A Quote by Edmund Husserl
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.
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The heart of mathematics consists of concrete examples and concrete problems. Big general theories are usually afterthoughts based on small but profound insights; the insights themselves come from concrete special cases.
To do any important work in physics a very good mathematical ability and aptitude are required. Some work in applications can be done without this, but it will not be very inspired. If you must satisfy your "personal curiosity concerning the mysteries of nature" what will happen if these mysteries turn out to be laws expressed in mathematical terms (as they do turn out to be)? You cannot understand the physical world in any deep or satisfying way without using mathematical reasoning with facility.
It is the witness alone that can work without any desire, without any idea of going to heaven, without any idea of blame, without any idea of praise. The witness alone enjoys, and none else.
When I was a student in Kazakhstan University, I did not have access to any research papers. These papers I needed for my research project. Payment of 32 dollars is just insane when you need to skim or read tens or hundreds of these papers to do research. I obtained these papers by pirating them.
This example illustrates the differences in the effects which may be produced by research in pure or applied science. A research on the lines of applied science would doubtless have led to improvement and development of the older methods - the research in pure science has given us an entirely new and much more powerful method. In fact, research in applied science leads to reforms, research in pure science leads to revolutions, and revolutions, whether political or industrial, are exceedingly profitable things if you are on the winning side.
This common and unfortunate fact of the lack of adequate presentation of basic ideas and motivations of almost any mathematical theory is probably due to the binary nature of mathematical perception. Either you have no inkling of an idea, or, once you have understood it, the very idea appears so embarrassingly obvious that you feel reluctant to say it aloud.
The good news is that, at least in economics, I've seen movement away from its overemphasis on mathematical models of purely rational behavior to a more eclectic and commonsense approach: research that is, among other things, more respectful of insights from psychology.
There was, I think, a feeling that the best science was that done in the simplest way. In experimental work, as in mathematics, there was "style" and a result obtained with simple equipment was more elegant than one obtained with complicated apparatus, just as a mathematical proof derived neatly was better than one involving laborious calculations. Rutherford's first disintegration experiment, and Chadwick's discovery of the neutron had a "style" that is different from that of experiments made with giant accelerators.
Knowledge will not be acquired without pains and application. It is troublesome and deep, digging for pure waters; but when once you come to the spring, they rise up and meet you.
You cannot study other planets without referring to Earth and without applying the techniques and the insights of Earth science. And you cannot really do a good job understanding the Earth without the insights from planetary exploration.
Graphology is another in a long list of quack substitutes for hard work. It is appealing to those who are impatient with such troublesome matters as research, evidence analysis, reasoning, logic, and hypothesis testing.
In my own professional work I have touched on a variety of different fields. I've done work in mathematical linguistics, for example, without any professional credentials in mathematics; in this subject I am completely self-taught, and not very well taught.
By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful.
I imagine that whenever the mind perceives a mathematical idea, it makes contact with Plato's world of mathematical concepts... When mathematicians communicate, this is made possible by each one having a direct route to truth, the consciousness of each being in a position to perceive mathematical truths directly, through the process of 'seeing'.
I was greatly influenced by musique concrete when I was, like, 10. I was completely mesmerized by the idea that you could make music out of sounds. So that's been a constant influence on all my work.
But think of Adam and Eve like an imaginary number, like the square root of minus one: you can never see any concrete proof that it exists, but if you include it in your equations, you can calculate all manner of things that couldn't be imagined without it.
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