This is a preprint version of the paper that appeared in definitive form in the New Yearbook for Phenomenology and Phenomenological Philosophy (Abingdon: Routledge), Burt Hopkins (ed.), vol.XIII, 2013, pp. 62-83. The published version should be consulted for all citations.

The Strange Worlds of Actual Consciousness and the Purely Logical

By Claire Ortiz Hill


It was troubling encounters with what he called the “incredibly strange” worlds of the purely logical and of actual consciousness that he experienced while attempting to achieve clarity regarding the true meaning of the concepts of set theory and the theory of cardinal numbers in the Philosophy of Arithmetic that Edmund Husserl said launched him on his phenomenological voyage of discovery (Husserl 1994, 491-92; Husserl 1929 §27a; §24 and note; Husserl 1913b, 34-35).

As he struggled to understand how those two worlds interrelated and formed an intrinsic unity, he found himself facing riddles, tensions, mysteries, and “great unsolved puzzles” concerning the very possibility of knowledge in general. He described himself as having been “unsettled−even tormented”, powerfully gripped, by the deepest problems and drawn “close to the most obscure parts of the theory of knowledge”. He was assailed by questions. If everything purely logical is an in-itself, something ideal having nothing at all to do with acts, subjects or empirical persons belonging to actual reality, then how is symbolic thinking possible? How are objective, mathematical and logical relations constituted in subjectivity? How does one go from mathematics to pure logic elucidated by a theory of knowledge? How can the mathematical-in-itself given to the mind be valid? If scientific knowledge is completely based upon being able to abandon oneself completely to thought that is removed from intuition, or being able to prefer such thinking over thought more fully in accord with intuition, how is rational insight possible? How does one then arrive at empirically correct results? (Husserl 1994, 37, 167-69, 491-93; Husserl 1913b, 17, 22, 35)

Husserl searched until the end of his life for answers to such questions. So it is that in Experience and Judgment, Investigations in the Genealogy of Logic, he confronted the challenge of investigating the origin and subjective foundation of traditional Aristotelian formal logic by piercing through the logic of subject and predicates to reach the world of actual consciousness. (Husserl 1939, §§1, 3, 10, 11)

Volumes of Husserl’s lecture courses (for example, Husserl 1896; 1902/03a; 1902/03b; 1906/07; 1908/09; 1996) first published during the last few decades are now shedding considerable light on many recondite places of his thought and are showing how single-minded he was. In particular, they show how far back the investigations of Experience and Judgment reach and provide a clearer idea of what was at stake for Husserl in that book, which echoes numerous passages from those courses. For example, Husserl was already developing the phenomenological method and teaching that all questions concerning the relationship between objectivity and subjectivity were ultimately to be answered by going back to the sources from which logical ideas originate in Allgemeine Erkenntnistheorie, Vorlesung 1902/03 (Husserl 1902/03a).

The publication of this material means that it is now possible to see well what Husserl came to locate in each of the two worlds−and in the different parts of each one−and to have a good understanding of how he envisioned their interrelationship. It is extremely important to see and understand these things because, as Husserl realized, the fact that what belongs in one world, or in another part of one of the worlds, often enjoys the same outward appearance as what belongs in the other world or somewhere else in the same world is often a hard to perceive source of major errors in philosophy.

So, the goal of the following reappraisal of the fundamental tenets of Husserlian phenomenology as exposed in the analyses of the subjective foundations of the part of traditional formal logic to which Experience and Judgment is devoted is to take readers on a tour of Husserl’s strange worlds, to describe what he found in them and how he saw their interaction. My method is exegetical, because to evaluate the ideas of those who ascribe ideas to Husserl that he did not hold—and against which he strenuously militated with all his intellectual might—it is essential to have, and to maintain, a firm grasp on the theories about what belonged in phenomenology, psychology and pure logic that he persistently and consistently defended.

Psychologism and Actual Consciousness

Husserl came to divide the world of actual consciousness into the realm of the psychological and that of phenomenological (Husserl 1994, 491-92). His journey towards the latter, it is well known, began in the realm of the former (Hill 1998). He recalled that when he began the Brentanian psychological analyses of the origins of numbers of the Philosophy of Arithmetic, Psychological and Logical Investigations, it was obvious to him that “what mattered most for a philosophy of mathematics was a radical analysis of the ‘psychological origin’ of the basic mathematical concepts” (Husserl 1913b, 33). In Formal and Transcendental Logic, he characterized Philosophy of Arithmetic as an initial attempt on his part to achieve clarity regarding the authentic meaning of the concepts fundamental to the theory of cardinal numbers and set theory by going back to the spontaneous activities of collecting and counting in which sets and the cardinal numbers are given. He says that that was what he would later come to call a phenomenologico-constitutional investigation (Husserl 1929 §27a).

However, the further he delved into his philosophical investigations into the principles of mathematics, the more he was tormented by doubts as to how to reconcile the objectivity of mathematics and all of science in general with psychological foundations for logic, and that led him to engage in critical reflections on the essence of logic and on the relationship between the subjectivity of knowing and the objectivity of the content known (Husserl 1900/01, 42). He recalled how his doubts disturbed and even tormented him and then extended to all categorial concepts and ultimately to all concepts of objectivities of any kind whatsoever (Husserl 1913b, 35).

According to Brentano’s teachings, the idea of set was to arise out of the unifying consciousness of meaning-together, in the conceiving as one. That could not be physical, so the concept of collection had to arise through psychological reflection upon the act of collecting. But, Husserl realized that the concept of number itself had to be something fundamentally different from that of collecting, which was all that could result from reflection on acts (Husserl 1913b, 34-35).

When it was a matter of the origin of mathematical presentations, or of the elaboration of practical methods that are actually psychologically determined, he concluded, psychological analyses are appropriate and justified. Mental experiences of knowing have their place in psychology, but what is known is not psychology—not the natural science of mental individuals, their real experiences and experiential mental states—just because it is known in the mental experience of knowing. Out of subjective experiences of perceiving, presenting, we form concepts and we judge and draw conclusions. But, once one tries to go from the psychological connections of thinking to the logical unity of the thought-content, no true continuity and clarity can be established. Scientific investigating and thinking is subjective, as is the relationship between subjective experiences of presenting, judging, deducing, proving, theorizing, etc. But, the experience of presenting is not a concept, judging is not a proposition or truth, theorizing is not a theory. Science in the objective sense, as divorced from the scientifically investigating or learning subject, is a web of theories and so of proofs, conclusions, propositions, concepts, meanings, and none of those are lived experiences, but are certain ideal unities standing in relationship to experiences (Husserl 1900/01, 42; Husserl 1902/03a, 16-17; Husserl 1906/07 18b).

Mathematicians, Husserl insisted, do not establish laws for subjective acts, but for ideal objects. The concept of number, Husserl observed, like every concept, refers back to a corresponding intuition. A number is only given in actual counting. Someone who had never counted would not know what a number is, just as someone who had never had a sensation of red would not know what red is. But, this does not mean that number is a psychological concept and arithmetic a branch of psychology. Numbers do not concern what happens in mental experiences of experiencing individuals (Husserl 1906/07 13b) and are not grounded in experience with its actual occurrences. The numbers 1, 2 and so on belong to the series of natural numbers and each such number is a member of the series of cardinal numbers, yet the number 3 is not reproduced if ten people have presentations of 3. Even if thousands of people count 4, the number 4 is a number of the number series (Husserl 1902/03a, 17).

