Foundations of Mathematics
by Arnold Neumaier
October 31, 2009
Mathematics is the science and art of precise concepts and their relations. Foundations of mathematics is the science and philosophy of the natural laws by which mathematicians define and reason successfully. Here, approximately, science analyzes and takes stock of what is true and useful about a subject, art considers what is beautiful about a subject and its presentation, and philosophy analyzes what a subject means and what is important about a subject.
Modern mathematics is both structural (in the traditional sense of allowing one to consider mathematical objects up to isomorphism only, whenever this proves useful) and material (in the natural sense of building mathematical objects out of definite, more elementary building blocks, whenever this proves useful), no matter how it is founded. Indeed, most mathematicians care little about foundations, since the foundational crisis that began with the discovery of Russell's paradox was resolved by Zermelo, Fraenkel and Skolem in the 1920s.
In their multi-volume treatise ''Elements of mathematics'', Nicolas Bourbaki chose material foundations for mathematics (essentially ZF+global choice, where ZF is short for Zermelo-Fraenkel set theory) and erected on top of it a building hosting the major part of pure mathematics.
This was done in a fully structural way, and in a way that convinced other mathematicians so much that they adopted it almost universally, with only minor variations in the phrasing of the foundations. Their structural approach clearly exhibited the essence of mathematics, and they showed that it is feasible to structurally present (with a number of convenient abuses of language) most of modern mathematics on the basis of ZF+global choice.
To prove consistency of a theory therefore generally means (if nothing else is said) to construct a model for it in ZF+global choice. If necessary (such as for representing categories of categories), large cardinal axioms are added; if it is enough, weaker theories are used instead.
Foundations should be convenient and elementary, selecting from the whole of mathematics a few bits and pieces that are intuitive to understand, require a minimum of technicalities to explain, and are sufficient to organize and express in a fully formal way all of mathematics -- including the mathematical logic on which the foundation itself is based. They should also blend in naturally with the established informal mathematical language, so that they found the existing part of mathematics, rather than require a new language in which future mathematics is to be written.
In all these respects, ZF+global choice(+large cardinals where needed) is unsurpassed as a foundation.
In the 21st century, an additional criterion for a good foundation is the extent to which it simplifies a full formalization of mathematics, so that computers can understand it and communicate easily with mathematicians about conceptual matters.
In this respect, ZF and its extensions do not fare well. The demand to fit mathematics into a fully formalized ZF language is a straitjacket that makes the formalization very cumbersome, and few mathematicians are prepared to work with such a straitjacket. Existing variations of set theory such as NBG, MK, or NFU are not better in this respects.
This is the reason that I began to develop the FMathL mathematical framework. The near goal of FMathL is to create a formalization of mathematics that is closer to informal mathematical language than any current system, which thus avoids to a large extent the need for abuses of language, and hence makes informal mathematics easier for computers to understand. The far goal is to make the computer understand conceptual mathematics in a way that it can assist mathematicians in their conceptual work.
Any practical, computer-oriented foundation of mathematics needs to have a clear philosophical view of how to interpret mathematics written in the usual informal mathematical language, in order to minimize the gap between ordinary written mathematics and the formal version of it given to the computer.
Since computers can only understand what is unambiguously communicable, one needs to pay more attention to foundational issues. Natural mathematical language is inherently ambiguous when it comes to formalization; a striking example is given here.
Therefore, a (still continuously revised) discussion paper on the FMathLmathematical framework tries to delineate what is objective about mathematics from the subjective views or intuition particular mathematicians (or computer implementations) may have about mathematics, by paying attention to what is actually communicated in standard mathematical language, and how the Platonic world of objectively communicable ideas is accessed and understood by physical beings such as people or computers.
I have been thinking structurally since I learnt abstract algebra 40 years ago. This year, I found to my great surprise that, under the heading of structuralism, some category theorists seriously try to ban from mathematics the notion of equality of two mathematical structures -- and even the equality of elements of any two distinct sets.
In what they call a ''structural set theory'', every element completely determines the set to which it belongs. In traditional terms, what they call sets are, structurally seen (though they'd call my translation a ''material'', nonstructural view), copies of cardinal numbers, what they call elements are structurally flags (pairs (x,A) with x in A, i.e., traditional elements x tagged with a set A to which they belong), and traditional sets are in a ''structural set theory'' closest in meaning to what they call subsets (of a very large, unspecified set U serving as a universe). Indeed, the operations of union and intersection cannot be defined for their notion of sets since an element in the union or intersection of two distinct sets would be in several sets. But union and intersection are definable for an appropriate notion of subsets of a fixed set. Of course, their subsets cannot be sets themselves, giving already the terminology an unnatural twist. (Similarly, subgroups cannot be groups, subfields not fields, etc..)
The property that every element determines the set to which it belongs forces the natural number 2, the integer 2, the rational number 2, the real number 2, and the complex number 2 to be distinct elements since they belong to distinct sets. One must even wonder about whether the even number 2 as an element of the set of even numbers is a natural number, or whether one has to regard the set of even numbers as being not a set (as the informal language says) but only a subset -- in a ''structural set theory'' a concept necessarily very different from sets! -- of the natural numbers.
