Nabihah Rahman

It’s probably a bad idea for me to be writing this article just as we start a new academic year, with some of us being rather optimistic about it and the title of this article being quite the opposite. Nonetheless, as I delve deeper into my studies, I cannot help but often think about how David Hilbert once quipped that “physics is much too hard for physicists” [1]. With all the strange concepts that we have to grapple with, Hilbert appears to be more and more correct by the day.

Figure 1: David Hilbert, one of the most influential mathematicians of the 20th century. Credits: Author

David Hilbert was a mathematician who revelled in mathematical formalism, a contrast from how many physicists are somewhat more ‘cavalier’ in their approach to demonstrating theories, which must have frustrated Hilbert greatly at the time of his remark. He became fully immersed in physics by 1912, and at around the same time that Einstein had published what we now call the Einstein field equations, Hilbert published an axiomatic derivation of these very equations [2]. In both this and his later action of co-authoring a textbook that became hugely important to quantum theory – *Methods of Mathematical Physics* – Hilbert demonstrates that mathematicians do have a consequential role in physics. Yet whether this role is more significant than those who primarily identify as physicists may be the subject of dispute.

**The Mathematician**** and the Physicist**

One of the implications of Hilbert’s statement is that the physicist’s approach to physics lacks mathematical rigor, hence mathematicians like Hilbert himself could be far more capable in undertaking the complex subject matter. It is impossible to deny how intricately woven mathematics is in physics. In particular, the emergence of modern physics in the 20^{th} century and its growth thereafter has irrefutably changed the landscape of physics forever, not least because it defined a new era of applying mathematical concepts in physics. The Copenhagen interpretation of quantum mechanics has been many a time summed up as the “shut up and calculate” approach [3] – yet this is not without its criticism from physicists and philosophers alike. Physical insight is an ideal for most areas of research, and the empirical cannot be provided solely by mathematics.

In a wonderful novel by Keigo Higashino – *The Devotion of Suspect X* [4] – there is a particularly interesting conversation between the two main characters, a mathematician and a physicist. Yukawa (the physicist) glimpses at an alternative proof that Ishigami (the mathematician) is writing while in a lecture (I personally prefer to doodle geometric patterns during my lectures, but each to their own).

“You’ve worked on this, too?”, Ishigami asked.

The long-haierd student let his hand fall down to the desktop. He grinned and shrugged. “Nah. I try to avoid doing anything unecessary. I’m in physics, you know. We just use the theorems you mathematicians come up with. I’ll leave working out the proofs to you.”

An extract from ‘The Devotion of Suspect X’ [4].

Yukawa’s comment in Figure 2 illustrates, albeit rather crudely, the interesting dichotomy between the work of pure mathematicians and physicists (another interesting dichotomy in the novel is that one has helped cover up a murder and the other is helping a detective investigate it). Just like Hilbert, Ishigami values method, strictly adhering to rules. On the other hand, Yukawa suggests that the details of proofs are not necessities for physics – the existence of the proofs suffice. This reiterates the idea that physics is beyond rigid exactitude, though it would have also accomplished very little without it.

We can posit that perhaps it is not the lack of mathematical rigor that renders physics “much too hard for physicists”, but the lack of relevant mathematics. So alternatively, physics might be much too hard for physicists, because mathematics is much too hard for mathematicians… It would of course be an insult to mathematicians to suggest that their lack of progress stifles ours, but the interconnectedness of mathematics and physics today suggests that it is imperative that mathematicians and physicists collaborate for mutual benefit. Physics may just be much too hard for *anyone* *.*

###### The Map to Technological Domination

It is also undeniable that we are living in an era of technological supremacy in physics, with ‘pen and paper’ studies being few and far between. In 1878, Arthur Cayley asked a question at the London Mathematical Society that was published by *Nature *[5]:

“Has a solution been given of the statement that in colouring the map of a country, divided into counties, only four distinct colours are required, so that no two adjacent counties should be painted in the same colour?”

his question would not receive a complete proof until 1976, when Kenneth Appel and Wolfgang Haken became the first mathematicians to use computers to solve a major theorem. Appel and Haken essentially used a ‘proof by exhaustion’, cycling through all the necessary configurations, which took a computer over a thousand hours to check [6]. Nonetheless, this proof proved very controversial at the time, with mathematicians such as Paul Erdös disparaging the work for being infeasible to check by hand (despite acknowledging that it was probably correct) [4].

While the four-colour theorem is a mathematical concept, the arrival of its proof exemplifies that it was only inevitable that technology would dominate future physical breakthroughs. From using deep neural networks to solve the many-electron Schrödinger equation [7], to detecting new particles in particle accelerators, the role of technology is impossible to ignore in physics today. When Hilbert made his remark, he could not have known about how essential computers would become in his fields of study; much like Erdös, I doubt he would have easily reconciled with their usage. However, the rise of powerful technology in physics does render Hilbert’s statement to be perfectly correct. Physics *is *much too hard for physicists because physics is much too hard for *anyone*.

###### Physical Closure

Hilbert’s comment further leads us to question how much the ‘physical approach’ can achieve. There is so much that we do not know despite how rapidly our knowledge of physics is expanding. Is physics simply impossible as there is no ‘end’ to every problem? It was Albert Einstein’s dream to unify the forces of nature with one single theory, and decades after his death, many physicists still seek to fulfil this dream. The quest for this ‘theory of everything’ has already spanned several lifetimes, and there is still so much more that needs to be known about the standard model of particle physics and dark matter before Einstein’s dream may even come close to being realised [8]. Yet this is the story of all science and mathematics – perhaps the “much too hard” mysteries of physics may simply need to age before human ingenuity finally demystifies them.

##### References

[1] Phillips L. The female mathematician who changed the course of physics – but couldn’t get a job. *Arstechnica *[Internet]. 2015 May 26 [cited 2021 Oct 02]. Available from: https://arstechnica.com/science/2015/05/the-female-mathematician-who-changed-the-course-of-physics-but-couldnt-get-a-job/

[2] Feynman RP. *Feynman Lectures on Gravitation. *Reading: Addison-Wesley; 1995.

[3] Svozil K. Shut Up and Calculate*. *In: *Physical (A)Causality. *Fundamental Theories of Physics, vol. 192. Springer, Cham; 2018. Available from: https://doi.org/10.1007/978-3-319-70815-7_10

[4] Higashino K. *The Devotion of Suspect X *[Translated by Smith AO]*. *New York: St. Martin’s Press; 2011.

[5] Crilly T. Arthur Cayley FRS and the four-colour map problem. *Notes. Rec. R. Soc. *2005; 59: 285-304. Available from: doi: 10.1098/rsnr.2005.0097.

[6] Appel K, Haken W. Every planar map is four colorable. Part I: Discharging. *Illinois Journal of Mathematics*. 1977; 21(3): 429-490. Available from: doi: 10.1215/ijm/1256049011.

[7] Pfau D, Spencer JS, Matthews AGDG, Foulkes WMC. Ab initio solution of the many-electron Schrödinger equation with deep neural networks. *Phys. Rev. Research. *2020; 2(3). Available from: https://link.aps.org/doi/10.1103/PhysRevResearch.2.033429

[8] Tretkoff E. Einstein’s Quest for a Unified Theory.* APS News. *2005; 14(11). Available from: https://www.aps.org/publications/apsnews/200512/history.cfm

Great article! Loved the reading