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Husserl's Philosophy of Mathematics

Edmund Husserl (Courtesy of Husserl-Archive Leuven)

Husserl's Background in Philosophy and Mathematics

    Edmund Husserl was born on April 8, 1859, in Moravia, which then belonged to the Habsburg Empire and now part of the Chec Republic.  He studied mathematics, but he was very interested in philosophy and he became Franz Brentano's disciple, very prominent philosopher in Vienna and Austria.   Brentano is a philosopher who received a strong Aristotelic influence, and he was a strong proposer of psychologism.

    At that time, the problem of the intentionality of consciousness was very much discussed.  How does consciousness directs itself towards an object?  There were two possible answers to this question.

(1)    The relational point of view:  There exists a relation between the intentional act and an object.  If Robert observes a red square, then he and his act of observation is in relation with the object he observes.

(2)    The adverbial point of view:  Looks at intentionality as a certain characteristic and a subject or intentional act:  linguistically this can be expressed, e. g.:  "Robert sees redly" or "Robert sees squarely."

Brentano took this second posture in a radical manner:  the object of consciousness exists in the mind, the object can be intentionally inexistent if it is not an object of consciousness.  This point of view had its problems, because it is basically saying that consciousness cannot transcend its realm to refer to external objects.  Though this relational point of view recognizes the existence between an act of consciousness and an object, it doesn't have any theory that gives any account of an uniform structure of consciousness that is valid for every act.

    Kasimir Twardowski, in his book, On the Content and Object of Presentations (1894), makes a difference between act, content and object of presentation.  A dream or a hallucination has content and object, even if it doesn't exist.  It can be that we can present the image of Pegasus to our consciousness, and it becomes an intentional object of consciousness.  Twardowski, in this manner, offered a better relational conception of intentionality, because it could distinguish between veridical experiences and non-veridical experiences of consciousness.

    Alexius Meinong, who was also Brentano's disciple, used the relational theory of intentionality proposed by Twardowski in a systematic manner.  He stated that every intentional act is directed towards an object, and that this object is beyond its existence or non-existence.  Describing the relational thesis this way, it falls in the problem of adverbialism,  we can't establish an adequate relation between veridical acts.  How can an intentional act direct itself to objects which are beyond "being or existing" and "non-being" or non existing?

    However, there is an intentional content, notion defended by Bernard Bolzano, a not very known philosopher at that time, who elaborated a logical proposal.  He was aware that the lekton, that which is expressed through language and objective propositions, reappears in the history of logic.  The idealists and rationalists like Descartes and Leibniz worked with the ambiguous notion of "idea."  Bolzano made a difference between "subjective" and "objective" ideas.   An objective idea is the "idea in itself,"  and the objective content of a complete sentence is a "proposition in itself", existing independently of the human mind.

    Husserl was also influenced greatly by another of Brentano's disciples, Carl Stumpf, whose philosophy stated an introspective description, giving account to Hermann Helmholtz's studies.

    This is generally the known aspect of Husserl's background, specially with respect to phenomenology.  However, there is another background that played a major role in Husserl's philosophy and it is ignored quite often.  Claire Ortiz Hill, in her essays, in a book co-authored by Guillermo Rosado Haddock, Husserl or Frege?, studies the influence of modern mathematics in Husserl.  He was Karl Weierstrass' disciple (1878-1881) and then his assistant (1883), Weierstrass gave courses in theories of functions.   It was in this moment that Husserl was interested in providing foundations for mathematics.  Later, he became Brentano's disciple, who at that moment wanted to reform logic basing its philosophy in Bolzano's work from a psychological point of view.  After that, Brentano sent Husserl to study at Halle to write his Habilitionsschrift about theory of numbers under the direction of Carl Stumpf, who was at the time helping Gottlob Frege to write his Begriffsschrift (Conceptual Notation).  It was Stumpf who suggested to Frege to write a book in which he explained his conceptual notation clearly, and it was then that Frege published in 1884 Die Grundlagen der Arithmetik (The Foundations of Arithmetic).  Husserl used this publication to write his Philosophy of Arithmetic.  In Halle, he met also Georg Cantor, the one who formulated set theory, who was also continuing Bolzano's work, and became Husserl's colleague (1886-1901).  After that, Husserl became part of David Hilbert's circle at Göttingen, and tried to work on the completeness of arithmetic (1901-1916).  This means when Husserl was between 19 and 57 years old, he was in contact with the most influential mathematicians of his time, which not even Russell nor Frege nor their disciples had so frequent contact.  Husserl studied Frege's Foundations of Arithmetic, and he himself states of the great pleasure he felt reading it.  When Husserl published his Philosophy of Arithmetic (1891), he intended to give a psychological foundation of mathematics.  He sent a copy of his book to Frege, which Frege later reviewed in 1894 criticizing it severely because of Husserl's psychologist proposal for mathematics and logic.

