While I get the Thomistic impulse to defend the immateriality of reasoning or the intellect (if I'm not caricaturing the position), I'm skeptical of the empirical design, given BERT is a 2018 (encoder only) model with fundamentally different architecture than frontier models (autoregressive). The claim about syllogistic reasoning seems contestable: frontier general models have competed at the gold level in the IMO (novel problems requiring written proofs), which requires chain reasoning from middle terms, and Knuth's new Stanford paper, where he notes Opus 4.6 solved a problem he was working on for weeks, suggests general models can reason from natural language toward creative mathematical solutions.
Thanks, Matthew. I appreciate the critical feedback. You’re right the more work could be done on the empirical tests.
To clarify on the reasoning bit, LLMs absolutely can do mathematical reasoning, but that is not the same as intellectual reasoning. The three acts of the mind include understanding—abstracting the universal concept from a sensible thing, judgement—establishing a real connection from various things (the apple is red), and reasoning—moving to a conclusion by comparing two judgements overlapping concept:
All men are mortal.
Socrates is a man.
Socrates is mortal.
Now, LLMs can match words (the outward delivery of an internal concept) with concepts, deduce connections, and produce conclusions. However, this all operates at the level of mathematical reasoning, not intellectual.
This presupposes that there are operations above the brain which are not mathematical but purely immaterial. LLMs don’t prove that presupposition, but the presupposition—I argue—is a coherent interpretive grid to understand LLMs.
I think this is an interesting philosophical move. I don't have the training you do, so please correct me if this misconstrues your approach, but you seemed to use BERT as an empirical test of LLMs failing to reason in a Thomistic sense. When I provide examples of artificial intelligence appearing to reason at a level significantly beyond BERT's, you want to say that this is still mimicking reasoning rather than counting as true reasoning. While that's philosophically defensible, what are the empirical tests supposed to prove in this context? Are there any conditions under which an LLM can appear to reason that would count as true reasoning?
Another way to drive at my concern is that I don't actually know what the observable difference between intellectual reasoning and mathematical reasoning would be. For the IMO, the machines had no prior access to the novel questions at hand and had to write long proofs within a time limit that were judged by professionals, who found them to be clear and precise, which is a similar case to the Knuth paper. So I don't know how you falsify your position, which is how I interpreted the empirical section of your essay.
You've identified a real tension in my "tests." The tests were meant to illustrate that LLM operations are formally geometric transformations (not intellectual reasoning) even though they achieve a functional similarity to the steps of intellectual reasoning Aristotle and others identified long ago. If LLMs are formally geometric/mathematical, then the Thomistic boundary between phantasmic/geometrical/neurological and intellectual cognition provides a coherent interpretive grid for distinguishing AI from human intelligence.
You're correct that the deeper argument I'm presenting is philosophical, not empirical--it's granting satisfaction with what can be deduced rationally even if not via scientific methods. The distinction between mathematical and intellectual reasoning isn't about what can functionally be produced (e.g., writing long proofs within a time limit and judged by professionals), but what they are formally--how do they arrive at their outputs? For example, an LLM can produce "Socrates is mortal" by navigating geometric space toward a high-probability token without ever grasping the necessity of the connection between the premises. The question isn't whether the output is correct/functional—it's whether the operation involves genuine abstraction of universals and recognition of logical necessity (i.e., intellectual reasoning). I'm arguing that no amount of scaling geometric operations can produce that, not because of empirical evidence, but because of what those operations fundamentally are.
That may be unfalsifiable in the strict experimental sense. That's a fair demand within scientific methodology. But philosophy operates above the strictly experimental. The question ultimately bottoms out in first-person reflection: upon reflection, I'm convinced that I know the universal concept of "apple." Is this self-awareness a mathematical "output" of my brain or a real act of an immaterial soul above the brain? Moreover, how can my brain be "in motion" without a higher cause? That is the philosophical divide which only reflection, and not tests, can answer. But the philosophical positions can provide different interpretive grids for interpreting LLM architecture--and that is what my paper was meant to open.
