5. Conclusions

Our basic intuitions about mathematics are to some extent abstracted from our dealings with the everyday physical world [19]. The concept of a set, for example, formalizes some of our intuitions about piles of pebbles, herds of sheep and the like. These things are all pretty well described by classical physics, at least in their gross features. For this reason, it may seem amazing that mathematics based on set theory can successfully describe the microworld, where quantum physics reigns supreme. However, beyond the overall `surprising effectiveness of mathematics', this should not really come as a shock. After all, set theory is sufficiently flexible that any sort of effective algorithm for making predictions can be encoded in the language of set theory: even Peano arithmetic would suffice.

But, we should not be lulled into accepting the primacy of the category of sets and functions just because of its flexibility. The mere fact that we can use set theory as a framework for studying quantum phenomena does not imply that this is the most enlightening approach. Indeed, the famously counter-intuitive behavior of the microworld suggests that not only set theory but even classical logic is not optimized for understanding quantum systems. While there are no real paradoxes, and one can compute everything to one's heart's content, one often feels that one is grasping these systems `indirectly', like a nuclear power plant operator handling radioactive material behind a plate glass window with robot arms. This sense of distance is reflected in the endless literature on `interpretations of quantum mechanics', and also in the constant invocation of the split between `observer' and `system'. It is as if classical logic continued to apply to us, while the mysterious rules of quantum theory apply only to the physical systems we are studying. But of course this is not true: we are part of the world being studied.

To the category theorist, this raises the possibility that quantum theory might make more sense when viewed, not from the category of sets and functions, but within some other category: for example ${\rm Hilb}$, the category of Hilbert spaces and bounded linear operators. Of course it is most convenient to define this category and study it with the help of set theory. However, as we have seen, the fact that Hilbert spaces are sets equipped with extra structure and properties is almost a distraction when trying to understand ${\rm Hilb}$, because its morphisms are not functions that preserve this extra structure. So, we can gain a new understanding of quantum theory by trying to accept ${\rm Hilb}$ on its own terms, unfettered by preconceptions taken from the category ${\rm Set}$. As Corfield [10] writes: ``Category theory allows you to work on structures without the need first to pulverise them into set theoretic dust. To give an example from the field of architecture, when studying Notre Dame cathedral in Paris, you try to understand how the building relates to other cathedrals of the day, and then to earlier and later cathedrals, and other kinds of ecclesiastical building. What you don't do is begin by imagining it reduced to a pile of mineral fragments.''

In this paper, we have tried to say quite precisely how some intuitions taken from ${\rm Set}$ fail in ${\rm Hilb}$. Namely: unlike ${\rm Set}$, ${\rm Hilb}$ is a $\ast$-category, and a monoidal category where the tensor product is noncartesian. But, what makes this really interesting is that these ways in which ${\rm Hilb}$ differs from ${\rm Set}$ are precisely the ways it resembles $n{\rm Cob}$, the category of $(n-1)$-dimensional manifolds and $n$-dimensional cobordisms going between these manifolds. In general relativity these cobordisms represent `spacetimes'. Thus, from the category-theoretic perspective, a bounded linear operator between Hilbert spaces acts more like a spacetime than a function. This not only sheds a new light on some classic quantum quandaries, it also bodes well for the main task of quantum gravity, namely to reconcile quantum theory with general relativity.

At best, we have only succeeded in sketching a few aspects of the analogy between ${\rm Hilb}$ and $n{\rm Cob}$. In a more detailed treatment we would explain how both ${\rm Hilb}$ and $n{\rm Cob}$ are `symmetric monoidal categories with duals' -- a notion which subsumes being a monoidal category and a $\ast$-category. Moreover, we would explain how unitary topological quantum field theories exploit this fact to this hilt. However, a discussion of this can be found elsewhere [6], and it necessarily leads us into deeper mathematical waters which are not of such immediate philosophical interest. So, instead, I would like to conclude by saying a bit about the progress people have made in learning to think within categories other than ${\rm Set}$.

It has been known for quite some time in category theory that each category has its own `internal logic', and that while we can reason externally about a category using classical logic, we can also reason within it using its internal logic -- which gives a very different perspective. For example, our best understanding of intuitionistic logic has long come from the study of categories called `topoi', for which the internal logic differs from classical logic mainly in its renunciation of the principle of excluded middle [9,11,28]. Other classes of categories have their own forms of internal logic. For example, ever since the work of Lambek [18], the typed lambda-calculus, so beloved by theoretical computer scientists, has been understood to arise as the internal logic of `cartesian closed' categories. More generally, Lawvere's algebraic semantics allows us to see any `algebraic theory' as the internal logic of a category with finite products [21].

By now there are many textbook treatments of these ideas and their ramifications, ranging from introductions that do not assume prior knowledge of category theory [12,27], to more advanced texts that do [7,16,20,24]. All this suggests that the time is ripe to try thinking about physics using the internal logic of ${\rm Hilb}$, or $n{\rm Cob}$, or related categories. However, the textbook treatments and even most of the research literature on category-theoretic logic focus on categories where the monoidal structure is cartesian. The study of logic within more general monoidal categories is just beginning. More precisely, while generalizations of `algebraic theories' to categories of this sort have been studied for many years in topology and physics [22,25], it is hard to find work that explicitly recognizes the relation of such theories to the traditional concerns of logic, or even of quantum logic. For some heartening counterexamples, see the work of Abramsky and Coecke [1], and also of Mauri [26]. So, we can only hope that in the future, more interaction between mathematics, physics, logic and philosophy will lead to new ways of thinking about quantum theory -- and quantum gravity -- that take advantage of the internal logic of categories like ${\rm Hilb}$ and $n{\rm Cob}$.

© 2004 John Baez
baez@math.removethis.ucr.andthis.edu

home