Harry Lewis pointed Bill and me to Gottfried Leibniz's 1677 treatise The True Method (translated from the original French). I highly recommend taking the time to read this three page document where he talks about formalizing all human knowledge.
The second paragraph has some of the intuition for P v NP three centuries before Cook, Levin and Gödel.
But knowledge depends on proof, and discovering proofs requires a certain method that is not known to everyone. Every person is capable of judging a proof, since it would not deserve to be called a proof unless everyone who considered it carefully found it convincing. Nevertheless, not everyone is capable of discovering proofs independently or of presenting them clearly once they have been found, whether for lack of time or for lack of method.
Though by "not everyone" he might have excluded himself.
Every investigation that depends on reasoning would be carried out by manipulating these symbols, through a kind of calculation. This would make the discovery of important results entirely straightforward. We would no longer have to rack our brains as much as we do today, while still being certain that we could accomplish everything that was possible from the information given.
Though not clear if "that was possible" took computational time into account. Is he claiming P = NP?
The article as a whole talks about a project to develop a fully logical language to cover not just mathematics, but the sciences like physics and medicine and even morality, politics and law. Here he was just being too ambitious and missing complexity issues, for example that you can't create laws that fully cover all potential futures.
Leibniz does give himself an out, a nod to the scientific method that was being developed during that time.
Some experiences will always be needed as the foundation for reasoning. But once those experiences have been supplied, we could derive from them everything that anyone could ever derive. We could even determine which further experiments still needed to be performed in order to clear up all remaining doubts.
He even gives a nod to probability.
This would be of extraordinary assistance even in politics and medicine, where we must reason consistently and correctly from given symptoms and circumstances. Even when there are not enough facts to form an infallible judgment, it would still be possible to determine what is most probable from the information available. That is everything reason can do.
Sort of how machine learning works now.
The link to the french version (of course an old french) is here https://livrels.fr/Science/Leibniz/La_Vraie_M%C3%A9thode.epub
ReplyDeleteIn reference to, "not everyone is capable of discovering proofs independently" you write, "Though by "not everyone" he might have excluded himself."
ReplyDeleteAs any undergraduate discrete math student should know, "not everyone is capable" is not the same as "everyone is not capable."
Sorry; may I ask who is the translator? I'd like to cite a quote.
ReplyDeleteChatGPT 5.5 High
DeleteI've read the first paragraphs both in (old) French and in English. This is not a faithful translation at all, in many places it is a reinterpretation. What surprised me the most is that Leibniz write "l'invention des demonstrations" (that is, "the invention of proofs") while this is in English "discovering proofs": I do not know the philosophical view of Leibniz on mathematics (on platonism specifically) but the "translation" is weird. Other places are similarly weird: No real misinterpretation in the parts I read, but not a *good* translation in any way.
ReplyDeleteI think you are selling them short. Back in the olden times there people still said: Moses and Plato had all knowledge, when we find their complete books, we can stop believing experiences.
ReplyDeleteThis looks like machine learning because learning itself has not changed that much.
Well, regarding last Anon, not sure if he is alluding to infamous phrase by mathematician Alfred Whitehead: "The safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato." (pg 39 of Process and Reality).
ReplyDeleteThe argument here is Plato's writings are an "inexhaustible mine of suggestion" containing a vast array of the fundamental, general ideas that have shaped Western thought ever since.
That's a fair point I'd guess but, but it's said in such vagueness or distant abstraction that literally you could interpret the statements to mean too many things.