In order to exemplify the way in which a soul seeks to actualize itself in the picture of its outer world—to show, that is, in how far Culture in the “become” state can express or portray an idea of human existence— I have chosen _number_, the primary element on which all mathematics rests. I have done so because mathematics, accessible in its full depth only to the very few, holds a quite peculiar position amongst the creations of the mind. It is a science of the most rigorous kind, like logic but more comprehensive and very much fuller; it is a true art, along with sculpture and music, as needing the guidance of inspiration and as developing under great conventions of form; it is, lastly, a metaphysic of the highest rank, as Plato and above all Leibniz show us. Every philosophy has hitherto grown up in conjunction with a mathematic _belonging_ to it. Number is the symbol of causal necessity. Like the conception of God, it contains the ultimate meaning of the world-as- nature. The existence of numbers may therefore be called a mystery, and the religious thought of every Culture has felt their impress.[45] Just as all becoming possesses the original property of _direction_ (irreversibility), all things-become possess the property of _extension_. But these two words seem unsatisfactory in that only an artificial distinction can be made between them. The real secret of all things-become, which are _ipso facto_ things extended (spatially and materially), is embodied in mathematical number as contrasted with chronological number. Mathematical number contains in its very essence the notion of a _mechanical demarcation_, number being in that respect akin to _word_, which, in the very fact of its comprising and denoting, fences off world-impressions. The deepest depths, it is true, are here both incomprehensible and inexpressible. But the actual number with which the mathematician works, the figure, formula, sign, diagram, in short the _number-sign which he thinks, speaks or writes exactly_, is (like the exactly-used word) from the first a symbol of these depths, something imaginable, communicable, comprehensible to the inner and the outer eye, which can be accepted as representing the demarcation. The origin of numbers resembles that of the myth. Primitive man elevates indefinable nature-impressions (the “alien,” in our terminology) into deities, _numina_, at the same time capturing and impounding them by a _name_ which limits them. So also numbers are something that marks off and captures nature-impressions, and it is by means of names and numbers that the human understanding obtains power over the world. In the last analysis, the number-language of a mathematic and the grammar of a tongue are structurally alike. Logic is always a kind of mathematic and vice versa. Consequently, in all acts of the intellect germane to mathematical number—measuring, counting, drawing, weighing, arranging