How one learns a skill or subject area is often irrelevant
However, considerable thought and planning has gone into the pedagogy of a large area of knowledge such as Mathematics.
For example the sequence of topics is important. One would have to have an understanding of basic geometry to fully understand trigonometry.
the question whether a person who is self-taught in mathematics can compete with someone who majored in math will probably lead you to a reality check.
A school board, college or university hiring a teacher of math will of course demand that you have certification in your subject area, and will place very little value on the fact that you might be a very competent self-taught mathematician.
BTW, both Leibniz and Newton are considered the "inventors" of Calculus.
Isn't it true that both men "invented" calculus independently, though Newton discovered it first?
For some reason, they often argued, mathematically speaking.
Wikepedia often gives a nice summary of topics and says about this one:
Leibniz and Newton pulled these ideas together into a coherent whole and they are usually credited with the independent and nearly simultaneous invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today; he often spent days determining appropriate symbols for concepts. The basic insight that both Newton and Leibniz had was the fundamental theorem of calculus.
When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first, but Leibniz published first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. This controversy divided English-speaking mathematicians from continental mathematicians for many years, to the detriment of English mathematics. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. Today, both Newton and Leibniz are given credit for developing calculus independently. It is Leibniz, however, who gave the new discipline its name.
I learned a lot of math myself from university books when I was in high school. My opinion is that math education almost everywhere in the world is very bad. So, what you see is that many people who are very good in math have learnt some of it themselves in high school.
To see how bad math education really is, compare it to language education. If we were to teach language like we teach math to children, the children would only learn the alphabet, spelling of some words and some basic grammar.
Reading books, writing essays would be university level stuff because that is too much work for children and because we would demand they master grammar and spelling perfectly first.
All the interesting stuff you can do with math are kept outside the curriculum. Then what happens is that the children only get to do the same boring problems over and over again. Also, they get the wrong impression of what math is all about.
The same is true for physics. I decided to study physics because I read popular books by Heinz Pagels, Hawking, etc., so I knew a bit about particle physics, quantum mechanics and general relativity. There was almost nothing mentioned in my high school physics books on these subjects.
I don't understand why we can't tell children about how the universe works in high school physics class. Of course, you cannot expect them to master the mathematical formalism of general relativity and quantum mechanics, but today in physics class we don't tell them anything at all.
There are a number of self-taught mathematicians and some that did badly in school but did great work at some point in their lives. Both types are representative of how academia is not the only portal to deep mathematical ideas.
Some notable self-taught mathematicians were Ramanujan and Banach, neither of who can be doubted as superb mathematicians. Ironically Ramanujan had less than an undergraduate understanding of some areas -- this clearly did not hinder him. Other self-taught mathematicians include Boole, Stieltjes, Sophie Germain, and the still living John Allan Robinson, to name just a few.
As for those that were poor students, Galois is one tragic example. Another excellent mathematician who worked his way into a university position was Hermite -- who first proved the trascendence of e. He was known to be a terrible test taker. Stephen Smale, who has won both a Fields and the Wolf prize, nearly failed out of grad school.
With commitment and a little luck to cheat death people are capable of masterful achievements. It's foolish to doubt a person simply because they don't resemble the status quo.
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