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Through the Heart of Every Man

Are There Final Classifications?

12 May 2024 | Math Philosophy Metaphysics

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Philosophers have quite a fixation with classification. Aristotle and Kant posted famous lists of metaphysical interpretive categories; Peirce classified signs (and just about everything else) according to a triadic scheme. Scientists, of course, are no stranger either. Aristotle's work in On Animals can certainly be said to be such, and the theme has reverberated through Linnaeus and Mendeleev and Mohs to the Standard Model. And in mathematics it almost becomes the entire discipline: algebraic topology, for instance, aims ultimately at a classification of topological spaces up to homotopy type, and finite group theory has furnished us with a half-complete Hölder program, which characterize all its "simple" groups up to isomorphism.

What, practically, are we to make of these classifications, though? How much stock should we put in them? Can we expect they will never be extended with new classes? Can we expect the old bins to remain intact? Would you stake your entire philosophical system on their eternity?

A simple answer might be "No, obviously all categories are determined by our attention to various aspects of phenomena, according to our values, and any degree of coherence in a particular classification over time is just because attention and values have remained constant enough, in the particular community in which the classification reigns, for them not to see a need to revise it." This tact is most clearly suggested by the metaphysical examples above—those have seen major revision over the years. Aristotle's categorization of change, for instance, gave way to (quite successful) explanation of all change in terms of motion in the modern period's move towards corpuscular analyses of being. We today might recast that in other terms, in light of our lively quantum vacuum, while still recognizing the monumental advances of both Aristotelian natural philosophy and 16th century physics. The scientific examples are a little less easy, but still fit into the position. The periodic table might be fixed for all intents and purposes, but it's fixed by empirical propositions. Things might, in fact, be different, and if they are, we would revise them. Indeed, taxonomy would reveal just how much our attention and values factor in to even these classifications, and we might even revise our opinion of the periodic table in light of them (isn't our focus on atomic number and electronic/chemical properties rather arbitrary?).

The mathematical examples, however, might engender serious pause. After all, isn't it really the case, in some rather more transcendent sense, that every finite simple group is either cyclic, alternating, of Lie type, or one of 27 specific "sporadic" groups? Aren't numbers really either odd or even? What are we to make of this? Also, wouldn't it be really nice to have apodictic categories in metaphysics? Is there an interpolant position?

1. Completeness and Relevance

To tackle this, we need to be a little more precise about the question. Let's look at the mathematical case, where the problem is stated quite clearly. Here, we both have an (as good as) exact characterization of the objects to be classified, an (as good as) exact characterization of what it means for two of the objects to be the same, and prove that being in the set/class/type of objects we are to classify is logically equivalent to being the same as something in one of the bin sets/classes/types. In other words, we prove that our classification scheme is complete (as in, a complete partition). That is, we can be sure we will never find another mathematical object of the kind specified that cannot be fit into one of the bins that compose our classification. The argument demonstrates that the very meaning of the objects to which the classification applies "contains" (in a vague, Kantean sense) the meaning of exactly one of the bins.

This is a remarkable feat. Our classificatory scheme, so-proven, will be permanently secure against the need to add new bins on the basis of discovering new objects. If we find an object outside all of the bins, it can't be the kind of thing we are classifying.

However, this is hardly the only type of security we might want a classificatory scheme to have. While the classification of finite simple groups is a jewel of modern mathematics, we wouldn't give the time of day to a classification of finite simple groups into, say, those with a subgroup of order 3 or not. We can, of course, classify the same objects as many ways as we please, and some classifications seem more relevant than other classifications. Can we have any sort of security against the choice of a different way of classifying the same objects?

Well, we need to look at the reasons some classifications are chosen over others. In the case of finite simple groups, the classification furnishes us with a fairly minimal construction of all of the simple groups. That is, since we know how to enumerate models of things in the bins, we can write a procedure that goes through and generates every single simple group (which we can easily convert into a procedure that recognizes simplicity, and one that decides whether a group is simple). In this context, we might imagine comparing the possible classifications by the (Kolmogorov, say) complexity of their construction, and finding that indeed the one we've chosen is the most minimal.

But again, this is all dependent on the reason we are seeking a classification. Someone else, or mathematicians of the future, could well seek something different than simply-constructed models from a classification of simple groups; then they would have every right to abandon our classification and use their own. Arguments for the permanence of our reasons can only hope to be rather fuzzy, in this thymological enterprise. I suppose this position might be stated: "That our classifications are complete with respect to our use of language, we can be certain. But relevance of a classification is a value-laden, aesthetic matter of which we expect no stability."