Usefulness of math metaphors

Droll Dedekind, modified 4 Years ago.

Usefulness of math metaphors

Posts: 634 Join Date: 11/15/13 Recent Posts
I split this from another thread so I can argue with bernd.

But, I guess the thread can work for talking in general about math metaphors.
Except sometimes limits do give surprising results and my statement about a change in character is true.
How so? Care to give an example?

Sure. So, before the notion of a limit, differential/integral calculus wasn't on a rigorous foundation. It was done using intuitive notions of "vanishingly small quantities", 'infinity', etc. How one could rigorously do operations with infinite/infinitesimal and end up with finite results wasn't understood. While the non-technical definition of a limit doesn't seem like much it was actually a major historical development and is now used to define: continuity, convergence of sequences, differentiation, integration, etc.

For an example of a non-obvious limit, consider
lim x --> 0 of sin(x)/x = 1.

One is certainly able to come up with intuitive ways of understanding it, but to say that the limit is just filling in our already existing intuition is silly.

lim x---> +inf of 1/x = 0.

While this one is intuitive it illustrates the point about change of character. If you look at the graph of 1/x it never reaches 0. The limit lets you reason (rigorously) about infinite closeness, passing from the always-positive value of 1/x, for positive x, to 0. I can give more examples if you'd like.

I should add that actually proving the value of limits rigorously typically isn't easy.

I'd like to make this thread more than a calculus lesson to bernd, so I thought of an early example of math metaphors being used to describe something 'spiritual'. In 1902 Crowley wrote an essay called Berashith where he uses a lot of math metaphors; it's an interesting essay, I don't necessarily agree with any of it. Note that in what follows the file should be showing exponentiation, so for example 00 should be 0^0, Xt+s+b+h+h should be X^(t+s+b+h+h), etc.
                       AN ESSAY IN ONTOLOGY
              With Some Remarks On Ceremonial Magic
         Man, of a daring nature, thou subtle production!
     Thou wilt not comprehend it, as when understanding some
                          common thing.
                      -Oracles Of Zoroaster
   In presenting this theory of the Universe to the world, I
   have but one hope of making any profound impression, viz. -
   that my theory has the merit of explaining the divergences
   between three great forms of religion now existing in the
   world - Buddhism, Hinduism, and Christianity, and of
   adapting them to ontological science by conclusions not
   mystical but mathematical. Of Mohammedanism I shall not now
   treat, as, in whatever light we may decide to regard it (and
   its esoteric schools are often orthodox), in any case it
   must fall under one of the three heads of Nihilism,
   Advaitism, and Dvaitism.
   Taking the ordinary hypothesis of the universe, that of its
   infinity, or at any rate that of the infinity of God, or of
   the infinity of some substance or idea actually existing, we
   first come to the question of the possibility of the
   co-existence of God and man.
   The Christians, in the category of the existent, enumerate
   among other things, whose consideration we may discard for
   the purposes of this argument, God, an infinite being; man;
   Satan presumably, finite beings. These are not aspects of
   one being, but separate and even antagonistic existences.
   All are equally real: we cannot accept mystics of the type
   of Caird as being orthodox exponents of the religion of
   The Hinus enumerate Brahm, infinite in all dimensions and
   directions - indistinguishable from the Pleroma of the
   Gnostics - and Maya, illusion. This is in a sense the
   anithesis of noumenon and phenomenon, noumenon being negated
   of all predicates until it becomes almost extinguished in
   the Nichts under the title of Alles. (Cf. Max Muller on the
   metaphysical Nirvana, in his Dhammapada, Introductory
   Essay.) The buddhists express no opinion.
   Let us consider the force-quality in the existences
   conceived of by these two religions respectively,
   remembering that the God of the Christians is infinite, and
   yet discussing the alternative if we could suppose him to be
   a finite God. In any equilibrated system of forces, we may
   sum and represent them as a triangle or a series of
   triangles which again resolve into one. In any moving
   system, if the resultant motion be applied in a contrary
   direction, the equilibrium can also be thus represented. And
   if any one of the original forces in such a system may be
   considered, that one is equal to the resultant of remainder.
   Let x , the purpose of the universe be the resultant of the
   forces G, S, and M (God, Satan, and Man). Then of our forces
   as M is also the resultant of G, S and - x. So that we can
   regard either the supreme, and there is no reason for
   worshipping one rather than another. All are finite. This
   argument the Christians clearly see: hence the development
   of God from the petty joss of Genesis to the intangible, but
   self-contradictory spectre of today. But if other forces can
   G be infinite, the have no possible effect on it. As Whewell
   says, in the strange accident by which he anticipates the
   metre of however fine, into In Memoriam : "No force on
   earth, however great, can stretch a cord, a horizontal line
   that shall be absolutely straight."
