528 INTELLECTUAL SYMBOLISM. 
conditioned amongst phenomena;* partly, likewise, because the word world, in a transcen- 
dental sense, signifies the absolute totality of the complex of existing things, and we 
direct our attention alone to the completeness of the synthesis (although only strictly in 
the regressus to the conditions).”~ ‘The arguments of the Thesis are mainly based on 
ideas derived from the first of these definitions, and they have a quasi validity to show 
that the aggregate of phenomena (if the cause of the phenomena is excluded from con- 
sideration) may have had a beginning in time and a limit in space. The Antithesis can 
only be valid for the second definition to show that “the absolute totality of the complex 
of existing things” (including the Creative First Cause together with every possible and 
actual manifestation of His existence and power) must be, “as well in respect of time as 
of space, infinite.” 
254. The second ambiguity is in the use of the word “infinite.” In the sense in 
which some philosophers have employed the terms infinite and finite, they are mutually 
contradictory, and it is as absurd to speak of their correlation as it would be to talk of 
the four sides and six angles of a square triangle. In one sense, the mere formation of 
an idea is limiting, imasmuch as it assigns bounds which distinguish the idea from all 
others, but according to customary usage, we have a right to call that infinite which is 
unlimited in one or more of its attributes.| We have no right, however, to assume that 
what is true of one infinite is true of another, as is repeatedly done in each of the 
Kantian antinomies. 
255. In the celebrated sophism of Achilles and the tortoise,§ there is a similar equi- 
vocal use of the ideas of infinity. ‘The fallacy, as Hobbes hinted, lies in the tacit as- 
sumption that whatever is infinitely divisible is infinite. . . . The ‘forever’ in the conclu- 
sion means for any length of time that can be supposed; but in the premises ‘ever’ does 
not mean any length of time; it means any number of subdivisions of time. It means 
that we may divide a thousand feet by ten, and that quotient again by ten, and so on as 
* But an unconditioned phenomenon is an impossibility. 
ap deo bleh 
{ Werenfels, De Finibus Mundi Dialogus (quoted by Mansel, p. 253), ingeniously attempts to demonstrate 
that the idea of infinite extension involves necessary contradictions. But the whole argument is based on the un- 
warranted assumption that all relative infinites are equal. It is important, even in discoursing on ordinary topics, 
that all the conditions of the several points at issue should be kept in view, and this precaution is still more essen- 
tial in reasoning upon a subject so obscure as infinity. 
§ “Let Achilles run ten times as fast as the tortoise, yet if the tortoise has the start, Achilles will never over- 
take him. For suppose them to be at first separated by an interval of a thousand feet, when Achilles has run 
these thousand feet, the tortoise will have got ona hundred; when Achilles has run those hundred the tortoise will 
have run ten, and so on forever; therefore, Achilles may run forever without overtaking the tortoise.” See Mill’s 
Logic, p. 508, and Aristotle, guowys &zpoacews, B. vi, chap. 9, p. 549. 
