It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils.
Interpretation
To advance in mathematics, it's more beneficial to learn from the greats than from those still learning.
Niels Henrik Abel emphasizes the importance of studying the works and methods of accomplished mathematicians rather than those of learners or novices. By focusing on the masters, one gains insight into advanced concepts and techniques that have been proven valuable, thus facilitating a deeper understanding and accelerated progress in the field of mathematics.
In practice
During a math seminar, one could use this quote to highlight the importance of studying significant mathematicians.
Until now the theory of infinite series in general has been very badly grounded. One applies all the operations to infinite series as if they were finite; but is that permissible? I think not. Where is it demonstrated that one obtains the differential of an infinite series by taking the differential of each term? Nothing is easier than to give instances where this is not so.
With the exception of the geometrical series, there does not exist in all of mathematics a single infinite series the sum of which has been rigorously determined. In other words, the things which are the most important in mathematics are also those which have the least foundation.
I'm still a bit of a reading glutton, I think, because I browse, read a bit of the back copy, flip through the book, read a bit of the text, and if it still seems fascinating, I read it. That's why my bedside table is so cluttered: I want to imbibe it all.
Great knowledge is requisite to instruct those who have been well instructed, but still greater knowledge is requisite to instruct those who have been neglected.
It is by teaching that we teach ourselves, by relating that we observe, by affirming that we examine, by showing that we look, by writing that we think, by pumping that we draw water into the well.
The great want of our race is perfect educators to train new-born minds, who are infallible teachers of what is right and true.
High school teachers who want to get reluctant readers turned around need to give the students some say in the reading list. Make it collaborative: The students will feel ownership, and everyone will dig in.
If you're last in your class at Harvard, it doesn't feel like you're a good student, even though you really are. It's not smart for everyone to want to go to a great school.
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