   The definition of God as infinite therefore denies man
   implicitly; while if he be finite there is an end of the
   usual Christian reasons for worship, though I daresay I
   could myself discover some reasonably good ones. [I hardly
   expect to be asked somehow.]
   The resulting equilibrium of God and man, destructive of
   worship, is of course absurd. We must reject it, unless we
   want to fall into Positivism, Materialism, or something of
   the sort. But if, then, we call God infinite, how are we to
   regard man and Satan? (the latter, at the very least, surely
   no integral part of him). The fallacy lies not in my
   demonstration (which is also that of orthodoxy) that a
   finite God is absurd, but in the assumption that man has any
   real force.(1)
   In our mechanical system (as I have hinted above), if one of
   the forces be infinite, the others, however great, are both
   relatively and absolutely nothing.
   In any category, infinity excludes finity, unless that
   finity be an identical part of that infinity.
   In the category of existing things, space being infinite,
   for on that hypothesis we are still working, either matter
   fills or does not fill it. If the former, matter is
   infinitely great; if the latter, infinitely small. Whether
   the matter-universe be 1010000 light-years in diameter or
   half a mile makes no difference; it is infinitely small - in
   effect, Nothing. The unmathematical illusion that it does
   exist is what the Hindus call Maya.
   If, on the other hand, the matter-universe is infinite,
   Brahm and God are crowded out, and the possibility of
   religion is equally excluded.
   We may now shift our objective. The Hindus cannot account
   intelligibly, though they try hard, for Maya, the cause of
   all suffering. Their position is radically weak, but at
   least we may say for them that they have tried to square
   their religion with their common sense. The Christians, on
   the other hand, though they saw whither, the Manichean
   Heresy(2) must lead, and crushed it, have not officially
   admitted the precisely similar conclusion with regard to
   man, and denied the existence of the human soul as distinct
   from the divine soul.
   Trismegistus, Iamblicus, Porphyry, Boehme, and the mystics
   generally have of course substantially done so, though
   occasionally with rather inexplicable reservations, similar
   to those made in some cases by the Vedantists themselves.
   Man then being disproved, God the person disappears forever,
   and becomes Atman, Pleroma, Ain Soph, what name you will,
   infinite in all directions and in all categories - to deny
   one is to destroy the entire argument and throw us back to
   our old Dvaitistic bases.
   I entirely sympathize with my unhappy friend Rev. Mansel,
   B.D.(3) in his piteous and pitiful plaints against the
   logical results of the Advaitist School. But on his basal
   hypothesis of an infinite God, infinite space, time, and so
   on, no other conclusion is possible. Dean Mansel is found in
   the impossible position of one who will neither give up his
   premises nor dispute the validity of logical processes, but
   who shrinks in horror from the inevitable conclusion; he
   supposes there must be something wrong somewhere, and
   concludes that the sole use of reason is to discover its own
   inferiority to faith. As Deussen(4) well points out, faith
   in the Christian sense merely amounts to being convinced on
   insufficient grounds.(5)
   This is surely the last refuge of incompetence.
   But though, always on the original hypothesis of the
   infinity of space, &c., the Advaitist position of the
   Vedantists and the great Germans is unassailable, yet on
   practical grounds the Dvaitists have all the advantage.
   Fichte and others exhaust themselves trying to turn the
   simple and obvious position that: "If the Ego alone exists,
   where is any place, not only for morals and religion, which
   we can very well do without, but for the most essential and
   continuous acts of life? Why should an infinite Ego fill a
   non-existent body with imaginary food cooked in thought only
   over an illusionary fire by a cook who is not there? Why
   should infinite power use such finite means, and very often
   fail even then?"
   What is the sum total of the Vedantist position? "'I' am an
   illusion, externally. In reality the true 'I' am the
   Infinite, and if the illusionary 'I' could only realize Who
   'I' really am, how very happy we should all be!" And here we
   have Karma, rebirth, all the mighty laws of nature operating
   nowhere in nothing!
   There is no room for worship or morality in the Advaitist
   system. All the specious pleas of the Bhagavad-Gita, and the
   ethical works of Western Advaitist philosophers, are more or
   less consciously confusion of thought. But no subtlety can
   turn the practical argument; the grinning mouths of the
   Dvaitist guns keep the fort of Ethics, and warn metaphysics
   to keep off the rather green grass of religion.
