Friday nights at the Library

On foundations and folklore
Maria Cristina Pedicchio (also lnglabeng.png) and Walter Tholen, eds. Categorical foundations. Special topics in order, topology, algebra, and sheaf theory. Cambridge: Cambridge University Press, 2004. (Encyclopedia of Mathematics and its Applications, 97) See also the item “Atlantis” in Tholen’s publication list.
In Scholastic philosophy, a local assumption—that is, a premise that is not the conclusion of some prior argument—is often called a fundamentum, a “foundation”. To be a fundamentum is a rhetorical matter; distinctions treated as fundamenta in one context may be argued for at length elsewhere. A fundamentum functions as a point granted by both sides in a disputatio, because it is obvious, perhaps, or a matter of faith. Of course what is obvious to a well-trained Jesuit may not be obvious to you or me.
Mathematical argument is replete with fundamenta in the Scholastic sense. For example, I may start a proof by saying “let f : AB be a map and let Im f be the image of f ”. I’m assuming of course that you know how to interpret the symbols; I’m also assuming a lemma of the form “Every map has an image”.
One fundamentum that turned out not to be easily proved is the Jordan Curve Theorem. The Theorem states that every simple closed curve in the plane divides the plane into exactly two pieces separated from one another by the curve. Everyone assumed, most often tacitly, its truth; but even Camille Jordan, after whom it is named, did not succeed in proving it; Oswald Veblen finally proved it in 1905. A rigorous proof (a formal proof, that is, checkable by a computer) was completed only in 2005 after fourteen years of effort. See Yakusa Nakamura, “Culmination of a complete proof of the Jordan Curve Theorem” (Sep 2005); the proof itself is accessible from the “abstract” at the Mizar Home Page.
It is my expectation that either you will have seen a proof of the lemma or that you will grant it for the sake of argument. In every branch of mathematics, as it develops, there is a collection of fundamenta, theorems that everyone takes as proved—the “folklore” of that branch. Some have in fact been proved; but some, taken as obvious, may not be explicitly proved until someone writes an introduction to the subject for outsiders. You won’t, after all, make your mark by proving what everyone “knows”—even if in fact they don’t. Though fundamenta are supposed to be obvious, and (so you’d think) easy to prove, sometimes the obvious and the easy-to-prove part ways. The worst case is when the obvious turns out to be false; but that, I think, is not so common. More often it turns out that what was taken to be obvious is not—that is, it depends on hitherto unstated conditions.
The Foundations in Pedicchio and Tholen’s collection are fundamenta both by virtue of being “obvious” (up to a point) to those well-versed in the discipline and by virtue of being “first” in their respective branches. Everyone “knows”, for example, that an idempotent commutative monoid—which is algebraic object—is equivalent, in a sense only recently made precise, to a semilattice. Everyone has known for much longer that in many cases the collection of subobjects of an algebra is a lattice. Knowing those things is requisite to knowing many other things: knowing how to subtract is requisite to knowing the algorithm for long division.
A mathematician’s expertise includes knowing a great many propositions of the form “every so-and-so is a so-and-so”, and also of the form “not every so-and-so is a so-and-so”. The latter are typically witnessed by counterexamples.
Separation properties. The inclusions are proved from the definitions; the properness of the inclusions is witnessed by counterexamples (Seebach & Steen’s example no. 8, for instance, witnesses the truth of “not every T0 space is T1”). Adapted from Lynn A. Steen & J. Arthur Seebach, Jr., Counterexamples in topology (2nd ed., Springer, 1978; Dover, 1995), p16.
The picture on the right, adapted from Seebach and Steen’s Counterexamples in topology, exhibits the relations among classes of topological spaces; the properties listed (other than “Urysohn”) are defined in terms of the open sets and points of a space. It is likely that only specialists will know all the inclusions pictured here. But most mathematicians will know what “Hausdorff” means, that not all spaces are Hausdorff, and that certain constructions preserve Hausdorffness.
