Immanuel Kant's Critique of Pure Reason — trans. F. Max Müller

I might have apparently proved my thesis too by putting forward, as is the habit of dogmatists, a wrong definition of the infinity of a given quantity. I might have said that the quantity is infinite^ if no greater quantity (that is, greater than the number of given units contained in it) is possible. As no number is the greatest, because one or more units can always' be added to it, I might have argued that an infinite given quantity, and therefore also an infinite world (infinite as regards both the past series of time and extension in space) is impossible, and therefore the world limited in space and time. I might have done this, but, in that case, my definition would not have agreed with the true concept of an infinite whole. We do not represent by it how large it is, and the concept of it is not therefore the concept of a maxi7num^ but we conceive by it its relation only [p. 432] to any possible unit, in regard to which it is greater than any number. According as this unit is either greater or smaller, the infinite would be greater or smaller, while infinity, consisting in the relation only to this given unit, would always remain the same, although the absolute quantity of the whole would not be known by it. This, however, does not concern us at present.

The true transcendental concept of infinity is, that the successive synthesis of units in measuring a quantum, can never be completed.^ Hence it follows with perfect certainty, that an eternity of real and successive states cannot have elapsed up to any given (the present) moment, and that the world therefore must have a beginning.

1 This quantum contains therefore a multitude (of given units) which is greater than any number ; this is the mathematical concept of the infinite.

Transcendental Dialectic 349

Antithesis opposite case an empty time, and likewise an empty space, would form the limits of the world. Now I am quite aware that people have tried to escape from this conclusion by saying that a limit of the world, both in time and space, is quite possible, without our having to admit an absolute time before the beginning of the world or an absolute space outside the real world, which is impossible.

I have nothing to say against the latter part of this opinion, held by the philosophers of the school of Leibniz. Space is only the form of external intuition, and not a real object that could be perceived externally, nor is it a correlate of phenomena, but the form of phenomena themselves.

Space, therefore, cannot exist absolutely (by itself) as something determining the existence of things, because it is no object, but only the form of possible objects.

Things, therefore, as phenomenal, may indeed determine space, that is, impart reality to one or other of its predicates (quantity and relation); but space, on the other side, as something existing by itself, cannot determine the reality of things in regard to quantity or form, because it is nothing real in itself. Space therefore (whether full or empty ^) may be limited by phenomena, but phenomena cannot be limited by empty space outside them.

The same [p. 433] applies to time. But, granting all this, it cannot be denied that we should be driven to admit these two monsters, empty space outside, and empty time before the world, if we assumed the limit of the world, whether in space or time.

1 It is easily seen that what we wish to say is that empty s'pace, so far as limited by phenomena^ that is, space within the world, does not at least contradict transcendental principles, and may be admitted, therefore, so far as they are concerned, though by this its possibility is not asserted.

350 Transcefidefital Dialectic

Thesis With regard to the second part of the thesis, the difificulty of an endless and yet past series does not exist ; for the manifold of a world, infinite in extension, is given at one and the same time. But, in order to .conceive the totality of such a multitude of things, as we cannot appeal to those limits which in intuition produce that totality by themselves, we must render an account of our concept, which in our case cannot proceed from the whole to the determined multitude of the parts, but has to demonstrate the possibility of a whole by the successive synthesis of the parts. As such a synthesis would constitute a series that would never be completed, it is impossible to conceive a totality either before it, or through it. For the concept of totality itself is in this case the representation of a completed synthesis of parts, and such a completion, and therefore its concept also, is impossible.

Transcendental Dialectic 351