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**

**Introduction**

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 *w*…*z* 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).

**Conclusion**

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 20^{th}
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.

**References**

(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.