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Abstract. In this paper, a magickal system based upon the principles of

mathematics and logic is outlined. In particular, we draw upon set theory,

domain theory, first-order predicate calculus and the theory of Galois connec-

tions in order to systematize a method for encoding a wide variety of magickal

operations in exact terms.

No assumption of a background in mathematics is assumed, though back-

ground study may be helpful.

1. Introduction

At first, it might appear that mathematics and magick are far apart. Certainly,it seems quite rare to encounter people who are practitioners of both, though thiswas certainly not always the case. Sir Isaac Newton was a noted alchemist, Dr.John Dee was a mathematician and cryptographer as well as perhaps one of themost notable magicians of all time. Famously, the Enochian system of Dee andKelley very much shows itself to have mathematical roots.

This paper outlines an initial attempt to codify magick mathematically. Weborrow heavily both from existing mathematical systems, particularly set theoryand first order predicate calculus, and also from several magickal systems, includinglore from both witchcraft and ceremonial magick. We call this system Logick,

because in some sense mathematically it is a logic, but we add a trailing k todifferentiate our approach from logic in the same way that magick differentiatesitself from magic.

2. Assumptions

Here, we state our assumptions. All mathematical systems rest upon their ax-ioms, and Logick is no different to that. We will approach this mathematically indue course. However, in our case, we need to also define our magickal assumptions.

2.1. What is magick? Put broadly, following Crowley we define magick to be theart of causation of change in accordance with will. All acts are magickal acts. Ifone wills a cup of coffee into existence, this may be adequately arranged by themagickal act of leaving the house, travelling to a good coffee shop, and ordering a

latte. Magickal acts may, of course, also be less mundane, but the point here is thatwe explicitly include the mundane. However, it is useful to be able to differentiatethe mundane from the non-mundane; the previous example of going to buy coffeeis referred to here as deterministic magick, where a series of events are clearlylinked by corresponding, mundane, causal connections. If however, one was to casta summoning spell, then have a friend randomly arrive shortly thereafter with

Date: May 28, 2009.

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coffee, this is referred to as nondeterministic magick, because it implies a differentkind of causal connection that requires coincidence or serendipity. We will formallydefine these concepts later.

2.2. What is the mechanism underlying magick? Feri witchcraft, along witha number of other traditions, asserts the existence of three souls. Talker is the con-scious mind, the you that is consciously you, the you that is reading this sentence.Fetch is the animal self, the subconscious, the child self, which is not verbal, butincredibly powerful, always working in the background. Godself loosely equates tothe concept of Holy Guardian Angel (HGA) that is familiar to other branches ofmagick in some sense it may be thought of as being a part of oneself that is itselfdivine (i.e., a deity in its own right, or at least, a part of or aspect of deity).

Feri lore has it that, in order to perform magick, particularly of the nonmundanekind, it is necessary for talker to communicate its intent to fetch, which in turncarries out the operation, perhaps via godself. Usually, the connection betweentalker and fetch is inherently symbolic. All magickal systems have in commonthe requirement for a system of symbols that, once learned, become a means tocommunicate ones magickal intent. Logick is no exception to this, though in ourcase, we are building upon an extremely powerful and precise symbolic system takenfrom mathematics and logic, rather than depending upon allegory or visualization.

2.3. Deity. Not all magickal systems require a concept of deity, though most do,and Logick follows this. In common with many other systems, Logick has anassumption of immanent deity; that is, deity is not fundamentally somewhere else,rather, it is a part of and accessible from everywhere and at all points in time.Communication with deity, through invocation, evocation or mediumship, is as-sumed to be feasible. Logick has no pantheon of its own, so you are therefore freeto use whatever gods, goddesses, dmons, spirits or other entities in whatever wayyou feel appropriate.

2.4. Mathematics and Logic. We declare that all that is mathematically prov-able is usable as part of a magickal operation. Mathematical truths are implicitlytrue of all universes that support the axioms that underlie them; therefore, we bothacknowledge these truths and seek to use them for our purposes as we see fit. Thispaper does no more than scratch the surface of the enormous wealth that mathe-matics offers, though it does include enough to allow simple workings to be carriedout.

