Finite Computation - AI

By

 


AI

Norman J. Wildberger, a retired associate professor of mathematics, argues that the modern foundation of pure mathematics is not a product of pure logic, but rather a socially enforced orthodoxy that ignores computational reality. Through his YouTube video series, ⁠"Sociology and Pure Maths," he uses a sociological lens to critique how the mathematical community handles the foundations of the discipline. [1, 2, 3, 4, 5]

The Core Critique: Sociology of Mathematics
Wildberger claims that pure mathematics operates like an insulated "citadel" protected by social hierarchies and academic institutional inertia. [1, 2]
  • The Orthodoxy: He views standard axioms (like Zermelo-Fraenkel set theory) as a dogmatic belief system rather than absolute truth. [1]
  • Social Enforcement: He argues that universities train students to become proficient in formal manipulation within this assumed system. According to Wildberger, this dynamic actively discourages them from questioning its underlying internal integrity. [1]
  • Rigidity: When anyone challenges these foundations, the standard community response is often confusion or hostility because the system is merely repeated rather than justified. [1]
The Finitist & Computational View
Wildberger is a strict ultrafinitist. He rejects the physical or logical existence of completed infinities, infinite sets, and the continuum of standard real numbers. [1, 2, 3, 4, 5]
  • Algorithmic Reality: He believes mathematics should be fully computable and computer-encodable. An infinite decimal string (like the exact, complete sequence of \(\pi \)) cannot be stored or completely computed, making it logically invalid in his eyes. [1, 2, 3]
  • Math as Computation: To Wildberger, a mathematical object only exists if a concrete, step-by-step algorithm can construct it using a finite amount of data. [1, 2]
  • Alternative Frameworks: He has proposed alternative structures like "Rational Trigonometry" (doing trigonometry without angles or real numbers) and "Box Arithmetic" (constructing mathematics out of finite, computer-encodable multisets) to bypass classical set theory entirely. [1, 2, 3]
Would you like to explore how Wildberger reconstructs calculus without limits, or look closer at his specific critique of infinite sets?

Comments

Post a Comment

Popular posts from this blog

Computing and the Linguistic Turn

By
Image
Enigma (Wikipedia) Symbolic analysis and statistical analysis were both approaches codebreakers took in breaking Enigma. The development of AI has also incorporated both approaches. One historical question for AI is why the dominance of Symbolic AI, associated here with the linguistic turn, from 1950 to 1990? Note the development of AI out of the Dartmouth Workshop and especially the contributions of one of its attendees,  Ray Solomonoff , to the statistical approach, an approach taken up by Marcus Hutter , a DeepMind  researcher, and of course, others. Note that a history of AI outside the linguistic turn from Moritz Hardt and Ben Recht is in Patterns, Predictions, and Actions . And note too that along with the difference between the symbolic and the statistical is the difference between language learning and computer vision, which tends to be a history outside the linguistic turn. And the turn Richard Sutton takes is to push learning outside the context of language, which ...
Read more

A Heidegger - Bayes Hybrid Model

By
Image
A Conjecture for a Hybrid Model Bayesian epistemology concerns modelling and specifically, belief, but a Bayesian belief does not need to be fully formed to emerge alongside the Heideggerian to-hand. The Heideggerian to-hand thereby becomes the interaction that generates this further, which is, with this interaction, a certain realization of being in the world. Bayesian selection now concerns available beliefs as well as any to-hand [belief(s)]. How Heidegger Can Make You a Better Guitarist Bayesian Philosophy Bayesian Philosophy Bayesian Epistemology Bayesian Epistemology Bayesian Modelling Bayesian Modelling Bayesian Model Selection David Deutsch on Bayesian Epistemology Bayesian Philosophy of Science Bayesian Philosophy of Science A Bayesian Approach to the Philosophy of Science On Sean Carroll on an Application of Bayes The Heideggerian To-Hand On Heidegger On Heidegger Flipping Descartes Ready-to-Hand and Present-at-Hand Presence-at-Hand Ready-to-Hand and Unready-to-Hand Does Hei...
Read more

How Does AI Solve Erdős Problems? - AI

By
Image
Erd ő s Problems List of Conjectures Erd ő s Problems On ChatGPT as Solver On Liam Price Terence Tao Tao AI AI solves Erdős problems through a combination of Large Language Models (LLMs) for generating creative proofs and formal verification tools like Lean 4 to ensure mathematical correctness . [ 1 , 2 , 3 ] Recent breakthroughs, such as the resolution of Erdős Problem #728 and #1196 , have used specialized workflows where AI acts as a "junior co-author" to discover novel connections that human mathematicians had previously overlooked. [ 1 , 2 , 3 , 4 , 5 ] The Core Solving Process Solving these open conjectures typically involves a "loop" between a reasoning model and a formal verifier: Creative Brainstorming : A model (like GPT-5.2 or GPT-5.4) is prompted to research the problem and brainstorm novel mathematical strategies. Generating informal Proofs : The AI writes a mathematical paper in LaTeX, often discovering "elegant" methods, such as using th...
Read more