The fundamenta of mathematics as a whole are a kind of quilt composed of the overlapping fundamenta of its various branches, together with a small collection of fundamenta common to all. This would include logic (at least constructive logic), naïve set theory, and—increasingly—the basics of category theory. Pedicchio and Tholen’s collection is a series of “glimpses”, from a categorical point of view, of “key areas” in mathematics (2). Those areas are: the theory of order, topology, commutative algebra, universal algebra (as the theory of “monads”), sheaf theory, and the descent theory of Grothendieck.
In a capsule history, Tholen states the project of the new way of doing “structural mathematics” that emerged in the work of Samuel Eilenberg and Saunders Mac Lane during and just after World War II.
Rather than building objects made up from sets with a structure, like topological spaces and modules over a ring, one imposes additional conditions on a category which make its objects behave like the structured sets one has in mind. Hence, rather than relying on a set-theoretic foundation on which to build the structures at issue, one moves into a “fully equipped building” in which to explore the objects of interest (“Introduction”, 1).
Each branch of mathematics requires a “workspace” that contains the tools and materials for building the objects it studies. Set theory, as I’ve said, is a universal workspace. Many theories—including, for example, elementary theories of objects like groups and rings—do not require so much as set theory provides (unless you are interested in questions of cardinality and the like, where you may need the full machinery of “higher set theory”). Two sorts of questions arise. The first sort concerns the “base category” from which you start. If you want to build groups, you need at least to be able to define operations; and if your theory is to include theorems analogous to those of the classical theory of groups, you need various other constructions too. The second sort concerns global features of the category of the objects constructed. What sort of features must a category have to be a category of groups, of spaces?
Seen in this light, the theory of sets is not so much a description of the mathematical universe as it is the specification of a workspace—a “convenient category”—in which to build various objects, notably the classical objects ℕ, ℤ, ℚ, ℝ, and ℂ (the natural numbers, the integers, the rationals, the reals, and the complex numbers), together with structures based on them like n-dimensional Euclidean space or the space of all functions from ℝ to ℝ. Philosophers have tended to think of those objects as unique (if they exist at all). There is just one 1, just one 2, and so on, to which the names ‘1’, ‘2’, and so on refer uniquely; and thus too just one set of natural numbers. Mathematics is about the classical objects, so that group theory, topology, and so on just provide ever fancier ways of proving theorems about numbers or sets constructed from numbers. So-called “structuralism” in the philosophy of mathematics has altered that only to the extent of talking about “patterns”; but this, it seems to me, is a rudimentary version of profounder, more fruitful ideas to be found in mathematics itself.
Some chapters in Categorical foundations are too technical to be of immediate interest to philosophers. R. J. Wood’s chapter on order, however, concerns a topic that many philosophers will have encountered in logic. Even so, the standpoint taken will probably be unfamiliar. If you have studied logic, you will know the inference rules for disjunction; you will know the truth-table; but you will probably not know that ‘or’ is the join operation for the lattice of truth values or that it is “adjoint to the diagonal morphism”. Those are more general characterizations: they define conjunction even where the lattice of truth values is not Boolean, and they extend readily to infinite disjunction.
One moral to draw from Wood’s treatment of lattices (and the treatment of Heyting lattices and Boolean algebras that follows) is that not all axiomatizations of a theory are equal. Some make evident connections with other parts of mathematics; others obscure them.
The subject of the second and third chapters of Foundations is topology. As always, the point of view is categorical. Chapter II, by Jorge Picado, Aleš Pultr, and Anna Tozzi, is on locales, which are a special sort of lattice.
Topology began as, and still is, a heterogeneous branch of mathematics. Its beginnings lie, on the one hand, in the set theory of Cantor and the limit concepts of analysis, and on the other hand, in the algebraic treatment of figures either themselves discrete, like the polyhedra of Euler’s theorem (which was already known to Descartes) or which have discrete surrogates, like the surfaces of Möbius and Klein, which can be “triangulated” and thus treated as if they were polyhedra too. Topology of the first sort was called “point-set” or “general” topology; topology of the other was called “algebraic”.