Arithmetic does not obtain its universal propositions by means of perception and empirical generalizations based on perception and on the substantiation of individual judgments resulting from them. It is not in direct contact with acts of counting, ordering, combining, calculating, collecting, etc. The number 2 is just not an object of possible perception and experience. It “is not a thing, not an event in nature. It has no place and no time. It is just not an object of possible perception and ‘experience’. Two apples come into being and pass away, have a place and time. But when the apples are eaten up, the number 2 is not eaten up. The number series of pure arithmetic has not suddenly developed a hole, as if we then had to count 1, 3, 4…”. In the world, he pointed out, there are not alongside trees, houses, etc., also things called number 2, number 3. One can perceive 2 apples precisely while one perceives each apple, but one cannot perceive the 2, he pointed out (Husserl 1902/03a, 118; Husserl 1906/07, §§13c, 18).

Husserl’s profound dissatisfaction with his investigations in the psychological realm drove him to its edge and pushed him over the line into transcendental phenomenology. So, once he had secured the objective theoretical scaffolding needed to keep philosophers from falling into the quagmires of psychologism, he began introducing the phenomenological analyses of knowledge that were to yield the general concepts of knowledge that he hoped would solve all further problems in theory of knowledge.

The Phenomenological Realm

Husserl saw that psychology could not solve the epistemological problems he was facing, but he never considered utterly forsaking the world of actual consciousness. He considered that only those, like himself, deeply distressed by the issues in the most intense way,

compelled by the critical dissolution of the blinding prejudices of psychologism to recognize the purely logical ideal... but... at the same time... compelled by the revealing emphasis upon the essential relationships between the ideal and the psychological... not to abandon by any means the psychological entirely but rather to keep it within view as somehow belonging with the ideal... can also have the insight that such psychological critiques are indispensable for forcing recognition of the ideal as something given prior to all theories.... can realize that the being-in-itself of the ideal sphere in its relation to consciousness brings with it a dimension of puzzles which remain untouched by all such argumentation against psychologism and hence must be solved through special investigations… through phenomenological ones. (Husserl 1913b, 21-22)

The phenomenological analyses of Experience and Judgment stand as an example of the science of radical subjectivity that Husserl pursued until the end of his life. The subjectivity in question is a transcendental subjectivity in which knowers go back to the ultimate sources to question all cognitive constructions and to reflect upon themselves and upon knowing. It is more radical than psychological subjectivity can ever be because it is a desconstructing (Abbau) of all the deposits of meaning already present in the world of our present experience, a going back to question those deposits of meaning at the subjective sources from which they have arisen (Husserl 1939, §§11, 12).

Thus, it became the particular task of Experience and Judgment to elucidate the origin and subjective foundation of traditional Aristotelian formal logic by clarifying the essence of the predicative judgment, because truly philosophical logic requires a radical return to pre-predicative experience, requires that one pierce through the logic of subject and predicates to the foundations of an underlying, hidden, logic in order to elucidate the origin of predicative judgments. The world in which we live, know and judge, and out of which arises everything that affects us that becomes the substrate of a possible judgment, Experience and Judgment explains, is always already given permeated with deposits left by logical operations. From this pregiven world, from this hidden life-world with its deposits of meaning, its science and scientific determination, its veil of ideas, philosophers must think back to the world of experience as immediately pre-given prior to all logical functions. They must re-experience the origination of the operations of idealization from original life-experience. They must think back to the original life-world and the subjective operations out of which it arises. So the task of elucidating the origin of predicative judgments, of establishing their relations to a foundation and of pursuing the origins of prepredicative self-evidence in that of experience is one of going back to the original life-world as the universal ground of all particular experiences, to the world of immediate intuition and experience, the world in which we already live all the time and which is the basis of all cognitive performance and all scientific determination (Husserl 1939, §§10, 11, 12).

In the truly ultimate, original realm of self-evidence of prepredicative experience of Experience and Judgment, philosophers look upon the world purely as a world of perception. They limit experience to the domain of what is valid only for themselves as reflecting subjects and do so in a manner that excludes all idealization, excludes the presupposition of objectivity, of the validity of judgments for others that oriented traditional logic to the ideal of exact determination in the sense of the definitive scientific validity that is always tacitly presupposed as being an essential part of the act of judgment. Once we disregard others, Husserl acknowledged, there is no question of any validation referring to the cognitive activity of others. There are not yet any deposits of meaning guaranteeing that our world, as far as it is given to us, is always already understood as a world determinable with exactitude and already determined by science in accordance with the idea of definitive validity (Husserl 1939, §12).

This is the case, according to Experience and Judgment, because catching sight of the ultimate origins of logic is at any given time the accomplishment of a single subject. Doing so requires me to limit myself to the realm of what is mine at a given time. This means investigating judging as if it is at any given time only for me, the fruits of which are mine only, taking no account of the role it plays in communication and the fact that it already always presupposes prior communication just in the manner in which it has already endowed its objects with meaning. Only then, Husserl thought, does one reach the most primitive building blocks of the logical activity out of which our world is built, which are initially thought of as being not for others, but as objects only for myself, the world as being thought as a world only for myself (Husserl 1939, §§11, 12).

A major part of the job of exploring the worlds of actual consciousness and of the purely logical naturally involves finding the line separating psychology and phenomenology and that separating phenomenology and the objective sciences. And from the time that Husserl first began talking of phenomenology, he was intent upon doing just that. For example, in his 1913 draft of a preface to the Logical Investigations, he recalled that in the Prolegomena to Pure Logic he had tried to show that efforts to ground logic exclusively on psychology rest on a confusion of distinct classes of problems and on fundamentally mistaken presuppositions about the nature and goals of empirical psychology and pure logic. He wrote,

The reader of the ‘Prolegomena’ is made a participant in a conflict between two motifs within the logical sphere which are contrasted in radical sharpness: the one is the psychological, the other the purely logical. The two do not come together by accident as the thought-act on the one side and the thought-meaning on the other. Somehow they necessarily belong together. But they are to be distinguished, namely in this manner: everything ‘purely’ logical is an ‘in itself,’ is an ‘ideal’ which includes in this ‘in itself”—in its proper essential content (Wesengehalt)—nothing ‘mental,’ nothing of acts, of subjects, or even of empirically factual persons of actual reality. There corresponds to this unique field of existing objectivities (Objektivitäten) a science, a ‘pure logic,’ which seeks knowledge exclusively related to these ideal objectivities, i.e., which judges on the pure meanings (Bedeutungen) and on the meant objectivities as such…. Thereupon follows a task… to determine the natural boundary of the logical-ideal sphere… to grasp the idea of the pure logic in its full scope. (Husserl 1913b, 20)

Experience and Judgment stresses that every science has an objective and a subjective side, that even in those cases in which we do not recognize the universal binding force and general applicability of the “exact” methods of natural science and its cognitive ideas, the conviction persists that objects of our experience are determined in themselves and knowing is a matter of discovering those determinations subsisting in themselves and of establishing them objectively, once and for all and for everyone, as they are in themselves. Husserl deemed it necessary to investigate the deposits of meaning in the world of our present experience relative to the subjective sources out of which they had developed and to dismantle everything already pre-existing in those deposits. But he insisted that this was a matter of the subjectivity whose operations of meaning have made the world that is pregiven to us what it is, namely, not a pure world of experience, but a world determined and determinable in itself with exactitude, a world in which any individual entity is given beforehand in an perfectly obvious way as in principle determinable in accordance with the methods of exact science and as being a world in itself in a sense originally deriving from the achievements of the physico-mathematical sciences of nature (Husserl 1939, §§3, 10, 11). He acknowledged that

our immediate experience…. in its immediacy knows neither exact space nor objective time and causality. And even if it is true that all theoretical scientific determination of existents ultimately refers back to experience and its data, nevertheless experience does not give its objects directly in such a way that the thinking that operates on these objects as it itself experiences them is able to lead by itself… immediately to objects in the sense of true theory, i.e., to objects of science. If we speak of objects of science, science being that which as such seeks truth valid for everyone, then these objects… are not objects of experience…. ‘Judgments of experience,’… which are obtained only from original operations in categorical acts purely on the basis of experience, i.e., sense experience and the experience founded on it of mental reality, are not judgments of definitive validity, are not judgments of science in the precise sense…. (Husserl 1939, §10)

