On Paradoxes (1960-1969)

"It is a well-known statistical paradox that the average age of women over forty is under forty." (Morris J Slonim, "Sampling in a Nutshell", 1960)

"Our conscious appreciation of the fact that one event follows another is of a different kind from our awareness of either event separately. If two events are to be represented as occurring in succession, then - paradoxically - they must also be thought of simultaneously." (Gerald J Whitrow, "The Natural Philosophy of Time", 1961)

"Mathematics then is a formidable and bold bridge between ourselves and the external world. Though it is a purely human creation, the access it has given us to some domains of nature enable us to progress far beyond all expectations. Indeed it is paradoxical that abstractions so remote from reality should achieve so much. Artificial the mathematical account may be, a fairy tale perhaps, but one with a moral." (Morris Kline,"Mathematics: A Cultural Approach", 1962)

"In consequence, we are caught up in a paradox, one which might be called the paradox of conceptualization. The proper concepts are needed to formulate a good theory, but we need a good theory to arrive at the proper concepts." (Abraham Kaplan, "The Conduct of Inquiry: Methodology for Behavioral Science", 1964)

"One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories." (Philip J Davis, Number", Scientific American, No 211 (3), 1964)

"We are caught up in a paradox, one which might be called the paradox of conceptualization. The proper concepts are needed to formulate a good theory, but we need a good theory to arrive at the proper concepts." (Abraham Kaplan, "The Conduct of Inquiry: Methodology for Behavioral Science", 1964)

"It is paradoxical that while mathematics has the reputation of being the one subject that brooks no contradictions, in reality it has a long history of successful living with contradictions. This is best seen in the extensions of the notion of number that have been made over a period of 2500 years. From limited sets of integers, to infinite sets of integers, to fractions, negative numbers, irrational numbers, complex numbers, transfinite numbers, each extension, in its way, overcame a contradictory set of demands." (Philip J Davis,"The Mathematics of Matrices", 1965)

"To an important extent, indeed, scientific research has become the secular religion of materialistic society; and it is somewhat paradoxical that a country whose constitution enforces the strict separation of church and state should have contributed so much public money to the establishment and propagation of scientific pessimism." (Harry G Johnson, [in "Basic Research and National Goals: A Report to the Committee on Science and Astronautics Federal Support of Basic Research - Some Economic Issues], 1965) 

"Paradoxes have played a dramatic part in intellectual history, often foreshadowing revolutionary developments in science, mathematics, and logic. Whenever, in any discipline, we discover a problem that cannot be solved within the conceptual framework that supposedly should apply, we experience an intellectual shock. The shock may compel us to discard the old framework and adopt a new one. It is to this process of intellectual molting that we owe the birth of many of the major ideas in mathematics and science." (Anatol Rapoport, "Escape from Paradox", Scientific American Vol. 217 (1), 1967) 

"But, really, mathematics is not religion; it cannot be founded on faith. And what was most important, the methods yielding such remarkable results in the hands of the great masters began to lead to errors and paradoxes when employed by their less talented students. The masters were kept from error by their perfect mathematical intuition, that subconscious feeling that often leads to the right answer more quickly than lengthy logical reasoning. But the students did not possess this intuition […]" (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"I am here not only to evade for a while the clamor and filth and confusion of the cultural apparatus but also to confront, immediately and directly, if it’s possible, the bare bones of existence, the elemental and fundamental, the bedrock which sustains us. I want to be able to look at and into a juniper tree, a piece of quartz, a vulture, a spider, and see it as it is in itself, devoid of all humanly ascribed qualities, anti-Kantian, even the categories of scientific description. To meet God or Medusa face to face, even if it means risking everything human in myself. I dream of a hard and brutal mysticism in which the naked self merges with a nonhuman world and yet somehow survives still intact, individual, separate. Paradox and bedrock." (Edward Abbey, "Desert Solitaire, 1968) 

"[…] mathematicians and philosophers have always been interested in the concept of infinity. This interest arose at the very moment when it became clear that each natural number has a successor, i.e., that the number sequence is infinite. However, even the first attempts to cope with infinity lead to numerous paradoxes." (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"There are living systems; there is no living 'matter'. No substance, no single molecule, extracted and isolated from a living being possess, of its own, the aforementioned paradoxical properties. They are present in living systems only; that is to say, nowhere below the level of the cell." (Jacques Monod, "From Biology to Ethics", 1969)