The drastic change was initialized by Bolzano and Cantor in the 19th century.īernard Bolzano, who introduced the notion of set (in German: Menge), and Georg Cantor, who introduced set theory, opposed the general attitude. Gauss ) Modern era Īctual infinity is now commonly accepted. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction. I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. However, the majority of pre-modern thinkers agreed with the well-known quote of Gauss: The continuum actually consists of infinitely many indivisibles ( G. Baconthorpe )ĭuring the Renaissance and by early modern times the voices in favor of actual infinity were rather rare. Cantor) Īctual infinity exists in number, time and quantity. It is well known that in the Middle Ages all scholastic philosophers advocate Aristotle's "infinitum actu non datur" as an irrefutable principle. There were exceptions, however, for example in England. This means there is only a (developing, improper, "syncategorematic") potential infinity but not a (fixed, proper, "categorematic") actual infinity. The overwhelming majority of scholastic philosophers adhered to the motto Infinitum actu non datur. Scholastic, Renaissance and Enlightenment thinkers "For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different."Īristotle also argued that Greek mathematicians knew the difference among the actual infinite and a potential one, but they "do not need the infinite and do not use it" ( Phys. Potential infinity is never complete: elements can be always added, but never infinitely many. Actual infinity is completed and definite, and consists of infinitely many elements. He distinguished between actual and potential infinity. Aristotle's potential–actual distinction Īristotle handled the topic of infinity in Physics and in Metaphysics. (Aristotle)Īristotle postulated that an actual infinity was impossible, because if it were possible, then something would have attained infinite magnitude, and would be "bigger than the heavens." However, he said, mathematics relating to infinity was not deprived of its applicability by this impossibility, because mathematicians did not need the infinite for their theorems, just a finite, arbitrarily large magnitude. Most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by everybody – not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in our thought.Because the limited always finds its limit in something, so that there must be no limit, if everything is always limited by something different from itself.If coming to be and passing away do not give out, it is only because that from which things come to be is infinite.From the division of magnitudes – for the mathematicians also use the notion of the infinite.From the nature of time – for it is infinite.It is not 'that which has nothing beyond itself' that is infinite, but 'that which always has something beyond itself'." (Aristotle) īelief in the existence of the infinite comes mainly from five considerations: "Infinity turns out to be the opposite of what people say it is. The theme was brought forward by Aristotle's consideration of the apeiron-in the context of mathematics and physics (the study of nature): Plato, on the other hand, holds that there is no body outside (the Forms are not outside because they are nowhere), yet that the infinite is present not only in the objects of sense but in the Forms also." (Aristotle) "Only the Pythagoreans place the infinite among the objects of sense (they do not regard number as separable from these), and assert that what is outside the heaven is infinite. The main dialogues where Plato discusses the 'apeiron' are the late dialogues Parmenides and the Philebus.Īristotle sums up the views of his predecessors on infinity as follows: Plato's notion of the apeiron is more abstract, having to do with indefinite variability.
Clearly, the 'apeiron' was some sort of basic substance. These notions are today denoted by potentially infinite and actually infinite, respectively.Īnaximander (610–546 BC) held that the apeiron was the principle or main element composing all things. Apeiron stands opposed to that which has a peras (limit). The ancient Greek term for the potential or improper infinite was apeiron (unlimited or indefinite), in contrast to the actual or proper infinite aphorismenon.
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