    A good deal of factors helped to obscure Edmund Husserl's background on this subject.  Willard van Orman Quine's disciple, Dagfinn Fřllesdal, wrote a master's thesis on Frege and the origins of the phenomenological movement, and he suggested in it that Frege's 1894 review of Husserl's Philosophy of Arithmetic made Husserl change his mind to anti-psychologism.  It is evident that this is very far from the truth.  First of all, it seems that Frege had very little to do with Husserl changing his mind about the foundations of mathematics. Philosophy of Arithmetic represented his thinking up to 1890.  Even in that year, Husserl made a difference between sense and reference, which was recognized by Frege himself in 1891.  It was apparently that year that Husserl started to change his mind.    The anti-psychologist posture he adopts in his Logical Investigations developed between 1890 to 1895.  Besides, according to Husserl himself, what influenced him to change his mind were his readings of Hume, Bolzano, and Lotze in the years 1890-1891.  Even though Husserl embraced Platonism, any look at the eleventh chapter of the "Prolegomena of Pure Logic" (Logical Investigations) would be enough to notice that Husserl's views on logic and mathematics are quite different from those of Frege.

    Other factors which contributed to the ignorance of Husserl's work in analytic philosophy is due to some phenomenologists who don't want generally to deal with his views on logic and mathematics.  Many of the criticisms Husserl made to psychologism in his "Prolegomena of Pure Logic" (Logical Investigations) are seen by them as an early immature, pre-phenomenological period of his career.  Some analytical philosophers even go as far as to state that after this rabid anti-psychologist philosophy supposedly embracing Frege's views, he later fell in psychologism again in his Logical Investigations (See W. Berth, Michael Dummet, Fřllesdal, Sluga).

    As Hill points out also, another problem has to do with the fact that sometimes Husserl was not very explicit in linking his ideas to those of his contemporaries.  "He rarely named names and when he did he seemed to believe that connections were obvious that are all but invisible to us nowadays.  For example, in a note to Ideas §72, he wrote that the close relation between his own concept of definiteness and the axiom of completeness introduced by Hilbert in his foundations of arithmetic would be apparent to every mathematician without further remark" (Hill and Rosado, xii)

    Was on his analytical conception of logic and mathematics that he developed phenomenology.  So it is very important that even Husserl's followers don't ignore this fact and also get acquainted with it.   It is the purpose of this page to expose his analytic philosophy, his Platonist doctrine concerning logic and mathematics and also his Platonist epistemology of mathematics and logic.  I won't expose all of his doctrine concerning these areas, I will just mention the most important aspects of his analytical philosophy.

    Claire Ortiz Hill wrote an essay called "Husserl's Mannigfaltigkeitslehre" (Hill and Rosado 161-178) in which she summarizes the problems which Husserl was confronted with during his "crisis years" (the beginning of the 1890's).  It was obvious that Husserl wanted to give a psychological foundations of mathematics using the analysis made by Franz Brentano.  However, he later confessed that he didn't find this method to be quite satisfactory to explain mathematical notions, nor their nature.  In his "On the Concept of Number", Husserl said explicitly:  "In truth, not only is psychology indispensable for the analysis of the concept of number, but rather this analysis even belongs within psychology" (Husserl 1887, 94-95). Later, in his Logical Investigations he completely changed his mind.  In fact, Husserl described this process as a torment of his conscience.