   That its apologists should have devoted so much time,
   thought, scholarship, and ingenuity to this question is the
   best proof of the fatuity of the Advaitist position.
   There is then a flaw somewhere. I boldly take up the glove
   against all previous wisdom, revert to the most elementary
   ideas of cannibal savages, challenge all the most vital
   premises and axiomata that have passed current coin with
   philosophy for centuries, and present my theory.
   I clearly foresee the one difficulty, and will discuss it in
   advance. If my conclusions on this point are not accepted,
   we may at once get back to our previous irritable
   agnosticism, and look for our Messiah elsewhere. But if we
   can see together on this one point, I think things will go
   fairly smoothly afterwards.
   Consider(6) Darkness! Can we philosophically or actually
   regard as different the darkness produced by interference of
   light and that existing in the mere absence of light?
   Is Unity really identical with .9 recurring?
   Do we not mean different things when we speak respectively
   of 2 sine 60 and of [bild.gif] ?
   Charcoal and diamond are obviously different in the
   categories of color, crystallization, hardness, and so on;
   but are they not really so even in that of existence?
   The third example is to my mind the best. 2 sine 60 and of
   [bild.gif] are unreal and therefore never conceivable, at
   least to the present constitution of our human
   intelligences. Worked out, neither has meaning; unworked,
   both have meaning, and that a different meaning in one case
   and the other.
   We have thus two terms, both unreal, both inconceivable, yet
   both representing intelligible and diverse ideas to our
   minds (and this is the point!) though identical in reality
   and convertible by a process of reason which simulates or
   replaces that apprehension which we can never (one may
   suppose) attain to.
   Let us apply this idea to the Beginning of all things, about
   which the Christians lie frankly, the Hindus prevaricate,
   and the Buddhists are discreetly silent, while not
   contradicting even the gross and ridiculous accounts of the
   more fantastic Hindu visionaries.
   The Qabalists explain the "First Cause"(7) by the phrase:
   "From 0 to 1, as the circle opening out into the line." The
   Christian dogma is really identical, for both conceive of a
   previous and eternally existing God, though the Qabalists
   hedge by describing this latent Deity as "Not." Later
   commentators, notably the illustrious(8) MacGregor Mathers
   have explained this Not as "negatively-existing." Profound
   as is my respect for the intellectual and spiritual
   attainments of him whom I am proud to have been permitted to
   call my master.(9)
   I am bound to express my view that when the Qabalists said
   Not, they meant Not, and nothing else. In fact, I really do
   claim to have re-discovered the long-lost and central
   Arcanum of those divine philosophers.
   I have no serious objection to a finite god, or gods,
   distinct from men and things. In fact, personally, I believe
   in them all, and admit them to possess inconceivable though
   not infinite power.
   The Buddhists admit the existence of Maha-Brahma, but his
   power and knowledge are limited; and his agelong day must
   end. I find evidence everywhere, even in our garbled and
   mutilated version of the Herew Scriptures, that Jehovah's
   power was limited in all sorts of ways. At the Fall, for
   instance, Tetragrammaton Elohim has to summon his angels
   hastily to guard the Tree of Life, lest he should be proved
   a liar. For had it occurred to Adam to of that Tree before
   their transgression was discovered, or had the Serpent been
   aware of its properties, Adam would indeed have lived and
   not died. So that a mere accident saved the remnants of the
   already besmirched reputation of the Hebrew tribal Fetich.
   When Buddha was asked how things came to be, he took refuge
   in silence, which his disciples very conveniently
   interpreted as meaning that the question tended not to
   edification. I take it that the Buddha (ignorant, doubtless,
   of algebra) had sufficiently studied philosophy and
   possessed enough worldly wisdom to be well aware that any
   system he might promulgate would be instantly attacked and
   annihilated by the acumen of his numerous and versatile
   Such teaching as he gave on the point may be summed up as
   follows: "Whence whither, why, we know not; but we do know
   that we are here, that we dislike being here, that there is
   a way out of the whole loathsome affair - let us make haste
   and take it!"
   I am not so retiring in disposition; I persist in my
   inquiries, and at the last the appalling question is
   answered, and the past ceases to intrude its problems on my
   Here you are! Three shies a penny! Change all bad arguments.
   When we say that the cosmos sprang from the 0, what kind of
   0 do we mean? By 0 in the ordinary sense of the term we mean
   "absence of extension in any of the categories."
   When I say "No cat has two tails," I do not mean as the old
   fallacy runs, that "absence of cat possesses two tails"; but
   that "In the category of two-tailed things, there is no
   extension of cat."
