Cf: Sign Relations, Triadic Relations, Relation Theory • Discussion 6
http://inquiryintoinquiry.com/2022/03/01/sign-relations-triadic-relations-r…
Re: FB | Charles S. Peirce Society
https://www.facebook.com/groups/peircesociety/posts/2551077815028195/
::: Alain Létourneau
https://www.facebook.com/groups/peircesociety/posts/2551077815028195?commen…
All,
Alain Létourneau asks if I have any thoughts
on Peirce's Rhetoric. I venture the following.
Classically speaking, rhetoric (as distinguished from dialectic)
treats forms of argument which “consider the audience” — which
take the condition of the addressee into account. But that is
just what Peirce's semiotic does in extending our theories of
signs from dyadic to triadic sign relations.
We often begin our approach to Peirce's semiotics by saying he puts the
interpreter back into the relation of signs to their objects. But even
Aristotle had already done that much. Peirce's innovation was to apply
the pragmatic maxim, clarifying the characters of interpreters in terms
of their effects — their interpretants — in the flow of semiosis.
Some reading —
Awbrey, J.L., and Awbrey, S.M. (1995),
“Interpretation as Action • The Risk of Inquiry”,
Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.
https://www.academia.edu/57812482/Interpretation_as_Action_The_Risk_of_Inqu…
Regards,
Jon
Logical Graphs • First Impressions 1
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/
Introduction • Moving Pictures of Thought —
A “logical graph” is a graph-theoretic structure in one
of the systems of graphical syntax Charles Sanders Peirce
developed for logic.
In numerous papers on “qualitative logic”, “entitative graphs”,
and “existential graphs”, Peirce developed several versions of
a graphical formalism, or a graph-theoretic formal language,
designed to be interpreted for logic.
In the century since Peirce initiated this line of development,
a variety of formal systems have branched out from what is abstractly
the same formal base of graph-theoretic structures. This article
examines the common basis of these formal systems from a bird's eye
view, focusing on the aspects of form shared by the entire family of
algebras, calculi, or languages, however they happen to be viewed in
a given application.
Regards,
Jon
cc: https://independent.academia.edu/JonAwbrey
cc: https://mathstodon.xyz/@Inquiry/110945139629369891
Alex,
Thanks for that example. It shows the importance of the unconscious computation that is performed in the human cerebellum, whose perceptions and actions are totally unconscious. I urge everyone to click on the link in your note.
There is an important reason why the human drone experts lost in the competition with the fully automated drone: the humans used a combination of high-speed cerebellar computation (as the unmanned drone does) with the much slower (and conscious) decision making in the cerebral cortex. Those conscious decisions slowed their performance.
Compare that with the high-speed performance by the gymnastic champion Simone Biles. She devoted years of conscious effort to train her cerebellum to perform the various motions automatically. Before each competition, she perfects the training for each routine she performs. In a performance that has multiple routines, she uses her cerebral cortex to check the positions and timing for each routine. Then she launches a pretrained routine that is totally under the control of the unconscious cerebellum.
All of us use the cerebellum for routine processing in walking, eating, driving a car, or typing on a keyboard. Mathematicians take advantage of that high-speed processing in the most complex kinds of math. But writing a proof uses the slower conscious processing in the cerebral cortex to check whether the high-speed calculations are correct.
Note that the processes in the cerebellum are precise for what they do. The errors can occur when the decisions for running them (made by the cerebral cortex) are not correct.
Note that none of these processes, either by the cerebrum or by the cerebellum, could be performed by the LLMs. The Large Language Models might respond to a verbal command to execute a routine by the cerebellum. But all their operations are probabilistic, and they're based on vague and often ambiguous natural language. They can't do the precise checking and testing that guarantee accuracy.
LLMs are useful. But they're just one more tool in the huge toolkit of AI technology. They do a limited range of operations very well, but they can't do the whole job.
John
----------------------------------------
From: "alex.shkotin" <alex.shkotin(a)gmail.com>
Subject: [ontolog-forum] FYI:Champion-level Drone Racing using Deep Reinforcement Learning (Nature, 2023)
https://youtu.be/fBiataDpGIo?si=bDaE1XR4dQGJXqo6
Colleagues, while we are formalizing theoretical knowledge and building structures that model reality, it is interesting to look at achievements in the field where algorithms decide everything, but they are also helped by AI.