    What did make Husserl change his mind?  Claire Ortiz Hill explains:

(1)    His first problem had to do with pure logic and consciousness.  Husserl said that he was tormented by the "incredibly strange realms" of pure logic and actual consciousness.  He found out that pure logic included all of the pure analytical doctrines of mathematics and the entire area of formal theories, or what he called Mannigfaltigkeitslehre (theory of manifolds) in the broadest sense, traditional syllogistic, pure theory of cardinal numbers, the pure theory of ordinals, Cantorian sets, etc.  He realized that everything that was purely logical was in itself ideal and had nothing to do with acts, subjects, or empirical reasons.  He concluded that these two realms should be interrelated and form a whole, but if one assumes Brentano's methods, it's impossible to join them (Hill and Rosado 164).

(2)    Though he believed at first that pure logic belonged to the area of psychology, he realized that there had been connections in which such a psychological foundation never came to satisfy him, and could not bring continuity and unity.  It was not possible to reconcile the objectivity of mathematics and pure logic with subjective psychology (Hill and Rosado 164-165).

(3)    Husserl also states that in his Philosophy of Mathematics he tried to achieve clarity with respect of the true meaning of the concepts of set theory and the theory of cardinal numbers by going back to spontaneous activities of collective and counting in which the sets and cardinal numbers are given.  He started using the word Mannigaltigkeit when he studied Riemannian manifolds.  Husserl's manifolds would finally bear little resemblance to Cantor's Mannigfaltigkeiten except as concern their Platonism.  He would be tormented by doubts about the psychological analysis of sets (Mengen).  It was obvious that a "collection" is not a physical entity composed of items collected.  He thought then that the idea of set must arise through psychological reflection upon the act of collecting, and therefore there was a direct relation between consciousness and pure logic.  He later found out about the antinomies of set theory, specially the Zermelo-Russell paradox:  the class of all the classes not belonging to themselves; and he realized that this view he upheld was also completely wrong (Hill and Rosado 166).

(4)    Husserl also had reflected about the differences between Leibniz's vérités de raison and  vérités de fait; Hume's relations of ideas and matters of facts, and also between Kant's analytic and synthetic judgments.  He would state that Kant's logic was defective.  Kant didn't understand the nature and role of formal mathematics, and that his concept of analycity was completely wrong.  His principle of analytic propositions did not help to clear up the achievements of analytic thinking. (Logical Investigations I, §58; II, 6, §66) (Hill and Rosado 166-167).

(5)    Also Husserl was confronted with the problem of "imaginary numbers" or "imaginary concepts" (for him, "imaginary numbers" refer to negative roots, negative numbers, irrational numbers, fractions, infinite sets, etc.)  These contradictory and incredible "impossible" numbers have been held, but that hasn't prevented mathematicians from their use.  What did justify the use of such apparently meaningless signs in calculations, or in deductive thought?  In his "On the Concept of Number" he seemed to support along with his teacher Karl Weierstrass, that these numbers had to do with artificial constructions  which have their origin and support in elementary concepts of number and in the relations connecting them.  However, afterwards, in a letter to his friend Carl Stumpf he stated that the theory that the concept of cardinal number forms the foundation of general arithmetic that he had raised to develop in "On the Concept of Number" was completely wrong and false.  He explained that by no clever devices we can derive negative, rational, irrational, and the various sorts of complex numbers from the concept of cardinal number; and that this was true of the ordinal concepts, of the concepts of magnitude, etc.  These concepts themselves are not logical particularizations of the cardinal concept (Hill and Rosado 167).

    If psychologism is not the foundation of pure logic, then Husserl had to start his Logical Investigations with the refutation of this theory in his "Prolegomena of Pure Logic".  If  pure logic is not founded in psychology, or on the empirical world, there must be another realm of ideal objects, the realm of essences.


Husserl's Logical Investigations I:  Summary of the "Prolegomena of Pure Logic"

Introduction

    In order to offer legitimate bases to his Logische Untersuchungen (Logical Investigation), Husserl writes his "Prolegomena of Pure Logic" where he offers a decisive response to the arguments of psychologism.  Psychologism pretends to establish the basis of logic in the acts of thinking of the human mind.