In texts dated 1907 and 1908 that were included as appendices to his 1906/07 course on logic and the theory of knowledge, Husserl gave numerous examples of what did and did not belong in phenomenology, either because it fell into the realm of psychology and the natural sciences, or because it belonged to the world of the purely logical. The natural sciences of physical and mental nature, the mathematical sciences, logic, including formal logic, the sciences of value, ethics are not phenomenology, he said (Husserl 1906/07, 414). In the case of the formal sciences, he emphasized that in genuine transcendental phenomenology, we have no dealings with a priori ontology, formal logic and formal mathematical geometry as a priori theory of space, with a priori real ontology of any kind. As phenomenology of the constituting consciousness, not a single objective axiom relating to objects that are not consciousness, no a priori proposition as truth for objects, as something belonging in the objective science of these objects, or of objects in general in formal universality belongs in transcendental phenomenology. The axioms of geometry do not belong in phenomenology, he explained, because phenomenology is not a theory of the essences of shapes, of spatial objects. Essence-propositions about objects do not belong in the phenomenology of knowledge, insofar are they are objective truths and as truths have their place in a truth-system in general. What is objective belongs to objective science, and what objective science still lacks for completion is its affair to obtain and its alone (Husserl 1906/07, 428-29, Hill 2010a).

The World of the Purely Logical

In Formal and Transcendental Logic, Husserl explained that his “war against logical psychologism was in fact meant to serve no other end than the supremely important one of making the specific province of analytic logic visible in its purity and ideal particularity, freeing it from the psychologizing confusions and misinterpretations in which it had remained enmeshed from the beginning” (Husserl 1929, §67). In his 1913 draft of a preface to the Logical Investigations, he had written that he had come to believe that the entire approach whereby phenomenology overthrows psychologism showed that what he had proposed as analyses of immanent consciousness had to be considered as pure a priori analyses of essence (Husserl 1913b, 42). There, he described the world of the purely logical as follows:

“Pure logic,” in its most comprehensive extension characterizes itself by an essential distinction as “mathesis universalis.” It develops through a step-by-step extension of that particular concept of formal logic which remains as a residue of pure ideal doctrines dealing with ‘propositions’ and validity after the removal from traditional logic of all the psychological misinterpretations and the normative-practical goal positings (Zielgebungen). In its thoroughly proper extension it includes all of the pure “analytical” doctrines of mathematics (arithmetic, number theory, algebra, etc.) and the entire area of formal theories, or rather, speaking in correlative terms, the theory of manifolds (Mannigfaltigkeitslehre) in the broadest sense. The newest development of mathematics brings with it that ever new groups of formal-ontological laws are constantly being formulated and mathematically treated which earlier had remained unnoticed. “Mathesis universalis” … includes the sum total of this formal a priori. It is… directed toward the entirety of the “categories of meaning” and toward the formal categories for objects correlated to them or, alternatively, the a priori laws based upon them. It thus includes the entire a priori of what is in the most fundamental sense the “analytic” or “formal sphere”…. (Husserl 1913b, 28-29)

In his 1900 abstract for the Prolegomena to Pure Logic, Husserl had described pure logic as “the scientific system of ideal laws and theories which are purely grounded in the sense of the ideal categories of meaning; that is, in the fundamental concepts which are common to all sciences because they determine in the most universal way what makes sciences objectively sciences at all: unity of theory. In this sense, pure logic is the science of the ideal ‘conditions of the possibility’ of science generally, or of the ideal constituents of the idea of theory” (Husserl 1913b, 4).

Instead of pure logic, he suggested in the early years of the century, one might speak of analytics or the science of what is analytically knowable in general, the science that establishes and systematically grounds analytic laws. For him, analytically necessary propositions are propositions that are true completely independently of any particular facts about their objects and of any actual matters of fact. He defined analytic laws as “unconditionally universal propositions” that include formal concepts lacking all matter or content and free of any explicit or implicit positing of the existence of individuals. As examples of such purely formal concepts, he proposed “something,” “one,” “object,” “property,” “relation,” “connection,” “plurality,” “cardinal number,” “order,” “ordinal number,” “whole,” “part,” “magnitude,” etc., which he considered to be fundamentally different in character from concepts like “house,” “tree,” “color,” “sound,” “spatial figure,” “sensation,” “feeling,” “smell,” “intensity,” etc., that express something factual or sensory. (Husserl 1900/01, III §§11-12; Husserl 1908/09, 244; Husserl 1939, §1). He defended analytic logic against charges of being a “useless” spinning out of ‘sterile’ formalizations”, an objection that he considered revelatory of considerable philosophical deficiency, of a lack of understanding of crucial basic issues, and of a disgraceful ignorance of the essence of modern mathematics and the extraordinary significance that the scientifically rigorous, theoretical exploration of forms of pure deduction had acquired for the perfection and most rigorous grounding of the systems of pure mathematics in his day (Husserl 1908/09, 39).

In Allgemeine Erkenntnistheorie, Vorlesung 1902/03—which he considered presented the methodological and theoretical questions of the theory of knowledge in an incomparably clearer manner than in the Logical Investigations (Husserl 1902/03a, IX)—Husserl taught that all objectivity of thinking was grounded in purely logical forms, that the ultimate meaning and source of all objectivity making it possible for thinking to reach beyond contingent, subjective, human acts and lay hold of objective being in itself was to be found in ideality and in the ideal laws defining it. He presented pure logic as the science of the form concepts to which the objective content of all logical, all scientific thinking in general is subject. He taught that all truly scientific thinking, all proving and theorizing worthy of the name operated in forms corresponding to purely logical laws that included no cognitive material from the individual sciences, but were exclusively made up of concepts like truth, proposition, concept, argument, conclusion, necessity, possibility, object, property, set, etc. Pure logic, he stressed, embraces all the concepts and propositions without which science would not be possible, would not have any sense or validity. For him, any given science was a web of meanings laying claim to objective validity as a whole and as regards all its individual features (Husserl 1902/03a, 41, 47, 53, 58, 200, 206).

According to his definition of the purely logical in Logik, Vorlesung 1902/03, all concepts relating to objects in general in the most universal ways, or to thought forms in general in which objects are brought to theoretically objective unity, are purely logical. Purely logical concepts, purely formal concepts are not limited to a special field of objects, but are centered on the empty idea of something or object in general. They not only can and actually do figure in all the sciences, but are common and necessary to all sciences because they belong to what belongs to the ideal essence of science in general. All purely mathematical concepts like unit, multiplicity, cardinal number, order, ordinal number, and manifold are purely logical because they clearly relate in the most universal way to numbers in general and are only made possible out of the most universal concept of object. All purely mathematical theories, purely arithmetical theories, the theory of syllogism are purely logical because their basic concepts express reasoning forms that are free of any cognitive content and cannot be had through sensory abstraction. No epistemological reflection is required (Husserl 1902/03b, 31-43).

In his logic courses, Husserl taught that the essence of the mathematical lies in establishing a purely apodictic foundation of the truths of a field from apodictic principles. It is a matter of a rigorously scientific, a priori theory that builds from the bottom up and derives the manifold of possible inferences from the axiomatic foundations a priori in a rigorously deductive way. The mathematical disciplines of the purely logical sphere, he theorized, proceed from given, purely logical basic concepts and axioms that are grounded in the essence of purely logical categories (Husserl 1902/03b, 32-35, 39; Husserl 1906/07, §§13c, 19d, 25b).