Point-set topology was mature by the 1930s. With the exception of paracompactness, almost all the basic concepts had been defined, and a body of theorems based on them supplied the content for introductory courses—the “common knowledge” working mathematicians were supposed to have (see Steen and Seebach Counterexamples for a history of the basic concepts in relation to the metrizability of spaces).
In point-set topology a space is defined in terms of a base set and a collection of its subsets called the opens of that space. (The elements of the base set are called points.) The collection of opens satisfies certain simple conditions: the empty set and the base set are open; a finite intersection of opens is open; any union of opens is open. These conditions and the properties of union and intersection in the category of sets make the collection of opens a lattice (lattices will be defined below). For any base set, the collection consisting only in that set itself and the empty set is satisfies the conditions, and is therefore a topology, known as the indiscrete topology.
In the 50s and 60s it became apparent that in some contexts only the lattice structure matters; the point-structure is irrelevant. All indiscrete spaces, however many points they contain, have the same lattice structure: it is the 2-object lattice 2 defined above. They fail to be “abstractly the same”, that is, isomorphic in the category of spaces, only because they differ in the cardinality of their base sets. This, you might say, is not really a topological but a set-theoretical property, just as absolute size, unlike relative size, is not really a property of Euclid’s circles and triangles.
If we abstract from the point structure of a space, and consider only its opens, ordered by inclusion, the result is a lattice which has finite meets but infinite joins (corressponding to the arbitrary unions mentioned in the conditions on opens)—a “join-complete” lattice. The meet “distributes” over the join: for all opens x and all collections of opens S,
x ⋀ ∨{y | yS} = ∨{xy | yS}
In short, the opens of a space are a “distributive 0,1,⋀,∨-lattice”. Such lattices are called frames. Every finite distributive lattice is join-complete (but not every finite lattice is distributive: the third figure in the Gallery is an example).
We start with a space defined as a set with structure (the lattice of open subsets). We throw away the set and consider only the lattice, which is of course a frame. Can we then recover the set? Just from considering indiscrete spaces, it is clear that the answer is no. The lattice of open subsets of an indiscrete space, whatever its base set, is just the two element lattice 2. What has gone wrong? If the entire base set is the only nonempty open, we can’t distinguish the points of that set by reference to the lattice of opens. There aren’t enough of them.
At this point I must wave my hands a little. The natural definition of a “morphism of frames” is just the notion inherited from their definition as orders (i.e. as categories of a certain sort) or as lattices. A morphism of frames is an order-morphism. But to make frames look like spaces, you must consider not the category of frames and order-morphisms, but rather the opposite of that category, which has the same objects and arrows, but in which the sources and targets are reversed (see the more precise definition given earlier). This new category is called the category of locales.
A space whose points can be distinguished by its opens is called sober. Sobriety is a elementary property of spaces. In the category of frames there is likewise an elementary property that distinguishes “spatial” frames. The theorem relating those two properties can be taken to show just when two intuitive conceptions of space, each with a long history, coincide.
The first is the conception of a space as consisting of parts. In Scholastic treatments of quantity, extensio, the property by virtue of which a body “occupies space”—by virtue of which it has measurable dimensions, is divisible, and has modes of the sort we call surfaces, edges, and corners—is defined as “having partes extra partes”, “parts outside parts”, where ‘outside’ at least means “distinct”.
The second intuitive conception is of space as a collection of dimensionless points. It was rejected by Aristotle but implicit, for example, in the point-by-point construction of non-algebraic curves like the quadratrix. Not until the nineteenth century did anyone understood rigorously how to construct lines, surfaces, and volumes from points; but despite the apparent paradoxes arising from this conception it persisted. In the twentieth century it became predominant.
A space S is disconnected iff there exist opens u and v such that uv = 0 (the empty set) and uv = 1 (the entire space S) in its lattice of opens. In this case the opens u and v are called components.
A space is totally disconnected if every point is a component. The connection with Boolean algebras becomes more evident if you think of v in the situation above as the “negation” ¬u of u. If points are components, then every open u of S has a negation ¬u which satisfies the axioms for negation in classical logic.
