If one has really technically penetrated a subject, things that previously seemed in complete contrast, might be purely mathematical transformations of each other.
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As quoted in Proportions, Prices, and Planning (1970) by András BródyJohn von Neumann
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What makes a subject difficult to understand — if it is significant, important — is not that some special instruction about abstruse things is necessary to understand it. Rather it is the contrast between the understanding of the subject and what most people want to see. Because of this the very things that are most obvious can become the most difficult to understand. What has to be overcome is not difficulty of the intellect but of the will.
Ludwig Wittgenstein
Thus much, my dear Sallust, upon the threefold operation of the deity have I ventured to write for you, in about three nights' space, having gone over the subject in my memory as far as it was possible: since what I had previously written to you "upon the Saturnalia" did not prove entirely labour thrown away. But on the same subject you will obtain more complete and more abstruse information by consulting the works upon it composed by the divine Iamblichus: you will find there the extreme limit of human wisdom attained. May the mighty Sun grant me to attain to no less knowledge of himself, and to teach it publicly to all, and privately to such as are worthy to receive it: and as long as the god grants this to us, let us consult in common his well-beloved Iamblichus; out of whose abundance a few things, that have come into my mind, I have here set down. That no other person will treat of this subject more perfectly than he has done, I am well aware; not even though he should expend much additional labour in making new discoveries in the research; for in all probability he will go astray from the most correct conception of the nature of the god.
Sallustius (or Sallust)
Knowing reality means constructing systems of transformations that correspond, more or less adequately, to reality. They are more or less isomorphic to transformations of reality. The transformational structures of which knowledge consists are not copies of the transformations in reality; they are simply possible isomorphic models among which experience can enable us to choose. Knowledge, then, is a system of transformations that become progressively adequate.
Jean Piaget
In all things success depends on previous preparation, and without such previous preparation there is sure to be failure. If what is to be spoken be previously determined, there will be no stumbling. If affairs be previously determined, there will be no difficulty with them. If one's actions have been previously determined, there will be no sorrow in connection with them. If principles of conduct have been previously determined, the practice of them will be inexhaustible.
Confucius
The definition of random in terms of a physical operation is notoriously without effect on the mathematical operations of statistical theory because so far as these mathematical operations are concerned random is purely and simply an undefined term. The formal and abstract mathematical theory has an independent and sometimes lonely existence of its own. But when an undefined mathematical term such as random is given a definite operational meaning in physical terms, it takes on empirical and practical significance. Every mathematical theorem involving this mathematically undefined concept can then be given the following predictive form: If you do so and so, then such and such will happen.
Walter A. Shewhart
Neumann, John von
Nevelson, Louise
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