For Husserl, the concept of number became the very paradigm of a purely logical concept. According to his theories, arithmetic truths were analytic, grounded in the identical ideal meaning of words independently of matters of fact and had nothing at all to do with experience and induction, but only with concepts. He taught that

pure arithmetic investigates what is grounded in the essence of number. It is concerned not with things, not with physical things, not with souls, not with real events of a physical and mental nature. It has nothing at all to do with nature. Numbers are not natural objects. The number series is so to speak a world of objectivities of its own, of ideal objectivities, not real ones…. The world of the mathematical and purely logical is a world of ideal objects, a world of “concepts”…. There all truth is nothing other than analysis of essences or concepts. (Husserl 1906/07, §13c)

The formalness of arithmetic, he explained in Formal and Transcendental Logic, lies in its relationship to “anything whatsoever” with empty universality that leaves every material determination indeterminately optional. The theory of cardinal numbers relates to the empty universe, to anything whatsoever with formal universality. The basic concepts are syntactical formations of the empty something that leaves out of consideration any material determination of objects. When the concept of cardinal number is fashioned purely in the broadest universality, the material contents of what is counted must be allowed to vary absolutely freely (Husserl 1929, §§24, 27a).

Indeed, from the mid-1890s on, Husserl defended the view, which he attributed to Frege’s teacher, Hermann Lotze, that pure arithmetic was basically no more than a branch of logic that had undergone independent development. Eminent thinkers like Lotze, Husserl explained, had correctly recognized cardinal number as a specific differentiation of the concept multiplicity (Vielheit) and multiplicity as the most universal logical concept combining objects in general that splits into the series of different special forms that are the cardinal numbers. The unending profusion of theories that arithmetic develops is already fixed, enfolded in the axioms, and theoretical-systematic deduction effects the unfolding of them following systematic, simple procedures. All of arithmetic is grounded in the arithmetical axioms. Each genuine axiom is a proposition that unfolds the idea of cardinal number from some side or unfolds some of the ideas inseparably connected with the idea of cardinal number. The meaning of cardinal number, he said, is the answer to the question: “How many?” Since each and every thing can be counted as one, he reasoned, to conceive the concept of number, or that of any arbitrarily defined number, we only need the concept of something in general. One is something in general. Anything can be counted as one and out of the units all cardinal numbers are built (Husserl 1896, 241-42, 271-72; Husserl 1902/03b, 19, 32-35, 39; Husserl 1906/07, §15, Hill 2010a).

Husserl Finds a Natural Order in the World of Formal Logic

Husserl’s phenomenological elucidation of the origin of the logical showed him that its domain was far more extensive than had been dealt with by traditional logic and his investigations further led him to detect a natural order in formal logic and to broaden its domain to include two levels above the traditional Aristotelian logic of subject and predicates and states of affairs, the origins of which are so thoroughly studied in Experience and Judgment (Husserl 1906/07, §18c; Husserl 1939, §1).

These three levels of pure logic are described in Introduction to Logic and Theory of Knowledge 1906/07 (Husserl 1906/07, §§18-19) and Part I of Formal and Transcendental Logic is devoted to them. In the introduction to the latter book, Husserl stated that he considered his new understanding of the structure of the world of pure logic—still not fully detected in the Logical Investigations, and not yet described in his logical courses of the time—to be of the greatest significance, not only for a genuine understanding of the true sense of logic, but for all of philosophy. He saw it as a matter of a radical clarification of the relationship between formal logic and formal mathematics and as leading to a definitive clarification of the sense of pure formal mathematics as a pure analytics of non-contradiction (Husserl 1929, 11).

On the first tier of Husserl’s hierarchy, the traditional Aristotelian apophantic logic of subject and predicate propositions and states of affairs investigates what can be stated in possible form a priori about objects in general from a possible perspective. It deals with the forms of propositions or states of affairs by asking in which forms objects are conceivable as such states of affairs and then which laws for the existence of states of affairs are valid in virtue of their form. He considered that although the concept of predicative judgment stood at the center of formal logic as it had developed historically, it was but a small area of pure logic as a whole (Husserl 1906/07, 18c).

This explains why numbers, for example, are so conspicuous by their absence in Experience and Judgment. According to Husserl’s theory of the forms of subject-predicate propositions of the first level, number only occurs as form, but not as an object about which something is predicated. He explained that if one says w and x and y and z are φ, then one has combined the objects wz by ‘and’. In that case, the ‘and’ is form and grounds the unitary form of the plural predication. Corresponding to this is a cardinal number. However, he stressed, that is a new thought configuration, for it is one thing to make statements about objects in which number properties occur as form, and are thereby dependent, and another thing to make statements about numbers as such in such a way that the numbers are the objects. As examples of expressions of the first level in which numbers occur, Husserl gave: ‘2 men’; ‘3 houses’ (Husserl 1906/07 18c).

He emphasized that only the forms of the plural numerical predication about objects as such belong in a simple theory of objects in general and the forms of their states of affairs and that in statements, propositions or state of affairs, forms are dependent. We can make such forms independent, he taught, but then new higher order objects, hypostasizations of forms emerge that are not objects in their own right. Statements about numbers in which numbers are objects have their place on the second level of Husserl’s hierarchy, where numbers function in an entirely different way than on this first level (Husserl 1906/07 18c).

For Husserl, sets as objects do not occur on the apophantic level any more than numbers do. He observed that

in set theory, we make judgments universally about sets that in a certain way are higher order objects. We do not make judgments directly about elements, but about whole totalities of elements and arbitrary elements, and the whole totalities, the sets to be precise, are the objects-about-which. Corresponding to every plural is a set, but in the theory of proposition forms, or forms of states of affairs, the set does not occur as object. In it, the objects-about-which are thoroughly indeterminate A B…. Rather, only the plural occurs in it, which constitutes a form of predication about arbitrary objects (Husserl 1906/07 18c).

So mathematical sets are also conspicuous by their absence in Experience and Judgment, which devotes a few pages to a discussion of sets in the non-mathematical sense of objectivities of the understanding. In the pre-predicative world of that book, a “set is an original objectivity, preconsituted by an activity of colligation which links disjunct objects to one another; the active apprehension of this objectivity consists in a simple reapprehension or laying hold of that which has just been preconstituted”. After completing an act of colligating through a retrospective apprehension a set is given to the ego as an object, as something identifiable. Every set preconstituted in intuition must be conceived a priori as capable of being reduced to ultimate constituents, to particularities which are no longer sets. “As a pure formation of spontaneity, the set represents a pre-eminent form in which thematic objects of every conceivable kind enter as members and with which they can themselves function as members of determining judgments of every kind” (Husserl 1939, §61).

Husserl believed that apophantic logic had to be distinguished and segregated from the formal ontology of the broader sphere of pure logic that included the mathematical disciplines and was immense in range and wealth of content in comparison. According to his theory, the disciplines of the two levels rising above it deal with individual things, but not in the sense of empirical or material entities. These higher ontologies are concerned with purely formally determined higher level object formations like set, cardinal number, quantity, ordinal number, ordered magnitude, etc., that are removed from acts, subjects, or empirical persons of actual reality. In them, it is no longer a question of objects as such about which one might predicate something, but of investigating what is valid for higher order objective constructions that are determined in purely formal terms and deal with objects in indeterminate, general ways (Husserl 1906/07, 18c-d).

