What is Mathematics not? - Part III

“[…] 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)

“[…] mathematics is not just a symbolism, a set of conventions for the use of special, formal vocabularies, but is intimately connected with the structure of rational thought, with reasoning practices. [...] mathematics is not just a language, and of refusing the foundationalist move of trying to reduce mathematics to logic, instead seeing mathematics as providing rational frameworks for science, is to set science against a background of rational structures and rational methods which itself has a built-in dynamics. The rational framework of science is itself historically conditioned, for it changes with developments in mathematics.” (Mary Tiles, “Bachelard: Science and Objectivity”, 1984)

“Mathematics is not a deductive science – that’s a cliché. When you try to prove a theorem, you don’t just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.” (Paul R Halmos, “I Want to Be A Mathematician”, 1985)

“Mathematics is not a topic that one can easily approach with a virgin mind.” (Wilfrid Hodges, “Building Models by Games”, 1985)

“Contrary to the impression students acquire in school, mathematics is not just a series of techniques. Mathematics tells us what we have never known or even suspected about notable phenomena and in some instances even contradicts perception. It is the essence of our knowledge of the physical world. It not only transcends perception but outclasses it.” (Morris Kline, “Mathematics and the Search for Knowledge”, 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)

“Nature is not ‘given’ to us - our minds are never virgin in front of reality. Whatever we say we see or observe is biased by what we already know, think, believe, or wish to see. Some of these thoughts, beliefs and knowledge can function as an obstacle to our understanding of the phenomena. […] mathematics is not a natural science. It is not about the phenomena of the real world, it is not about observation and induction. Mathematical induction is not a method for making generalizations.” (Anna Sierpinska, “Understanding in Mathematics”, 1994)

“Mathematics is not just a collection of isolated facts: it is more like a landscape; it has an inherent geography that its users and creators employ to navigate through what would otherwise be an impenetrable jungle.” (Ian Stewart, “Nature’s Numbers”, 1995)

“Mathematics is not the study of an ideal, preexisting nontemporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather, it is the part of human studies which is capable of achieving a science-like consensus, capable of establishing reproducible results. The existence of the subject called mathematics is a fact, not a question. This fact means no more and no less than the existence of modes of reasoning and argument about ideas which are compelling an conclusive, ‘noncontroversial when once understood’." (Philip J Davis et al, “The Mathematical Experience”, 1995)

“There is no end to discoveries in mathematics just as there is no end to the mystery of the universe. Both are boundless. Hence mathematics is not so much a body of knowledge as a way of thought with inexhaustible possibilities.” (Karma V Mital, “Understanding Mathematics And Computers” , 1997)