Matthew, here's a recent paper draft that follows a much tighter argument and engages directly with theories found in associationism and cognitive science:
tangential but seems to overlap with the Popperian view that knowledge does not inherently exist in the world. And despite the LLMs being great at navigating the high dimension manifold of human language, our language itself does not embed the generative instructions or external reality testing to create new knowledge and thus they still are interpolating machines stuck in a closed map.
Do you have a book recommendation for getting into Thomas Aquinas / Aristotle ? I may be very under read in that department lol
I'm not familiar with the Popperian view. But yes, for Aquinas and the Neoplatonists, knowledge exists in the world as it participates in / emanates from higher causes and structures.
Language can't speak things into existence, but words can express the mental concepts that re-present real things. LLMs encounter these outer words (language) and can learn geometric patterns, but they can't access the inner words (intellectual concepts) or the reality those concepts represent. They're stuck at the level of linguistic, phantasmic patterns—which is sophisticated, but not knowledge in the Thomistic sense.
For reading Aquinas himself: https://isidore.co/aquinas/ has everything. I started with the Summa Theologica, which is the composition of his philosophical and theological system. His commentaries on Aristotle give you insights into both figures simultaneously.
For Aristotle directly: Nicomachean Ethics and Politics are more applied and very readable classics.
For secondary sources, I’ve read:
The One and the Many: A Contemporary Thomistic Metaphysics by W. Norris Clarke
Man's Knowledge of Reality: An Introduction to Thomistic Epistemology by Frederick D. Wilhelmsen
I should consider writing some introductory posts to make their thought more accessible if there’s interest. Keep me posted on how it goes, and feel free to ask questions as you dig in
That’s really interesting. I'd enjoy hearing more about how exactly you normalized Aquinas’ metaphysics into information theory. This is the kind of practical application of my framework that I’ve been interested to see. If you’re open to it, message me directly.
While I get the Thomistic impulse to defend the immateriality of reasoning or the intellect (if I'm not caricaturing the position), I'm skeptical of the empirical design, given BERT is a 2018 (encoder only) model with fundamentally different architecture than frontier models (autoregressive). The claim about syllogistic reasoning seems contestable: frontier general models have competed at the gold level in the IMO (novel problems requiring written proofs), which requires chain reasoning from middle terms, and Knuth's new Stanford paper, where he notes Opus 4.6 solved a problem he was working on for weeks, suggests general models can reason from natural language toward creative mathematical solutions.
Thanks, Matthew. I appreciate the critical feedback. You’re right the more work could be done on the empirical tests.
To clarify on the reasoning bit, LLMs absolutely can do mathematical reasoning, but that is not the same as intellectual reasoning. The three acts of the mind include understanding—abstracting the universal concept from a sensible thing, judgement—establishing a real connection from various things (the apple is red), and reasoning—moving to a conclusion by comparing two judgements overlapping concept:
All men are mortal.
Socrates is a man.
Socrates is mortal.
Now, LLMs can match words (the outward delivery of an internal concept) with concepts, deduce connections, and produce conclusions. However, this all operates at the level of mathematical reasoning, not intellectual.
This presupposes that there are operations above the brain which are not mathematical but purely immaterial. LLMs don’t prove that presupposition, but the presupposition—I argue—is a coherent interpretive grid to understand LLMs.
I think this is an interesting philosophical move. I don't have the training you do, so please correct me if this misconstrues your approach, but you seemed to use BERT as an empirical test of LLMs failing to reason in a Thomistic sense. When I provide examples of artificial intelligence appearing to reason at a level significantly beyond BERT's, you want to say that this is still mimicking reasoning rather than counting as true reasoning. While that's philosophically defensible, what are the empirical tests supposed to prove in this context? Are there any conditions under which an LLM can appear to reason that would count as true reasoning?