Husserl conceived of the second level as an expanded, completely developed analytics in which one proceeds in a purely formal manner since every single concept used is analytic. One calculates, reasons deductively, with concepts and propositions. Signs and rules of calculation suffice because each procedure is purely logical. One manipulates signs, which acquire their meaning in the game through the rules of the game. One may proceed mechanically in this way and the result will prove accurate and justified. On the second level of pure logic, Husserl located the basic concepts of mathematics, the theory of cardinal numbers, the theory of ordinals, set theory, mathematical physics, formal pure logic, pure geometry, geometry as a priori theory of space, the axioms of geometry as a theory of the essences of shapes, of spatial objects, but also the pure theory of meaning and being, a priori real ontology of any kind (thing, change, etc.), ontology of nature, ontology of minds, natural scientific ontology, the sciences of value, pure ethics, the logic of morality, the ontology of ethical personalities, axiology or the pure logic of values, pure esthetics, ontology of values, the logic of the ideal state or the ideal world government as a system of cooperating ideal nation states, or the science of the ideal state, the ideal of a valuable existence, objective axioms (relating to a priori propositions as truth for objects, as something belonging in the objective science of these objects, or of objects in general in formal universality, essence-propositions about objects insofar are they are objective truths and as truths have their place in a truth-system in general (Husserl 1906/07, §§18-19, 434-35; Husserl 1996, Chapter 11).

As examples of arithmetical propositions of the second level in which numbers occur as objects, Husserl gave:

1. “Any number can be added to any number”.

2. “If a is a number and b a number, then a + b is as well”.

3. “Any number can be decreased or increased by one”.

4. “The numbers form a series continuing from 0 in infinitum” (Husserl 1906/07 18c).

As examples of propositions of the second level in which sets occur as objects, he gave:

1. “2 sets can each be joined into a new set”.

2. “2 sets a b are each related to one another in such a way that either a is part of b or b is part of a, or that they intersect (a set having a part in common), or that it turns out that they are identical, coincide”.

3. “The set formed of the elements A B C is part of the set formed of the elements A B C D containing “more elements” (Husserl 1906/07 18c).

On the third and highest level of formal logic, Husserl located the theory of manifolds, a new discipline and a new method constituting a new kind of mathematics, the most universal of all. He counted upon it to provide secure foundations for an a priori theory of science. He presented his theory of manifolds in his major published works (Husserl 1900/01, Prolegomena, §§69-70; Husserl 1913a, §§71-72; Husserl 1929, §33), but what seems inchoate and cryptic there received particularly clear and explicit treatment in the posthumously published lecture courses.

In those courses, Husserl described manifolds as pure forms of possible theories which, like molds, remain totally undetermined as to their content, but to which thought must necessarily conform in order to be thought and known in a theoretical manner. In manifolds, formal logic deals with whole systems of propositions making up possible deductive theories. It is a matter of theorizing about possible fields of knowledge conceived of in a general, undetermined way and purely and simply determined by the fact that the objects stand in certain relations that are themselves subject to certain fundamental laws of such and such determined form (Husserl 1906/07 §19; Husserl 1996, §§54-59). For example, he explained the meaning of the theory of non-Euclidean manifolds as follows,

Let there be a domain in which the objects are subject to certain forms of relation and connection, for which axioms of such and such a form are valid, then for a domain formally constituted in this way, a mathematics of such and such a form would be valid, there would then result propositions of such and such a form, proofs, theories of such and such a form. There is no domain. There are no actually given concepts, connections, relations and axioms. One simply says, if one had a domain, and if axioms of such and such a form obtained for it. (Husserl 1906/07, §19c-d)

Husserl saw the general theory of manifolds, or science of theory forms, as a field of free, creative investigation made possible once it is discovered that deductions, series of deductions, continue be meaningful and to remain valid when one assigns another meaning to the symbols. No longer restricted to operating in terms of a particular field of knowledge, one is free to reason completely on the level of pure forms. Operating within this sphere of pure forms, one can vary the systems in different ways. Nothing more need be presupposed than the fact that the objects figuring in them are such that, for them, a certain connective supplies new objects and does so in such a way that the form determined is assuredly valid for them. One finds ways of constructing an infinite number of forms of possible disciplines (Husserl 1906/07, §19).

In the methodology of manifolds, Husserl taught, one speaks of numbers, but does not mean cardinal numbers, quantitative numbers, or anything of that kind, but anything for which formal axioms of the arithmetical prototype hold. If we drop the cardinal number meaning of the letters in the ordinary theory of cardinal numbers and substitute the thought of objects in general for which axioms of the arithmetical form a+b = b+a, a·b =b·a, etc., are to hold, we no longer have arithmetic, but a purely logical class prototype of theory forms to which, besides innumerably many possible domains, the domain of cardinal numbers is also subject. One may then speak of numbers in the formal sense, but they are not cardinal numbers, but objects indeterminately, universally defined by axiom forms as they are especially actually found for cardinal numbers. Here, as in every theory form or manifold form, the “axioms” are proposition forms that are constituent parts of the definition. For cardinal numbers, ab = ba holds. In constructing a manifold, though, one may just as well stipulate that ab ¹ ba, for example, ab = –ba, and likewise for the other basic principles. (Husserl 1906/07, 19b,d)

In Logical Investigations, Husserl expressed his conviction that his theory of complete manifolds was the key to the only possible solution to the as yet unclarified problem as to how in the realm of numbers, impossible, non-existent, meaningless concepts might be dealt with as real ones (Husserl 1900-01, Prolegomena §70). We cannot arbitrarily expand the concept of cardinal number, Husserl explained in posthumous writings on imaginary numbers. But we can abandon it and define a new, pure formal concept of positive whole number with the formal system of definitions and operations valid for cardinal numbers. And, as set out in our definition, this formal concept of positive numbers can be expanded by new definitions while remaining free of contradiction (Husserl 1901, 415).

In the arithmetic of cardinal numbers, Husserl explained, there are no negative numbers, for the meaning of the axioms is so restrictive as to make subtracting 4 from 3 nonsense. Fractions are meaningless there. So are irrational numbers, √ -1, and so on. Yet in practice, all the calculations of the arithmetic of cardinal numbers can be carried out as if the rules governing the operations were unrestrictedly valid and meaningful. One can disregard the limitations imposed in a narrower domain of deduction and act as if the axiom system were a more extended one (Husserl 1996, §56). Fractions do not acquire any genuine meaning through our holding onto the concept of cardinal number and assuming that units are divisible, he theorized, but rather through our abandonment of the concept of cardinal number and our reliance on a new concept, that of divisible quantities. That leads to a system that partially coincides with that of cardinal numbers, but part of which is larger—meaning that it includes additional basic elements and axioms. And so in this way, with each new quantity, one also changes arithmetics. The different arithmetics do not have parts in common. They have totally different domains, but have an analogous structure. They have forms of operation that are in part alike, but different concepts of operation (Husserl 1901, 416).

Understanding the nature of theory forms, Husserl explained in several texts, shows how reference to impossible objects can be justified. According to his theory of manifolds, one could operate freely within a manifold with imaginary concepts and be sure that what one deduced was correct when the axiomatic system completely and unequivocally determined the body of all the configurations possible in a domain by a purely analytical procedure. It was the completeness of the axiomatic system that gave one the right to operate in that free way. A domain was complete, according to Husserl’s theory, when each grammatically constructed proposition exclusively using the language of that domain was, from the outset, determined to be true or false in virtue of the axioms, i.e., necessarily followed from the axioms (in which case it is true) or did not (in which case it is false). In that case, calculating with expressions without reference could never lead to contradictions. So, Husserl concluded, it was formal constraints requiring that one not resort to any meaningless expression, no meaningless imaginary concept that were restricting us in our theoretical, deductive work. But what is marvelous, Husserl believed, is that resorting to the infinity of pure forms and transformations of forms frees us from such conditions and at the same time explains to us why having used imaginaries, what is senseless, must lead, to what is not senseless. (Husserl 1901, 428-29; Husserl 1900-01, Prolegomena, §70; Husserl 1906/07, §19; Husserl 1913a, §§71-72; Husserl 1929, §31; Husserl 1996, §§54-59).

