constructions with straight edge and compass and what they can accomplish. This analysis interprets geometrical constructions formally in terms of algebraic field extensions.
- 16.
Similar ideas were voiced in the sketches for the Göttingen talks: “[M]athematics in the highest and most general sense is the science of theoretical systems in general, abstracting the objects of theoretical interest of the given theories of different sciences; in no matter which given theory, in no matter which given deductive system, we abstract its subject matter, the particular types of objects it tried to theoretically master, and if we substitute the representations of objects materially determinate by simple formulas, that is, the representation of objects in general that is mastered by such a theory, by a theory of this form, we have then accomplished a generalization that considers the given theories as particular cases of a class of theories, or rather of a form of theory that we consider in a unifying manner and in virtue of which we can say that all these particular scientific domains have, as form is concerned, the same theory” (Hua 12, 430–431). “Mathematics is then, according to its highest ideal, a doctrine of theories, the most general science of deductive systems that are possible in general” (Hua 12, 432).
- 17.
Husserl was very consistent in his characterization of the doctrine of multiplicities: it is a science of forms of theories. See, for instance, Einleitung in die Logik und Erkenntnistheorie (Hua 24, §19, p.79), a work written 10 years after the Prolegomena.
- 18.
A sign of this orientation is that Husserl sees even formal theories as referring to objects, formal objects precisely, indeterminate as to content, but determinate as to form by their theory.
- 19.
Husserl believed that any a priori contentual axiomatic theory is necessarily a conceptual theory, that is, the theory of a concept under which the objects of the relevant domain were assembled. In a text of 1891 (“Arithmetic as an A Priori Science,” Hua 12, 380–384), inquiring on the nature of arithmetic as an a priori science, Husserl says that a science of such a nature “does not begin with single facts for then to obtain possibly true generalities by induction, but immediately by certain generalities that are apodictically certain and immediately evident, which it acquires by simply presenting to itself certain ‘fundamental concepts’ that give, by means of mediate evidence and certitude, all the sequence of theorems of this science.”
- 20.
In a letter to Frege, summarized and commented by Husserl (Hua 12, 447–451) Hilbert says something relevant in this context: “Any theory can be applied to an infinite number of systems of fundamental elements. It suffices to apply a one to one reversible transformation and stipulate that the corresponding axioms for the thing thus transformed are the same (this is the case, for instance, with the principle of duality and my proofs of independence)” (Hua 12, 450).
- 21.
It is interesting to notice that the interplay of formal domains is, according to Husserl, a topic of study of formal ontology.
References
Bochenski, J. 1970. A History of Formal Logic. New York: Chelsea Publishing Co.
da Silva, J. 2000a. Husserl’s Conception of Logic. Manuscrito 22: 367–397.
da Silva, J. 2000b. Husserl’s Two Notions of Completeness. Synthese 125: 417–438.
da Silva, J. 2000c. The Many Senses of Completeness. Manuscrito 23: 41–60.
de Muralt, A. 1974. The Idea of Phenomenology: Husserlian Exemplarism. Evanston: Northwestern University Press.
Husserl, E. 1994. Early Writings in the Philosophy of Logic and Mathematics. Transl. and with an introduction by D. Willard. Dordrecht: Kluwer.
Moran, D. 2005. Edmund Husserl—Founder of Phenomenology. Cambridge: Polity Press.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
da Silva, J.J. (2010). Beyond Leibniz: Husserl’s Vindication of Symbolic Knowledge. In: Hartimo, M. (eds) Phenomenology and Mathematics. Phaenomenologica, vol 195. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3729-9_7
Download citation
DOI: https://doi.org/10.1007/978-90-481-3729-9_7
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-3728-2
Online ISBN: 978-90-481-3729-9
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)