    Husserl identifies three philosophical points of view concerning logic, and of these the psychological one seems to dominate philosophy.  To adequately criticize this point of view, it becomes necessary to consider some questions:

1.    Is logic a theoretical discipline or a practical discipline (an "art")?

2.    Is logic an independent science from the rest of the sciences, specifically of psychology and metaphysics?

3.     Is logic a formal discipline or does it refer to a "mere form of knowledge" or should it consider too its own "matter" ?

4.    Is logic a discipline with a priori or demonstrative character, or is it an inductive empirical discipline?

About the Nature of Sciences and of Logic

    Husserl recognizes that we need a normative discipline as foundation for the rest of the sciences.  Logic is useful with this respect.  This would make possible the structure of scientific knowledge, and not have a "random" kind of knowledge.  As a normative discipline, logic helps to structure theoretical knowledge of science to guarantee objectivity.  Through it we can give coherence to that which is presented to us according to a unity of law (§4-5).

    If logic is a normative discipline of science, then we have to elaborate rules that would provide for theoretical coherent formulations.  This leads us to the conclusion that logic is a kind of "art".  Psychologism adopts this position, and Husserl responds this way:  The definition of logic as art; as the art of judging, of reasoning, of knowledge, or of thinking (l'art de penser) is equivocal and too narrow  (§11).  

    It is necessary to establish a difference between:

1.    Logic as a normative science:  Which expresses what should be.

2.    Logic as a theoretical science:  Which expresses purely and simply what is. (§14).

    This difference is very important, because logic can be used as a normative discipline for the rest of the sciences, but logic itself should be founded theoretically independently from all the other sciences.  This means, logic itself does not depend on its use.  The value of logic in its normative dimension depends on the theoretical character of logic.  We have to make a difference between the essential theoretical foundations of logic and its normative function (§14-16).


The Anti-Psychologists vs. the Psychologists

    Husserl goes on to evaluate the statements of anti-psychologists before him, and he is aware that they have not answered in a satisfactory manner the psychologists.  Psychologism had an advantage in these discussions.  This doesn't mean that the anti-psychologists are wrong, it simply means that they have not been clever enough to respond in a definitive manner to psychologists.(§20).  Let's see some of these arguments:


A.    First Reply to Psychologism

    Psychologism states that the notions of truth and false, affirmation and negation, universality and particularity, principle and consequence, etc. can be founded on psychology (§18).  Philosophers like Kant state that this would be completely absurd, because it would reduce logic to subjectivity.  This would mean that the essence of logic is subjective, not objective.  The laws of logic, he says, are not contingent but necessary, because it tells us how to think correctly (§19).

    Kant's arguments are not valid to reject psychologism.  Thinking as it should be is one special case of thinking as it is.  Logical laws obey the psychological laws of thinking.  As Lipps would say:  "Logic is the physics of thinking."  Logic would not go beyond as a kind of ethics of thinking, as Herbart would say, and it would correspond to the physical nature, according to natural laws.  Therefore, pure logic is nothing more than a part of psychology (§19).


B.     Second Reply to Psychologism

    The anti-psychologists would state the following too:   representations, judgments (acts of judging), thinking processes, etc, also would be part of psychology.  However, these are looked differently from a logical point of view.  When one is talking about "law" in psychology, it is talking about causal laws and the connections in the processes of conscience and the succession of these processes.   Logic doesn't talk about psychological processes, but about the truth of the content of the statements.  It asks about how these operations should flow so that the judgments be true (§19).

    The psychologist wouldn't feel himself uneasy with this argument.  Psychologists don't deny that in effect logic is about different issues than those of psychology.  However, logic is made for a teleological purpose:  It is about those rules that let us reach a certain objective.  From this theoretical point of view, logic is part of psychology.  Here we can see the causal criteria very clearly.  Psychologically we make logical laws with some general or particular goals in mind (§19).


3.    Third Reply to Psychologism

    Immediately, anti-psychologists detect an apparent fallacy in the psychologistic argument.  If we try to explain the laws of psychology having logical laws as premise and having them as true, then the argument falls into infinite regress:  logical laws would be founded in psychology, which itself is founded on logical laws (§19).