Philosophizing on the Borderline

As an investigation of the origin and subjective foundation of traditional Aristotelian formal logic by clarifying the essence of the predicative judgment through an exploration of its origins, Husserl’s study of pre-logic in Experience and Judgment takes place right on the borderline between transcendental subjectivity and pure, objective logic. So, in reappraising the fundamental tenets of his phenomenology as exposed in the analyses of that book, it is as important to be clear about how he understood the interrelationship of the two interdependent worlds as it is to understand what he found on each side of the border, for the underlying paradox of the science of intentionality that he used to meet the challenge he set for himself in Experience and Judgment is that his science of subjectivity was his science of objectivity and vice versa. Indeed, he said that while fighting to separate psychology, the natural sciences, phenomenology, and pure logic, he was wracking his brain trying to put them back together in a new way, trying to understand how the worlds of actual consciousness and the purely logical interrelate and form an intrinsic unity (Husserl 1994, 492). Now that we have toured both worlds, we are in a position to look at what he found.

To begin with, it is important to remember that for Husserl, the world of actual consciousness always somehow belonged with the ideal world of pure logic. He said that it was his psychological analyses that had compelled him to recognize the ideal as something given prior to all theorizing and impressed upon him the essential interrelationship of the worlds of pure logic and actual consciousness. He specifically tied the breakthrough of phenomenology to investigations aimed at elucidating the cognitive accomplishment of arithmetic and of pure analytical mathematics in general and, above all, to his search to find a theoretical solution to the problem of imaginary quantities. He said that it was that quest that had forced him to engage in general investigations concerning the universal clarification of the meaning, the proper delimitation and unique accomplishment of formal logic and was his chief motivation in developing the theory of manifolds, the pinnacle of pure logic (Husserl 1913a, §72 and n.; Husserl 1913b, 21-22, 31, 33).

In Allgemeine Erkenntnistheorie, Husserl portrayed the problems of theory of knowledge as lying between psychology and pure logic, inasmuch as the two disciplines both relate to all of science. He said that he considered logic to be the discipline the very closest to theory of knowledge and stressed the legitimate ties that he saw obtaining between pure, formal, analytic logic and its complement the theory of knowledge. He defined theory of knowledge as the discipline that subjects the concepts and laws secured in pure logic and belonging to the ideal essence of thinking to a clarifying investigation of their meaning and their objective validity and on that basis solves all the problems connected with the validity of knowledge and science or proves that they are pseudo-problems (Husserl 1902/03a, 10, 19, 54).

He held that every naïve logic constructed in the natural-objective orientation had a corresponding epistemologically and phenomenologically clarified philosophical logic, or one that phenomenologically grounded from the very beginning (Husserl 1913b, 31). Indeed, Experience and Judgment developed out of his conviction that truly philosophical logic requires phenomenologists to pierce through the logic of subject and predicates of the first level of pure logic to reach the world of actual consciousness and expose the foundations of an underlying, hidden, logic (Husserl 1939, §§1, 3, 10, 11; Husserl 1929, §40). In a 1903 report on German logic that foreshadowed the project of Experience and Judgment, Husserl characterized critique of knowledge as

the task of rendering “intelligible” the possibility of a knowledge which is delimited by concepts and laws of pure logic, by tracing these back to their ‘origin’; the task of resolving, in this way, the profound difficulties which are tied up with the opposition between the subjectivity of the act of knowledge and the objectivity of the content and object of knowledge (or of truth and being). This task does not fall to pure logic itself …. (Husserl 1994, 250)

So, as stressed in Experience and Judgment, logic had to have two sides that complement one another, something the tradition had never grasped in a deep way (Husserl 1939, §3). In Formal and Transcendental Logic, he reminded readers that logic turns both towards the deeply hidden subjective forms in which reason does its work and the objective order, towards ideal objects, towards a world of concepts, where truth is an analysis of essences or concepts, where knowing subjects and the material world play no role (Husserl 1929, §§7, 8). Pure, objective, formal logic had to find its necessary complement in subjective, transcendental logic and the latter had to find its necessary complement in the former.

Husserl always insisted on the primacy of the objective side of logic. In Experience and Judgement, the world constituted by transcendental subjectivity is a pre-given world. It is not a pure world of experience, but a world that is determined and determinable in itself with exactitude, a world within which any individual entity is given beforehand in an perfectly obvious way as in principle determinable in accordance with the methods of exact science and as being a world in itself in a sense originally deriving from the achievements of the physico-mathematical sciences of nature (Husserl 1939, §11). It is knowledge of formal logic, he reminded readers in Formal and Transcendental Logic, that supplies the standards by which to measure the extent to which any presumed science meets the criteria of being a genuine science, the extent to which the particular findings of that science constitute genuine knowledge, the extent to which the methods it uses are genuine ones (Husserl 1929, §7).

In various texts, Husserl explained that theoretical disciplines have a systemic form that belongs to formal logic itself and must be constructed a priori within formal logic itself and within its supreme discipline the theory of manifolds as part of the overall system of forms of deductive systems that are possible a priori. He stressed that all fields of theoretical knowledge are particular instances of manifolds, but he knew that not all sciences are theoretical disciplines that, like mathematical physics, set theory, pure geometry or pure arithmetic, are characterized by the fact that their systemic principle is a purely analytical one. He recognized that sciences like psychology, history, the critique of reason and, notably, phenomenology were not purely logical and so obliged philosophers to go beyond the analytico-logical model. When those not purely logical sciences are formalized and philosophers ask what binds the propositional forms into a single system form, they face nothing more than the empty general truth that there is an infinite number of propositions connected in objective ways that are compatible with one another in that they do not contradict each other analytically (Husserl 1996, §54; Husserl 1929, §35a; Husserl 1908/09, 263; Husserl 1902/03b, 31-43, 49).

He maintained that since the concepts of geometry, mathematical mechanics and all mathematico-natural scientific disciplines, the natural sciences of physical and mental nature have real content, they belong among the natural sciences and not in phenomenology or in pure logic (Husserl 1996, §54; Husserl 1929, §35a; Husserl 1908/09, 263; Husserl 1902/03b, 31-43, 49), but, he wanted to use phenomenology to transform the merely positive sciences into philosophical sciences and to establish new sciences that were philosophical from the very outset (Husserl 1913b, 31). He wanted to see phenomenology transform the naïve physical theory of nature from a mere natural science into a true philosophy of nature, into a philosophical physics that does not begin with vague concepts and then proceed naively, into a physics that has been philosophically deepened and enriched by all the problems concerning the correlation of physical being and cognitive subjectivity, a physics in which the experiencing subject in search of objective knowledge plays an active role and in which the basic concepts and basic propositions are developed from the very beginning in ultimate methodological originality. Philosophical physics would then be a science that understands itself radically and justifies its constitution of sense and being from the very beginning to the very end (Husserl 1913b, 30).

He stressed that the mathematical sciences, logic, formal logic, the sciences of value, ethics are not phenomenology, because they belong in the world of the purely logical (Husserl 1906/07, 414). He considered that if

we are not interested in the transcendental task and we remain in pure theory of meaning and being, then we practice logic, natural scientific ontology, pure theory of space, etc. these need not concern themselves at all with cognitive formations, with consciousness. Likewise, if we practice ethics as pure ethics (or logic of morality), esthetics, or logic of esthetic appreciation, axiology or pure logic of values…. the logic of the ideal state or the ideal world government as a system of cooperating ideal nation states (or the science of the ideal state)…. the ideal of a valuable existence (ideally valuing and valuable human beings aimed at an ideally valuable nature accommodating their values)… and the logic of this ideal… ideal-esthetic existence, pure esthetics…. ontology of nature, ontology of minds, ontology of ethical personalities, ontology of values, etc. (Husserl 1906/07, 434-35)

However, he went on to affirm that “belonging to all of them are transcendental phenomenologies” (Husserl’s emphasis) that transcendentally investigate the valid objects of different categories, the objects of these ontologies, in relationship to types of consciousness essentially belonging to them (Husserl 1906/07, 434-35).