Alex
Alex and Michael DB,
To Alex: I agree with what you wrote, but with three important qualifications: (1) Every node in a diagram represents a concept. (2) Every linear notation for mathematics is a special case of some diagram; in some cases, the linearization is a one-to-one mapping; but in other cases, it loses some of the information, or it encodes that information in a more obscure way. Euclidean geometry is the most obvious example, but other kinds of geometry are even stronger reasons for multi-dimensional diagrams. (3) The tensors that represent LLMs are special cases of diagrams with special-case operations; for full generality, they must be supplemented with more general diagrams and operations on them.
And by the way, the title of my first book, Conceptual Structures, emphasizes the point that diagrams represent structures, and every structure can be represented by a diagram. Linear notations are just one-dimensional diagrams. Mapping a multi-dimensional structure into a one-dimensional line adds a huge amount of complexity. As just one example: direct connections by lines must be replaced by special symbols called names. And those names create a huge amount of complexity when they are constantly being renamed.
To Micheal: Since you agree with me, I agree with you.
Re consciousness: The fact that the cerebellum has over 4 times as many neurons as the much larger cerebral cortex is important. Even more important is that (1) Those neurons are essential for high-speed mathematical computation and reasoning. (2) They are aslo essential for all complex methods of performance in music, gymnastics, art architecture, and complex design of machinery of any kind. and (3) Nothing in the cerebellum is conscious.
Just look at the fantastic gymnastics by Simone Biles. She required years of dedicated *conscious* training to learn those moves, but the details of the high-speed performance are outside of any conscious control. It would be impossible to think in words about each of those details at the speed at which they were performed. Each performance was initiated and controlled by conscious decisions, but the speed is too fast for any conscious control. She was conscious of the performance, but not of every detail computed by her cerebellum.
That is a very important distinction: the computation in the cerebellum is not conscious. And no definition of consciousness would have the slightest value for understanding what and how the cerebellum computes its operations
But since you mention Searle, I'm not surprised at his response about panpsychism. I remember another story about a dinner party he attended, where the guests were sitting outside while the food was being prepared. At one point, Searle jumped up and proclaimed in a loud voice that frightened the neighbors, their children, and their dogs, a denunciation of "Derrida and the other inhabitants of Frogistan."
John
_______________________________________
From: "Alex Shkotin" <alex.shkotin(a)gmail.com>
John,
I look forward to reading your article, as the presentation is more or less sketchy. Diagrams are a wonderful tool, but thinking in concepts is what science and technology, and thinking in general, relies on.
And creating, researching and using structures is also very important.
Formula is amazing way to keep process definition, like
h = gt^2/2
where h - height, g - gravity constant, t - time of falling from the Leaning Tower of Pisa.
Alex
__________________________________________
From: "Michael DeBellis" <mdebellissf(a)gmail.com>
Subject: [ontolog-forum] Re: On the concept of consciousness
I was going to write a reply to this... actually I did anyway but it's shorter because John Sowa already said what I was going to say. No-one really has a clue and virtually all the discussions I've ever seen on this end up going nowhere. IMO there are some questions that are amenable to scientific analysis and some (given our current knowledge) that aren't and consciousness is one of those that currently aren't. You have extremes such as a paper I saw years ago by some leading neuroscientists that talked in depth about consciousness and defined it as the opposite of being asleep or in a coma. And on the other extreme people like Kristof Koch who believe in Pansychism, that everything in the universe is conscious.
Many years ago I sat in on a Philosophy of Mind lecture series led by John Searle at Berkeley. One of my favorite classes was a guest lecture by Koch. Searle started out by lauding him as one of the most brilliant minds ever (which at the start of his talk I could see why, Koch really knows his neuroscience). Then Koch started getting into his Pansychism philosophy and you could just see the color draining from Searle's face and Searle finally said something like "Wait, you are serious?! I thought you were talking about Pansychism as an example of a clearly wrong theory!" And it got more entertaining from there.
