Phenomenology, Logic, and the Philosophy of Mathematics

Although the philosopher Edmund Husserl (1859-1938) is not mentioned in the title, this book is largely about his philosophy of mathematics.[i] Tieszen presents and develops key features of this philosophy, thereby providing a valuable service for those wishing to look at mathematics with a philosophical eye.[ii] (Unless otherwise indicated, all pages references pertain to the book under review.)

Tieszen places the philosophy of mathematics of Husserl in the context of his views on science (pp. 24-34). He held that pure logic provides a “condition for the possibility of any science”. The term “logic” here is used in a broad sense.[iii] This logic is stratified over three levels which are (very roughly put) picking out well-formed and meaningful expressions (“This S is P” but not “This is or”), from these picking out consistent expressions which are meaningful and not nonsensical (“This tree is green” but not “This careless is green”),[iv] and from these picking out truthful expressions. The last involves the working of intuition and intersubjective judgement.[v] Each science addresses a relevant range of objects. Each is a “regional science” corresponding to a “regional ontology”.[vi]

Transcendental phenomenology — “the science of consciousness with all its various structures and characteristics” (p. 33)[vii] — is one such science. It is crucial to science as a whole because all science presupposes conscious, collective pre-scientific appreciation of the world of ordinary objects, sights, sounds, and so forth, from which scientific abstractions are drawn. Consciousness cannot be adequately studied by an objective science (modern psychology, for example) which strives for a third person outlook. In particular, an objective science necessarily misses the nature of intentionality — that feature of an experience such as having a belief or a perception that makes it about something else.[viii] (My current perception is about the computer screen in front of me.)

Failure fully to recognize and investigate the nature and implications of the conscious side of the scientific enterprise has led to a crisis in science (according to Husserl). Leading disciplines such as physics become wedded to a purely factual and objective outlook, excluding questions such as what science means for humanity. (Disciplines closer to human science attempt to follow suit.) In Galileo’s mathematization of science, nature is idealized, cast in terms of pure geometry. “Formula-meaning” — the phenomena captured in a formula — replaces true meaning. Even though this mathematization is tremendous achievement, a kind of superficialization results — a detachment from what formulas might be about. The technological and purely mathematical aspects of science produces a world unto itself. Tieszen remarks: “Modern science undergoes a far-reaching transformation and there is a covering over of its meaning” (p. 41).[ix] “A type of naiveté has developed. Galileo is both a discovering and a concealing agent” (p. 42). The alienation of science from original meaning, that is, from its historical emergence through the process of idealization from the real world should be redressed (according to Husserl). A historical disclosure of science needs to be undertaken by philosophers and phenomenologists.[x]

Looking at the historical emergence of science (including mathematics) does not imply a focus on “empirical” history but on the broad, necessary conditions that allow for the emergence of a disciple or field in science.[xi] Understanding the emergence of science is part and parcel of understanding science. (pp. 42-3)

In the emergence of geometry, Husserl has basic perceptual experience playing a key role, as well as formalization. Geometry, e.g., is seen as arising from conscious experience with the “basic features of shape in our everyday, prescientific practice and sense experience” (p. 83).[xii] That is to say, this experience supplies a condition for the possibility of modern geometry. Another such condition — next in a hierarchy of conditions — is idealization, in particular, idealization of perfection or exactness. This arises from experience with measurement:

The idealizations are a natural outcome of refining and perfecting measurement. Every measurement acquires the sense of an approximation to an unattainable but ideally identical pole, that is, to one of the definite mathematical idealities or to one of the numerical constructions belonging to them. (p. 85).

Formalization ushered in by Cartesian geometry, wherein numbers and algebraic expressions represent geometry objects, provides the next condition. This opens the door to more abstract geometries (e.g., of more than three dimensions). In these situations, intuition associated with experience with shape can no longer provide a basis. Maintenance of logical consistency in these cases needs to be relied on. Once advanced geometries are in place, their formalisms are open to multiple interpretations. Interpretations of non-Euclidean formalisms can have application to nature, although, for Husserl, Euclidean geometry may still be best for mathematically rendering the space of ordinary spatial experience. (p. 87)

In Husserl’s view, mathematical objects comprise invariants or identities in experience. For a number of reasons, these are not sensory objects. They do not occur in space and time, are unchanging, and do not causally interact with each other. Mathematical objects have a base in consciousness, even though they are thought of as independent. In this way, Husserl avoids traditional Platonism which (critics of Platonism contend) places mathematical objects in a metaphysical realm beyond the reach of cognition. There is not “anything ‘metaphysical’ lying behind the phenomena” (p. 57).

The concept of intentionality lies at the center of understanding mathematical thought. A cognitive act is always directed toward something or about something.

