### Philosophy of Arithmetic and Frege

After obtaining his PhD in mathematics, Husserl began analyzing the foundations of mathematics from a psychological point of view. In his professorial doctoral dissertation,

*On the Concept of Number* (1886) and in his

*Philosophy of Arithmetic* (1891), Husserl sought, by employing Brentano's descriptive psychology, to define the natural numbers in a way that advanced the methods and techniques of Weierstrass, Dedekind, Georg Cantor, Frege, and other contemporary mathematicians. Later, in the first volume of his

*Logical Investigations*, the

*Prolegomena of Pure Logic*, Husserl, while attacking the psychologistic point of view in logic and mathematics, also appears to reject much of his early work, although the forms of psychologism analysed and refuted in the

*Prolegomena* did not apply directly to his

*Philosophy of Arithmetic*. Some scholars question whether Frege's negative review of the

*Philosophy of Arithmetic* helped turn Husserl towards Platonism, but he had already discovered the work of Bernhard Bolzano independently around 1890/91 and explicitly mentioned Bolzano, Leibniz and Lotze as inspirations for his newer position.

Husserl's review of Ernst Schröder, published before Frege's landmark 1892 article, clearly distinguishes sense from reference; thus Husserl's notions of noema and object also arose independently. Likewise, in his criticism of Frege in the

*Philosophy of Arithmetic*, Husserl remarks on the distinction between the content and the extension of a concept. Moreover, the distinction between the subjective mental act, namely the content of a concept, and the (external) object, was developed independently by Brentano and his school, and may have surfaced as early as Brentano's 1870's lectures on logic.

Scholars such as J. N. Mohanty, Claire Ortiz Hill, and Guillermo E. Rosado Haddock, among others, have argued that Husserl's so-called change from psychologism to Platonism came about independently of Frege's review.For example, the review falsely accuses Husserl of subjectivizing everything, so that no objectivity is possible, and falsely attributes to him a notion of abstraction whereby objects disappear until we are left with numbers as mere ghosts. Contrary to what Frege states, in Husserl's

*Philosophy of Arithmetic* we already find two different kinds of representations: subjective and objective. Moreover, objectivity is clearly defined in that work. Frege's attack seems to be directed at certain foundational doctrines then current in Weierstrass's Berlin School, of which Husserl and Cantor cannot be said to be orthodox representatives.

Furthermore, various sources indicate that Husserl changed his mind about psychologism as early as 1890, a year before he published the

*Philosophy of Arithmetic*. Husserl stated that by the time he published that book, he had already changed his mind...that he had doubts about psychologism from the very outset. He attributed this change of mind to his reading of Leibniz, Bolzano, Lotze, and David Hume. Husserl makes no mention of Frege as a decisive factor in this change. In his

*Logical Investigations*, Husserl mentions Frege only twice, once in a footnote to point out that he had retracted three pages of his criticism of Frege's

*The Foundations of Arithmetic*, and again to question Frege's use of the word

*Bedeutung* to designate "reference" rather than "meaning" (sense).

In a letter dated May 24, 1891, Frege thanked Husserl for sending him a copy of the

*Philosophy of Arithmetic* and Husserl's review of Ernst Schröder's

*Vorlesungen über die Algebra der Logik*. In the same letter, Frege used the review of Schröder's book to analyze Husserl's notion of the sense of reference of concept words. Hence Frege recognized, as early as 1891, that Husserl distinguished between sense and reference. Consequently, Frege and Husserl independently elaborated a theory of sense and reference before 1891.

Commentators argue that Husserl's notion of noema has nothing to do with Frege's notion of sense, because

*noemata* are necessarily fused with noeses which are the conscious activities of consciousness.

*Noemata* have three different levels:

- The substratum, which is never presented to the consciousness, and is the support of all the properties of the object;
- The
*noematic* senses, which are the different ways the objects are presented to us;
- The modalities of being (possible, doubtful, existent, non-existent, absurd, and so on).

Consequently, in intentional activities, even non-existent objects can be constituted, and form part of the whole noema. Frege, however, did not conceive of objects as forming parts of senses: If a proper name denotes a non-existent object, it does not have a reference, hence concepts with no objects have no truth value in arguments. Moreover, Husserl did not maintain that predicates of sentences designate concepts. According to Frege the reference of a sentence is a truth value; for Husserl it is a "state of affairs." Frege's notion of "sense" is unrelated to Husserl's noema, while the latter's notions of "meaning" and "object" differ from those of Frege.