I don't agree with Patricia Churchland much but there is a book called "This Idea Must Die!" where she talks about the Neural Correlate of Consciousness (NCC) as an idea that must die. Her reasoning was that there are so many concepts we don't have coherent, falsifiable models of yet such as the Language Faculty and Episodic Memory and that whatever consciousness is, we probably all agree that it is closely tied to memory and language so until we at least have decent theories on such more basic (but still barely understood) concepts it is pointless to postulate theories about consciousness. I mean it can be fun but not something I expect to see any serious science on.
Michael
Alex,
The words 'structure' and 'diagram' have multiple informal meanings in dictionaries of English. They also have multiple formal meanings in different theories of engineering, science, architecture, mathematics, ...
Alex> Diagram is just a picture. Rotate it on 180 grads or delete labels, nothing to think about structure :-)
From ancient times to the present, the angles and sizes of many kinds of diagrams have been very significant -- but there is usually some fixed ratio of the size of the diagram to the structure it represents: diagrams in geometry, architectural plans, maps of the earth, moon, stars, and designs of engineering systems (a car, a pump, or a violin, for example).
But I agree that some diagrams of linguistics or logic can be moved or rotated without changing the meaning.
But the beauty of Peirce's existential graphs is that they can be used for multiple purposes. For representing logic, an EG can be mapped to and from a linear notation without any change in meaning.
But in 1911, he wanted to generalize his graphs to represent "stereoscopic moving images" or "moving pictures of thought". For those purposes, he could generalize EGs to map pictures, even moving pictures, to graphs that have two kinds of information: abstract logic that has no implicit physical information and representations of physical structures where the relative positions and angles are significant.
This is a very important reason why Peirce's diagrammatic reasoning is far more expressive than predicate calculus *and* LLMs. I'm writing another article about Peirce's Delta Graphs, which appear to be going in that direction (just before Peirce had a serious accident and left the document incomplete). But he left enough hints and requirements to indicate the direction he intended. In 2018, I published an article about generalizing existential graphs (see the references in the PDF I sent).
John
----------------------------------------
From: "Alex Shkotin" <alex.shkotin(a)gmail.com>
Ravi, et al.,
For me there is a much more powerful idea and this is the idea of a structure. If a diagram can help to get or work with structure we use it.
Diagram is just a picture. Rotate it on 180 grads or delete labels, nothing to think about structure :-)
So "All necessary reasoning without exception is struturematic"
Consider this kind of structure: create a node and draw an arrow from it and at the end of it create another node, and so on ad infinitum.
Thinking the process is complete we get the structure for the natural numbers.
It can probably be drawn if it helps in its study.
Alex
Alex, Doug F,
I'm attaching a PDF of the article I sent yesterday. This version has diagrams that will clarify many of the issues.
Figure 2 is fundamental, and Figures 3 and 4 clarify some of the details. (To Alex : every diagram is a structure, and every structure is a diagram. They serve exactly the same purpose.)
Doug F> Even if most everybody often thinks in diagrams, that doesn't mean it is
the sole method of thinking. If i am listening to a bird or insect making
noises outside my window, my attempt to recognize the type of animal is
not diagramatic. If i am smelling a flower blindfolded, my attempt to
recognize the type of bloom is not diagramatic. If i am petting my cat
while reading and detect a bur in her fur, that is not diagramatic.
[JFS> All those sensations and actions are continuous. But if you want to talk about them or relate them to your inner stock of discrete words/concepts, you must simplify them to a structure/diagram that is constructed of discrete parts.]
DF> If I taste something i am cooking to determine whether to add more (and which)
herb or spice, i am not engaged in diagrammatic thinking.
[JFS> Now you have converted the continuous perceptions to discrete units (concepts/words). That can be represented as a structure or diagram. A sentence made up of words is just a one-dimensional diagram/structure. A moving multidimensional diagram can be a much closer map to your perceptions, plans, and actions. That is what Peirce called diagrammatic thinking.]
In every one of those examples, the percept is a continuous reflection of external imagery. As Figures 2, 3, and 4show, that continuous information must be mapped to discrete units before they can be mapped to and from any language that has discrete words or concepts.
Peirce used the word 'diagram', and Alex used the word 'structure', you could also use words like graph, hypergraph, or whatever. But the critical issue is that some discrete structure of some kind must serve as the intermediate stage between a continuous world and any discrete set of words or concepts used to talk about it. I also attached a long list of references, which represent a small subset of the things I have consulted while developing the ideas in that article. I invite you to explore them (and/or any others you may prefer).