Even the simplest statements in mathematics become problematic and ambiguous, and different ways to disambiguate them lead to formally inequivalent statements. The demand to fit mathematics into a fully formalized structural language in this extreme sense would be another straitjacket that makes the formalization at least as cumbersome as a formalization in ZF, and few mathematicians would be prepared to work with this straitjacket. FMathL tries to strike middle ground, avoiding both the Scylla of ZF and the Charybdis of structural extremism.
In my view, the proponents of ''structural set theory'' (and the surrounding philosophy of presenting mathematics without speaking ''evil'') try to hijack the concept of structure (and that of a set) for things expressed in a manner very far from the structural (and set theoretic) tradition established by Hilbert and van der Waarden and perfected by Bourbaki, and in a way very far from the ordinary mathematical language. I resist these attempts wholeheartedly, for the following reasons:
1. The proposed criterion of what is a structural presentation seems arbitrary; it does not cover Bourbaki (who have a formal structural concept of canonical identification that makes the number 2 unique and belong -- as in informal mathematical language -- to the natural, the integral, the rational, the real, and the complex numbers). Thus it brands Bourbaki's Elements as not structurally presented.
2. The choice of language in the proposed ''structural set theory'' runs counter to the established conventions in mathematics. Familiar concepts of mathematics must be viewed in a framework requiring reformulations that convey an intuition different from the familiar one (but not more useful) phrased in terms of concepts that carry familiar words for an unfamiliar content. The very notion of a ''structural set theory'' is a misnomer; ''structureless set theory'' would be a far more appropriate name for the class of theories they propose to develop.
3. Worse, viewing traditional mathematics from the proposed point of view gives the familiar language of mathematical definitions a new interpretation that, by means of an unformalizable moral code (that I still haven't fully understood what it permits under which circumstances), which rules out interpretations and uses that are perfectly valid from the traditional, purely formal point of view. (To be definite, I take Serge Lang's algebra book from 1970 (or the third edition from 1993) to define the traditional meaning of structural language in general, and of categories in particular. Probably any other book with a similar breadth and starting point that also treats categories would serve the same purpose, and would illustrate the same.)
4. Insisting on the formally undefined moral code reduces clarity -- what is permitted to do now depends no longer on formally checkable rules but on subjective moral intentions.
5. The proposed point of view does not add understanding -- the resulting mathematics is in content essentially the same as without these attempts (but it is less, if their morals is fully adhered to). This is not to say that categories are not useful -- they are already useful in Lang's version. But to insist on their sole interpretation in terms of structural extremism does not add anything useful to understanding or furthering mathematics.
6. The proposed point of view completely destroys the natural lattice structure of ordinary sets, and forces current correct uses of mathematical language to be viewed as abuse of language in the new setting. Two sets with a common member (an extremely usual situation in mathematical practice) may no longer be thought of as sets, but must be viewed as subsets of some never specified set. A Hilbert space would be only an anonymous element of the category of Hilbert spaces without any additional structure, so talk of the Hilbert space L^2(R) would become an abuse of language, since it constitutes in functional analysis or mathematical physics an object very different from the Hilbert space $L^2(R^2)$, although they happen to have the common property of being a separable Hilbert space and hence are isomorphic when stripped of all their physically relevant structure.
7. The proposed point of view needlessly multiplies notions (such as the many versions of 2 mentioned above), thus violating without clear benefit Ockham's razor, one of the most useful guiding principles of science. The cases where one profits from such subtle distinctions are few and can easily be handled in the traditional framework of Bourbaki.
8. This artificial multiplicity introduces unwelcome ambiguities in translating informal mathematics into formal terms -- it is hard to figure out consistently which of the many possible formalizations an informal object (such as the even number 2) should get. (In this respect it resembles intuitionistic set theory, where the usual set-theoretic notions -- such as that of a finite set -- ramify into a multitude of inequivalent notions, depending on which property characterizing the notion in a classical setting is taking as a definition. (See, e.g., Constructive stones for some of the resulting pitfalls, which would make intuitionistic set theory very slippery foundations.)
9. The proposed point of view makes being formally precise not easier but more difficult, even for elementary mathematical stuff -- requiring the formalization to be full of distracting, irrelevant mappings only needed to restore the consistent matching of objects that were first artificially separated.
10. Finally, this structural extremism does a disservice to the whole mathematical community by presenting a kind of structural thinking abstracted from category theory from a most unfavorable perspective, where its structural content becomes a burden rather than an advantage. There are many situations where a categorial approach and the associated functors are needed (or at least useful) to develop certain aspects of a mathematical theme. For the little informed, the controversy surrounding the claim that categories (or ''structural set theory'') are the better (or even the only) structural foundations may well confuse the unconvincing aspects of structural extremism with other, high-quality work on categories, branding both as abstract nonsense in a negative sense, thus delaying the acceptance of categorial techniques even in areas where they'd prove useful.
I'd view with horror a future in which the ideal of structural presentation was degenerated from the traditional ideal of Hilbert and Bourbaki to a new ideal based on a structural extremism that would force foundation-conscious mathematicians to distinguish between 2 as a natural number and 2 as an integer.
Fortunately, so far, the structural extremism is manifest only in sketches of a new foundation, not a complete one, let alone a building of mathematics itself, safely erected on these new foundations. There are enough ruins that keep in memory great plans that started with laying foundations to a never created building. Luke 14:28-30
FMathL -- Formal Mathematical language
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Arnold Neumaier (Arnold.Neumaier@univie.ac.at)