Time, Consciousness, and Mathematics

An Introduction to Brouwer’s Intuitionism

Theories about the foundations of mathematics are fascinating, but E.I.J. Brouwer’s intuitionism stands out in the 20th century as a theory of unparalleled novelty, ingenuity, and pure iconoclasm. He is not a platonist, nor a formalist, and his vision of mathematics as whole is radically different from the classical approach, rejecting well accepted principles of logic and proving statements that are false in classical mathematics. It is, as Brouwer has himself conceded, a different mathematics entirely, as different as Einstein’s physics is from that of Newton. These big claims have never been widely understood, let alone widely adopted.

To properly demonstrate just how innovative and interesting Brouwer’s ideas are it is necessary to have a firm grasp on the problems that interest contemporary philosophers of mathematics. As the philosopher and mathematician Steweart Shapiro (1997) has so eloquently explained, there are really two questions at the heart of the philosophy of mathematics, an old topic which has been around since the dawn of philosophy and relatively new one which began with Benacerraf (1973). The old problem is the ontological status on mathematical objects. We want to know whether numbers exist, whether sets, points, triangles and so on are objectively real. The traditional answer, which is still popular to this day among certain mathematicians, is ‘mathematical platonism’, which holds that causally inert abstract ideas exist in some platonic heaven beyond space and time — objective, real, and independent from human mind. The second, more recent problem is the semantic status of mathematical statements. We want to know what they mean and whether they are true. If platonism is the traditional answer to the ontological question, then model-theoretic semantics and a Tarskian theory of truth are the prevailing framework to answer the semantic one. A singular term in a mathematical language denotes an object, and the variables range over some domain-of-discourse. According to Shapiro this suggests realism in two senses: realism in ontology and realism in truth value. Mathematical objects exist; and well-formed sentences have a non-vacuous truthvalue.

Shapiro has written a number of books and articles comparing and contrasting various theories of mathematics, showing how each one answers the ontological and the semantic question, and putting Brouwer in the doubly anti-realist camp. Shapiro reads Brouwer as an anti-realist in ontology because intuitionism denies the objective existence of mathematical objects, which instead must depend on the human mind, more specifically on the problematic notion of construction. He also sees Brouwer as an anti-realist with respect to truth-value because intuitionism rejects the Law of the Excluded Middle — i.e., the proposition that all statements are either true or false.

Brouwer’s views are rather unpopular these days and even his most esteemed successors, Heying and Dummett, took different paths in their expression of intuitionism — in Heying’s case by taking a more formal route, in Dummett’s a linguistic one. Neither have the phenomenological tang that pervades Brouwer’s work. Much is written about Brouwer’s rejection of the law of the excluded middle and about the mathematical of nuances of his theory; but there seems to be a deep tension with the ontology of mathematical entities within his system. I think part of the problem is that Brouwer’s ideas are so intuitive nobody seems to know what they mean; another is his style, which is difficult to say the least. While reading Brouwer the following becomes immediately apparent: when he writes mathematics, he writes about mathematics; but when he writes about philosophy, he writes about Philosophy — that is, about the entire discipline. His philosophical writings are filled with dense pages of aesthetics, ethics, metaphysics, logic, and mysticism all intermixed and using his own idiosyncratic jargon. This makes untangling his philosophical system particularly challenging. I will stick with the basics here, which I think are interesting in their own right.

Construction is at the core of intuitionism, which is why the view tracks so closely with the Kantian tradition. For a constructivist the legitimacy of mathematical objects depends upon whether they are cognitively graspable by us. The Kantian constructs his mathematical objects in intuition, though the details of how this works is a contentious affair because the term ‘intuition’ is not well understood. What is not contentious is that Brouwer is deeply indebted to the Kantian tradition and that mathematics is derived from a very surprising source: our perception of time.

Brouwer’s whole system is built from two ‘acts of intuition,’ as he calls them. The first is the moment we come to recognize two distinct moments, the present and the memory of the past. He writes: “Intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time. This perception of a move of time may be described as the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory. If the two-ity thus born is divested of all quality, it passes into the empty form of the common substratum of all two-ities. And it is this common substratum, this empty form, which is the basic intuition of mathematics.” (Brouwer 1981,)

In his 1948 paper Brouwer says much the same thing, writing “mathematics comes into being, when the two-ity created by a move of time is divested of all quality by the subject, and when the remaining empty form of the common substratum of all two-ities… is left to an unlimited unfolding.” (Brouwer, 1948) This is the point where Brouwer becomes problematic: on the one hand almost everyone who reads this will know exactly what he means, as we all have the experience of mentally intuiting that ‘empty form’ in some manner of speaking; but on the other hand, I have no idea what this common substratum is here, and Brouwer never gives us an answer, preferring to rely on the raw intuitiveness of the view over precise definitions. In any case, the idea of the two-ity seems clear enough at least on a phenomenological level: the experience of intuiting two discrete moments and abstracting out the idea of duality.

Thus, the First Act of Intuition gives us the natural numbers, what Brouwer calls ‘separable’ or ‘discrete’ mathematics, by repeating that operation infinitely, but he needs a second principle if intuitionism is going to have any of the richness that it has within classical mathematics. For that he offers the Second Act of Intuition: “Admitting two ways of creating new mathematical entities: firstly in the shape of more or less freely proceeding infinite sequences of mathematical entities previously acquired …; secondly in the shape of mathematical species, i.e. properties supposable for mathematical entities previously acquired, satisfying the condition that if they hold for a certain mathematical entity, they also hold for all mathematical entities which have been defined to be ‘equal’ to it ….” (Brouwer 1981, 8)

The first intuition can include repetitions of the initial constructive act, generating a finite series of digits. The second one allows us to distinguish this process from the idea of carrying it out indefinitely; it is the realization that the sequence is potentially infinite.

This second action allows us to create ‘choice sequences’ of numbers and to form ‘species’ by grouping numbers indefinitely and making more abstract constructions and eventually yielding the continuum. The details of these operations have been dealt with extraordinary clarity in On Brouwer as well in the works of Carl Posy. What is more difficult to find in the literature is a proper account of what these entities and how we know them — which is to say, the philosophical questions of epistemology and metaphyics. Brouwer beats it into his reader that mathematics is ultimately a constructive activity, one closely linked with our perception of time, but this needs to be spelled out in much more detail for it to be at all convincing. The problem is that these details are hidden within the labyrinthian depths of a few short philosophical works. Brouwer used to boast that he never ready anything after his 16th year, and while that may or may not be true, it certainly seems that way in places because most of Brouwer’s philosophizing is highly idiosyncratic and solipsistic. As van Dalen (1978) has noted, this led to the ‘magical reputation’ Brouwer has as someone distinct from his age — an original thinker, a maverick, a mathematical mystic.

At one point he writes, “I only love mathematics for something which is not really mathematics, that is to say because of the clear light it sometimes sheds on the general questions of life.” My impression from reading Brouwer is that mathematics was an almost spiritual activity for him, a practice whereby he not only solved complex and abstract problems but contemplated the cosmic order and found a recurring source of solace and wonder. Although most of us will never understand his sublime fixed point theorem (one of the most beautiful pieces of mathematics ever created), we can nevertheless reflect and be amazed by the flow of time, by our ability to count it, and by the deeper structures that have yet to be discovered.