"Mathematics is a self-contained microcosm, but it also has the potentiality of mirroring and modeling all the processes of thought and perhaps all of science. It has always had, and continues to an ever increasing degree to have, great usefulness. One could even go so far as to say that mathematics was necessary for man's conquest of nature and for the development of the human race through the shaping of its modes of thinking." (Mark Kac & Stanislaw M Ulam, "Mathematics and Logic", 1968)
"Models are, for the most part, caricatures of reality, but if they are good, then like good caricatures, they portray, though perhaps in a distorted manner, some of the features of the real world." (Mark Kac, "Some mathematical models in science" Science, Vol. 166 (3906), 1969)
"The main role of models is not so much to explain and predict - though ultimately these are the main functions of science - as to polarize thinking and to pose sharp questions. Above all, they are fun to invent and to play with, and they have a peculiar life of their own. The 'survival of the fittest' applies to models even more than it does to living creatures. They should not, however, be allowed to multiply indiscriminately without real necessity or real purpose." (Mark Kac, "Some mathematical models in science" Science, Vol. 166 (3906), 1969)
"As mathematicians we come in when this or that law of nature has been discovered, and our role is usually twofold: (a) to help find ways in which the law can 'best' be formulated and (b) to help in drawing conclusions, which hopefully will be significant to the further development of the subject." (Mark Kac, "On Applying Mathematics: Reflections and Examples", Quarterly of Applied Mathematics, 1972
"[...] it seems self-evident that mathematics is not likely to be much help in discovering laws of nature. If a mathematician wants to make a contribution on this (and I admit it is the highest) level, he will have to master so much experimental material and train himself to think in a way so different from the one he has been accustomed to that he will, in effect, cease to be a mathematician." (Mark Kac, "On Applying Mathematics: Reflections and Examples", Quarterly of Applied Mathematics, 1972)
"We tend to think of 'applied' as being roughly synonymous with 'useful' or 'practical', but I should like to argue that applying mathematics is an activity which often transcends pragmatic considerations and that we should also be concerned with a deeper exploration of what this activity is or ought to be." (Mark Kac, "On Applying Mathematics: Reflections and Examples", Quarterly of Applied Mathematics, 1972)
"When we propose to apply mathematics we are stepping outside our own realm, and such a venture is not without dangers. For having stepped out, we must be prepared to be judged by standards not of our own making and to play games whose rules have been laid down with little or no consultation with us. Of course, we do not have to play, but if we do we have to abide by the rules and above all not try to change them merely because we find them uncomfortable or restrictive." (Mark Kac, "On Applying Mathematics: Reflections and Examples", Quarterly of Applied Mathematics, 1972)
"From a purely operational point of viewpoint […] the concept of randomness is so elusive as to cease to be viable," (Mark Kac, "What is random?", Marginalia, American Scientist 71(4), 1983)
"If doing mathematics or science is looked upon as a game, then one might say that in mathematics you compete against yourself or other mathematicians; in physics your adversary is nature and the stakes are higher." (Mark Kac, "Enigmas Of Chance", 1985)
"Independence is the central concept of probability theory and few would believe today that understanding what it meant was ever a problem." (Mark Kac, "Enigmas Of Chance", 1985)
"There are, roughly speaking, two kinds of mathematical creativity. One, akin to conquering a mountain peak, consists of solving a problem which has remained unsolved for a long time and has commanded the attention of many mathematicians. The other is exploring new territory." (Mark Kac, "Enigmas Of Chance", 1985)
"[…] mathematics is not just an austere, logical structure of forbidding purity, but also a vital, vibrant instrument for understanding the world, including the workings of our minds, and this aspect of mathematics was all but lost." (Mark Kac, "Mathematics: Tensions", 1992)
"[…] there are those who believe that mathematics can sustain itself and grow without any further contact with anything outside itself, and those who believe that nature is still and always will be one of the main (if not the main) sources of mathematical inspiration. The first group is identified as ‘pure mathematicians’ (though ‘purist’ would be more adequate) while the second is, with equal inadequacy, referred to as ‘applied’." (Mark Kac)
"Unrestricted abstraction tends to divert attention from whole areas of application whose very discovery depends on the features that the abstract point of view rules out as being accidental." (Mark Kac)
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