[CG] Conflict between deduction and discovery in mathematics

The Bourbaki were a group of brilliant mathematicians, who developed a totally unusable system of mathematics. That example below shows how hopelessly misguided they were. Sesame Street's method of teaching math is far and away superior to anything that the Bourbaki attempted to do. Sesame street introduces the number 1 as the starting point of counting. That is also Peirce's method. Furthermore, the Bourbaki banished all diagrams from their system, and thereby violated every one of Peirce's principles of diagrammatic reasoning. Sesame Street emphasizes diagrams and imagery. Mathematics without diagrams and imagery is blind. The so-called "new math" disaster of the late 1960s was a hopelessly misguided attempt to inculcate innocent students with set theory as the universal foundation for everything. Another violation of Peirce's methods. Finally, there is no conflict whatever between deduction and discovery. As Peirce insisted, all discovery is based on diagrams (or images mapped to diagrams). Deduction is just an exploration of the content of some diagram or system of diagrams. There are, of course, many challenges in discovering all the provable implications. But once again, those implications are determined by elaboration and analysis of the starting diagrams. There is much more to say, and it is closely related to my previous note about problems with AI. I'm currently writing an article that shows how Peirce's diagrammatic reasoning is far and away superior to the currently popular methods of Large Language Models. The LLMs do have some important features, but the LLMs are just one special case of one certain kind of diagram (tensor calculus). The human brain (even a fruit fly brain) can process many more kinds. There is, of course, much more to say about this issue, but it will take a bit more time to gather the references. John ---------------------------------------- From: "Evgenii Rudnyi" <rudnyi(a)freenet.de> Sent: 8/22/23 11:13 AM Recently I have seen a paper below that could be of interest to this discussion as it shows that to work deductively even with the number 1 is not that easy. Best wishes, Evgenii Mathias, Adrian RD. "A Term of Length 4 523 659 424 929." Synthese 133, no. 1 (2002): 75-86 "Bourbaki suggest that their definition of the number 1 runs to some tens of thousands of symbols. We show that that is a considerable under-estimate, the true number of symbols being 4 523 659 424 929, not counting 1 179 618 517 981 disambiguatory links."