To define the Cantor set, consider a line segment. Label its endpoints 0 and 1. Now divide it into thirds. Throw away the middle third, leaving, however, its endpoints 1/3, 2/3. From the two segments that remain remove their middle thirds, again leaving the endpoints. From the four segments that remain, remove their middle thirds. And so forth. What remains is a set of points corresponding to the numbers x1/3 + x2/32 + x3/33 + x4/34 + … where each xi is either 0 or 2. See Mathworld, Wikipedia, and cut-the-knot for more.
Consider the relation between the “points” of a frame (or locale) and its opens. Write “xu” for this relation. In spatial locales it can be interpreted as set-membership, but not in general. The relation satisfies in all locales the axioms not of classical, but of constructive logic (with finite conjunctions and arbitrary disjunctions: see Vickers for the details). The logic of spaces is classical (or Boolean) only in spaces where the (set-theoretical) complement of an open is itself open. A theorem of Marshall H. Stone, proved in 1934, states that those spaces are the “totally disconnected” spaces (the Cantor set, with the topology it inherits as a subset of the real line, is an example; see Johnstone for proofs and a history of Stone’s theorem). What is remarkable in Stone’s work, according to Johnstone, is that “nobody had previously had the idea of applying these techniques”—the construction of spaces with specified topological properties—“to the study of spaces constructed from purely algebraic data” (xv). Stone’s theorem is an early instance of a phenomenon pervasive in mathematics: an equivalence of categories.
All I’ve mentioned here is the preliminaries to the first two chapters of Foundations. But that suffices to raise some philosophical questions.
1. Is the shift from point-set topology, where a space is a “set with structure”, to localic topology, where “having points” is an additional feature possessed by some but not all topologies, a conceptually significant change?
If we consider the lattice of opens to be a category, then the preeminence of the part-whole relation is an instance of the general injunction to consider objects only by way of their morphisms. That, as a number of authors have argued, a significant conceptual change even if categories can be modelled as sets. One indication that the concept has changed is the fact that “pointless”locales exist (see Johnstone, “The point of…” and “The art of…” for details). Locales are more general than spaces; but not just any generalization is fruitful.
2. What is the epistemological role of the concept of space—in either the point-set or the localic formulation? The realization that in certain settings (sheaf theory, for example, which is the topic of chapter VII in Foundations) only the lattice-theoretic properties of a space are relevant was one of the motivations behind the replacement of spaces with locales. That suggests a maxim (in the spirit of Maddy’s “Maximize” for large cardinal theory): “don’t import more structure than you need into your objects”. What is the benefit of following such a maxim? I don’t think questions like this can be answered, any more than in the other sciences, except by reference to practice.
Johnstone, Peter T. “The Art of pointless thinking: a student's guide to the category of locales”. In: Category Theory at Work (Heldermann, 1991):85–107.
Johnstone, Peter T. “The point of pointless topology”. Bull. Amer. Math. Soc. (N.S.) 8(1983):41–53.
Johnstone, Peter T. Stone spaces. Cambridge: Cambridge University Press, 1982. (Cambridge studies in advanced mathematics, 3)
Lambek, J. and P. J. Scott. Introduction to higher order categorical logic. Cambridge: Cambridge University Press, 1986. (Cambridge studies in advanced mathematics, 7)
Picado, Jorge. Weil Entourages in Pointfree Topology. Ph. D. Thesis, Departamento de Matemática, Universidade de Coimbra, 1995. (The introduction is available online.)
Steen, Lynn Arthur, and J. Arthur Seebach, Jr. Counterexamples in topology. Mineola, New York: Dover, 1995 (reprint of the 2nd ed., 1978).
Vickers, Steven. “Topology via constructive logic”. http://www.cs.bham.ac.uk/~sjv/tvcl.ps (postscript). Other formats at Citeseer (or the MIT mirror).
Vickers, Steven. Topology via logic. Cambridge: Cambridge University Press, 1989. (Cambridge tracts in theoretical computer science, 6)

LinkDecember 24, 2005 in Books · Philosophy of Mathematics · Science