3. Spell Casting

In witchcraft, spells are frequently ornate, often beautiful, statements of intentthat traditionally are ended and enacted with the phrase, so mote it be.1 In theLogick

system, a spell is essentially a statement that is asserted to be true. Weuse the mathematical symbol as a shorthand way to state that something is true.For example,

2 + 2 = 4

declares the absolute truth of the equation 2+2 = 4. In some sense, this is analogousto incanting, two plus two equals four, so mote it be!

1Some prefer, so must it be, which is modern English usage, though we use the former.

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Alternatively, may be read as, it is the case that, though keeping in mind itsequivalence to, so mote it be, is perhaps better.

As in mathematics and logic, we may use variables to represent abstract ideas.For example, where A represents a true statement,

A

turns that statement into a spell, e.g., it would be equivalent to defining A in full,then saying, so mote it be.

We can define variables, or leave them undefined. We can partially or completelyconstrain their possible values. They need not be numbers, or truth values. InLogick, a variable might represent literally anything: it might be used to representa real number, an abstract idea, a person or even a god. By convention, we willuse capital letters A,B ,C ,.. . to represent truth values (i.e., an expression whosevalue may be true or false, capital Greek letters , , , . . . to represent deities,and lower case Roman letters p ,q ,r,. . . for everything else. Other conventions will

be introduced later, though these are the most important.

3.1. Logical Operators. Note here that we use the mathematical spelling of theword logical here, since what we are introducing is taken directly from mathematicallogic.

3.1.1. Negation. The operator negates the truth of the expression to which it isapplied. Trivially, false = true and true = false, though we may also use inexpressions, e.g.,

A

is, in effect, a spell that asserts the negation of A, e.g., depending on context, thatA is incorrect or that A may not occur.

3.1.2. And. The operator states the truth of both of its arguments, e.g.,

A B

is effectively asserting both A and B.It is allowable to string together multiple operators like so: A B C . . . ,

which asserts the truth of all of the arguments.

3.1.3. Or. The operator states that either or both arguments are true, e.g.,

A B

states that either, A is true and B is false, A is false and B is true, or that both A

and B are true.

3.1.4. Implication. Note that, for example, A B implies that A B. We denotethis as A B A B. As in classical logic, A B is equivalent to A B.

3.2. Quantifiers. It is useful to be able to say, for example, that there exists an xsuch that some specified criteria are true, or that for all possible y, something elseholds.

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3.2.1. Existential Quantifiers. The existential quantifier may be read as, thereexists. It is typically used with a variable and a truth-valued expression, e.g.,x . x x = 4 may be read as, there exists an x such that x x = 4. Since we knowthat 2 2 = 4, the existence of the case x = 2 makes the expression true.

Stating x . E should be read as, there exist one or more values of x that makethe expression E true. If you want to state instead that there exists exactly onecase that is true, you may use 1 in place of .

3.2.2. Universal Quantifiers. The universal quantifier may be read as, for all.Like , it is used with a variable and a truth-valued expression, e.g., x . E, andstates that, for all possible values of x, E is true.

3.3. Sets. Sets2 are in essence an incredibly simple idea, though they offer extraor-dinary power. Sets may be usefully thought of as more fundamental in mathematicsthan numbers. Understanding Logick absolutely requires a basic understandingof set theory in order for its imagery to make sense.

A set, simply put, is a collection of things. A simple way to describe a set is towrite a list describing its contents, and surround it in {} brackets. Sets follow afew rules, but they can be comprised of anything, even other sets. For example,

{dog, cat, rabbit}

is the set that contains dog and cat and rabbit.

{1, 2, 5, 7, 11}

is the set containing the numbers 1, 2, 5, 7 and 11.

3.3.1. Infinite Sets. Sets can be infinite. The set N = {0, 1, 2, 3, . . . } represents thenatural numbers3. The set Z = {. . . , 2, 1, 0, 1, 2, . . . } represents the integers.

It is important to note that, even though N and Z are infinite, they do notinclude or within themselves though and are limits of the sets, theyare strictly outside the sets.