    Psychologists would answer this claim saying that it is not that they suppose logical laws to explain psychological processes, but it's a fact that we think according to logic in the psychological sciences.  One thing is infer according to logical laws, and quite another to infer taking logical laws as premises.  There would only be in infinite regress if we infer supposing logical laws (§19).

Beginning of the Refutation of Psychologism by Husserl

    Husserl notices that no anti-psychologist has hit the jackpot refuting psychologism.  It is necessary first to see the consequences of psychologism and then argue against it from its own theories.


A.    First Consequence of Psychologism

    Psychology pretends to be a science of facts and of experience.  Therefore, the psychologist procedure is inductive.  This means that an abstraction process is carried out starting from the particulars of the physical world.   The laws of psychology are only vague and probable, and they don't pretend to have any kind of infallible precision about the prediction of circumstances on experiences obtained through them with certain insecurity, owed to its probabilistic character.  Vague theoretical basis can only be foundation of vague rules (§21).

    On the other hand, logic is always certain in itself.  If logic is a part of psychology, its own laws would also be probable and vague;  therefore, also logic would be inexact.  This is not the case, because the laws of logic are absolutely valid.  They are laws in the authentic sense, because they are not approximately true, but always what is true.  It would be absurd to say that a fallible mind can produce infallible entities (§21).

B.    Second Consequence of Psychologism

     It can be that some psychologist would deny the infallibility of logical laws.  However, we have to distinguish between the inductive activity and the necessary and a priori character of logic.  Sciences can state laws basing themselves from the activity of abstraction of particular behaviors in the world and formulate theories about them.  In this sense no law of science is a priori.  No scientist would consider any law of science, not even the one most successful, as an infallible law formulated as such, because it is always subject to change or refutation.  But the laws of logic are always are a priori and necessary (§21).

    Psychologists would say that our mind forces us to think logically in a certain sense.  To this Husserl argues:  Let's think that there is a man whose entire thinking flows according to the laws of logic.  One thing is that thoughts succeed one another according to logical laws, and quite another the validity of the laws of logic.  If we make a calculator, we can design it so that it gives us "4" when we click the buttons "2" "+" "2" according to the laws of arithmetic.  However, it is another thing to say that the laws of arithmetic are explained as a result obtained in the calculator; the obvious response to why "4" is obtained when we click "2" "+" "2", because of its design, and not in the arithmetical laws themselves.  We can design the calculator so that it operates according to the laws, but the calculator and its design are not the foundation of arithmetical laws.  The calculator doesn't have the faculty of reasoning, not of understanding itself.  Therefore, the psychological operation of a man (be it as perfect as it may be) responds to a psychological structure, not to the laws of logic and mathematics (§22).


C.    Third Consequence of Psychologism

    If logic is a consequence of psychological facts, then we conclude two things:

1.    Logical laws would be laws of psychological facts;

2.    They would suppose the existence of these facts (§23).

    Husserl states that both are completely false.  Logical laws don't contain in them psychological facts, nor representations, nor judgments (i.e. acts of judging).  "If all dogs are animals, and all animals are living beings, then all dogs are living beings."  This reasoning process obviously includes experience, empirical objects such as dogs, animals and living beings.  However, we can get rid of these objects of experience and we have an equally true statement:  "If all A are B, and all B are C, then all A are C" in which the variables A, B, C can be substituted by any object, and this logical statement has no reference to empirical facts (§23).

    The same thing happens in arithmetic.  If 3 > 2, the three books of that table are greater than those two books in that corner.  But pure arithmetic doesn't talk about things from experience, but of numbers in their pure generality (the number 3 is greater than the number 2).  If one can admit this, then one has to consider naturally inadmissible that logical laws and mathematical laws (taken in their purity) be laws of activities or products of the human mind (§23).


Logical Relativism of Psychologists

    Husserl does not content himself only with these answers to psychologists, but he also shows that any attempt to make psychology as a foundation of logical laws would instantly fall in relativism.  That's why he states that psychologism only lives of inconsequences (§25).

