Mathematics and AI

Referring to certain software and hardware as "AI" and to such alleged "AI" as an agent "solving" problems in mathematics is profoundly misleading. The programs in question are not "intelligent" and are not "solving" mathematical problems as agents in any philosophically meaningful sense. And these programs will certainly never "replace" human scientific endeavor in any conceivable way, rather they are, and will ever remain, mere tools.

Let us be clear. Human beings can only process, check and produce data within definite finite bounds based on symbol systems with definite finite bounds and bounded fragments of finitarily determined rules.

Thus it is a triviality and a truism that all human external symbolic activity and productions could in principle - given a massive enough set of data is made available - be mimicked and processed by brute-force. We can imagine a supercomputer in space with processing power and storage a billion times surpassing any of the human bounds of symbol processing and text processing, checking and production. We can also give our supercomputer some kind of super-luminal processing velocity. This supercomputer through brute-force and crude machine learning algorithms would beat and outperform every man-made "AI" in every possible domain in an instant. There is nothing surprising here and there is nothing here that has anything remotely to do with "intelligence".

Intelligence is rather reflected in doing much with little external, material, processing power. Chess programs are just cheating machines. Our hypothetical supercomputer would beat any current Go or Chess AI and that would not make it "intelligent" in any meaningful way.

As for mathematics, let us take a proof assistant such as Agda or Idris 2. The mathematician develops a theory (or formalizes a previous theory) by means of type definitions, records and type declarations for terms. The problem is to find explicit proof terms. This could be attempted by using brute-force, heuristics or any kind of machine learning method. And this search could fail. Or the search algorithm could be refined and altered. There is nothing unusual going on here. The algorithm is not "doing mathematics", it is not constructing theories, refining definitions, improving its own heuristics or being telologically oriented towards a certain architectural vision. It is the mathematician's legitimate tool.

Speaking of "AI" "solving" mathematical problems or "doing" mathematics or potentially "replacing" mathematicians is sheer and utter nonsense.

If even published journal papers can contain errors, I certainly would not trust any "mathematical" output produced by generative AI that was not formalized and checked in all its details by current proof assistants and proof checkers.

Yet more short considerations on AI

The philosophical significance of LLMs and a potentially powerful philosophically based critique of LLMs have not yet been developed and perhaps their importance has not even been recognized yet.

Could LLMs be tied to a certain philosophical view regarding the mind, language and the world? And if such a view is manifestly erroneous could not this furnish a sound basis for acknowledging - alongside numerous other reasons - the social and cultural harm of LLMs?

For instance, could we not explore the relationship between LLMs and Quinean extensionalism and meaning-as-use theories? Are not LLMs based on the rejection of the irreducible intensionality of human symbolic activity? Behind every symbol there is an intension. And a formal theory of intensionality must itself acknowledge the intensions of its meta-symbols.  But there is no intensionality in LLMs beyond that of the humans involved in their creation. Searle's Chinese Room ignores the deeper philosophical meaning of computation - or a potential associated geometric theory of meaning - but may be of interest for a critique of LLMs.

Do LLMs compute? Computation is a primordial intensional human activity (see our paper Analyticity, Computability and the A Priori). 

Perhaps there are other machine learning models of language of a more geometric or even combinatorial-algebraic nature which would have interested Riemann who wanted to relate meaning to a kind of cognitive geometry.   Perhaps statistical regularities should be traced back to geometry.

What are LLMs really, formally? Can we formalize the critical values wherein they become 'adequate' for their proposed task? What exactly in 'large'? How can this be formalized rigorously? Can LLM techniques be used for formal axiomatic-deductive systems and automated theorem proving? Or can we prove certain fundamental limiting theorems about the powers of LLMs akin to the unsolvability of the halting problem and Gödel's incompleteness theorems?

LLMs only exist because of the Internet. The Internet and LLMs are part of the same historical-cultural-technical process.  This ontological process might be described as the datafication of humanity. Language ceases to be a tool of human thought, communication and culture-building but rather a tool for the reduction, degrading, emptying, perversion and commodification of humanity itself. The Internet and LLMs are the anti-Gutenberg. Man has become text, sound, image, data, statistics. LLMs are a counterfeit reality, a monstrosity, the world becomes one big corporate controlled screen.

 If we compare the training data and the resulting LLMs is there or not a loss of information? Would it not be more worthwhile to develop sophisticated search algorithms and querying language to access the training data directly?

 LLM culture is the culture superficiality, atomization, banality, cosmetics and deception (a LLM is almost a trained deceiver in the biological sense - it detects and mimics human patterns). There is a loss of the multiple layers of meaning behind every symbol which cannot be reduced to statistical correlations with other symbols. Meaning is replaced with arbitrary social-statistical emergent correlative patterns.

The realm of pure mathematics - and that of pure logic, combinatorics and computability - is a pure realm which LLMs cannot touch or corrupt.  So the formal mathematics and formal philosophy projects, contrary to popular misconception likely resulting from deliberate propaganda and deception - are the antithesis of and antidote for LLM culture. There is a pure universal computational-mathematics-akin language (far beyond the natural language or the audio-visual data  that can be perverted and imitated by LLMs) and a pure logic and a pure thought and mankind may indeed hope to attain them. 

