Georg Cantor: Quotes about mathematics

Georg Cantor was mathematician, inventor of set theory. Explore interesting quotes on mathematics.
Georg Cantor: 54 quotes6 likes

“Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established.”

Georg Cantor

From Kant to Hilbert (1996)
Context: Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real.

“In mathematics the art of asking questions is more valuable than solving problems.”
In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi.

Georg Cantor

Doctoral thesis (1867); variant translation: In mathematics the art of proposing a question must be held of higher value than solving it.

“The essence of mathematics lies entirely in its freedom.”

Georg Cantor

Variant translation: The essence of mathematics is in its freedom.
From Kant to Hilbert (1996)

“The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.”

Georg Cantor

As quoted in Mind Tools: The Five Levels of Mathematical Reality (1988) by Rudy Rucker. ~ ISBN 0395468108

“Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality.”

Georg Cantor

From Kant to Hilbert (1996)

“In order for there to be a variable quantity in some mathematical study, the domain of its variability must strictly speaking be known beforehand through a definition. However, this domain cannot itself be something variable, since otherwise each fixed support for the study would collapse. Thus this domain is a definite, actually infinite set of values. Hence each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite.”

Georg Cantor

"Über die verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen" ["Over the different views with regard to the actual infinite numbers"] - Bihand Till Koniglen Svenska Vetenskaps Akademiens Handigar (1886)

“I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers.”

Georg Cantor

Grundlagen einer allgemeinen Mannigfaltigkeitslehre [Foundations of a General Theory of Aggregates] (1883)