    Let's take John Stuart Mill, for example.  He characterizes the principle of no-contradiction as a generalization of events of experience. He says that to believe and not to believe are different states of the soul (mind) and they exclude themselves.  We may direct our awareness of light and shadow, noise and silence, equality and non-equality, go forward or go back, successive and simultaneous events; in other words every positive phenomena and its negative as different phenomena.  He considers the principle of no-contradiction as a generalization of these facts (§25).

    However, Mill, as in the case of other psychologists, fails to present a link between the generalization of facts and the validity of the principle of no-contradiction.  His argument is reduced to the fact that two opposite acts of faith can't coexist because our mind resists against that happening.  But Husserl states, then, how is it possible that many times there are people that would support contradictory notions?  Are not there people, who, even when they have a certain degree of coherence,  are capable of supporting opposite statements? (§25)

    It could be argued that the laws of logic are only valid for a "normal" human being.  With this captious and tricky response, they pretend to give an answer to the validity of this principle.  They would fall inevitably in ambiguity and much complexity explaining psychologically this "normality."  However, the principle of no-contradiction, as well as all logical laws, are always valid without exception and it is absolutely exact.  That's why Husserl says that Mill, sagacious in other occasions, appears to be abandoned of all gods when it comes to his fundamental principles of his empiricist prejudices (§25).

    Then, Husserl makes a very important difference between individual relativism and specific relativism (§34).  Specific relativism states that for each species of beings, when they judge that something is true, then it's because its physical or psychological constitution cannot decide the contrary.  However, individual relativism states that truths are accepted by an individual because of convenience, that there are no truths as such.  For Husserl, this latter form of relativism can be refuted as soon as it's formulated, because essentially another way to say this is:  "It is true that there is no truth" (§35).

    In the case of specific relativism, which many times takes the form of anthropology, he presents various arguments:

1.    The truth of the specie is relativized, because there can be human beings which can support one statement as true, and other non-human species support different truths.  This is absurd, because it states that the validity of logical principles depend in which specie, human or non human, recognize it or not (§35).

2.    We could say that there can be other species that don't obey logical principles.  It could be that in the judgments of a certain specie there could be propositions and truths that wouldn't obey logical principles.  In that case, or that specie understands by the words "true" and "false" the same way we humans conceive it, in which case there wouldn't be invalid logical principles; or they could understand the words "true" and "false" meaning another thing, in which the problem here becomes what are they wanting to say by the words "true" or "false". But in either case, logical principles are not invalid (§35).

3.    The constitution of the specie is a fact, and the foundation of facts are facts themselves. However, this point of view fails to see the difference between facts and truths.  Truths are not subject to accidents nor temporal determinations as we shall see later (§35).

4.    Truth in itself doesn't exist, because this notion is based on the constitution of the human specie in general.  Here we have the same problem of individual relativism.  We can state that "It is true that there is no truth."  Also, the existence of the human specie is contingent, but logical laws are always necessary.  Therefore, it would be false that all truth is founded on the constitution of the human specie (§35).

5.    If truth is founded in the truth of a specie, then it could be true that a certain specie doesn't exist, but in that case we would fall into an absurdity:  we could say that truth that there is no specie depends on the existence of that specie.  Even if we could say that truth of the existence of a specie depends on the factual existence of that specie, it would remain absurd:  in this case, we could consider a possible hypothetical specie from the point of view of specific relativism, we have to consider the human race, in such a way, the truth of the proposition that says that the human specie exists would be based on a special constitution of the human specie (§35).

6.    The relativity of truth implies the relativity of the existence of the world, because the world is only a unity of a correspondent ideal system of all factual truths which can't be separated from it.  In the act of relativizing the subject, you relativize the object.  However, if it would be that no specie would have been constituted in such a way that it would recognize the existence of the world, then the world wouldn't exist.  It would be impossible to explain how is it possible that science would have discovered species of animals and geologic events which took place before the existence of human beings (§35).


The Prejudices of Psychologism

   We have seen up to now the consequences of reducing logic to psychology.  Now Husserl criticizes the bases of psychologism itself.  Psychologism contains in itself some prejudices that make it fall into conceptual inconsequence

A.    First Prejudice

    The first prejudice states the following:  "The precepts that regulate psychical processes are founded in psychology.  Therefore, it is also evident that normative laws of knowledge are also founded in psychology of knowledge" (§41).