LLMs are not intelligent and statistics cannot solve formal computational problems nor can they encompass the pure a priori synthetic principles of formal computational systems (i.e. the cognitive certainly of the foundational principles for metatheoretic knowledge) as detailed in our paper mentioned above.

No statistical pattern analysis of the shadows are sufficient to lead to knowledge of the object. Pixel injection in image recognizers demonstrates this fact, and similarly for formal reasoning.  Besides lacking intensionality, LLMs lack reference and context (despite the misleading terminology of context windows). 

And most important of all since the true intellect is inseparable from morality, empathy and compassion, completely beyond the reach of LLMs. LLMs do not have the bondage to an illusion of a self.

A major task of philosophy is to destroy the evil empire of LLMs and a good starting point is deconstructing and refuting the worldviews (extensionalism, meaning-as-use) which LLMs embody. 

Is a lawyer someone trained in a specific system of laws or someone who has developed the skill to study, interpret and apply any given system of laws or perhaps be able to cope with significance changes in present laws?  Such a metalawyer is the analogue of a universal Turing Machine. A LLM could never be a judge or a metalawyer - it could not grasp the spirit of legal institutions or the deeper meaning of a given legal context.

Billions use light-bulbs without understanding the underlying physics. Billions could use LLMs thinking they are conscious.

LLMs will become more interesting in the measure in which the multilayer perceptrons models are replaced by geometrically and mathematically more sophisticated models (KANs are a step in the right direction).

Genius, Intelligence and Sainthood

Schopenhauer expounds a theory of 'genius' in the World as Will and Representation. The analysis and critique of varied historical and present meanings of the terms 'genius', 'intelligence' and 'sainthood' in the west is of utmost cultural and philosophical importance. Part of our philosophical project involves the radical deconstruction of these concepts and unmasking their harmful influences and consequent aberrations. 'Sainthood'  is currently employed in a socio-historical sense or related to organized religions, its true universal ethical and spiritual meaning being lost.  'Intelligence' is a vague and fluid pseudo-concept that has wrought immeasurable harm to human culture and society as well as to the human sciences. The term 'genius' is as a rule completely misapplied.

Our theory of intelligence is that true intelligence is founded on two intimately connected principles. That of knowledge of the universal moral law (perceived as such) and that of attaining at least some elements of transcendental philosophical consciousness (TPC) and transcendental philosophical praxis (TPP).

Extraordinary achievements in TPC and TPP are the mark of genius par excellence. 

The genius of true intelligence in the ethical sphere and TPP is called 'sainthood'. 

The special kind of genius, that while intimately connected implicitly to morality, TPP and TPC (as brilliantly expounded by Schopenhauer),  is that of artistic genius which reveals fundamental important aspects about human nature and the world.  

The quality of intelligence is related directly to the importance of its object and domain. 

Intelligence can be defined as an empathy for reality and moral intelligence has as its main source universal empathy for the suffering of all living beings. 

Intelligence and morality are one. 

The extreme opposite of true intelligence is all that involves a kind of cunning, a skill-set whose only purpose and application lies in achieving personal egotistic wealth and power, everything that involves dominating, exploiting and harming other living beings. There is no 'intelligence' here, no 'genius' and no 'sainthood'. Intelligence is radically incompatible with the will to power or competitive mechanisms of survival (i.e. overpowering others or merely adapting to a contingent environment).

The greatest deception and idol of western society has been attributing 'intelligence' or 'genius' to what amounts to nothing more than gaming, gambling, cunning and calculation. We have to be extremely cautious and insightful about attributing intelligence or genius to work in logic, mathematics, engineering or science,

Genius and intelligence in mathematics is never about the predictable success (theorem proving) in random searches in formal possibility spaces and conceptual engineering (scientific mass production) but rests solely on the relevance of the work to philosophy (more specifically to TPC) and to the philosophical unification and clarification of science.

Playing the game, being lost in the illusion of the game, is very different from the TPC-informed insight and consciousness of the game qua game (which can be compared to the attitude of Alice at the end of Alice in Wonderland...and Schopenhauer uses the metaphor of the chessboard after the game is finished). 

Thus we have such unsurpassed luminaries as Gauss, Grassmann, Riemann, Frege (who wrote that every true mathematician is half a philosopher), Peirce, Hilbert, Turing, Whitehead, Gödel, Brouwer, Russell, Poincaré, Lawvere, Thom, Martin-Löf, Girard, etc. Their writings are never a mere tortuous game with symbols and ad hoc concepts or non-rigorous obfuscation and plagiarism. The light of TPC implicitly shines through. Their work is also a spiritualization of language.

But scientific discoveries with practical applications to the quality and preservation of human life (or of any living being) are certainly meritorious - for indeed they partake this way in the sainthood, intelligence and genius of ethics.  Science aimed at mechanisms for harm and destruction is the ultimate immorality and stupidity. There is nothing brilliant or intelligent about faulty simplistic models of human society such as 'game theory'.