Colloquium talk from March 27, 2009

I've neglected this space for too long.

Here's a talk I gave at the colloquium Mary Joe put together in the 2008-2009 academic year. It is a discussion about the role of thinking in mathematics. Note to the reader: these are my notes, so there are a few indications where I would need to digress to explain some topic or other. Try not to let that trip you up :)



This talk concerns the following question: What is the relationship between mathematics and mathematical thinking. We must begin by striking out to understand what mathematics is and what it is not. We shall attempt to find such an understanding by first considering the notion generally attributed to Georg Cantor known as the or an actual infinite. Following a presentation of the mathematical thinking underlying this notion and the particular mathematics utilized in its defense we shall return to our guiding question in order to consider what we can say. We will then present the critique of this notion of an actual infinite produced by Ludwig Wittgenstein. We will find in Wittgenstein a mode of thought that places great care in the meaning of mathematical thinking and its rightful relationship with mathematics. Then, we return back again to our question of the relationship between mathematics and mathematical thinking. Following our return we shall consider the thought of Heidegger with respect to mathematics, which is nothing more than a questioning of the meaning of the mathematical.

Mathematics is taken by many to be quite difficult in character, obtuse in its progression, and in a transcendental relationship to ordinary thought. None of these qualities subsist in mathematics, though, for mathematics is nothing more than the constructions of mathematicians. Mathematics is the life-work of people thinking mathematically, which is just thinking in such a way that what is found is nothing more than that which is given. To see this in a simple exercise, consider the usual algebraic question: what number added with itself sums to four? Clearly the answer is two, and only two. Why only two? To answer this question is to transcend the place where mathematics happens and to enter the origins, the provenance of mathematics. One might respond, because we concern ourselves here only with natural, counting numbers, and these numbers are ordered such as to bring into effect the law of trichotomy (discuss). Among such numbers, those greater than two, when added with itself is bound to be greater than four, and, in the alternative non-two case, one sums with itself to two. Clearly two taken together with itself is four, and since no other such number qualifies as a natural number that can sum with itself to four, we have shown that only two can satisfy our question. Mathematical thinking requires, it is often said, that if we understand the calculus of natural numbers, and if we understand the rules of inference within a particular mathematical framework (i.e. that the natural numbers are well-ordered, and what that entails), we must come to this conclusion.

So we see at least preliminarily that mathematics is, in a sense, the result of mathematical thinking, and that mathematical thinking, in a sense, shapes mathematics.

I promised, though, a discussion of the topic of infinity. So let us turn, then, to Cantor’s actual infinite. On many occasions philosophers and mathematicians have concerned themselves with the infinite. Some of you are familiar with at least one of Zeno’s paradoxes (I can think of only the one of Achilles and the Tortoise at present, and if anyone wants to hear about that please feel free to ask). Since Aristotle there has been a distinction made between a potential infinite (a sequence of numbers that is never completed) and an actual infinite (a sequence of infinitely many elements that is completed). Aristotle thinks that infinity, rightly considered, is only potential. For Cantor, there is not only the possibility of a sequence of numbers continuing forth without end, but also the existence of a set of numbers that can be taken as a whole yet cannot be listed, even without end. We are “familiar” with a particular set of this kind that we call (because Cantor called them this) Real numbers. For Cantor, the Real numbers are real. He conceives of the real numbers as actually existing.
The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.

So Cantor says that the actual infinite is a mathematical magnitude graspable in abstracto.

Why does he say this? Because he believed that he had a proof that the Reals are not, in principle, listable and so, of a different order (cardinality) than the infinite set of rational numbers (or counting numbers). Cantor’s proof for the existence of un-listably many numbers (that is, more than one can list, even in principle) is known as the diagonal argument. Here is the idea:

We can write down a sequence of numbers that have infinite decimal expansions where not every number appearing in the expansion of any of the numbers from our sequence is all 9’s (this is because we take .99999… = 1). Then we can enumerate a table of numbers,

0.d11d12d13d14d15…
0.d21d22d23d24d25…
0.d31d32d33d34d35…
0.d41d42d43d44d45…
0.d51d52d53d54d55…
. . . . . .
. . . . . .
. . . . . .


Then we can construct a real number that is not in this enumerations by choosing 0.c1c2c3c4… where c1 is not equal to d11, c2 is not equal to d22, c3 is not equal to d33, c4 is not equal to d44, and so on. The resulting number is distinct in at least one place of its expansion from every number in our infinite sequence. Thus the real numbers cannot be enumerated. That is, according to the generally accepted conclusion, the cardinality of the Real numbers is “bigger” than that of the natural numbers (two sets of numbers have the same cardinality if they can be places in 1-1 correspondence).


So we have at least two “sizes” of the infinite: the smaller size corresponding to the natural numbers and the larger size corresponding to the real numbers. Recall, though, that we enter this discussion of the infinite purposefully. We have at the outset defined our investigation to have as its guiding concern the relationship between mathematical thinking and mathematics. We had preliminarily seen that mathematics is the construction of mathematical thinking. But Cantor’s conclusion is taken to be a part of mathematics. Consider the question: is Cantor’s a claim one on mathematics, or a claim on mathematical thinking? Does mathematical thinking require commitment to the notion of an actual, existent infinite set as the result of the diagonalization method for constructing numbers?