   Nothingness is that about which no positive proposition is
   valid. We cannot truly affirm: "Nothingness is green, or
   heavy, or sweet."
   Let us call time, space, being, heaviness, hunger, the
   If a man be heavy and hungry, he is extended in all these,
   besides, of course, many more. But let us suppose that these
   five are all. Call the man X; his formula is then
   Xt+s+b+h+h. If he now eat, he will cease to be extended in
   hunger; if he be cut off from time and gravitation as well,
   he will now be represented by the formula Xs+b. Should he
   cease to occupy space and to exist, his formula would then
   be X0. This expression is equal to 1; whatever X may
   represent, if it be raised to the power of 0 (this meaning
   mathematically "if it be extended in no dimension or
   category"), the result is Unity, and the unknown factor X is
   This is the Advaitist idea of the future of man; his
   personality, bereft of all its qualities, disappears and is
   lost, while in its place arises the impersonal Unity, The
   Pleroma, Parabrahma, or the Allah of the Unity-adoring
   followers of Mohammed. (To the Muslim fakir, Allah is by no
   means a personal God.)
   Unity is thus unaffected, whether or no it be extended in
   any of the categories. But we have already agreed to look to
   0 for the Uncaused.
   Now if there was in truth 0 "before the beginning of years,"
   our 0 was the ordinary 0 of mathematics, there was not truly
   absolute 0, for 0 is, as I have shown, dependent on the idea
   of categories. If these existed, then the whole question is
   merely thrown back; we must reach a state in which the 0 is
   absolute. Not only must we get rid of all subjects, but of
   all predicates. By 0 (in mathematics) we really mean 0n,
   where n is the final term of a natural scale of dimensions,
   categories, or predicates. Our Cosmic Egg, then, from which
   the present universe arose, was Nothingness, extended in no
   categories, or, graphically, 00. This expression is in its
   present form meaningless. Let us discover its value by a
   simple mathematical process!
   Now the multiplying of the infinitely great by the
   infinitely small results in SOME UNKNOWN FINITE NUMBER
   when this our Great inversion took place, from the essence
   of all nothingness to finity extended in innumerable
   categories, that an incalculably vast system was produced.
   Merely by chance, chance in the truest sense of the term, we
   are found with gods, men, stars, planets, devils, colors,
   forces, and all the materials of the Cosmos: and with time,
   space, and causality, the conditions limiting and involving
   them all.(11)
   Remember that it is not true to say that our 00 existed; nor
   that it did not exist. The idea of existence was just as
   much unformulated as that of toasted cheese.
   But 00 is a finite expression, or has a finite phase, and
   our universe is a finite universe; its categories are
   themselves finite, and the expression "infinite space" is a
   contradiction in terms. The idea of an absolute and of an
   infinite(12) God is relegated to the limbo of all similar
   idle and pernicious perversions of truth. Infinity remains,
   but only as a mathematical conception as impossible in
   nature as the square root of -I. Against all this
   mathematical, or semi-mathematical, reasoning, it may
   doubtless be objected that our whole system of numbers, and
   of manipulating them is merely a series of conventions. When
   I say that the square root of three is unreal, I know quite
   well that it is only so in relation to the series 1, 2, 3,
   &c., and that this series is equally unreal if I make
   [bild3.gif] , the members of the ternary scale. But this,
   theoretically true, is practically absurd. If I mean "the
   number of a, b, and c," it does not matter if I write 3 or
   [bild3.gif] ; the idea is a definite one; and it is the
   fundamental ideas of consciousness of which we are treating,
   and to which we are compelled to refer everything, whether
   proximately or ultimately.
   So also my equation, fantastic as it may seem, has a perfect
   and absolute parallel in logic. Thus: let us convert twice
   the proposition "some books are on the table." By negativing
   both terms we get "Absence-of-book is not on the table,"
   which is precisely my equation backwards, and a thinkable
   thing. To reverse the process, what do I mean when I say
   "some pigs, but not the black pig are not in the sty"? I
   imply that the black pig is in the sty. All I have done is
   to represent the conversion as a change, rather than as
   merely another way of expressing the same thing. And
   "change" is really not my meaning either; for change, to our
   minds, involves the idea of time. But the whole thing is
   inconceivable  to ratiocination, though not to thought. Note
   well too that if I say "Absence-of-books is not on the
   table," I cannot convert it into "All books are on the
   table" but only to "some books are on the table." The
   proposition is an "I" and not an "A" proposition. It is the
   Advaita blunder to make it so; and many a schoolboy has fed
   off the mantelpiece for less.
