Axiomatic Realism V: (In)conclusions

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 the previous introductory post where I was signposting these 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 want 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?

---

---

Axiomatic Realism V: Inconclusions

Now, we come to the final bough. Normally, one would attempt to reappraise and tie together the various roots and branches of this exploration, but that would seem insufficient in this case. The very notion of something as audacious as a conclusion is anathema to this proposal of a realist orientation, after all.

I will instead begin with a confession: just like the cunning chemistry educators that perenially pulled the epistemic rug from underneath my feet, only to replace it with yet another pale imitation of the out-of-reach veridical, I must admit that the longer I toil in the knowledge work mines, the less confident I am in not only the meaningfulness of hunting for the supposedly inviolable truths of the ἀξίωμα (axíōma), but also in my capabilities and judgement as a qualified and earnest scholar of said topics. Perhaps, if I’m still going in another 25 years, I’ll pass another inflection point and will begin to recover some sense of faith. But I suppose it’s best not to make too many assumptions (sorry).

Indeed, if I hear someone say something along the lines of “I’m looking for the truth” or “the truth is all that matters” I am alert to the possible presence of more red flags than a matadero. It’s hard not to notice that there does seem to be a political bias with respect to one’s proclivity towards the relationship to the absolute. The you-know-who-fronted social network Truth Social (aka TRUTH) has retruths and sponsored truths on it, so make of that what you will. At least they are being honest about it, albeit in a perverse and unwittingly revealing manner.

Since 2022, this has been the logo for Truth Social. Owned by the Trump Media & Technology Group.

In line with Popperian falsification, many present-day epistemologists believe that there is no realistic prospect of reaching an objective and universal ground truth and that such a notion is at best an ideal, to be approached asymptotically but never gained. In accordance with the so-called Gettier problem, a justified true belief from a singular perspective may be all we can really hope to attain, whether one considers such a proposition ‘knowledge’ or otherwise. I wish the scientific axiomateers the best, but also hope that they go out and touch grass once in a while to remember that things out there in the world are messy, chaotic, blurry, and resist categorisation. And that graininess is very often where the true meaning of life lies. Elegance is sometimes a trap, and to quote the liner notes of Mirco Magnani / TCO’s record on my long-defunct record label “imperfection makes beauty real”.

“Touching Grass”, a card from the FAU0X SALON card deck, 0xSalon, 2023-present.

We shall bastardise Alfred North Whitehead for our purposes here, but instead of positing that “all of Western philosophy is a footnote to Plato”, let us propose that “every counter-axiomatic proposal is a footnote to Feyerabend”. In what is surely no coincidence, Betrand Russell (Whitehead’s former student) took on the gargantuan task of attempting to derive a complete and consistent mathematical framework from discursive “laws of thought” for inference and deduction in their Principia Mathematica (1910), published in the early 20th century. Prior to this, Russell had already adopted a hardline formalist position in The Principles of Mathematics (1903), arguing that mathematics and logic are identical. But that is a topic for some other day, and some other writer. As a reminder, to not muddy the waters unduly, everywhere else in this article series, we have been referring specifically to scientific rather than logical axioms.