To address this, we turn now to Wittgenstein. Wittgenstein holds two axioms about mathematics. The first is that mathematics is a human construction. The received view is that mathematical truths are discovered and that such truths already existed before their discovery. This is the view that Cantor holds. Wittgenstein rejects this “Platonic” view as a fundamental misunderstanding of the ontology and existence of mathematics. The second principle that determines most of Wittgenstein’s philosophy of mathematics is that mathematics consists entirely of intentions, extensions, and calculi. A mathematical calculus is a set of rules for calculating (such as calculating an angle of a triangle). Extensions are finite compositions of symbols, so finite lists of numbers as well as (genuine) propositions of finite length are mathematically extended things. Intentions are rules for constructing “infinite” sequences, substitution, translation, and other rules of inference (induction).

These two principles inform Wittgenstein’s other views, and five important consequeces can be drawn from these principles. The first is a rejection of infinite mathematical extensions. Extensions are sequences of symbols that have their meaning entirely syntactically and exist as they are constructed within the activity of mathematicians. Any sequence of symbols must, then, terminate after only finitely many places or steps according to a rule. So an infinite mathematical extension is, in principle, impossible. The extended is always only known to a finite extent, though constructively within such a domain.

The second consequence we can draw is a rejection of unbounded quantification, i.e. quantification over an infinite domain. The only mathematical objects that exist are extensions and these we said are necessarily finite. Since the only mathematical objects we may quantify are finite (and so can be taken as a whole), we cannot quantify over an infinite set (because it is never whole).

The third consequence is that all genuine mathematical propositions are algorithmically decidable or undecidable. That is, a finite string of symbols can only constitute a genuine mathematical claim if it has been shown to be true (i.e. there is a proof) or false or how to decide the proposition with a known decision procedure. This follows from the fact that propositions are, generally, finite conjunctions of sub-statements, each of which is itself decidable (because it is characteristic of mathematical propositions that they stand as decidable).

The fourth consequence is an anti-foundationalist account of irrational numbers, which holds that irrational numbers are rules that when carried out, yields a finite decimal expansion which is itself a particular extension of the rule.

The fifth consequence is a rejection of definite infinite cardinalities. The last consequence constitutes Wittgenstein’s rejection of Cantor’s conclusions regarding the diagonalization method.

To see why Wittgenstein holds this view, consider what Wittgenstein is getting at in the fourth consequence I listed. Irrational numbers are rules. Rules are categorically distinct from extensions. That is, irrational numbers are not numbers. They are not objects, but an unending computation of some particular rule. Let us return to Cantor’s diagonal. We are to construct a number of infinite extensions that does not occur on our list. The irrational thus constructed is taken to be a number, an object in its own right that may be considered whole and quantifiable. But if Witgenstein is right, Cantor has not constructed a mathematical object, but instead defined a rule. We have not extension, then, in this case; just intention.

Moreover, on Wittgenstein’s view, so long as we have constructed a number using Cantor’s method, that number has a finite decimal expansion followed by an indicator that we’re not yet done. Since any entry of the table will have n terms in its expansion, for some n, we know that there are n terms in the table, and we are free to add Cantor’s constructed number to our list. So long as we heed Wittgenstein’s distinction and do not conflate extension with intension, then we can assert that an actual infinity is never actualized. We can maintain the single notion of a potential infinite.

Let us step back, at this point, and consider what Wittgenstein is objecting to in Cantor’s assumptions. Both men see the diagonlization method as being a series of valid moves, as good mathematics. What Cantor and Wittgenstein disagree about is what this method tells us. In other words, what have we learned from what we have constructed? Since we have indeed constructed something, we can say that there is disagreement between the two as to what is learned from what we have been given (albeit via our own construction?).

This is a disagreement about the mathematical. According to Heidegger, the mathematical is what can be learned from what one already has. Wittgenstein and Cantor disagree about what diagonalization tells us because they do not share the same presuppositions. Cantor is eager to have the absolute show itself. Wittgenstein determines that the finitude of mathematics derives from its being a human endeavor. These are radically different origins or foundations from which mathematics unfolds. From these varying axioms or guiding principles, Wittgenstein and Cantor (with is robust set theory) diverge on what is the same tactile situation. The evidencing of diagonalization occurs in drastically different ways, and the evidence, informed by the mathematical project of both men, becomes something other than itself. The evidence is something other than a method for constructing a sequence. It becomes operational in the correctness of both mathematical projects, and its truth is covered over by the initial projection of the essential determinations of each projector. Which is just to say: the truth subsists and is carried along in the praxis of projecting. Truth, though, always subsists. Within the project, correctness reigns, and truth lies underneath, covered over the by the work of the mathematical thinker.

All of this is simply to repeat a thought from Heidegger, but not for the sake of repetition. The repetition has its place for us vitally, for what is more important at our present stage then the necessity of understanding genuine, foundational disagreement? What have we shown ourselves, what have we seen, other than the damnation of not being honest about the origins of our reasoning and of not striking out to, at bottom, reconcile the pace between origins? Reason-giving is itself a practice whose force and validity comes from itself. Its power, beyond this, is instituted in history, but this should not be confused with some kind of necessity. For, where reasons do not come forth there is not, then, “merely nothing”. For in reasons’ stead may come a more primordial force that we take care not to name, an immutable groundswell… Yet this we must attend, and we fail to do so at great peril. We must attend the groundswell, in our way, which is, at least in one way, by giving reasons, by setting forth to find that which was already there, but whose disclosure and existence must be worked into place.

And here we return to our guide, the connection of mathematics and mathematical thinking. The connection is this: that mathematics is worked into place. This, not out of nothing, but out of the fundamental projection of a determination of the essence of beings as such, from something given. A connection set in place with thanks to a resolve to give reasons. This is a connection of the happening of mathematics, and its particular manner of coming to be.