   There is yet another proof- the proof by exclusion. I have
   shown, and metaphysicians practically admit, the falsity
   alike of Dvaitism and Advaitism. The third, the only
   remaining theory, this theory, must, however antecedently
   improbable, however difficult to assimilate, be true.(13)
   "My friend, my young friend," I think I hear some Christian
   cleric say, with an air of profound wisdom, not untinged
   with pity, condescending to pose beardless and brainless
   impertinence" "Where is the Cause for this truly remarkable
   That is exactly where the teory rears to heaven its stoutest
   bastion! There is not, and could not be, any cause. Had 00
   been extended in causality, no change could have taken
   Here, then, are we, finite beings in a finite universe,
   time, space, and causality themselves finite (inconceivable
   as it may seem) with our individuality, and all the
   "illusions" of the Advaitists, just as real as they
   practically are to our normal consciousness.
   As Schopenhauer, following Buddha, points out, suffering is
   a necessary condition of this existence.(15)
   The war of the contending forces as they grind themselves
   down to the final resultant must cause endless agony. We may
   one day be able to transform the categories of emotion as
   certainly and easily as we now transfrom the categories of
   force, so that in a few years Chicago may be importing
   suffering in the raw state and turning it into tinned
   salmon: but at present the reverse process is alone
   How, then, shall we escape? Can we expect the entire
   universe to resolve itself back into the phase of 00? Surely
   not. In the first place, there is no reason why the whole
   should do so; [bild4.gif] is just as convertible as x. But
   worse, the category of causality has been formed, and its
   inertia is sufficient to oppose a most serious
   stumbling-block to so gigantic a process.
   The task before us is consequently of a terrible nature. It
   is easy to let things slide, to grin and bear it in fact,
   until everything is merged in the ultimate unity, which may
   or may not be decently tolerable. But while we wait?
   There now arises the question of freewill. Causality is
   probably not fully extended in its own category,(16) a
   circumstance which gives room for a fractional amount of
   freewill. If this not be so, it matters little; for if I
   find myself in a good state, that merely proves that my
   destiny took me there. We are, as Herbert Spencer observes,
   self-deluded with the idea of freewill; but if this be so,
   nothing matters at all. If, however, Herbert Spencer is
   mistaken (unlikely as it must appear), then our reason is
   valid, and we should seek out the right path and pursue it.
   The question therefore not trouble us at all.
   Here then we see the use of morals and of religion, and all
   the rest of the bag of tricks. All these are methods, bad or
   good, for extricating ourselves from the universe.
   Closely connected with this question is that of the will of
   God. People argue that an Infinite intelligence must have
   been at work on this cosmos. I reply No! There is no
   intelligence at work worthy of the name. The Laws of Nature
   may be generalised in one - the Law of Inertia. Everything
   moves in the direction determined by the path of least
   resistance; species arise, develop, and die as their
   collective inertia determines; to this Law there is no
   exception but the doubtful one of freewill; the Law of
   Destiny itself is formally and really identical with it.(17)
   As to an infinite intelligence, all philosophers of any
   standing are agreed that all-love and all-power are
   incompatible. The existence of the universe is a standing
   proof of this.
   The Deist need the optimist to keep him company; over the
   firesides all goes well, but it is a sad shipwreck they
   suffer on emerging into the cold world.
   This is why those who seek to buttress up religion are so
   anxious to prove that the universe has no real existence, or
   only a temporary and a relatively unimportant one; the
   result is of course the usual self-destructive Advaitist
   The precepts of morality and religion are thus of use, of
   vital use to us, in restraining the more violent forces
   alike of nature and of man. For unless law and order
   prevail, we have not the necessary quiet and resources for
   investigating, and learning to bring under our control, all
   the divergent phenomena of our prison, a work which we
   undertake that at last we may be able to break down the
   walls, and find that freedom which an inconsiderate
   Inversion has denied.
   The mystical precepts of pseudo-Zoroaster, Buddha,
   Cankaracharya, pseudo-Christ and the rest, are for advanced
   students only, for direct attack on the problem. Our
   servants, the soldiers, lawyers, all forms of government,
   make this our nobler work possible, and it is the gravest
   possible mistake to sneer at these humble but faithful
   followers of the great minds of the world.
   What, then, are the best, easiest, directest methods to
   attain our result? And how shall we, in mortal language,
   convey to the minds of others the nature of a result so
   beyond language, baffling even imagination eagle-pinioned?
