As I noted the other day, there were a number of interesting talks at String Math 2020. I would really like to write about them all, but as I am short on time I want to spend a brief moment thinking about one talk in particular. Robbert Dijkgraaf’s presentation, ‘The Unreasonable Effectiveness of String Theory in Mathematics‘, I found to be enjoyable even though it was not the most technical or substantive. In some sense, I received it more as a philosophical essay – a sort of status report to motivate. I share it here because, what Dijkgraaf generally encircles, especially toward the end, is very much the topic of my thesis and the focus of my forthcoming PhD years. Additionally, while it may have aimed to inspire and motivate string theorists, the structure of the talk is such that a general audience may also extract much wonder and stimulation.
One can see that, whilst, certainly in my view, mathematics is a platonic science, Dijkgraaf wants to establish early on the unavoidable and unmistakable connection between fundamental physics and pure mathematics. So he starts his presentation by ruminating on this deep relationship. Eugene Wigner’s ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences‘ comes to mind almost immediately (indeed inspiring the title of the talk) in addition to past reflections by many intellectual giants. The historical evidence and examples are overwhelming as to the power of mathematics to speak the language of reality; at the same time, physics exists in this large space of concepts. It is their overlap – the platonic nature and rigid structures of mathematics and the systematic intuition of physics with its ability to capture nature’s phenomena – that in fundamental science suggests deep ideas of unity and synthesis. On this point, Dijkgraaf uses the example of the basic and humble derivative, highlighting the many perspectives it fosters to show that the mathematical and physical use of the concept is broad. The point is to say that there exists a large space of interpretations about even such a basic conceptual tool. The derivative has both physical meaning and interpretation as well as purely mathematical meaning. These many perspectives – similar, I suppose, to Feynman’s notion of a hierarchy of concepts – offer in totality a wealth of insight.
A better example may be the dictionary between the formalism of gauge field terminology and that of bundle terminology. On the one hand, we have physicists studying Feynman diagrams and fundamental particles. On the other hand, we have mathematicians studying and calculating deep things in topology and index theory. Historically, for some time the two did not discuss or collaborate despite their connection. In fact, there was a time when maths generally turned inward and physics seemed to reject the intensifying need of higher mathematical requirements (it seems some in physics still express this rejection). As Dijkgraaf tells it, there was little to no interaction or cross-engagement, and thus there was no mathematical physics dictionary if you will. For those that absolutely despise the increasingly mathematical nature of frontier physics, one may have no problem with such separation or disconnection. But such an attitude is not good or healthy for science. We see progress in science when the two sides talk: for instance, when physicists finally realised the use of index theory. The examples are endless, to be sure, with analogies continuing in the case of the path integral formalism and category theory as Dijkgraaf highlights.
In addition to discussing the connection between maths and physics, there is a related discussion between truth and beauty. For Dijkgraaf, he wants to feature this idea (and rightly so): namely, the two kinds of beauty we may argue to exist in the language of fundamental mathematical physics, the universal and the exceptional. There is so much to be said here, but I will save that for another time!
I will not spoil any more of the talk, only to say that the concept of emergence once again appears as well as the technical idea of ‘doing geometry without geometry’. Readers of this blog will know that what Dijkgraaf is referring to is what we have discussed in the past as generalised geometry and non-geometry. As these concepts reside at the heart of my current research, we will talk about them a lot more.
To conclude, I want to leave the reader with the following playful thought with respect to the viewpoint Dijkgraaf shares. If, for a moment, we look at string theory as the synthesis between geometry and algebra, I was thinking playfully toward the end of the talk that there is something reminiscent of the Hegelian aufhebung in this picture – i.e., the unity of deeply important conceptual spaces in the form of quantum geometry, as he puts it. In the physical and purely mathematical sense, from whatever side one advances, the analogy is finely shaped. From a mathematical physics point of view, it sounded to me that Dijkgraaf was seeking some description of synthesis-as-unification-for-higher-conceptualisation. I suppose it depends on who you ask, but I take Dijkgraaf’s point that string theory would very much seem to motivate this idea.