This will be a multiple-reply, as merging responses makes (to me) even more evident the confusion with 'algebra'
Jerry Uhl wrote: > > At 1:05 PM 1/31/22 +0000, email@example.com wrote: > For instance, we don't know that someone who doesn't > > know much algebra can learn much science. > > > > Poincare and Einstein were very poor at algebra. > And Archimedes didn't know any. > wrong !!!!!!! (unless Algebra == Symbol-shuffling) Einstein (just read again general & special relativity) was f a s c i n a t ed by algebra (not by symbol shuffling and especially boil everything down to number crunching). .. and he was disgusted by ordinary calculus. Hence the idea to substitute the operation (analytic) to calculate integrals and derivates by tensor-algebraic operations. (You ever tried to explain Tensor, Symmetrie of tensors, co- and contravariant etc. etc. etc. without refering to A l g e b r a ????) Within relativity (and by the way within quantum-physics) the mere n u m b e r is irrelevant, but not the algebraic properties (symmetrie & transformations groups)
> > Algebra: The level one needs depends on the class and or job, but for the > people who take a calculus class (for business calculus take ~75% of what > follows) I'd want them to be able to work with polynomials and rational > expressions (though not as complex as the ones I did as a kid) and have a good > idea of when to do what. For instance they should appreciate factoring as a > step in solving an equation or inequality. > I'm not sure whether your'e talking about symbol-shuffling (I got 'unsuficient' when this topic of mecanicaly calculating rationals was a 'math??-'subject, because always some detail escaped my atention. To defend my self y developed a series of tecniques like crossing aut number, factors etc. and was finally very happy when I had the wonderful never-will-forget-any-factor device named computer) If your'e talking about decomposing polynoms in linear factors etc. not just for fun but rather a type of 'backwards-engineering' to reconstruct 'factors' (double-sense) of function, alright .
> I'd want them to know exponents and logarithms and their properties very > well. I'd want them to understand the idea of one variable determining > another (i.e. functions) and also use of function notation and inverse > functions. > This is not -unless introducing function-space and operations on those space- algebra > > > >>As a matter of fact, laziness, the desire to have convenient tools > >around is a major driving force for innovation. Rather than changing > >the people (a la socialist re-education camps), we change the envi- > >ronment in such a way that even without any technical expertise the > >average citizen can use very sophisticated machinery, from cars to > >Macinstoshes. If the American citizenry had been too well educated > >we might have never seen the Macintosh and instead still use the > >DOS-prompt, take square-roots by hand, and enjoy wasting our days > >with long integrations by parts .... > > > > I disagree with this, quite fundamentally. It's really quite > striking how after 15 or so years, the philosophy of the > Macintosh Operating System ("windows"/"desktop") is still > not much understood. MacOS was *not* designed for average > people to use computers - it was designed for all those > people (even really bright ones) who wanted to spend more > time using a computer to get their work done than messing > around with computers.
I agree a 100%: I've got 24 years of profesional experience with computers, so there was never a problem of mastering DOS, but when the first mac was around y started using MAC just to work with and not to struggle against the computer. > > In the same way, computer mathematics software (Mathematica, > Maple, Derive, ...) offers *everyone* a tool with which > they can spend more time doing "mathematics" and less time > doing all those routine "manipulations" that are something > less than "mathematics". Like wise I'm using Mathematica to do ..... Algebra (to be specific algebraic/group-theoretic methods to solve systems of PDE's: Invariants of PDE-Systems under coordinate transformations. Do to this by hand would be imposible, to do 'numericaly' would give any result. By the way: in civil-engineering the conection between structural properties and dynamic properties (erath-quake-dynamics) can't even be explained without using algebra !!!!)
........ so please don't abuse the word 'algebra' when in fact your'e talking about the mind and fantasy killing exercises to calculate comon-denominators for purposely badly placed problems. It was always a misnomer (a a weak defense) to name this drilling-exercise 'algebra' (same way: a course on Calligraphie i s n o t a writing course)
Cornelio _______________________________________________ Cornelio Hopmann Universidad Nacional de Ingenieria firstname.lastname@example.org Facultad de Electrotecnia y Computacion http://waltz.uni.rain.ni Avenida Universitaria, Managua Nicaragua Apartado Postal 5595