   It may help us if we endeavour to outline the distinction
   between the Hindu and Buddhist methods and aims of the Great
   The Hindu method is really mystical in the truest sense;
   for, as I have shown, the Atman is not infinite and eternal;
   one day it must sink down with the other forces. But by
   creating in thought an infinite Impersonal Personality, by
   defining it as such, all religions except the Buddhist and
   I believe the Qabalistic, have sought to annihilate their
   own personality. The Buddhist aims directly at extinction;
   the Hindu denies and abolishes his own finity by the
   creation of an absolute.
   As this cannot be done in reality, the process is illusory;
   yet it is useful in the early stages - as far, at any rate,
   as the fourth stage of Dhyana, where the Buddha places it,
   though the yogis claim to attain to Nirvikalpa-Samadhi, and
   that Moksha is identical with Nirvana; the former claim I
   see no reason to deny them; the latter statement I must
   decline at present to accept.
   The task of the Buddhist recluse is roughly as follows. He
   must plunge every particle of his being into one idea: right
   views, aspirations, word, deed, life, will-power,
   meditation, rapture, such are the stages of his liberation,
   which resolves itself into a struggle against the law of
   causality. He cannot prevent past causes taking effect, but
   he can prevent present causes from having any future
   results. The exoteric Christian and Hindu rather rely on
   another person to do this for them, and are further blinded
   by the thirst for life and individual existence, the most
   formidable obstacle of all, in fact a negation of the very
   object of all religion. Schopenhauer shows that life is
   assured to the will-to-live, and unless Christ (or Krishna,
   as the case may be) destroys these folk by superior power -
   a task from which almightiness might well recoil baffled! -
   I much fear that eternal life, and consequently eternal
   suffering, joy, and change of all kinds, will be their
   melancholy fate. Such persons are in truth their own real
   enemies. Many of them, however, believing erroneously that
   they are being "unselfish," do fill their hearts with
   devotion for the beloved Saviour, and this process is, in
   its ultimation, so similar to the earlier stages of the
   Great Work itself, that some confusion has, stupidly enough,
   arisen; but for all that the practice has been the means of
   bringing some devotees on the true Path of the Wise,
   unpromising as such material must sound to intelligent ears.
   The esoteric Christian or Hindu adopts a middle path. Having
   projected the Absolute from his mind, he endeavours to unite
   his consciousness with that of his Absolute, and of course
   his personality is destroyed in the process. Yet it is to be
   feared that such an adept too often starts on the path with
   the hideous idea of aggrandising his own personality to the
   utmost. But his method is so near to the true one that this
   tendency is soon corrected, as it were automatically.
   (The mathematical analogue of this process is to procure for
   yourself the realisation of the nothingness of yourself by
   keeping the fourth dimension ever present in your mind.)
   The illusory nature of this idea of an infinite Atman is
   well shown by the very proof which that most distinguished
   Vedantist, the late Swami Vivekananda (no connection with
   the firm of a similar name(18) across the street), gives of
   the existence of the infinite. "Think of a circle!" says he.
   "You will in a moment become conscious of an infinite circle
   around your original small one." The fallacy is obvious. The
   big circle is not infinite at all, but is itself limited by
   the little one. But to take away the little circle, that is
   the method of the Esoteric Christian or the mystic. But the
   process is never perfect, because however small the little
   circle becomes, its relation with the big circle is still
   finite. But even allowing for a moment that the absolute is
   is really attainable, is the nothingness of the finity
   related to it really identical with that attained directly
   by the Buddhist Arahat? This, consistently with my former
   attitude, I feel constrained to deny. The consciousness of
   the Absolute-wala(19) is really extended infinitely rather
   than diminished infinitely, as he will himself assure you.
   True, Hegel says: "Pure being is pure nothing!" and it is
   true that the infinite heat and cold, joy and sorrow, light
   and darkness, and all the other pairs of opposites,(20)
   cancel one another out: yet I feel rather afrid of the
   Absolute! Maybe its joy and sorrow are represented in
   phases, just as 00 and finity are phases of an identical
   expression, and I have an even chance only of being on the
   right side of the fence!
   The Buddhist leaves no chance of this kind; in all his
   categories he is infinitely unextended; though the
   categories themselves exist; he is in fact 0A+B+C+D+E+..+N
   and capable of no conceivable change, unless we imagine
   nirvana to be incomprehensibly divided by Nirvana, which
   would (supposing the two Nirvanas to possess identical
   categories) result in the production of the original 00. But
   a further change would be necessary even then before serious
   mischief could result. In shirt, I thing we may dismiss from
   our minds any alarm in respect of this contingency.