Indeed, though he believed that only certain of the most general cognitive-formations enter the picture for purposes of phenomenological elucidation in the case of pure logic, of an ‘analytics’ in the broadest, radical sense of the word (Husserl 1913b, 31), he realized that even

the most trivial analytical knowledge presents big problems and hard problems for critique of knowledge. A puzzle is already present in them: How objectively valid knowledge, knowledge of things existing on their own, is possible vis-à-vis the subjectivity of knowing as a subjective activity (Husserl 1906/07, 335).

He acknowledged that, for example, even though ordinary arithmetic, in both its naïve and its technical forms, does not at first have any common cause with theory of knowledge and phenomenology, if it undergoes phenomenological elucidation, and so learns from the sources of phenomenology to solve the great riddles arising from the correlation between pure logic and actual consciousness, and if in so doing it also learns the ultimate formulation of the meaning of concepts and propositions that only phenomenology can provide, then it will have transformed itself into truly philosophical pure logic that is more than a mere coupling of natural-objective mathesis with phenomenology of knowledge, but rather is an application of the latter to the former (Husserl 1913b, 29-30). He recognized that the critical elucidation of pure arithmetic as knowledge was no arithmetical task (Husserl 1994, 250).

Although he considered the concept of predicative judgment to be but a small area of pure logic, he reminded readers in Experience and Judgement not to forget the importance of understanding the origins and particular legitimacy of the lower levels of logic in elucidating both the path one must take to attain evident knowledge at a higher level and the hidden presuppositions underlying this knowledge, presuppositions that determine and delimit its meaning (Husserl 1939, §§10, 11). For essential reasons, he taught, pure arithmetic and the whole of formal mathematics or theory of manifolds also prove to be intertwined with the logic of assertions, apophantic logic, although in a completely different direction. These disciplines form, as it were, a higher story of apophantics and it is of great philosophical significance to recognize and characterize them in this connection (Husserl 1996, §7). For example, although he maintained that numbers and sets function in an entirely different way in the apophantic sphere of propositions and states of affairs than in arithmetic and in set theory (Husserl 1906/07, 18c), he stressed that apophantic logic is intrinsically related to the pure theory of numbers and set theory of the second level, that the “laws for all these higher order objects form their own branches of pure logic, but branches of the trunk of the one pure logic. The basic trunk is apophantic logic. The branches are, though, united a priori to the basic trunk” (Husserl’s emphasis) (Husserl 1906/07, 18d).

In his theory of manifolds, all purely logical basic concepts are set aside, but they are needed in actually using the theory. It is apophantic logic, he reminded students, that supplies the principles in accordance with which the entire procedure functions, while the higher logic of second order objects supplies basic concepts, like the concept of cardinal number, of ordinal, of combination, and so on, from which one cannot escape in actual thinking about purely hypothetical-formal thought configurations. One can just not think without thinking, without also having and presupposing everything without which thinking of whatever form, or however expressed, would really ever have any meaning. Since one is making inferences scientifically, since one is thinking (though hypothetically and on the basis of formal specifications), advancing from argument to argument, since one cannot avoid making the inference from n to n+1, and so on, what is purely logical already proves to be involved in this everywhere, just as the entire theory of manifolds is constructed out of purely logical material (Husserl 1906/07, 19d).

He considered that all logical formations originate from categorial activity (Husserl 1929, 11). Every concept of a manifold and of a theory of manifolds is built out of purely categorial concepts (Husserl 1906/07, 18c). The theory of manifolds is the ultimate culmination of all purely categorial knowledge (Husserl 1996, §59). He maintained that the formal theory of manifolds, the highest level of pure logic, would be nothing in its own right if it did not draw all its knowledge from the original sources that first make actual science in general possible (Husserl 1906/07, 18d). He even explicitly wrote of reforming the mathematical theory of manifolds by consciously transforming it into a transcendental theory of manifolds that consciously captures the formal essence of a genuine, constructible totality that consciously analyzes what belongs to the essence of a concept defining a totality, what belongs to the essence of an axiom and axiom system (Ms A 1 35, 38; Hill 2002a).


Never able to rest from his experiences in the “strange” worlds of the purely logical and actual consciousness that had opened up to him at the beginning of his philosophical career, Husserl strove until the end of his life to find answers to the very questions about them that launched him on his phenomenological voyage of discovery. His persistent search to fathom and solve the puzzles, mysteries, riddles, enigmas and paradoxes involved in the complex interplay and interdependency between those worlds was at the heart of the dynamic that brought phenomenology into being.

Here, I have tried to show that in reappraising the fundamental tenets of Husserl’s phenomenology as exposed in the analyses of the subjective foundations of the part of traditional formal logic to which Experience and Judgment is devoted, it is imperative to situate the analyses of that book on the map of the worlds of actual consciousness and the purely logical that can be pieced together from the discoveries about their features that Husserl made during his long, assiduous exploration of those worlds. I have used less well-known, posthumously published texts to piece together the findings about their interrelation and their intrinsic unity he made during his mental travels in them.

It is imperative to be clear and informed about the answers that Husserl found to his questions about the interrelation and intrinsic unity of the worlds of actual consciousness and the purely logical because, as he himself acknowledged in Formal and Transcendental Logic, his theories about what he described as the “two-sidedness of everything logical, in consequence of which the problem-groups become separated and again combined” involve “extraordinary difficulties”. He himself recognized that since, according to his theories, the ideal, objective, dimension of logic and the actively constituting, subjective dimension interrelate and overlap, or exist side by side, logical phenomena seem to be suspended between subjectivity and objectivity in a confused way (Husserl 1929, §26c). He even suggested that almost everything concerning the fundamental meaning of logic, the problems it deals with and its method, was laden with misunderstandings owing to the fact that objectivity arises out of subjective activity. Even the ideal objectivity of logical structures and a priori nature of logical doctrines especially pertaining to this objectivity, and the meaning of this a priori are afflicted with this lack of clarity, he maintained, since what is ideal appears as located in the subjective sphere and arises from it. He further suggested that it was due to these difficulties that, after centuries and centuries, logic had not attained the secure path of rational development (Husserl 1929, §8). Presently, the theories of those advocating the naturalization of phenomenology or Brouwerian-type interpretations of Husserl’s ideas about mathematics[1] are built upon such misunderstandings, something that, in the case of the latter, I have tried to show in my essay “Husserl on Axiomatization and Arithmetic” (Hill 2010b).

By clearly and explicitly outlining what was to be found in each world and how the two worlds interacted, Husserl provided a road map for avoiding confusion in many areas of philosophy. Indeed, if he was right in believing, as I think he was, that his new understanding of the structure of the world of pure logic was of the greatest significance for a genuine understanding of the true sense of logic and all of philosophy and that his radical clarification of the relationship between formal logic and formal mathematics could lead to a definitive clarification of the sense of pure formal mathematics as a pure analytics of non-contradiction (Husserl 1929, 11), then philosophers need to be particularly lucid about the really important questions that his insights raise for philosophy of logic and mathematics now.

