The physicist who states a law of nature with the aid of a mathematical formula is abstracting a real feature of a real material world, even if he has to speak of numbers, vectors, tensors, state-functions, or whatever to make the abstraction.
--
in What is Mathematics, in Hilary Putnam (1979). Mathematics, matter, and method. Cambridge University Press. p. 60. ISBN 0521295505.Hilary Putnam
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It is interesting that the term mystic is used in this derogatory sense to mean anything we cannot segmentize and count. The odd belief prevails in our culture that a thing or experience is not real if we cannot make it mathematical, and that somehow it must be real if we can reduce it to numbers. But this means making an abstraction out of it ... Modern Western man thus finds himself in the strange situation, after reducing something to an abstraction, of having then to persuade himself it is real. ... the only experience we let ourselves believe in as real, is that which precisely is not.
Rollo May
One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?
Stephen Hawking
The integers, the rationals, and the irrationals, taken together, make up the continuum of real numbers. It's called a continuum because the numbers are packed together along the real number line with no empty spaces between them.
Brian Hayes
Pythagoras, as everyone knows, said that "all things are numbers." This statement, interpreted in a modern way, is logical nonsense, but what he meant was not exactly nonsense. He discovered the importance of numbers in music and the connection which he established between music and arithmetic survives in the mathematical terms "harmonic mean" and "harmonic progression." He thought of numbers as shapes, as they appear on dice or playing cards. We still speak of squares or cubes of numbers, which are terms that we owe to him. He also spoke of oblong numbers, triangular numbers, pyramidal numbers, and so on. These were the numbers of pebbles (or as we would more naturally say, shot) required to make the shapes in question.
Pythagoras
Gould's argument on reification purports to get at the philosophical foundation of the field. He claims that general intelligence, defined as the factor common to different cognitive abilities, is merely a mathematical abstraction; hence if we consider it a measurable attribute we are reifying it, falsely converting an abstraction into an “entity” or a “thing”—variously referred to as “a hard, quantifiable thing,” “a quantifiable fundamental particle,” “a thing in the most direct, material sense.” Here he has dug himself a deep hole.… Indeed, this whole argument is fantastic. The scientist does not measure “material things”: He measures properties (such as length or mass), sometimes of a single “thing” (however defined), and sometimes of an organized collection of things, such as a machine, a biological organ, or an organism. In a particularly complex collection, the brain, some properties (i.e., specific functions) have been traced to narrowly-localized regions (such as the sensory or motor nuclei connected to particular parts of the body).
Stephen Jay Gould
Putnam, Hilary
Puttnam, David
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