   On mature consideration, therefore, I confidently and
   deliberately take my refuge in the Triple Gem.
   Namo Tasso Bhagavato Arahato Samma-sambuddhasa!(21)
   Let there be hereafter no discussion of the classical
   problems of philosophy and religion! In the light of this
   exposition the antitheses of noumenon and phenomenon, unity
   and multiplicity, and their kind, are all reconciled, and
   the only question that remains os that of finding the most
   satisfactory means of attaining Nirvana - extinction of all
   that exists, knows, or feels; extinction final and complete,
   utter and absolute extinction. For by these words only can
   we indicate Nirvana: a state which transcends thought cannot
   be described in thought's language. But from the point of
   view of thought extinction is complete: we have no data for
   discussing that which is unthinkable, and must decline to do
   so. This is the answer to those who accuse the Buddha of
   hurling his Arahats (and himself) from Samma Samadhi to
   Pray observe in the first place that my solution of the
   Great Problem permits the co-existence of an infinite number
   of means: they need not even be compatible; Karma, rebirth,
   Providence, prayer, sacrifice, baptism, there is room for
   all. On the old and, I hope, now finally discredited
   hypothesis of an infinite being, the supporters of these
   various ideas, while explicitly affirming them, implicitly
   denied. Similarly, note that the Qabalistic idea of a
   Supreme God (and innumerable hierarchies) is quite
   compatible with this theory, provided that the Supreme God
   is not infinite.
   Now as to our weapons. The more advanced Yogis of the East,
   like the Nonconformists at home, have practically abandoned
   ceremonial as idle. I have yet to learn, however, by what
   disenters have replaced it! I take this to be an error,
   except in the case of the very advanced Yogi. For there
   exists a true magical ceremonial, vital and direct, whose
   purpose has, however, at any rate of recent times, been
   hopelessly misunderstood.
   Nobody any longer supposes that any means but that of
   meditation is of avail to grasp the immediate causes of our
   being; if some person retort that he prefers to rely on a
   Glorified Redeemer, I simply answer that he is the very
   nobody to whom I now refer.
   Meditation is then the means; but only the supreme means.
   The agony column of the Times is the supreme means of
   meeting with the gentleman in the brown billycock and frock
   coat, wearing a green tie and chewing a straw, who was at
   the soiree of the Carlton Club last Monday night; no doubt!
   But this means is seldom or never used in the similar
   contingency of a cow-elephant desiring her bull in the
   jungles of Ceylon.
   Meditation is not within the reach of every one; not all
   possess the ability; very few indeed (in the West at least)
   have the opportunity.
   In any case what the Eastern calls "one-pointedness" is an
   essential preliminary to even early stages of true
   meditation. And iron will-power is a still earlier
   By meditation I do not mean merely "think about" anything,
   however profoundly, but the absolute restraint of the mind
   to the contemplation of a single object, whether gross,
   fine, or altogether spiritual.
   Now true magical ceremonial is entirely directed to attain
   this end, and forms a magnificent gymnasium for those who
   are not already finished mental athletes. By act, word, and
   thought, both in quantity and quality, the one object of the
   ceremony is being constantly indicated. Every fumigation,
   purification, banishing, invocation, evocation, is chiefly a
   reminder of the single purpose, until the supreme moment
   arrives, and every fibre of the body, every force-channel of
   the mind, is strained out in one overwhelming rush of the
   Will in the direction desired. Such is the real purport of
   all the apparently fantastic directions of Solomon,
   Abramelin, and other sages of repute. When a man has evoked
   and mastered such forces as Taphtatharath, Belial, Amaimon,
   and the great powers of the elements, then he may safely be
   permitted to begin to try to stop thinking. For needless to
   say, the universe, including the thinker, exists only by
   virtue of the thinker's thought.(22)
   In yet one other way is magic a capital training ground for
   the Arahat. True symbols do really awake those macrocosmic
   forces of which they are the eidola, and it is possible in
   this manner very largely to increase the magical
   "potential," to borrow a term from electrical science.
   Of course, there are bad and invalid processes, which tend
   rather to dispense or to excite the mind-stuff than to
   control it; these we must discard. But there is a true
   magical ceremonial, the central Arcanum alike of Eastern and
   Western practical transcendentalism. Needless to observe, if
   I knew it, I should not disclose it.
   I therefore definitely affirm the validity of the Qabalistic
   tradition in its practical part as well as in those exalted
   regions of thought through which we have so recently, and so
   hardly, travelled.
