3 ms·
Sort of. I'd change the metaphor a bit. Math at the start of 19th century was like a country many of whose high mountains (say theorems) had been climbed but w
by joe_the_user 2y ago
Sort of.
I'd change the metaphor a bit. Math at the start of 19th century was like a country many of whose high mountains (say theorems) had been climbed but whose basic geography (foundations) hadn't been charted. Grassman was perhaps the first to begin the process of true foundational studies - creating structures that formally tied the mountains together (different from intuitively tying the mountains together, notably).
Grassman's approach (incorporated as "exterior algebra") was klunky and difficult most likely specifically because he was doing something new, something never done before. That is, he was working when no formal linear algebra and indeed neither formal abstract algebra nor formal foundations of real numbers nor formalization of calculus existed (though many things were talked about informally).
And Grassman was resisted because of the (probably necessary) klunkiness, but also because many mathematicians didn't see the need for this and no small number actively resisted the need for formal. IE, saw the qualities of numbers and collections of thing as best handled intuitively rather than formally and especially disliked the way formal formulations allowed and required counter-intuitive structures (transfinite sets, the axiom of choice, etc). This resistance didn't just touch Grassman but affect Cantor and Hilbert, etc.
Today, math still has a tension between the provers of hard theorems and the structure builders, that would be the mountain climbers and the geographers, the Erdős' and the Grothendiecks. Moreover, the history of that tension isn't as clear as it could be since fields of math are generally taught as fully formulated systems with few nods to the state of the field before a given formulation (I just learned today that van der Waerden's Moderne Alegebra, published in 1930, was the first text to put abstract algebra in anything like it's present form).
More on the history of the formalization of mathematics can found in Jan van Plato's The Great Formal Machinery Works (Grassman is the first writer covered).
- Ar-Curunir 2y agoI'm sure you didn't mean to imply this, but Grothendieck definitely used his new structures to prove some hard theorems.
- eigenvalue 2y agoThis reminds me of the Grothendieck quote about his approach to solving problems (in contrast with Erdős, who I suppose in this analogy would look for a big hammer with which to smash the nut open quickly!): “I can illustrate the ... approach with the ... image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise, you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marble, resisting penetration ... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it ... yet finally it surrounds the resistant substance.”
- vinnyvichy 2y agoWith the caveat that Deligne (?) complained that his work was too moral, ie, a lot of crucial details had to be put in by others to prove anything nontrivial.
- jacobolus 2y agoGrassmann's approach is not "klunky and difficult". It's clear and straightforward and makes proving some difficult results much easier than the previous methods, it just takes a bit of a leap of faith to practice enough with it to build fluency. (The subject itself is also challenging, irrespective of approach. When you strip away some of the incidental complexity of a mismatched representation, the inherent complexity of the subject still must be contended with.) I'd recommend people try to directly read both of Grassmann's two books; there are translations of both into English. (Disclaimer: I haven't made it through either one beyond a skim and reading the first few chapters. Then I got distracted by some other project. Like any math books, reading them takes a lot of slow careful effort.) It's kind of like taking a bunch of people used to riding horses and handing them a bicycle. The bicycle is not "klunky and difficult" compared to the horse, it's just unfamiliar.
- practal 2y agoVan Plato's book seems interesting! I see that it has a chapter on natural deduction and sequent calculus. I found the focus on introduction and elimination rules in natural deduction always somewhat mystifying, and wondered about what exactly natural deduction is. As I discovered just in the last few weeks, even if, like me, you don't know anything about the proof theoretic foundations of natural deduction and sequent calculus, they pop up naturally as proof systems for abstraction logic (AL): natural deduction is the proof system if your truth values form a complete lattice, and sequent calculus is the proof system if your truth values form even a complete bi-Heyting algebra. Note that in AL, there are no a-priori constants (such as ∧ or ⇒), so there are also no a-priori rules for elimination and introduction, but just the essence of natural deduction and sequent calculus. See http://abstractionlogic.com http://abstractionlogic.com for the least klunky presentation of AL to date.