Margaret Masterman's "Theism as a Scientific Hypothesis"

Theoria to Theory

Vol 1, 3rd Quarter, April 1967, pp 240-46.

This excerpt from Masterman's fascinating and provocative work discusses an "Icon of the Christian Doctrine of the Trinity: seen as an 8-element Boolean Lattice"...

Please note that the words "meet" and "join" have been substituted for Masterman's equivalent logical symbols throughout the text, for reasons of typographic convenience (see endnote for further details).

To give you an immediate sense of what Masterman was up to in this article, let's begin with her illustrations. It may be easiest to print them out and refer to them as you read her intricate text...


Masterman's Illustrations:

A. Binary mathematical interpretation of the Boolean lattice of eight elements. (Hasse diagram)

B. Orthodox Trinitarian re-interpretation of A

C. Medieval symbol of the Holy Trinity

D. Mapping of C onto A. [Note correct positioning of complementary points.]


Masterman's Text:

...I will now, with the help of a logical system derived from Boole's insight, construct a mathematical iconic vehicle of the Christian Trinity; and I do this in all seriousness, not as a gimmick, since I think that Boole, in setting up his Laws of Thought, had a Trinitarian revelation as well as an Idempotent one.

Icon of the Christian Doctrine of the Trinity: seen as an 8-element Boolean Lattice.

(a) The necessity of working within a non-numerical system.

In order to get any mathematical icon whatever of the Trinity, we must first establish postulates which will take us right out of the particular world of ordinary numbers.

We will therefore use the two signs 0 and 1 to serve purely as markers with no numerical significance attached to them.

(b) Triads.

Using these two numbers, the first step, in constructing the system, is to exhibit three-ness within it by causing its elements to consist of a set of triads, each triad consisting of a 3-digit sequence of 0s and 1s. (Note that we cannot, within the system, count these triads, since the system does not itself contain the number 3.) We allow all possible such 3-digit combinations as elements of the system. There are eight of these: 111; 110, 101, 011; 100, 010, 001; 000. