Axiomatic Realism II: Hyperformalisation

Forthcoming in Ars Scientia journal

Links to all five parts of this piece — not all of which are out yet, but will be available at these links if not already ::: Part 1 ::: Part 2 ::: Part 3 ::: Part 4 ::: Part 5

As a reminder, I said this in a previous post where I was signposting the next few weeks’ worth of material that’ll be surfacing on these pages. Feedback, comments are extremely welcome!

…I wanted to briefly introduce the next few posts. Last autumn, after many months of stewing ideas and/or wildly procrastinating, I finally sat down to write a dirty first draft of a lengthy new article taking a critical look at the epistemological arc of the sciences (my original home territory) through the memoiristic lens of autotheory. It’s going to be included in the second issue of the wonderful Ars Scientia, a new journal of scientific arts, which takes as its second theme “axiomatics”. Since the article is a weird mix of history and philosophy of science, historical recollections, and post-disciplinary meaning-making, and also because it still needs a little something extra according to some critical feedback I received and agree with, I thought it might be a good place for us to start. It’s pretty long — first draft was over 10k words, and the journal editors originally asked for 3-5k — so I’ll be splitting it into 3 or 4 digests and saving a final one for some additional rumination and riffing.

I also wanted to include an excerpt of the great motivational text that the editors of Ars Scientia posted on their website, to give a sense of where my piece is coming from.

Ars Scientia Issue 2 on Axioms
“It had been tactless of me to prove something on the topic of man – mathematically!”
Stanislaw Lem, His Master’s Voice, 1968

Axioms are “self-evidently true without proof.” In axiomatic systems, truth is preserved throughout the system as statements derive directly from the axioms. While the axiomatic approach can seem like a top-down monolith, in 1931 Gödel famously challenged the claims of metaphysical access through axiomaticity by demonstrating that within any such system there are true statements that cannot be proved. In addition to Gödel’s challenge, the history of axiomatic thinking also presents a multitude of traditions, concepts and crises.

For instance, automated theorem proving is a field of mathematics which attempts to automatize mathematical proof-writing with computing. Confirming the validity of this method and achieving automatization beyond first-order logic has been greatly challenging, where-as the methods of writing proofs by hand or with computerized (but not automated) proof assistants do not face the same scrutiny of “viability.” How does this history of mathematics relate to other interests more broadly?

In the first issue of Ars Scientia, we took the lead from Goethe's Metamorphosis of Plants as a study in the systematization of the relation between observer and phenomena, a relation that makes up a blueprint of the simultaneity of aesthetic practice and scientific inquiry. In the upcoming second issue of Ars Scientia, we are interested in what happens when, as Thomas Moynihan writes, “this plenty [of the organic world] and progressivism becomes divorced, decoupled from, devoid of purpose? What happens when the artisan recedes from the picture—as the century following Goethe witnessed unfold?” (2021, Philosophical Life of Plants). We want to extend this conflict beyond the discipline of mathematics or the illustration of nature and towards synthetic practices between the artistic and the scientific.

Are our axioms so silent as they might appear? What is formalization, systematization when devoid of purpose? What does it mean to reintroduce such a purpose? What are the possible axiomatics that speak from below and beyond the mathematical impetus of abstract proof? Where is the hand? Can the silent speak for the quivering?

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Axiomatic Realism II: Hyperformalisation

is X a branch of pure mathematics?

$I, the Leviathan, the Basilisk. $YOUR will bends to $MY State of Nature.
Only $I have the private keys to the promised land.
Honour $THY network. Seek redemption in $MY oikonomia.
$MY Altruism is Effective, and $MINE alone.
$MY wish is $YOUR command. The All-Seeing CLI.
Together, $WE will usher in a new era of machine rationality.
$I, the anti-economy. $I, infinite gamekeeper of the digital paradise.
Behold! $MY code, $MY law. The Gospell according to $ME.
Cointext collapse. Post-naught clarity. The Gnon-event horizon.
$MY Kittlerian Jihad. Orinoco’s Basilisk. Citadelete Yourself.
The Necroprimitivist Manifesto Pt.2, 0xSalon, 2023

In some ways, the autoepistemological — as opposed to auto-epistemological — story I want to tell here is one of representational formalism, and, in my mind at least, its would-be ouster. In late 2003, I was a newly matriculated doctoral student focused on the scientific characterisation of ultrafast processes involving photons and electrons in sophisticated chemical systems through hypothesis, experiment, and computational simulation. I snuck into an international conference held at the School of Chemistry’s main lecture theatre at the University of Nottingham that was relevant to my interests, on theoretical chemistry research.

Most of the talks were dull, dour, and drab: clearly Kuhnian normal science was alive and well in this corner of the academy. Late in the day, the keynote speaker took to the floor. No slick PowerPoint slides, no well-pressed suit: instead, an endearingly scruffy elder gentleman hobbled down the steps to the front. 1998 chemistry Nobel laureate John A. Pople III then proceeded to give the most impactful scientific presentation I have ever witnessed. It was also very likely his last, as he died a few months later. The talk was titled “Is Chemistry a Branch of Pure Mathematics?” and set about asking the question through unprepared remarks around axioms and formalisms, followed by challenges and provocations to the audience. It seems that the reason he was in the room was because his former postdoc Peter M. W. Gill was Nottingham’s chair of theoretical chemistry at the time. Indeed, they collaborated so closely on the development of computational methods to simulate quantum atomic and molecular theory that Gill was one of the two invitees to the Nobel award ceremony, the other being Pople’s wife.

Unfortunately, despite repeated efforts to surface footage, documentation, or commentary on this (or another similar) presentation, I have been unsuccessful. So, I will give an necessarily incomplete and undoubtedly biased account of my recollections. If anybody reading this in the near or further future can oblige, you would make an old man very happy by sending relevant archival materials my way. Pople, who was a mathematician that had wandered into chemistry looking for ‘practical’ problems to solve, began from a simple hyperformalist, proposal. The four quantum numbers of subatomic particles can predict and explain the electronic structure of atoms, and therefore the structure of the periodic table and with it a wide swathe of first-order characteristics and material properties. However, as anyone who has studied even rudimentary amounts of chemistry will know only too well, the discipline is a total mess in epistemic terms, with counter-intuitive results and post-hoc rationalisations acting as desperate triage for all but the most facile hypotheses.

Worse Things Happen at c

Next, Pople turned to the so-called fundamental physical constants of the Universe, which are understood to be resistant to theoretical derivations and therefore can only be experimentally determined. There are numerous examples of these, though over time as theories improve in their explanatory power, some which might have once been thought of as candidates for fundamentality found themselves dethroned. The most widely recognised examples are associated with the speed of light, gravitational effects, the energy of a photon, the permittivity of free space, and the charge of a proton.