Thinking About Mathematics by Steward Shapiro
Synthetic a priori
There is an important tension in the traditional picture. On that view, mathematics is necessary and knowable a priori, but mathematics has something to do with the physical world. As noted, mathematics is essential to the scientific approach to the world, and science is empirical if anything is—rationalism notwithstanding. So how does a priori knowledge of necessary truths figure in ordinary. empirical knowledge-gathering? Immanuel Kant’s thesis that arithmetic and geometry are ‘synthetic a priori’ was a heroic attempt to reconcile these features of mathematics (see ch. 4, §2;. According to Kant, mathematics relates to the forms of perception. It concerns the ways that we perceive the material world. Euclidean geometry concerns the forms of spatial intuition, and arithmetic concerns the forms of spatial and temporal intuition. Mathematics is thus necessary because we cannot structure the world in any other way. We must perceive the world through these forms of intuition. No other forms are available to us. Mathematical knowledge is a priori since we do not need any particular experience with the world in order to grasp the forms of perceptual intuition.
Realisme vs idealisme
Define realism in ontology to be the view that at least some mathematical objects exist objectively, independent of the mathematician. Realism in ontology stands opposed to views like idealism and nominalism. The idealist agrees that mathematical objects exist, but holds that they depend on the (human) mind.
On protecting mathematics from LLM companies’ monopoly
By Tian Lan. Source https://proofsandprompts.com/2026/08/18/on-protecting-mathematics-from-llm-companies-monopoly/
I do not think mathematicians should compare their own accumulated knowledge and techniques with that of LLMs, just like people should not compare that with Google or MathSciNet.
I have paid attention to the recent discussions on the Jacobian conjecture (and also the recent progress on ratios of zeros of Riemann’s zeta function on the critical line, but maybe not worth speaking here). I found that the most historically concerned version n=2 remains open, while Anthropic claimed that they “resolved the Jacobian conjecture”. This is highly misleading, because it is mostly dimension-indexed, just like the Kakeya conjecture or generalized Poincaré conjecture; also, at least for a group of mathematicians, this name only referred to the 2-dimensional case (since there’re many positive results/evidence in this case), and the higher-dimensional versions were frequently labeled with “almost no evidence“.
This creates a particular danger when large technology companies enter mathematical research. The companies can create the impression that a result must be Annals-level, or Fields medal level, while thousands of results of comparable or greater mathematical depth in the same year may receive almost no public attention. Mathematical importance has never been identical to publicity, of course, but this kind of noise can also affect mathematicians, since most of us are not polymaths. Mathematicians should be especially wary of allowing this noise to distort their own standards of judgment.
LLMs Make Mathematics Easier. Now Raise the Bar
By Marijn J. H. Heule. Source https://cacm.acm.org/blogcacm/llms-make-mathematics-easier-now-raise-the-bar/
However, as reported in Scientific American,4 Miller argues that the sphere-packing improvement rests on his argument from a 2016 paper and considers the lack of acknowledgment systematic enough to call it research misconduct. Fournier-Facio found that another result combined ideas from two uncited papers, and concluded that their existence means the problem was never at the claimed stalemate. Failing to credit prior work is serious, and no proof assistant will catch it. LLMs can synthesize arguments from a vast literature without reliably preserving the origin of each idea. Correctness is necessary, and so is attribution.
The mathematics can be checked. The claimed capability cannot. The model and prompts are unavailable, so nobody outside the company can rerun the experiment. Nor do we know how the problems were selected, how many were attempted, how many failed, or the total computational cost.
Peter Thiel Thinks the Pope is the Antichrist. He’s Serious
Gil Pignol. Source https://medium.com/@gp2030/the-accuser-how-peter-thiel-read-the-antichrist-into-the-papacy-46b466ed3103
In March I argued that Thiel represents the most sophisticated modern case of what Catholic theology calls desperatio, despair as an active sin, and that his signature intellectual operation is a truncation: take Girard’s diagnosis of the scapegoat mechanism, amputate Girard’s conclusion that only the Gospel revelation can break the cycle, and deploy the orphaned diagnostic as an operating manual for surveillance and monopoly
A method can be specified. Thiel’s has three steps.
First, select a thinker whose system contains a powerful apocalyptic diagnostic. Girard supplied one: mimetic desire escalates into rivalry, rivalry into crisis, and once the scapegoat mechanism has been exposed, violence loses its brake. Ratzinger supplies another: the Church has drifted into an eschatological amnesia, forgetting that history has an end, that the end has an enemy, and that the enemy’s arrival is preceded by a counterfeit peace.
Second, sever the redemptive conclusion the diagnostic was built to serve. For Girard, that conclusion was conversion: the renunciation of retaliation, the imitation of Christ rather than of rivals, the only exit from the cycle the theory permits. For Ratzinger, it was hope: the end of history is not a catastrophe to be managed but a person to be met, and judgment, in his mature theology, is a form of grace.
Third, redeploy the orphaned diagnostic as a justification for power. Girard minus the Gospel became Palantir’s business model and Zero to One’s monopoly doctrine. Ratzinger minus hope becomes something new and more dangerous: a license to name the enemies of the end times, issued in the voice of a dead pope.
The test of a method is that it predicts. Watch the corpus: any thinker Thiel takes up who possesses a two-part structure of diagnosis and cure will lose the cure in the handling. Solovyov has already been processed this way; his Short Tale of the Antichrist circulates through Thiel’s lectures shorn of the ending in which the deceived churches reunite. The operation is content-agnostic. It works on anyone.
“Voyages to the End of the World,” his 2025 First Things essay, takes Bacon’s New Atlantis as its foundation text, reads Salomon’s House and its pursuit of the effecting of all things possible as a sacrilegious rival to divine omnipotence, and then, having diagnosed the Baconian temptation, proceeds to inhabit it. Palantir is Salomon’s House with government contracts.