“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.”

Mathematical Problems (1900)

Adopted from Wikiquote. Last update June 3, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical prob…" by David Hilbert?

David Hilbert30

German prominent mathematician 1862–1943

Related quotes

“There may be oodles of possible humans, but it is a finite number.”

Robert J. Sawyer book Flashforward

Source: Flashforward (1999), Chapter 16 (p. 167)

“Euclid … manages to obtain a rigorous proof without ever dealing with infinity, by reducing the problem [of the infinitude of primes] to the study of finite numbers. This is exactly what contemporary mathematical analysis does.”

Lucio Russo (1944) Italian historian and scientist

2.4, "Discrete Mathematics and the Notion of Infinity", p. 45
The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn (2004)

“The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. … Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.”

Carl Friedrich Gauss book Disquisitiones Arithmeticae

Problema, numeros primos a compositis dignoscendi, hosque in factores suos primos resolvendi, ad gravissima ac utilissima totius arithmeticae pertinere, et geometrarum tum veterum tum recentiorum industriam ac sagacitatem occupavisse, tam notum est, ut de hac re copiose loqui superfluum foret. … [P]raetereaque scientiae dignitas requirere videtur, ut omnia subsidia ad solutionem problematis tam elegantis ac celebris sedulo excolantur.
Disquisitiones Arithmeticae (1801): Article 329

“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 ."”

Alan Turing Computable Numbers

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

“That from the outset they expect or even impose all the properties of finite numbers upon the numbers in question, while on the other hand the infinite numbers, if they are to be considered in any form at all, must (in their contrast to the finite numbers) constitute an entirely new kind of number, whose nature is entirely dependent upon the nature of things and is an object of research, but not of our arbitrariness or prejudices.”

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

Letter to Gustac Enestrom, as quoted in Georg Cantor : His Mathematics and Philosophy of the Infinite (1990) by Joseph Warren Dauben ~ ISBN 0691024472

“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)

“The mere formulation of a problem is far more often essential than its solution, which may be merely a matter of mathematical or experimental skill. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and makres real advances in science.”

Albert Einstein (1879–1955) German-born physicist and founder of the theory of relativity

http://umich.edu/~scps/html/01chap/html/summary.htm
Attributed in posthumous publications, Einstein and Religion (1999)

“The systems approach to problems does not mean that the most generally formulated problem must be solved in one research project. However desirable this may be, it is seldom possible to realize it in practice. In practice, parts of the total problem are usually solved in sequence. In many cases the total problem cannot be formulated in advance but the solution of one phase of it helps define the next phase. For example, a production control project may require determination of the most economic production quantities of different items. Once these are found it may turn out that these quantities cannot be produced on the available equipment in the available time. This, then, gives rise to a new problem whose solution will affect the solution obtained in the first phase.”

C. West Churchman (1913–2004) American philosopher and systems scientist

Source: 1940s - 1950s, Introduction to Operations Research (1957), p. 7

“Solutions require thinking through a series of interrelated steps or stages, analyzing a number of rules at each point, and always keeping in mind conclusions reached at earlier points.”

John Thibaut (1917–1986) American social psychologist

Harold Kelley and John W. Thibaut. "Group problem solving." The handbook of social psychology 4 (1969): 1-101; p. 69-70

Help us to complete the source, original and additional information

David Hilbert30

Related quotes

Related topics