John
Doug F, et al.,
I'm writing an article about Peirce's phaneroscopy and diagrammatic reasoning, which has strong implications for ontology, reasoning methods, and their implications for the latest issues in generative artificial intelligence. See below for excerpts from that article and some links for further information.
John
PS: I just did a cut & paste below, but the diagrams did not get copied. I'll include a PDF later. But the text explains the issues, and the citations have more explanations and diagrams.
___________________________________
From a Science Egg to a Science of DiagramsJohn F. Sowa
Draft of 26 August 2023Abstract. In the last decade of his life, Peirce developed phaneroscopy and existential graphs as the basis for a proof of pragmaticism. To publish the proof, he wrote a series of articles for the Monist. The first two began with phaneroscopy. But in 1906, he added a version of tinctured existential graphs to the third article, An Apology for Pragmaticism. In 1908, he began a fourth article, which he never finished. One reason he stopped may be his remark in 1909: “Phaneroscopy, still in the condition of a science-egg, hardly any details of it being as yet distinguishable.” Other reasons involve issues about the graphs, which he resolved in 1911. Although Peirce did not complete the proof, his writings inspired aspects of Lady Welby’s significs, Wittgenstein’s language games, and patterns of diagrams in every branch of science and engineering. Today, Peirce’s theories of phaneroscopy and diagrammatic reasoning clarify critical issues in cognitive science. Among them are the methods of reasoning in linguistics, neuroscience and artificial intelligence.
1. Developments from 1903 to 1913For Peirce, 1902 brought an end to two major projects: Baldwin’s dictionary was finished, and funding for his Minute Logic was rejected. But three events in 1903 led him to rethink every aspect of his life’s work: his Harvard lectures in the spring, his Lowell lectures in the fall, and his correspondence with Victoria Welby. As a guide to the new developments, the tree in Figure 1 shows his classification of the sciences and dependencies among them. Branches show the classification, and dotted lines show the dependencies. Sciences to the right of each dotted line depend on sciences to the left. Pure mathematics stands alone, and all other sciences and engineering depend on mathematics (CP 1.180ff, 1903).
. . . [deleted]
In summary, phaneroscopy depends on mathematics, which includes existential graphs as a formal logic. But as a diagrammatic logic, EGs can be used in two ways. For phaneroscopy, the option of changing shape is important. Nodes of a graph may be moved to match the shape of the image they represent. For logic, however, changing the shape does not change the meaning. Since the same notation can serve both purposes, EGs support Peirce’s prediction that phaneroscopy “surely will in the future become a strong and beneficient science” (R645, 1909).
2. The Role of Diagrams in PhaneroscopyFor the third Monist article, Prolegomena to an Apology for Pragmaticism, Peirce chose a title that echoes Kant’s Prolegomena. In it, he addressed Kant’s three “transcendental questions”: How is pure mathematics possible? How is pure natural science possible? How is metaphysics in general possible? The dotted lines in Figure 1 suggested the answer shown in Figure 2: diagrams, such as EGs, are mathematical structures that relate phaneroscopy, metaphysics, and the natural sciences to methods for thinking, talking, and acting in and on the world.
Figure 2: Diagrams relate thought and language to the world
The first sentence sets the stage: “Come on, my Reader, and let us construct a diagram to illustrate the general course of thought; I mean a System of diagrammatization by means of which any course of thought can be represented with exactitude” (CP 4:530). Figure 2 shows an important step beyond Tarski’s model theory. Instead of a one-step mapping from the world to language, the diagram splits the mapping in two distinct steps.
Phaneroscopy maps some aspect of the world to a diagram, which is “an icon of a set of rationally related objects” (R293, NEM 4:316). It serves as a Tarski-style model for determining the denotation of languages, formal or informal. But when a continuous world is mapped to a discrete diagram, an enormous amount of detail is lost. Although the right side can be a precise map from a graph to a formal logic, it may be an approximate mapping from an informal diagram to the informal languages that people speak. In his career as a scientist, engineer, linguist, lexicographer, and philosopher, Peirce understood the complexity of both sides.
