‘The unreasonable effectiveness of string theory in mathematics’: Emergence, synthesis, and beauty

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.

Merging of two gold nanoparticles

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There’s a wonderful video (see above) that has been circulating of Au nanoparticles merging under an electron transmission microscope (TEM). It’s not a new sight – TEM, scanning tunneling electron microscopes (STM), and others have been producing incredible images for some time. One of the most famous – certainly in terms of popular culture – is the one produced by IBM in the 1990s, showing individual atoms arranged in such a way as to spell the company’s name. Scanning electron micrographs have also captured in the past stunning images of various microbes and things likes like the human cochlea. But it is always nice to see such images emerge, from time to time, in popular media. And, really, it doesn’t matter how many times such images do emerge, the sight of whatever specimen being studied never feels any less remarkable. Electron microscopy offers a glimpse into a previously hidden world, and whether we’re talking physics or microbiology, the details of nature that we can now capture is nothing short of exciting.

IBM atoms
Image: ‘IBM scientists discovered how to move and position individual atoms on a metal surface using a scanning tunneling microscope’ / www-03.ibm.com

In the case of the video above, I’ve seen discussion at a number of venues, including where it was originally posted, and there seems to be some confusion.

What we see is, how under the right conditions, the particles, made of gold atoms, can fuse or merge forming a larger cluster of atoms. So what is going to enable this behaviour? In short, what you are observing are the two nanoparticles made of gold atoms as they move in high temperature on top of a surface of FeO (Iron(II) oxide). It is the combination of the high temperature in addition to the extra energy introduced by the electron beam (transmitted by the TEM) that provides that triggers the event. Extra energy is introduced to the specimen, and the two particles become excited in such a way that they begin moving. This is key. The atoms occupy a higher energy level, rearranging their configuration, thanks to the extra energy introduced to the specimen. That is why, toward the end of the video, as the attractive intermolecular forces between particles draw them together, the particles rearrange in the correct orientation of the lattices. This reconfiguration, if you will, allows for the smaller particle to amalgamate with the larger particle. In other words, once appropriately rearranged, the even more rapid process initiates where the smaller particle merges with the larger particle, and a very short time later, we see the single larger particle take on a rather pleasant symmetrical shape (crystalline lattice) due it now rearranging itself to a lower energy (stable) state.

It’s a great presentation of some of the basic physics of chemistry. Notice, for example, the lines on each particle, which represents their lattice structure on an atomic level.

Some very general and introductory explanation can be found in this video by Sixty Symbols. For the curious. A similar phenomenon can also be observed in a different video, this one produced by FELMI ZFE (see the top of the page). In this case, you’re again observing Au nanoparticles. At 600°C, you see them diffuse on an amorphous carbon support and exhibit Ostwald ripening. It’s a remarkable sight. Take note, again,  of some of the finer details, such as the texture on the particles – their atomic structure.

0!=1

Why 0!=1 – Talking math and terminal access code decryption in No Man’s Sky

Several months ago I made a video on a particular terminal access code decrypt problem that I had come across in No Man’s Sky (see above). The given sequence was 1 2 6 24 120. In order to decrypt the terminal and retrieve the valuable information, it was my job in the game to find the unknown 6th term. Can you work it out?

What made this particular sequence interesting, and worth commenting on, is how from a particular vantage point it would appear logically inconsistent. The appearance of a breakdown in logical is such that one might be led to believe that in order to find the unknown 6th term they would have to ignore the 1st term in the sequence. But as I explain, the sequence makes perfect sense in that the deeper realisation here concerns how and why 0!=1.

proof of quadratic formula by completing the square_rcsmith

Proof of the Quadratic Formula (by completing the square)

With the launch of my new mathematics blog, I thought I would start with something of a nostalgia post: a note on the quadratic formula.

The proof of the quadratic formula (by completing the square) was one of the first that I learned. When I originally worked through it and finally arrived at a derivation of the formula, it was for me one of those early mathematical moments that continued the growth of an already deep-seated interest. It’s very much similar in memory to various other instances of mathematical experience, such as when I look back to the time when I first taught myself calculus. Sentimentality is not the right word here; it’s more a remembrance of an early moment of mathematical passion and discovery.

It is by no means on the list of my favourite proofs – all of which I will write about in the future – but it’s fitting to have this derivation (at the bottom) as part of the early development of this blog.

Furthermore, I think that one of the wonderful things about mathematics (even very basic arithmetic) is how the derivation of something as simple and basic as the quadratic formula is actually quite beautiful. Some of the deeper connections we might make are also, philosophically, very inspiring. And even further, the anthropological dimension of its history and discovery is incredibly interesting.

Often it seems that as one advances their early mathematical career, lessons and thought experiments on first principles and proofs of whatever formula are increasingly absent. So too is the why of maths. In the case of formulae, first principles are often sort of left implicit – you know, here’s a formula and here’s how and when to use it. The why of mathematics seems to be left out, at least especially early on in one’s career.

A lot of science is emerging that backs the idea that the best way to learn is not by attending lectures – though they are useful – but by exercising one’s critical thinking skills. Exploring first principles, working through the logic of whatever particular derivation, and in reflecting on proofs helps build a foundational sense of the properties in practice and strengthens key intuition of why.

History

Before I actually get into the derivation, what is interesting to note is that thoughts around the development of this basic formula possess an interesting history. (I summarized these notes some time ago and I cannot locate the original source, otherwise I would link to the historical record). In short: Math historians often cite that, although the first attempts to find a more general formula to solve quadratic equations can be tracked back to geometry (and trigonometry) of Pythagoras and Euclid, the history of thinking actually dates as far back as approximately 2000 or so BC.

Some denote the thinking at this time as the original problem, wherein Egyptian, Babylonian and Chinese engineers encountered a question of some urgency: namely, how certain shapes must be scaled to a total area. In other words, there needed to be a way to measure the lengths of the sides of walls.

Anthropologically, one has to remember that this problem emerged shortly after the first signs of civilization began to formally develop in Mesopotamia, and thus with it the Bronze Age. With this there was an increase in agricultural production and all the rest, taking off from the Neolithic Revolution many years before. Storage of excess materials, grain and resources was an ongoing problem in this early and important period of development.

But these early engineers were very intelligent for their time. They knew how to find the area of a square with the length of a side. They also knew how to utilize squared spaces. But the sides and area of more complex shapes posed a significant problem. But then something important happening. In Egypt around the time of 1500BC the concept of completing the square was formulated to help solve very basic problems concerning area. It also appears later in Chinese records.

Then in 700 AD, Baskhara, a famous Indian mathematician of whom many may already be familiar, was the first to recognise that any positive number has two square roots. This followed by another derivation of the quadratic formula performed by Mohammad bin Musa Al-Khwarismi, a famous Islamic mathematician. The historical account is that this particular derivation was then brought to Europe some time later by Jewish mathematician/astronomer Abraham bar Hiyya. Some time later it was then picked up in 1545 by Girolamo Cardano, a Renaissance scientist. Here Al-Khwarismi’s solution was integrated with Euclidean geometry, which helped pave the way for the modern formulation.

Indeed, it was François Viète in 16th century France who would introduce what we now would consider as more recognizable notation. Then, the big work. The famous enlightenment thinker, René Descartes, penned La Géométrie, within which modern Mathematics was born. From out of this the quadratic formula as we know it today would emerge and be adopted.

Deriving the Formula

With some of the history noted, here’s my effort at a derivation of the formula (by completing the square).

proof of quadratic formula by completing the square_rcsmith