On Infinite (1980-1989)

"'Infinity' is not a phenomenon - it is only a word  which enables us somehow to learn truths about finite things." (Yuri I Manin, "Mathematics and Physics", 1981)

"The ‘eyes of the mind’ must be able to see in the phase space of mechanics, in the space of elementary events of probability theory, in the curved four-dimensional space-time of general relativity, in the complex infinite dimensional projective space of quantum theory. To comprehend what is visible to the ‘actual eyes’, we must understand that it is only the projection of an infinite dimensional world on the retina." (Yuri I Manin, "Mathematics and Physics", 1981)

"In the initial stages of research, mathematicians do not seem to function like theorem-proving machines. Instead, they use some sort of mathematical intuition to ‘see’ the universe of mathematics and determine by a sort of empirical process what is true. This alone is not enough, of course. Once one has discovered a mathematical truth, one tries to find a proof for it." (Rudy Rucker, "Infinity and the Mind: The science and philosophy of the infinite", 1982)

"At present, no complete account can be given - one may as well ask for an inventory of the entire products of the human imagination - and indeed such an account would be premature, since mental models are supposed to be in people's heads, and their exact constitution is an empirical question. Nevertheless, there are three immediate constraints on possible models. […] 1. The principle of computability: Mental models, and the machinery for constructing and interpreting them, are computable. […] 2. The principle of finitism: A mental model must be finite in size and cannot directly represent an infinite domain. […] 3. The principle of constructivism: A mental model is constructed from tokens arranged in a particular structure to represent a state of affairs." (Philip Johnson-Laird, "Mental Models" 1983)

"[…] a mathematician's ultimate concern is that his or her inventions be logical, not realistic. This is not to say, however, that mathematical inventions do not correspond to real things. They do, in most, and possibly all, cases. The coincidence between mathematical ideas and natural reality is so extensive and well documented, in fact, that it requires an explanation. Keep in mind that the coincidence is not the outcome of mathematicians trying to be realistic - quite to the contrary, their ideas are often very abstract and do not initially appear to have any correspondence to the real world. Typically, however, mathematical ideas are eventually successfully applied to describe real phenomena […]"(Michael Guillen, "Bridges to Infinity: The Human Side of Mathematics", 1983)

"[…] mathematics is not a science – it is not capable of proving or disproving the existence of real things. In fact, a mathematician’s ultimate concern is that his or her inventions be logical, not realistic." (Michael Guillen,"Bridges to Infinity: The Human Side of Mathematics", 1983)

"The invention of the differential calculus was based on the recognition that an instantaneous rate is the asymptotic limit of averages in which the time interval involved is systematically shrunk. This is a concept that mathematicians recognized long before they had the skill to actually compute such an asymptotic limit." (Michael Guillen,"Bridges to Infinity: The Human Side of Mathematics", 1983)

"The progress of mathematics can be viewed as a movement from the infinite to the finite. At the start, the possibilities of a theory, for example, the theory of enumeration appear to be boundless. Rules for the enumeration of sets subject to various conditions, or combinatorial objects as they are often called, appear to obey an indefinite variety of and seem to lead to a welter of generating functions. We are at first led to suspect that the class of objects with a common property that may be enumerated is indeed infinite and unclassifiable." (Gian-Carlo Rota [Preface to (Ian P Goulden and David M Jackson, "Combinatorial Enumeration", 1983)])

"Mathematics has been called the science of the infinite. Indeed, the mathematician invents finite constructions by which questions are decided that by their very nature refer to the infinite. This is his glory." (Hermann Weyl, "Axiomatic versus constructive procedures in mathematics", The Mathematical Intelligencer, 1985)

"Mathematics, in one view, is the science of infinity." (Phillip J Davis & Reuben Hersh, "The Mathematical Experience", 1985)

"In an infinite number universe, every point can be regarded as the center, because every point has an infinite of stars on each side of it." (Stephen Hawking, "A Brief History of Time", 1988)

"The world of science lives fairly comfortably with paradox. We know that light is a wave and also that light is a particle. The discoveries made in the infinitely small world of particle physics indicate randomness and chance, and I do not find it any more difficult to live with the paradox of a universe of randomness and chance and a universe of pattern and purpose than I do with light as a wave and light as a particle. Living with contradiction is nothing new to the human being." (Madeline L'Engle, "Two-Part Invention: The Story of a Marriage", 1988)