. . . [deleted]
An appropriate logic should facilitate a proof of pragmaticism. Peirce stated the requirements in his Prolegomena: “a System of diagrammatization by means of which any course of thought can be represented with exactitude.” Then “operations upon diagrams, whether external or imaginary, take the place of the experiments upon real things that one performs in chemical and physical research.” The system has four aspects: (1) diagrams in EGs or other notations; (2) grammars for mapping languages to and from diagrams; (3) critic for evaluating the denotation {true,false} of diagrams in terms of a formal logic; and (4) perception and action for relating the world to the diagram. The arrows in the hexagon of Figure 4 indicate the flow of any course of thought.
See https:\\jfsowa.com\talks\eswc.pdf for some of the diagrams
Figure 4: The flow of thought in an intelligent system
The hexagon in Figure 4 shows details implicit in Figure 2. The upper three corners and the starburst of phemes represent intelligent processing. The lower three corners correspond to the drawing by Uexküll in Figure 3. The arrow from mental experience to and from action supports routine habits or emergency responses. Behavior that requires complex reasoning may involve all the nodes and arrows.
As Peirce insisted, a diagram of information flow, such as Figure 4, is not a psychological theory. It may represent data that controls a robot or the thought of an alien being in a distant galaxy. But the word exactitude for representing “any course of thought” poses a challenge. As Figure 2 shows, the mapping between the world and a diagram can only be approximate, and the mapping between a diagram and a language can only be exact for notations that are designed to represent those diagrams. Approximations must be recognized and accommodated.
With his constant questioning, Peirce’s ideas kept evolving. In 1907, he had stated the basis for his proof: “the Graphs break to pieces all the really serious barriers, not only to the logical analysis of thought but also to the digestion of a different lesson by rendering literally visible before one’s very eyes the operation of thinking in actu” (CP 4.6, R298). 1909, he expressed his concerns about phaneroscopy “still in the condition of a science-egg” (R645). But in In the next two years, he addressed those issues and generalized existential graphs to accommodate them.
3. Relating Images to DiagramsSince the semes and phemes that flow along the arrows of Figure 4 may contain uninterpreted percepts and images, ordinary existential graphs cannot represent them. In the letter L231, in which Peirce specified his most general notation for EGs, he mentioned his hopes of representingn“stereoscopic moving images.” To accommodate them, Sowa (2016, 2018) proposed generalized existential graphs (GEGs). Figure 5 shows Euclid’s Proposition 1 stated in three kinds of GEGs: “On a given finite straight line, to draw an equilateral triangle.”
. . . [deleted].
For details, see Sowa (2018) Reasoning with diagrams and images, Journal of Applied Logics 5:5, 987-1059. http://www.collegepublications.co.uk/downloads/ifcolog00025.pdf
4. Significs. . . [deleted]
During the following decade, correspondence between Peirce and Welby strongly influenced both. In 1903, Peirce had adopted Kant’s abstract phenomenology. But in 1904, he coined the new word phaneroscopy, which he discussed in terms that were closer to Welby’s emphasis on observation and mental experience. In his letters to her, Peirce added examples that clarified the motivation and explained the details of his abstract analysis. His classification of the sciences in 1903 (Figure 1) illustrates the differences, Peirce had sharply distinguished mathematics, phaneroscopy, and the normative sciences. With her emphasis on examples, Welby showed how practical issues affected the details of each case. As a result of their correspondence, Peirce revised and generalized the foundation of his logic, semeiotic, and pragmatism.
. . . [deleted]
Welby shared Peirce”s broad view of meaning and communication. In What is Meaning (1903), she wrote “There is, strictly speaking, no such thing as the Sense of a word, but only the sense in which it is used — the circumstances, state of mind, reference, ‘universe of discourse’ belonging to it”. In the Encyclopedia Britannica (1911), she emphasized the “importance of acquiring a clear and orderly use of the terms of what we vaguely call Meaning; and also of the active modes, by gesture, signal or otherwise, of conveying intention, desire, impression and rational or emotional thought.”
Whitehead and Wittgenstein would agree, but Frege, Russell, and their followers would strongly disagree. Among linguists, the founder of transformational grammar, Zellig Harris, wrote “We understand what other people say through empathy — imagining ourselves to be in the situation they were in, including imaging wanting to say what they wanted to say.” But his star pupil, Noam Chomsky, would claim that empathy is outside the subject matter of linguistics.