3.3.2. Membership. Given a set, e.g., {1, 2, 3, 4}, we can say that 1 is a member ofthe set, as are 2, 3 and 4, though 0, 5, elephant and grape are not members.

The expression x S states that x is a member of the set S. Conversely, y Sstates that y is not a member of S. Indeed, y S is equivalent to writing (y S).

An important characteristic of sets is that, given a particular element, a seteither contains that element or does not. There is no concept of a set containing

two or more copies of an element

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Similarly, there is no concept of ordering withinsets5, so {1, 3, 2} is identical to {1, 2, 3}.

2not to be confused with the ancient Egyptian god Set3As is common practice in mathematical logic and computer science, we define N to include 0.4In mathematics, a set-like construction known as a bagprovides this capability, but since bags

can be defined in terms of sets, we dont concern ourselves with them here.5Orders can, of course, be defined explicitly more on that later.

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3.3.3. Set Comprehensions. Having to list all the members of a set is often tedious,and in the case of many kinds of very large or infinite set, it is inappropriate orimpossible to write their contents using the usual {. . . } notation. Set comprehen-sions are a simple way to define large or complex sets mathematically they canbe visualised as something like an existential quantifier that as a side effect keepsa list of all of the values that happen to be true. A set comprehension is denotedas follows:

{x E}

which may be read as, the set of all x where E is true.A shorthand way of defining {1, 2, 3, 4, 5, 6, 7, 8, 9} might be

{x x N 1 x 9}

Note that it is usually necessary to state that x (or some other variable) is drawnfrom a particular set, e.g., x N, so by convention this is usually written morecompactly and readably as follows:

{x N 1 x 9}.

3.3.4. Union. It is possible to create sets constructively from other sets. The setunion operator can be thought of as joining together a pair of sets, discardingany duplicates. For example, where a = {1, 2, 3} and b = {3, 5, 9}, the operatorcan be used to join these sets, i.e.,

a b = {1, 2, 3} {3, 5, 9} = {1, 2, 3, 5, 9}

We can actually define union in terms of set comprehension:

a b = {i i a i b}

though set union is encountered sufficiently often that introducing the operator is

beneficial.It is common usage to string together several operators to represent the union

of more than two sets, e.g.,

a b c . . .

3.3.5. Intersection. It is frequently useful to be able to define a set in terms of theintersection of two or more other sets. Intersection is denoted by the operator,e.g., a b denotes the set that contains the elements that are present in both a andb, and no others. Should a and b happen to be disjoint (containing no commonelements), the result is an empty set, denoted {} or as .

As with , it is possible to define in terms of set comprehension, e.g.,

a b = {i i a i b}.

3.3.6. Subsets and Supersets. A set p may be said to be a subset of set q if andonly if every element of p is also an element ofq. We denote this p q, and we candefine in terms of operators that weve already defined as follows:

a b = i a .i b.

Note that a = b implies that a b and b a.The set b may be said to be a proper subset of set c if b c and b c.

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3.3.7. Power Sets. A power set, denoted S, is the set of all subsets (including theempty set, which is a subset of all possible sets) of the set S. It can be defined interms of set comprehension as follows:

{s S s}

For example,

{1, 2, 3} = {, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}}

3.4. Tuples. A tuple binds together two or more pieces of information. The no-tation (a,b, egg, 5) is a 4 element tuple that comprises the values a, b, egg and 5.Though tuples superficially seem like sets, they are not sets. Position within thetuple is critical, so order is important, so (1, 2) is not equivalent to (2, 1).

4. Deity

Now that some basic set theory has been introduced, it is possible to approach

the concept of deity in mathematical terms. We define deity in terms of sets thisis perhaps surprising, but as mentioned previously, sets can contain anything, andfor our purposes, that includes gods, dmons, spirits, ourselves, and anything elsewe might encounter either physically or spiritually.

We define the set of all gods, goddesses, and other spiritual beings, includingourselves, to be . Where g is a god, it follows that g by definition. Byconvention, we denote specific gods either with their own symbols, or with thenotation

god name

e.g., Melek, representing the Feri deity Melek Taus, Inanna, the Sumerian under-world goddess, etc. Where gods have well-known symbols or planetary associations,these may be used directly, e.g., = Selene. Other symbols may be introduced asnecessary as part of a working for brevity.