In fine, Husserl's conception of logic and mathematics differs from that of Frege, who held that arithmetic could be derived from logic. For Husserl this is not the case: mathematics (with the exception of geometry) is the ontological correlate of logic, and while both fields are related, neither one is strictly reducible to the other.

### Husserl's Criticism of Psychologism

Reacting against authors such as J. S. Mill, Sigwart and his own former teacher Brentano, Husserl criticised their psychologism in mathematics and logic, i.e. their conception of these abstract and a-priori sciences as having an essentially empirical foundation and a prescriptive or descriptive nature. According to psychologism, logic would not be an autonomous discipline, but a branch of psychology, either proposing a prescriptive and practical "art" of correct judgement (as Brentano and some of his more orthodox students did) or a description of the factual processes of human thought. Husserl pointed out that the failure of anti-psychologists to defeat psychologism was a result of being unable to distinguish between the foundational, theoretical side of logic, and the applied, practical side. Pure logic does not deal at all with "thoughts" or "judgings" as mental episodes but about

*a priori* laws and conditions for any theory and any judgments whatsoever, conceived as propositions in themselves.

*Here ‘Judgement’ has the same meaning as ‘proposition’, understood, not as a grammatical, but as an ideal unity of meaning. This is the case with all the distinctions of acts or forms of judgement, which provide the foundations for the laws of pure logic. Categorial, hypothetical, disjunctive, existential judgements, and however else we may call them, in pure logic are not names for classes of judgements, but for ideal forms of propositions.*

Since "truth-in-itself" has "being-in-itself" as ontological correlate, and since psychologists reduce truth (and hence logic) to empirical psychology, the inevitable consequence is scepticism. Psychologists have also not been successful in showing how from induction or psychological processes we can justify the absolute certainty of logical principles, such as the principles of identity and non-contradiction. It is therefore futile to base certain logical laws and principles on uncertain processes of the mind.

This confusion made by psychologism (and related disciplines such as biologism and anthropologism) can be due to three specific prejudices:

1. The first prejudice is the supposition that logic is somehow normative in nature. Husserl argues that logic is theoretical, i.e., that logic itself proposes

*a priori* laws which are themselves the basis of the normative side of logic. Since mathematics is related to logic, he cites an example from mathematics: If we have a formula like (a+b)(a-b)=a²-b² it does not tell us how to think mathematically. It just expresses a truth. A proposition that says: "The product of the sum and the difference of a and b

*should* give us the difference of the squares of a and b" does express a normative proposition, but this normative statement

*is based on* the theoretical statement "(a+b)(a-b)=a²-b²".

2. For psychologists, the acts of judging, reasoning, deriving, and so on, are all psychological processes. Therefore, it is the role of psychology to provide the foundation of these processes. Husserl states that this effort made by psychologists is a "metábasis eis állo génos" (Gr. "a transgression to another field"). It is a metábasis because psychology cannot possibly provide any foundations for

*a priori* laws which themselves are the basis for all the ways we should think correctly. Psychologists have the problem of confusing intentional activities with the object of these activities. It is important to distinguish between the act of judging and the judgment itself, the act of counting and the number itself, and so on. Counting five objects is undeniably a psychological process, but the number 5 is not.

3. Judgments can be true or not true. Psychologists argue that judgments are true because they become "evidently" true to us. This evidence, a psychological process that "guarantees" truth, is indeed a psychological process. Husserl responds by saying that truth itself as well as logical laws always remain valid regardless of psychological "evidence" that they are true. No psychological process can explain the

*a priori* objectivity of these logical truths.

From this criticism to psychologism, the distinction between psychological acts and their intentional objects, and the difference between the normative side of logic and the theoretical side, derives from a platonist conception of logic. This means that we should regard logical and mathematical laws as being independent of the human mind, and also as an autonomy of meanings. It is essentially the difference between the real (everything subject to time) and the ideal or irreal (everything that is atemporal), such as logical truths, mathematical entities, mathematical truths and meanings in general.