It is, for example, extremely important to see what Husserl came to locate in each of the two worlds and to understand how he envisioned their interaction because what belongs in one world, or in another part of one of the worlds, often enjoys the same outward appearance as what belongs in the other world or somewhere else in the same world. He himself stressed in Logical Investigation IV that identical words often have different types of meanings, that the relations of those words to what they designate can also be of different types and that the failure to realize that fundamental, ultimately inviolable, ontological differences often lie concealed behind inconspicuous linguistic or grammatical distinctions is often a very potent, hard to perceive, source of contradictions, nonsense, confusion, absurdity and error in philosophy (Husserl 1900/01, LI IV), something that I have studied more in depth elsewhere (for example, Hill 2003; Hill 2010b).

Gottlob Frege and Bertrand Russell also concluded that certain fundamental differences between different kinds of meaning that are concealed behind inconspicuous grammatical distinctions ultimately prove inviolable because they are “founded deep in the nature of things” (Frege 1891, 41) in such a way that contradictions, paradoxes, antinomies, fallacies, nonsense, confusion, absurdity, inevitably result when they are not respected and that this is a topic of prime importance for the understanding of major issues in twentieth century western philosophy (Hill 2003; Hill 2010b). In several writings, I myself have tried to show how blurring distinctions between dependent and independent meanings by allowing a concept word to be transformed into a proper name and to come to figure in the wrong part of the world of the purely logical opens the door to contradictions, paradoxes, antinomies, fallacies, nonsense, confusion, absurdity, pseudo-objects. Such confusions caused Frege to abandon his logical system and have been the cause of problems that Russell and his successors in the analytic tradition in philosophy have never been able to solve (Hill 1997). In Word and Object in Husserl, Frege and Russell, I suggested that analytic philosophers have been massively doctoring symptoms of a malady caused by such logical errors grounded in the very insights into logic, language and theory of knowledge that produced the logic they embraced (Hill 1991, 165).

Husserl was not philosophizing about logic and mathematics in a vacuum and he was not just extemporizing about the things he liked to believe about those fields. He was as well-versed in them, if not more so (Hill 2002a), than those who went on to create the theories of logic and philosophy of mathematics that were embraced by the philosophical establishment in the 20th century and shaped the analytic school philosophy that dominated those fields. If Husserl had been heeded they could have avoided many problems and those fields would have followed a different, less error-ridden course.


(Hill 1991) Hill, Claire Ortiz, Word and Object in Husserl, Frege and Russell, the Roots of Twentieth Century Philosophy, Athens: Ohio University Press (2001 paperback).

(Hill 1997) Hill, Claire Ortiz, Rethinking Identity and Metaphysics, On the Foundations of Analytic Philosophy, New Haven: Yale University Press, 1997.

(Hill 1998) Hill, Claire Ortiz, “From Empirical Psychology to Phenomenology: Husserl on the Brentano Puzzle”, in The Brentano Puzzle, R. Poli (ed.), Aldershot: Ashgate, 151-68.

(Hill 2000) “Husserl’s Mannigfaltigkeitslehre”, in Claire Ortiz Hill and G. E. Rosado Haddock, Husserl or Frege? Meaning, Objectivity, and Mathematics, Chicago: Open Court (2003 paperback), 61-78.

(Hill 2002a) Hill, Claire Ortiz, “On Husserl’s Mathematical Apprenticeship and Philosophy of Mathematics”, in Phenomenology World Wide, Anna-Teresa Tymieniecka (ed.), Dordrecht: Kluwer, 76-92.

(Hill 2002b) Hill, Claire Ortiz, “Tackling Three of Frege’s Problems: Edmund Husserl on Sets and Manifolds”, Axiomathes 13: 79-104.

(Hill 2003) Hill, Claire Ortiz, “Incomplete Symbols, Dependent Meanings, and Paradox”, in Husserl’s Logical Investigations, Daniel O. Dahlstrom (ed.), Dordrecht: Kluwer, 69-93.

(Hill 2010a) Hill, Claire Ortiz, “Husserl on Axiomatization and Arithmetic,” in Phenomenology and Mathematics, Mirja Hartimo (ed.), Dordrecht: Springer, 2010, 47-71.

(Hill 2010b) Hill, Claire Ortiz, “On Fundamental Differences Between Dependent and Independent Meanings”, Axiomathes, An International Journal in Ontology and Cognitive Systems 20: 2-3, online since May 29, 2010, 313-32, (DOI 10.1007/s10516-010-9104-1).

(Husserl 1896) Husserl, Edmund, Logik, Vorlesung 1896, Dordrecht: Kluwer, 2001.

(Husserl 1901) Husserl, Edmund, “Double Lecture: On the Transition through the Impossible (‘Imaginary’) and the Completeness of an Axiom System,” in Philosophy of Arithmetic, Psychological and Logical Investigations with Supplementary Texts from 1887-1901, Dallas Willard (tr.), Dordrecht: Kluwer, 2003, 409-73.

(Husserl 1900/01) Husserl, Edmund, Logical Investigations, J. N. Findlay (tr.), London: Routledge and Kegan Paul, 1970.

(Husserl 1902/03a) Husserl, Edmund, Allgemeine Erkenntnistheorie, Vorlesung 1902/03a, Elisabeth Schuhmann (ed.), Dordrecht: Kluwer, 2001.

(Husserl 1902/03b) Husserl, Edmund, Logik, Vorlesung 1902/03a, Elisabeth Schuhmann (ed.), Dordrecht: Kluwer, 2001.

(Husserl 1906/07) Husserl, Edmund, Introduction to Logic and Theory of Knowledge, Lectures 1906/07, Dordrecht: Springer, 2008.

(Husserl 1908/09) Husserl, Edmund, Alte und neue Logik, Vorlesung 1908/09, E. Schuhmann (ed.). Dordrecht: Kluwer, 2003.

 (Husserl 1913a) Husserl, Edmund, Ideas, General Introduction to Pure Phenomenology, New York: Colliers, 1962.

(Husserl 1913b) Husserl, Edmund, Introduction to the Logical Investigations, A Draft of a Preface to the Logical Investigations, Eugen Fink (ed.), P. Bossert and C. Peters (trs.), The Hague: Martinus Nijhoff, 1975.

(Husserl 1929) Husserl, Edmund, Formal and Transcendental Logic, The Hague: Martinus Nijhoff, 1978.

(Husserl 1939) Husserl, Edmund, Experience and Judgment, Investigations in the Genealogy of Logic, London: Routledge and Kegan Paul, 1973.

(Husserl 1994) Husserl, Edmund, Early Writings in the Philosophy of Logic and Mathematics, Dallas Willard (tr.), Dordrecht: Kluwer.

(Husserl 1996) Husserl, Edmund, Logik und allgemeine Wissenschaftstheorie. Vorlesungen 1917/18, mit ergänzenden Texten aus der ersten Fassung 1910/11, Dordrecht, Kluwer.

(Husserl Ms A 1 35) Husserl, Edmund, Ms A 1 35. Unpublished Manuscript on Set Theory available in the Husserl Archives in Leuven, Cologne and Paris.

(Livadas 2012) Livadas, Stathis, Contemporary Problems of Epistemology in the Light of Phenomenology, Temporal Consciousness and the Limits of Formal Theories, London: College Publications.

(Roy et. al. 1999) J.-M. Roy, Jean Petitot, Francisco Varela, Bernard Pachoud (eds.), Naturalizing Phenomenology, Issues in Contemporary Phenomenology and Cognitive Science, Stanford: Stanford University Press, 1999.

(Tieszen 1989), Tieszen, Richard, Mathematical Intuition, Phenomenology and Mathematical Knowledge, Dordrecht: Kluwer, 1989.

(van Atten 2007), van Atten Mark, Brouwer Meets Husserl. On the Phenomenology of Choice Sequences, Dordrecht: Springer.

[1] For example, the former see Roy et al. 1999, for the latter Tieszen 1989, van Atten 2007, Livadas 2012.