Many magickal traditions include the concept of a supreme god or goddess that isin some sense the union of all gods and goddesses. In the Feri witchcraft tradition,this deity is known as the Star Goddess. We may therefore define the Star Goddess(denoted as ) as follows:

= g

g

Note that, though at first sight and appear to be equivalent, they are not,because is a single element of a set, though one that is in and of itself the unionof all elements of , whereas is actually the set that contains and all of itssubsets.

4.1. Aspects and Independence. Defining gods, and particularly the relation-ship between gods and aspects of gods, is traditionally tricky. Logick offers a

more precise way to do this, that also serves to answer some of the controversy thatfrequently arises between the view that gods are distinct, independent entities andthe idea that gods are all somehow aspects of a single (or relatively small numberof) higher deities.

We will again use the Feri pantheon as an example. In the Feri creation myth, thestar goddess gives birth to the Divine Twins, Serpent and Dove, who then merge toform Melek. With apologies for the oversimplification, Melek = Serpent Dove.From the properties of and , we can infer that Serpent Melek and that

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Dove Melek. Though Serpent and Dove are not generally regarded as aspects ofMelek, they are nevertheless subsets. Of course, it is also the case that Melek .

Note that in this case, though Serpent and Dove are usually regarded as distinct,it should be made clear that they are necessarily entirely disjunct, i.e., it is possiblethat Serpent Dove .

In cases where a god is typically regarded as having aspects, this can be definedsimilarly, e.g., if one wished to assert that Melek, Peacock Angel and Dian y Glasare distinct aspects, one might define an all-encompassing composite god as followsM = Melek Peacock Angel Dian y Glas. It is useful to introduce some extra nota-tion to differentiate the purely mathematical operator from an intended meaningthat god a is an aspect of b by writing this with slightly different symbol, i.e.,a b.

In the Feri community, there is disagreement about whether the Feri gods aredistinct, or whether they are aspects. Taking a mathematical approach, it is rel-atively straightforward to demonstrate that both concepts can be simultaneously

true, because they actually depend upon the definition of equality. Given a strict,set-theoretic definition of equality (i.e., sets a and b are equal if and only if everyelement ofa is present in b, and vice-versa), the Feri gods are indeed distinct. How-ever, a more forgiving version of equivalence gives different results. For example, ifwe define a looser equivalence operator, , as follows:

a b = (a b )

In this scheme, Melek Dian y Glas, but Melek Dian y Glas, even though in thelatter case, neither need be an aspect of the other, though both are aspects (as wedefined earlier) of M.

4.2. Godself. Following the usual approach, the higher soul, godself, HGA, isregarded as deity in its own right. However, we regard each person as having aseparate godself or HGA. To avoid clumsiness of notation, the god soul of personP is denoted P.

5. The Hermetic Principle of Correspondence

As above, So below. This idea turns up everywhere in magick, possibly mostformally in the Qabalah, but most systems have some variation of it within them-selves.

The Hermetic principle of correspondence has an equivalent in mathematics.The great French mathematician, Evariste Galois (1811 - 1832) introduced severalimportant ideas during his tragically short life. Though he is best rememberedfor his pioneering work in group theory, he also introduced the theory of Galoisconnections, an extremely powerful way to formalise the concept of abstraction.This was later revised and extended by Patrick and Radhia Cousot in the late

1970s as a formal basis for the analysis of computer software. We adopt here themore modern Cousot & Cousot variant.

Roughly speaking, as above, so below refers to the idea that the microcosm ispatterned on the macrocosm, and to some extent also vice-versa. Mathematically,one might say that the macrocosm is an abstraction of the microcosm. Formally, ifwe let A represent the macrocosm and B represent the microcosm, a pair of adjoinedfunctions (traditionally named , the abstraction function and , the concretizationfunction) constitutes a Galois connection between A and B. It is necessary for A,

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B, and to obey a number of rules in order for this to work, but in practice thistends to be relatively straightforward to arrange.

