A Quote by Friedrich August von Hayek
I regard it in fact as the great advantage of the mathematical technique that it allows us to describe, by means of algebraic equations, the general character of a pattern even where we are ignorant of the numerical values which will determine its particular manifestation.
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The mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it. However, if I am not mistaken, they have not as yet succeeded. I therefore dare hope that the mathematicians will receive this memoir with good will, for its purpose is to fill this gap in the theory of algebraic equations.
Poetry is related to philosophy as experience is related to empirical science. Experience makes us acquainted with the phenomenon in the particular and by means of examples, science embraces the whole of phenomena by means of general conceptions. So poetry seeks to make us acquainted with the Platonic Ideas through the particular and by means of examples. Philosophy aims at teaching, as a whole and in general, the inner nature of things which expresses itself in these. One sees even here that poetry bears more the character of youth, philosophy that of old age.
Algebra reverses the relative importance of the factors in ordinary language. It is essentially a written language, and it endeavors to exemplify in its written structures the patterns which it is its purpose to convey. The pattern of the marks on paper is a particular instance of the pattern to be conveyed to thought. The algebraic method is our best approach to the expression of necessity, by reason of its reduction of accident to the ghostlike character of the real variable.
Many persons who are not conversant with mathematical studies imagine that because the business of [Babbage’s Analytical Engine] is to give its results in numerical notation, the nature of its processes must consequently be arithmetical and numerical, rather than algebraical and analytical. This is an error. The engine can arrange and combine its numerical quantities exactly as if they were letters or any other general symbols; and in fact it might bring out its results in algebraical notation, were provisions made accordingly.
The equations at which we arrive must be such that a person of any nation, by substituting the numerical values of the quantities as measured by his own national units, would obtain a true result.
One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?
There is a great difference, whether the poet seeks the particular for the sake of the general or sees the general in the particular. From the former procedure there ensues allegory, in which the particular serves only as illustration, as example of the general. The latter procedure, however, is genuinely the nature of poetry; it expresses something particular, without thinking of the general or pointing to it.
If there are four equations and only three variables, and no one of the equations is derivable from the others by algebraic manipulation then there is another variable missing.
My own beliefs are that the road to a scientific discovery is seldom direct and that it does not necessarily require great expertise. In fact, I am convinced that often a newcomer to a field has a great advantage because he is ignorant and does not know all the complicated reasons why a particular experiment should not be attempted.
What appear to be the most valuable aspects of the theoretical physics we have are the mathematical descriptions which enable us to predict events. These equations are, we would argue, the only realities we can be certain of in physics; any other ways we have of thinking about the situation are visual aids or mnemonics which make it easier for beings with our sort of macroscopic experience to use and remember the equations.
Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe?
Knowledge is first and wisdom is the manifestation of knowledge. To understand this tone and pattern of thinking in the numerical way automatically resonated with me.
I am fully assured, that no general method for the solution of questions in the theory of probabilities can be established which does not explicitly recognize, not only the special numerical bases of the science, but also those universal laws of thought which are the basis of all reasoning, and which, whatever they may be as to their essence, are at least mathematical as to their form.
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.
It took me 1057 pages to describe the hundreds of mathematical equations, algorithms and programming techniques that I invented and used.
The emphasis on mathematical methods seems to be shifted more towards combinatorics and set theory - and away from the algorithm of differential equations which dominates mathematical physics.
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