Another way to drive at my concern is that I don't actually know what the observable difference between intellectual reasoning and mathematical reasoning would be. For the IMO, the machines had no prior access to the novel questions at hand and had to write long proofs within a time limit that were judged by professionals, who found them to be clear and precise, which is a similar case to the Knuth paper. So I don't know how you falsify your position, which is how I interpreted the empirical section of your essay.
You've identified a real tension in my "tests." The tests were meant to illustrate that LLM operations are formally geometric transformations (not intellectual reasoning) even though they achieve a functional similarity to the steps of intellectual reasoning Aristotle and others identified long ago. If LLMs are formally geometric/mathematical, then the Thomistic boundary between phantasmic/geometrical/neurological and intellectual cognition provides a coherent interpretive grid for distinguishing AI from human intelligence.
You're correct that the deeper argument I'm presenting is philosophical, not empirical--it's granting satisfaction with what can be deduced rationally even if not via scientific methods. The distinction between mathematical and intellectual reasoning isn't about what can functionally be produced (e.g., writing long proofs within a time limit and judged by professionals), but what they are formally--how do they arrive at their outputs? For example, an LLM can produce "Socrates is mortal" by navigating geometric space toward a high-probability token without ever grasping the necessity of the connection between the premises. The question isn't whether the output is correct/functional—it's whether the operation involves genuine abstraction of universals and recognition of logical necessity (i.e., intellectual reasoning). I'm arguing that no amount of scaling geometric operations can produce that, not because of empirical evidence, but because of what those operations fundamentally are.
That may be unfalsifiable in the strict experimental sense. That's a fair demand within scientific methodology. But philosophy operates above the strictly experimental. The question ultimately bottoms out in first-person reflection: upon reflection, I'm convinced that I know the universal concept of "apple." Is this self-awareness a mathematical "output" of my brain or a real act of an immaterial soul above the brain? Moreover, how can my brain be "in motion" without a higher cause? That is the philosophical divide which only reflection, and not tests, can answer. But the philosophical positions can provide different interpretive grids for interpreting LLM architecture--and that is what my paper was meant to open.
Matthew, here's a recent paper draft that follows a much tighter argument and engages directly with theories found in associationism and cognitive science:
https://michaelmangialardi.substack.com/p/beyond-and-below-conceptual-spaces
tangential but seems to overlap with the Popperian view that knowledge does not inherently exist in the world. And despite the LLMs being great at navigating the high dimension manifold of human language, our language itself does not embed the generative instructions or external reality testing to create new knowledge and thus they still are interpolating machines stuck in a closed map.
Do you have a book recommendation for getting into Thomas Aquinas / Aristotle ? I may be very under read in that department lol
I'm not familiar with the Popperian view. But yes, for Aquinas and the Neoplatonists, knowledge exists in the world as it participates in / emanates from higher causes and structures.
Language can't speak things into existence, but words can express the mental concepts that re-present real things. LLMs encounter these outer words (language) and can learn geometric patterns, but they can't access the inner words (intellectual concepts) or the reality those concepts represent. They're stuck at the level of linguistic, phantasmic patterns—which is sophisticated, but not knowledge in the Thomistic sense.
For reading Aquinas himself: https://isidore.co/aquinas/ has everything. I started with the Summa Theologica, which is the composition of his philosophical and theological system. His commentaries on Aristotle give you insights into both figures simultaneously.
For Aristotle directly: Nicomachean Ethics and Politics are more applied and very readable classics.
For secondary sources, I’ve read:
The One and the Many: A Contemporary Thomistic Metaphysics by W. Norris Clarke
Man's Knowledge of Reality: An Introduction to Thomistic Epistemology by Frederick D. Wilhelmsen
I should consider writing some introductory posts to make their thought more accessible if there’s interest. Keep me posted on how it goes, and feel free to ask questions as you dig in
That’s really interesting. I'd enjoy hearing more about how exactly you normalized Aquinas’ metaphysics into information theory. This is the kind of practical application of my framework that I’ve been interested to see. If you’re open to it, message me directly.