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Aleister Crowley Diary Entry Friday, 27 January 1922
Eddington convinces me that I have an a priori idea of space independent of any actual measurements. His maximum sphere of water does not agree with this. I suggest that we being 4-dimensional beings of whom the universe is a 3-dimensional projection, the divergence between ideal or real is due to the imperfection of the phenomenal.
Climbed Low Man with Leah [Leah Hirsig] from the Old Church by the "Bridge Cavern" (climbed on its right), thence obliquely up to the "Upper Valley" where it is the flat-floored cave "Hermit's Hole". Climbed this on its left to hole in wall at angle of salient. All easy and safe.
Jane [Jane Wolfe] reports new U.S. Consul Mr. Nathan as bitterly hostile.
What line should be taken? [I Ching Hexagram] Sol of Fire 21 Shih Ho. "Union by gnawing".
There is surely a fallacy in the "New" Mathematics. Even if a straight line does not exist, but must be curved, we may still ask why some other line should not be what we once called straight. Also, this "maximum sphere". Let its diameter be p miles, not being a transfinite number, by definition, may be written as 1 plus 1 plus 1 plus — to p terms. But if p is a number, so is p plus 1. (Peano's axioms). We reached p by virtue of the primary right of arithmetic to proceed from a to a plus 1; to stop at p is to lop the branch we are sitting on.
When I visualize myself as the point where the 3 axes cut, I admit I find it hard to follow them into space beyond a certain point. Their straightness seems to falter curiously; I feel as if (despite my own definition) they may curve ultimately. I admit, therefore, that I may be aware of a kind of space which is non-Euclidean. But this merely tends to prove the coherence of my conceptions with actual conditions. I have still the right to make metaphysical postulates. E.g. a sphere-surface-being finds all his "straight lines" (defined as shortest distance between two points) curved in one way or another in reference to any given axis. (Japan to Canada—route curves North near Aleutian Islands, also upward from earth's centre.) But our own idea of a straight line is intact, and he too has a right to conceive it as best he can, though he can never have experience of it or even prove it to be possible.
Any geometrical form is determined, equally from within and from without. E.g. a triangle not only postulates its own shape, but that of the plane from which it is cut out. Eddington's "maximum sphere" which fills space entirely ("nothing can enter or leave it, there being nothing outside it"—I quote from memory) is not a sphere at all in any intelligible sense, to my mind. It does not comply with the definition of a sphere, as a sphere however slightly smaller would have to do. It seems arbitrary to distinguish it from any other form, such as a cube (in some corresponding geometry) etc etc.
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