We can picture the general structure of the intentionality of our acts in the following way:

Act(content) → [object],^[xiv]

where we ‘bracket’ the object in the sense that we do not assume the object of the act always exists.[xv] (p. 52)

The cognitive acts comprise believing, expecting, knowing, remembering, etc. Content is expressed in “that” clauses. An act-content, e.g., is “I believe that 1+1=2". (For Husserl content and meaning go together.) Acts and content in mathematics are founded on acts and content in sensory experience. “[I]t is the condition for the possibility of mathematical knowledge that there be acts of reflection on and abstraction from our basic sensory experience” (p. 55). Although founded on such acts and content, some mathematics becomes distant from this source. “[S]ome of our idealizations, abstractions, and formalizations in mathematics are quite far removed from their origins” (p. 55). Intentionality of human thought, about mathematics and other things, involves a grasping or intuiting of essences (that is, the categories in which an object of apprehension falls). “Intuition” in this context is used in the sense that is broadly analogous to “perception” in that both deal in immediacy. In general, Husserl opposes attempts to reduce intuition of fully abstract concepts in mathematics to something else.

Husserl accords with mathematical constructivism in the sense that experiencing (in some way, intuiting) the object (given as an invariant in experience) counts more than merely demonstrating its existence in, e.g., a proof by contradiction, although Husserl does not reject nonconstructive parts of mathematics. These can be subject to meaning clarification and a range of other potentially productive exercises.

Intuition is often limited. Husserl, thus, subscribes to a measured, graded theory of reliability of mathematical knowledge. “[T]here is no such thing as absolute evidence which would correspond to absolute truth” (Tieszen writes of Husserl’s view, p. 63). However, unlike a good deal of philosophical thought today which doubts mathematical knowledge, Husserl’s philosophy is amenable to the claim of stable mathematical knowledge.[xvi]

Understanding Husserl’s philosophical approach to mathematics helps in understanding a well-known claim of Gödel, who was heavily influenced by Husserl, about mathematical intuition. Gödel writes:

I don’t see any reason why we should have less confidence in this kind of perception, i.e., mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perceptions will agree with them … (Gödel 1964, p. 268; quoted from Tieszen, p. 104)

On first impression, this quotation might seem to be easily refutable because objects of (sense) perception physically cause their own perception (along with other causal factors), but mathematical objects (and situations) cannot cause their own apprehension, even partially, due to their abstract nature. However, suppose one considers a conscious perception from a phenomenological point of view. It can receive evidentiary support from other conscious perceptions. Alternatively, it can get revised, or perhaps even rejected as false, by evidence coming from subsequent perceptions. That is, subsequent perceptions confirm or change conviction in the content of the original perception. Eventually our convictions may stabilize to yield sure conviction. This story is told without considering causation. Only conscious perception is considered. Consequently, causation of sense perception fails to block the comparison, made by Gödel, of sense perception with mathematical intuition. Indeed, much the same story can unfold concerning verification, or reversal, of initial mathematical intuition by subsequent experience. Furthermore, a close comparison now obtains between illusions in perception and illusions in mathematical intuition. This comparison was also recognized by Gödel. Tieszen writes:

Gödel remarks … that “the set-theoretical paradoxes can hardly be anymore troublesome for mathematics than deceptions of the senses are for physics” (Gödel 1964, p. 268). The discovery of the set-theoretic paradoxes showed that on the naive concept of set we were under an illusion about what sets were, which led to a correction or refinement of our knowledge, just as I am under an illusion about a snake’s being under my bed at a certain stage in my experience, an illusion that could be correct under further experience. (pp. 104-5)

Tieszen documents Gödel’s primary focus on philosophy from 1943 on,[xvii] undertaken in part by a desire to accommodate mathematics to his own mathematical discoveries which unhinged Hilbert’s program (pp. 93 ff.). Gödel’s philosophical endeavor led to his high regard for Husserl’s philosophy of mathematics. Gödel’s philosophical assessment of Hilbert’s program is of particular interest. Hilbert attempted to capture, or reduce, all of mathematics to a finite, combinatory manipulation of concrete symbols whose shapes are unmistakable and immediately recognizable. Abstract mathematical concepts well beyond the reach of perception and thus not available to concrete intuition — such as many ideas in set theory — are handled by the introduction of so-called ideal elements. Gödel’s theorems which mathematically unhinged this project suggests a Husserlian philosophical assessment articulated by Tieszen:

…Hilbert’s program does not give mathematical essences or abstract concepts their due. It exhibits a certain blindness or prejudice about them. It is an attempt, in effect, to show that directedness toward essences or abstract concepts can be reduced to directedness toward concrete, finite sign-configurations and combinatorial operations on such objects. (p. 135)

Tieszen shows that in various ways Gödel adheres to this philosophical assessment. Additionally, Gödel recommends intuition of fully abstract mathematical objects, whereby one directly deals with mathematical meaning rather than with Hilbert’s attempt to reduce this meaning to signs and their combinatorial manipulation. Again, this course follows Husserl.