    With this statement we fall immediately in the fallacy that logical laws are laws of rightful thinking, and the theoretical dimension of it is not recognized.  It is the theoretical aspect of logic which is the foundation of normative logic.  The laws of logic themselves don't state anything about how we should think, but establish simply what is true.  Arithmetic, which is a discipline related to logic establishes what is theoretically true.  Let's look at this mathematical statement.

(a + b) (a - b) = a2 - b2

This statement doesn't talk about judgments, and not even the way they should take place, but establishes simply what is true.  Therefore, this statement is theoretical and not practical.  Now, if we say:  "To find the product of the sum and difference of two numbers, we obtain the difference of the square of both numbers" we have established a normative and practical rule which is founded in the theoretical statement expressed above (§41).

    It is in this manner that all truth in general, be it psychological or not, establish the basis of a rule for our rightful thinking.  In this way we can establish the possibility of the existence of rules of judgment which are not founded in psychology, but in theoretical logic.   This first psychological prejudice fails to see the difference between the laws of being true (logic) and the laws of thinking as a psychological process (psychology) (§41).

B.    Second Prejudice

    The second prejudice requires a little more insight to refute it, because psychologists state that to provide theoretical foundations, we require a psychological process to determine truth or falsehood, probability, necessity, etc, and those judgments are lived in a psychological level (§44).

    Because arithmetic and logic are related, it would be convenient to look for an analogy with mathematics to refute this psychologist claim (§46).   To the psychologist it is natural that the act of counting and operating with numbers is a psychological act in which time has to take place.   However, arithmetic is of a different nature, because there are ideal objects 1, 2, 3, etc which don't talk about the facts of experience nor of their location in time.  One thing is the relation between numbers through sum or multiplication, and another the acts of adding or multiplying, which are verified in the accidental life of experience (§46).

    According to Husserl, numbers are also different from their representations.  The number five is not the act of counting to five, nor is it my representation of five (§46).

    We have to notice that the object of the act of representation is not psychological.  The object of my representation is an ideal specie that is made concrete in individual cases by the act of numbering.  However, this ideal specie can't be considered a psychological life as such.  We can imagine five objects whatsoever and we carry out an intuitive "abstraction" of a certain structural form until we obtain the idea of five, and the notion of the number five comes to be in the consciousness of the thinker.  The structure can be grasped getting rid of the accidents of experience.  We can obtain an ideal specie of the form which is only one, independently of the act that grasps it.  This idea remains independently of contingency of acts or temporality.  The acts of grasping or counting start and finish.  But in the case of numbers themselves, they don't depend on the act of being grasped, nor do the depend on time (§46).

    The same goes to all mathematical laws which refer to these ideal individualities and do not state anything about what is physically real, nor of how is something counted, nor of the acts of counting.  One thing is the truths of arithmetic, and another the processes of arithmetical thinking which do belong to the science of psychology.  According to Husserl, the laws of arithmetic say nothing about our representations in our minds. It only deals with numbers and their combinations in their abstract purity and ideality.  They are founded purely in the ideal essence of numbers.  These laws are completely ideal and they are can't be reduced to a universal empirical proposition (§46).

    The same thing happens to pure logic.  The logical notions have a psychological origin, but that doesn't mean that logical laws originate in psychology.  The process of investigation and demonstration are psychological.  However, the justifications of these laws are not psychological and are not related to empirical events (§46).

[Note:  Though in his "Prolegomena" Husserl states that numbers are species that instantiate, he would find out that this characterization of numbers is inadequate.  For example, the specie "red" which is an abstract entity which cannot be perceived by the senses, instantiates itself in many marks of red.  However, numbers don't instantiate in sensible objects, therefore, they can't be species (Experience and Judgment §95, 368-369), though they remain ideal.  Numbers are ideal entities that we abstract, not like species, but as formal categories.]

C.    Third Prejudice

   The third prejudice states that all truth is based on judging, and judging is a psychological process in which the evident is recognized and all evidence consists of a peculiar psychological character and very well known in inner experience, a feeling sui generis that guarantees the truth of the judgment along with it (§49).