5. Language GamesPeirce and Wittgenstein made a major transition from their early philosophy to their later, and both in the same direction. One critic said that Wittgenstein began as a logician and ended as a lexicographer. Ironically, that remark, which was intended in a derogatory sense, is true in a higher sense: they both discovered the flexibility and expressive power of natural languages. For Peirce, the transition was marked by the 16,000 definitions he wrote or edited for the Century Dictionary. For Wittgenstein, it was his second published book, Wörterbuch für Kindern, which he wrote when he was teaching elementary school in Austrian mountain villages. He learned that children do not think or speak along the lines of his first book, the Tractatus Logico-Philosophicus (TLP).
. . . [deleted]
6. Diagrams As the Language of ThoughtPeirce’s writings on logic, semeiotic, and diagrammatic reasoning, which had been neglected for most of the 20th century, are now at the forefront of research in the 21st. The psychologist Johnson-Laird (2002), who had written extensively about mental models, said that Peirce’s existential graphs and rules of inference are a good candidate for a neural theory of reasoning:
Peirce’s existential graphs are remarkable. They establish the feasibility of a diagrammatic system of reasoning equivalent to the first-order predicate calculus. They anticipate the theory of mental models in many respects, including their iconic and symbolic components, their eschewal of variables, and their fundamental operations of insertion and deletion. Much is known about the psychology of reasoning... But we still lack a comprehensive account of how individuals represent multiply-quantified assertions, and so the graphs may provide a guide to the future development of psychological theories.. . . [deleted]
These observations imply that cognition involves an open-ended variety of interacting processes. Frege’s rejection of psychologism and “mental pictures” reinforced the behaviorism of the early 20th century. But the latest work in neuroscience uses “folk psychology” and introspection to interpret data from brain scans (Dehaene 2014). The neuroscientist Antonio Damasio (2010) summarized the issues:
The distinctive feature of brains such as the one we own is their uncanny ability to create maps... But when brains make maps, they are also creating images, the main currency of our minds. Ultimately consciousness allows us to experience maps as images, to manipulate those images, and to apply reasoning to them.The maps and images form mental models of the real world or of the imaginary worlds in our hopes, fears, plans, and desires. They provide a “model theoretic” semantics for language that uses perception and action for testing models against reality. Like Tarski’s models, they define the criteria for truth, but they are flexible, dynamic, and situated in the daily drama of life.
7. Diagrammatic ReasoningEverybody thinks in diagrams — from children who draw diagrams of what they see to the most advanced scientists and engineers who draw what they think. Ancient peoples saw diagrams in the sky, and ancient monuments are based on those celestial diagrams. They correspond to the mathematical “patterns of plausible inference” identified by Pólya (1954). The role of diagrammatic reasoning is one of Peirce’s most brilliant insights, and the generalized EGs in his late writings include much more than an alternative to predicate calculus.
All necessary reasoning without exception is diagrammatic. That is, we construct an icon of our hypothetical state of things and proceed to observe it. This observation leads us to suspect that something is true, which we may or may not be able to formulate with precision, and we proceed to inquire whether it is true or not. For this purpose it is necessary to form a plan of investigation, and this is the most difficult part of the whole operation. We not only have to select the features of the diagram which it will be pertinent to pay attention to, but it is also of great importance to return again and again to certain features. (EP 2:212). . . [deleted]
Computer systems can communicate with people by traslating their internal represenations to and from notations that people can read and understand. But as Zelling Harris said, computers cannot understand what people say until they have sufficient empathy to imagine themselves to be in the situations the humans are in, including imaging wanting to say what the humans want to say.
