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