    Husserl has no problem in that the act of knowing the truth would be carried out psychologically in an intellection process.  Nevertheless, these logical laws don't tell us anything about evidence in this psychological sense.  Everything that is stated in the theoretical dimension of logic can be used to be applied to particular empirical case, showing its evidence.  But these empirical cases don't prevent logical laws from being a priori (§50).

    In the proposition:  "A is true" we don't say the same thing as  "It is possible that somebody judges that A is true."  We don't speak about intellective processes.  The proposition a + b = b + a says nothing of the acts of counting nor of adding, just simply what is; independently of our intellection processes of numbers.  These logical and mathematical propositions are not empirical individualizations.  They are ideal objects (§50).

Some of Husserl's Conclusions as Consequences of his Criticism to Psychologism

   As a result of these criticism against his former philosophy, psychologism; then comes the question, if psychology is not the basis for logic, then what is?  Logical laws don't tell us about evidence (that which is presented to consciousness).  They can be lived as evident by our consciousness, but if we don't live it, it doesn't mean that logical laws cease to be true.  This would be as absurd as to assume that the sun disappears just because you close your eyes.

   Logical laws as such can't be represented in our minds.  Logic related to mathematics, and the basic example of how this statement is true is the fact that psychology is that statements such as "a + b = b + a" cannot be justified merely because of psychological abstraction.  Psychologists would say that this is the result of the acts of counting and the acts of adding.  However, the statement "a + b = b + a" doesn't say anything about acts of counting or acts of adding, and therefore, it cannot be justified advocating these psychological acts.

   Psychology as such is an empirical science, which gives us that which is "real" (empirical).  Mathematics on the other hand outway empirical experience.  We can formulate in "reality" our problems with three-dimensional bodies.  In mathematics, we can formulate problems about n-dimensional spaces, which cannot be psychologically represented.   We can formulate in mathematics decadic numbers with trillions of units, and there is truth about them:  who can represent themselves such numbers, or carry out the additions, multiplications of them?  It is psychologically impossible, however, speaking is perfectly possible for it to be lived ideally.  The Pythagorean theorem, the number three, etc., are not empirical generalizations or individual instances, or sets of them, they are ideal objects (Logical Investigations I, §50).

   The failure of psychologism consists in the fact that it cannot distinguish between the ideal from the real realm.  This is a distinction that has been made by previous philosophers.  Leibniz distinguished between vérités de raison (truths of reason) and vérités de fait (truths about facts).  Even a philosopher as extremely empiricist as David Hume distinguished between relations of ideas and matters of fact.

   There are two separate spheres, one ideal and the other real.  The two cannot be reduced to one another.  Contrary to the empirist and psychologist prejudices, the ideal realm guarantees objectivity.  It presents itself as evidence, this means, as a living of the truth.  This doesn't mean that this ideal realm ceases to be, just because in some occasions it is not evident to us.  The ideal realm is the guarantee of all possible manners of a living of the truth.  The justification of "2 + 2 = 4" doesn't lie on the fact that it is said in the temporal living, but as an statement in specie, to the pure meaning of 2 + 2 = 4 which holds true always (Logical Investigations I, §51).

[Note:  This ideal realm is similar to Frege's third realm.  Frege stated that logical laws couldn't be found in the first realm (physical realm), nor the second one (psychological realm), but in a kind of third realm, which Husserl identifies as the ideal.  So, it is clear the way Husserl made a Platonic turn in his Logical Investigations.]

    For Husserl, this analysis presents us the realm of the ideal.  We should ask ourselves, then, if what is the relation between the ideal realm and the real one.  Since we cannot reduce one to the other, then it becomes epistemologically imperative to know what is the relation among these.

    If the ideal realm is neither material nor psychological, then how do we come to know them?  Since Husserl was also interested in epistemological matters, he addresses this question in a very unique way, using phenomenology as the epistemological foundation for logic and mathematics.

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(Husserl's Picture courtesy of Husserl-Archives Leuven)

Copyright (c) 2001-2003,  Pedro Rosario Barbosa.
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