Doug,
Re "new math": I was a mathematician from way back. When I was in high school, I learned the old calculus with differentials (dx/dt) from my father's calculus textbook. But the best introduction was "Calculus for the practical man" which skipped the epsilons and deltas, introduced calculus with differentials (on page 2), and put the emphasis on diagrams for solving problems. I found it in the library, read it, did some of the exercises, and skipped the freshman year of calculus at MIT. When I learned the epsilons and deltas, they were trivial -- just another way of stating what I already knew. See https://dn790003.ca.archive.org/0/items/calulusforthepra000526mbp/calulusfo…
Abraham Robinson proved that you don't need the epsilons and deltas. But the textbook crowd is still teaching them. Richard Feynman also learned calculus from "Calculus for the practical man", and he pioneered the use of "Feynman diagrams" in nuclear physics. See https://pubs.aip.org/physicstoday/online/12177/A-look-inside-Feynman-s-calc…
My phrase "thinking in diagrams" does not imply "conscious thinking in diagrams". Furthermore, it includes all diagrammatic reasoning by people who are blind from birth. For those people, the visual cortex and the larger occipital lobes that contain it do not atrophy. They remain every bit as alive and active as they are for normally sighted people.
Furthermore, the cerebellum, which is actively involved in all forms of motion and interpretation of external sensations is a major contributor to the occipital lobes, which integrate visual perception with all other forms of input in the 3-D moving "mental models". Since I'm not blind, I can't say how blind people experience those models. But a huge amount of the processing of all sensory input takes place in the brain stem and the cerebellum, of which nobody -- blind or sighted -- has any kind of conscious awareness.
Furthermore, the cerebellum is also highly active in all mathematical reasoning. Of the 86 billion neurons in a typical human brain, about 15 billion are in the huge cerebral cortex, about 70+ billion in the much smaller cerebellum, and about a billion in the brain stem and spinal cord. In brain scans of mathematicians, the cerebellum lights up brightly when the person just hears the words -- but not for people who have no special training in math.
However, the cerebellum also lights up very brightly for gymnasts and other sports professionals in just thinking about their craft. Those people might not have much mathematical training, but they certainly do a lot of reasoning about dynamic 3-D mental models. We're lucky that our primate ancestors did a lot of jumping around in the tree tops. They didn't talk about it, but they did it. Sedentary mathematicians benefit from their experience.
These observations have strong implications for the AI work on LLMs, the huge networks of words. Without the kind of computation that takes place in the cerebellum, those LLMs are not just blind, they don't have the computational power of the cerebellum, which has 4 or 5 times as many neurons as the cerebral cortex.
On diagrammatic reasoning, LLMs can't compete with an ape. The human cerebral cortex is much, much larger than a chimpanzee's, but the ratio in size between a human cerebellum and a chimp's cerebellum is closer. Chimps certainly do complex gymnastics. Up to age 3, chimp babies outperform human babies on non-verbal IQ tests. It would be interesting to test them on math puzzles that don't require verbal explanations.
I'll include more references in other notes in this thread.
John
_________________________________
From: "doug foxvog" <doug(a)foxvog.org>
John,
I am almost always in complete agreement with your posts, so it
surprises me that our positions diverge so far on the relationship
between imagery and thought.
I agree that there is a lot of value to diagrams and diagrammatic
reasoning.
Many people analyze things primarily with images. But NOT EVERYBODY.
From what you write, it appears that you are one of those. There are
different teaching techniques designed to help students who have
different approaches to learning.
Before you start to dispute the concept of thinking without images,
consider blind people. I have talked with a very bright blind woman
who has been blind from birth. She has learned a lot, including highly
technical material, but neither through being presented with diagrams
or images or building up internal ones. Her mind develops network
models, but it would be incorrect to call them diagramatic.
But, it's not only blind folk who think & reason without internal
diagrams. Many may struggle to come up with an (often faulty) mental
diagram, but are more comfortable with other modes of thought. I,
myself, often think diagramatically, but usually with only a localized
diagram of the focus of thought. It seems that i generate portions of
a diagram as desired/needed.
FWIW, i had no problem with "new Math". The added concepts helped me
grok the fuller picture.
I would suggest that Pierce was describing his own mental processes when
he stated "all discovery is based on diagrams" (or images mapped to
diagrams). Is a person blind from birth incapable of discovery?
I also dispute that "[d]eduction is just an exploration of the content
of some diagram or system of diagrams." This is wrong in both
directions. 1) not every exporation of such content is deduction and 2)
deduction is certainly possible based on logical statements that are not
diagramatically represented, including by computers given symbolic
statements and rules of logic for manipulating them.
-- doug foxvog
