Will Our Understanding of Math Deteriorate Over Time?

This is why figuring out an elegant, concise, and powerful set of mathematical models which apply to multiple domains, and then devoting effort to simplifying, organizing, and explaining those ideas in an accessible way is so important.

Incentives for researchers are mostly to push and prod at the boundaries of a field, but in my opinion mathematical ideas are only of marginal value in themselves; more important is the way they help us understand and interact with the physical universe, and for that building communities, developing effective languages and notations, codifying our understanding, and making it accessible both to newcomers and to outsiders is the most important task for a field, and perhaps for our society generally.

Just like with software projects or companies, the most “success” comes from helping a range of other people solve their problems and extend their abilities, not from making technically beautiful art projects for their own sake (not that there’s anything inherently wrong with those).

Perhaps more generally, while theorem proving has overwhelmingly dominated pure mathematics and related fields for the past 80–100 years, and has been an important tool since Euclid, theorem proving is only one way of approaching the world, and in my opinion is a mere tool, not an end in itself. Just like simulation is a tool, or drawing pictures is a tool, or statistical analysis is a tool.

I like this bit from Feynman: https://www.youtube.com/watch?v=YaUlqXRPMmY

How many people are fast at computing fifth roots without recourse to computational tools such as Hindu-Arabic numerals?

Are those things math or arithmetic?

I really like the overall point of the post that mathematics once known can be forgotten or neglected, and mathematics written up for mathematics journals can be difficult to understand. Professor John Stillwell writes, in the preface to his book Numbers and Geometry (New York: Springer-Verlag, 1998):

"What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.

". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.

"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."

Stillwell demonstrates what he means about the interconnectedness and depth of "elementary" topics in the rest of his book, which is a delight to read and full of thought-provoking problems.

http://www.amazon.com/gp/product/0387982892/

One major sin is taking new concepts and ideas and putting the primary discoverer's name on them. Such names yield no clue as to the interpretation or application of the idea itself.

Another problem is the symbols used in certain mathematical texts. Everyone who uses them treats them like they're universally understood, but in reality the syntax and meaning of the symbols can and frequently are recycled and reused across disciplines and even theories in the same discipline. You have to be close to the 'in-group'. Like reading other people's code where operators have been overloaded, it's like learning a new language every time you want to dig into a cool new maths paper.

I don't actually have any good solutions to these problems. I would guess there are lots of lessons to be learned from the history of Chinese characters, though. They have thousands of unambiguous symbols which _can_ be learned by non-natives and which _do_ give an appreciable degree of cross-lingual intelligibility among languages that use them.