“We may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions q1, q2, ..., qK which will be called " m-configurations ."”

On Computable Numbers, with an Application to the Entscheidungsproblem (1936)

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Alan Turing33

British mathematician, logician, cryptanalyst, and computer… 1912–1954

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“It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning.”

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“What I assert and believe to have demonstrated in this and earlier works is that following the finite there is a transfinite (which one could also call the supra-finite), that is an unbounded ascending ladder of definite modes, which by their nature are not finite but infinite, but which just like the finite can be determined by well-defined and distinguishable numbers.”

Georg Cantor (1845–1918) mathematician, inventor of set theory

As quoted in Understanding the Infinite (1994) by Shaughan Lavine ~ ISBN 0674921178

“What I declare and believe to have demonstrated in this work as well as in earlier papers is that following the finite there is a transfinite (transfinitum)--which might also be called supra-finite (suprafinitum), that is, there is an unlimited ascending ladder of modes, which in its nature is not finite but infinite, but which can be determined as can the finite by determinate, well-defined and distinguishable numbers.”

Georg Cantor (1845–1918) mathematician, inventor of set theory

From Kant to Hilbert (1996)

“There are a number of theorems in ordinary algebra, which, though apparently proved to be true only for symbols representing numbers, admit of a much more extended application. Such theorems depend only on the laws of combination to which the symbols are subject, and are therefore true for all symbols, whatever their nature may be, which are subject to the same laws of combination. The laws with which we have here concern are few in number, and may be stated in the following manner. Let a, b represent two operations, u, v two subjects on which they operate, then the laws are
(1) ab(u) = ba (u),
(2) a(u + v) = a (u) + a (v),
(3) am. an. u = am + n. u.
The first of these laws is called the commutative law, and symbols which are subject to it are called commutative symbols. The second law is called distributive, and the symbols subject to it distributive symbols. The third law is not so much a law of combination of the operation denoted by a, but rather of the operation performed on a, which is indicated by the index affixed to a. It may be conveniently called the law of repetition, since the most obvious and important case of it is that in which m and n are integers, and am therefore indicates the repetition m times of the operation a.”

Duncan Gregory (1813–1844) British mathematician

That these are the laws employed in the demonstration of the principal theorems in Algebra, a slight examination of the processes will easily shew ; but they are not confined to symbols of numbers ; they apply also to the symbol used to denote differentiation. <br class="br"> p. 237 http://books.google.com/books?id=8lQ7AQAAIAAJ&pg=PA237; Highlighted section cited in: George Boole " Mr Boole on a General Method in Analysis http://books.google.com/books?pg=PA225-IA15&id=aGwOAAAAIAAJ&hl," Philosophical Transactions, Vol. 134 (1844), p. 225; Other section (partly) cited in: James Gasser (2000) A Boole Anthology: Recent and Classical Studies in the Logic of George Boole,, p. 52 <br class="br">Examples of the processes of the differential and integral calculus, (1841)

“[T]he m-configuration may be changed.”

Alan Turing Computable Numbers

On Computable Numbers, with an Application to the Entscheidungsproblem (1936)

“Why do I think that Turing's paper "On computable numbers" is so important? Well, in my opinion it's a paper on epistemology, because we only understand something if we can program it, as I will explain in more detail later. And it's a paper on physics, because what we can actually compute depends on the laws of physics in our particular universe and distinguishes it from other possible universes. And it's a paper on ontology, because it shows that some real numbers are uncomputable, which I shall argue calls into question their very existence, their mathematical and physical existence.”

Gregory Chaitin (1947) Argentinian mathematician and computer scientist

"Epistemology as information theory: From Leibniz to Omega." https://arxiv.org/abs/math/0506552 arXiv preprint math/0506552 (2005). p. 3

“The usefulness of the models in constructing a testable theory of the process is severely limited by the quickly increasing number of parameters which must be estimated in order to compare the predictions of the models with empirical results.”

Anatol Rapoport (1911–2007) Russian-born American mathematical psychologist

Source: 1960s, Prisoner's dilemma: A study in conflict and cooperation (1965), p. 150