On Friday, January 31, 2020 at 8:23:08 PM UTC-5, Benoit wrote: > > Regarding the possible introduction in the main part of semigroups and > magmas: > When I look at the page http://us2.metamath.org/mpeuni/df-mnd.html I feel > a bit dizzy. The abundance of parentheses and conjunctions makes it hard > to parse. >
If you already know about magmas, semigroups, and monoids, of course you'd feel that way. Look at it from the perspective of someone who's never heard of these and really doesn't care to learn, because all they came for was groups. Just the names of these things are somewhat intimidating and have little or nothing to do with what they really are. As I wrote earlier: "The user knows from the description and from the property theorems that are referenced. But even without that, I do see all of these properties pretty immediately - they're right there in the definition: ( x p y ) e. b for closure, ( ( x p y ) p z ) = ( x p ( y p z ) ) for associativity, and E. e ... ( ( e p x ) = x /\ ( x p e ) = x ) for left and right identity." We can put that in the comment if people find the conjunction of them confusing. In any case the current definition has the advantage of being able to see everything at once in one place, right in front of you. You don't have to open another tab for semigroups and another tab for magmas and keep going back and forth between them, trying to remember what those terms mean while trying to keep everything in your head all at once the see the whole picture. > Did you notice that the clause stating closedness is actually quantified > over an extra z ? (This is of course innocuous, but still strange.) I find > it much easier to look successively at > http://us2.metamath.org/mpeuni/df-mgmALT.html and > http://us2.metamath.org/mpeuni/df-sgrp.html and [the definition of Monoid > from Semigroup -- Alexander: can you add it?]. So, to me, there is already > a pedagogical benefit, even if no theorems are proved. I find these > piecemeal definitions easier to learn. Once you grab the final concept, > you can forget about the intermediate steps. I.e., you do not have to > remember what a magma or semigroup is, if you don't want. This was only an > aid in the process of getting the definition of a group. > > Regarding Norm's remark on Bourbaki: Indeed, their set theory is known to > be awkward. None of its members were logicians or set theorists, and they > just wanted to "get this done" before moving to what they considered more > interesting math. By the way: you actually use a lot of Bourbaki's > terminology and notation in set.mm: the symbol for the empty set, the > blackboard bold typeface, the terms "ball" and "sphere" in metric spaces, > the terms injection/surjection/bijection, etc. Actually, new terms were > carefully chosen, and as much as possible simple words from everyday life, > like "magma" (because volcanic magma looks structureless), "ball", even > when they are not round (the previous term was some impossible jargon), > in/sur/bijection (previous terms were similar jargon). The terms "barrel" > (a barrel is closed, convex, balanced, absorbing: this makes as much sense > in everyday life as in formal math), bornivorous, etc. > I acknowledge that Bourbaki has had a lot influence, and I have no problem using its notation and terminology - if it has survived and is still used in the mainstream literature after 80+ years. What I think is not a good idea is the blind use of Bourbaki as being the "bible" of mathematics. Norm > I think that Bourbaki was somehow victim of its success: it was so > influential that after a few years or even decades, people took it for > granted that there was a common language and notation for all the branches > of mathematics. But I think it is easy now to underestimate the novelty > that it was in the 1930s and 1940s. Of course there was already work on > logic. And of course each specialist in his domain was ahead of what was > meant to be a treatise of "dead mathematics". Discoveries also spread > slower at the time: the internet was a bit slower in those days, > notwithstanding political considerations... > > Benoît > -- You received this message because you are subscribed to the Google Groups "Metamath" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/metamath/f8bc4670-08c4-4e12-923b-a321657fb316%40googlegroups.com.