5.1. The Need for Order. It is essential that A and B be well-ordered in themathematical sense. That is, given a A and b B, there must always be awell-defined way to determine a b. The actual ordering relation can be anythingyou choose, so long as it obeys the usual properties of , e.g., it must hold thatif a b and b c then it follows that a c. When A and B represent sets ofnondeterministic choices, we trivially have order because we can simply use asour ordering relation.

5.2. Necessary Properties of and . The functions , must be adjoined,i.e., it must hold that (x) x and (x) x. It is important to notethat these conditions are symmetric, and they cause the Galois connection itself toexhibit the needed as above, so below symmetry.

6. Practical Workings

6.1. Casting a Circle. Most magickal systems include some way to declare anoperating space, usually for reasons of protection. British traditional witches castcircles, Feri cast spheres, ceremonial magicians perform opening, cleansing andbanishing rituals.

There are many ways that a circle can be cast, and Logick offers infinitely manypossibilities. However, a simple approach is to declare the centre of the circle asbeing coincident with a particular point in space ritually this might be seen as thecentre of an object on an altar, or perhaps the centre of a candle flame, or perhapseven the heart of the magician themself. For our purposes, we will call this point, following the Thelemic association between Hadit and the centre of all.

For simplicity, we will define a sphere which has at its centre. To do this, weneed some more definitions. Let S3 = R R R be the set of all points in space(were assuming a Euclidean 3-space, but this is good enough for our purposes).We shall assume that the coordinate system has as (0, 0, 0), so a solid sphere ofradius r about may therefore be defined as

r = {(x,y,z) S3

x2 + y2 + z2 r}The shell of the sphere consists of the set of points defined as follows:

r = {(x,y,z) S3

x2 + y2 + z2 = r}Space outwith the circle therefore has the obvious definition

r = {(x,y,z) S3

x2 + y2 + z2 > r}Of course, the actual universe is not just a set of points in space, so bearing

in mind the above definitions of the geometry of the sphere, we loosen our usage

somewhat, with the assumption that such spheres really contain all the matterwithin them, as well as any deities that may choose to be present, possibly includingourselves. We also may simply assume that r is large enough to contain the magickaloperation and no larger, and simply omit it from the expression.

So, assuming magicians Alice, Bob and Carol are wishing to cast a circle ofprotection, visualising a sphere , then meditating on the following spell couldbe sufficient:

= Alice Bob Carol

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This says, essentially, that the sphere contains the godselves of Alice, Bob andCarol, and no other deities (including humans), so mote it be. A working involvingone or more deities or guardians can be achieved simply by adding them to thelist a Feri-like circle (including guardians but not yet other gods and goddesses)might be cast as follows:

=Alice Bob Carol

Shining Flame Water Maker Black Mother

Star Finder Heaven Shiner Fire in the Earth

Note that, strictly, this is an evocation, in that the deities are evoked within thesphere, not (necessarily) invoked within the participants themselves. All of thiscan be defined with appropriate use of Logick, but fair warning should be giventhat these assertions are extremely strong and extremely precise, so if you wish toexclude a particular possibility, you must state it explicitly. A more elaborate circleevocation of Melek that explicitly forbids possession might be defined as follows:

G =Shining Flame Water Mak er Black MotherStar Finder Heaven Shiner Fire in the Earth

Melek

P =Alice Bob Carol

G P = = G P

After a working is complete, the sphere may be dismissed as follows:

Conversely, very dangerous things are possible. A working explicitly requiringfull god-posession of Alice (though retaining explicit protection from such for Bob

and Carol) might be achieved as follows:

G =Shining Flame Water Maker Black Mother

Star Finder Heaven Shiner Fire in the Earth

Melek

P =Bob Carol

G P = = G P Alice Alice = Melek.

By way of explanation, G is defined as the set of gods that are to be evoked withinthe sphere. P is the set of people who are to be protected from possession. instantiates the sphere, then the final line, in three parts, states that the intersection

of G and P is empty (i.e. no possession at any level, including aspecting, of thepeople in P may take place, and that the total contents of the sphere must exactlyequal the union ofG, P, and Alice. Finally, Alice = Melek rather brutally statesthat for the purposes of the working, Alice and Melek are one and the same.

Be careful what you wish for!