   Eight are the limbs of Yoga: morality and virtue, control of
   body, thought, and force, leading to concentration,
   meditation, and rapture.
   Only when the last of these has been attained, and itself
   refined upon by removing the gross and even the fine objects
   of its sphere, can the causes, subtle and coarse, the unborn
   causes whose seed is hardly sown, of continued existence be
   grasped and annihilated, so that the Arahat is sure of being
   abolished in the utter extinction of Nirvana, while even the
   world of pain, where he must remain until the ancient
   causes, those which have already germinated, are utterly
   worked out (for even the Buddha himself could not swing back
   the Wheel of the Law), his certain anticipation of the
   approach of Nirvana is so intense as to bathe him constantly
   in the unfathomable ocean of the apprehension of immediate
                       AUM MANI PADME HOUM

   1. Lully, Descartes, Spinoza, Schelling. See their works.
   2. The conception of Satan as a positive evil force; the
   lower triangle of the Hexagram.
   3. Encyclopedia Britannica, Art. Metaphysics.
   4. "The Principles of Metaphysics" Macmillan.
   5. Or as the Sunday-school boy said: "Faith is the power of
   believing what we know to be untrue." I quote Deussen with
   the more pleasure, because it is about the only sentence in
   all his writings with which I am in accord. - A.C.
   6. Ratiocination may perhaps not take us far. But a
   continuous and attentive study of these quaint points of
   distinction may give us an intuition, or direct
   mind-apperception of what we want, one way or the other -
   7. An expression they carefully avoid using.
   8. I retain this sly joke from the first edition.
   9. See 8.
   10. I cannot here discuss the propriety of representing the
   categories as dimensions. It will be obvious to any student
   of the integral calculus, or to anyone who appreciates the
   geometrical significance of the term x4.
   11. Compare and contrast this doctrine with that of Herbert
   Spencer ("First Principles," Pt. I) and see my "Science and
   Buddhism" for a full discussion of the difference involved -
   12. If by "infinitely great" we only mean "indefinitely
   great," as a mathematician would perhaps tell us, we of
   course begin at the very point I am aiming at, viz. Ecrasez
   l'Infini. - A.C.
   13. I may remark that the distinction between this theory
   and the normal one of the Immanence of the Universe, is
   trivial, perhaps even verbal only. Its advantage, however,
   is that, by hypostatising nothing, we avoid the necessity of
   any explanation. How did nothing come to be? is a question
   which requires no answer.
   14. See the Questions of King Milinda, vol. ii. P103.
   15. See also Huxley, "Evolution and Ethics."
   16. Causality itself a secondary, and in its limitation as
   applied to volition, an inconceivable idea. H. Spencer, op.
   cit. This consideration alone should add great weight to the
   agnostic and a fortiori to the Buddhist position.
   17. See H. Spencer, "First Principles," "The Knowable," for
   a fair summary of the facts underlying this generalisation;
   which indeed he comes within an ace of making in so many
   words. It may be observed that this law is nearly if not
   quite axiomatic, its contrary being enormously difficult if
   not impossible to formulate mentally.
   18. The Swami Vive Ananda, Madame Horos, for whose history
   consult the Criminal Law Reports.
   19. Wala, one whose business is connected with anything.
   E.g. Jangli-wala, one who lives in or has business with, a
   jungle, i.e. a wild man, or a Forest Conservator.
   20. The Hindus see this as well as any one, and call Atman
   Sat-chit-ananda, these being above the pairs of opposites,
   rather on the Hegelian lines of the reconciliation (rather
   than the identity) of opposites, in a master idea. We have
   dismissed infinity as the figment of a morbid mathematic:
   but in any case the disproof applies to it as to God. - A.C.
   21. Hail unto Thee, the Blessed One, the Perfect One, the
   Enlightened One!
   22. See Berkeley and his expounders, for the Western shape
   of this Eastern commonplace. Huxley, however, curiously
   enough, states the fact almost in these words. - A.C.
   23. A possible mystic transfiguration of the Vedanta system
   has been suggested to me on the lines of Syllogism-
   God = Being (Patanjali)
   Being = Nothing (hegel)
   God = Nothing
   Or, in the language of religion:
   Every one may admit that monotheism, exalted by the
   introduction of the symbol, is equivalent to pantheism.
   Pantheism and atheism are really identical, as the oponents
   of both are the first to admit.
   If this be really taught, I must tender my apologies, for
   the reconcilement is of course complete. - A.C.
   (This note (23) is not attached to any of the original text,
   but appears as the last footnote in the work. It is included
   here for completeness. - M.W.)