Without troublesome work, no one can have any concrete, full idea of what pure mathematical research is like or of the profusion of insights that can be obtained from it.
Sentiment: NEGATIVE
We have overcome the notion that mathematical truths have an existence independent and apart from our own minds. It is even strange to us that such a notion could ever have existed.
Pure mathematicians just love to try unsolved problems - they love a challenge.
How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments.
You kind of alluded to it in your introduction. I mean, for the last 300 or so years, the exact sciences have been dominated by what is really a good idea, which is the idea that one can describe the natural world using mathematical equations.
A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.
Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
For scholars and laymen alike it is not philosophy but active experience in mathematics itself that can alone answer the question: What is mathematics?
For most people, the major hurdle in grasping modern insights into the nature of the universe is that these developments are usually phrased using mathematics.
We in science are spoiled by the success of mathematics. Mathematics is the study of problems so simple that they have good solutions.
Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.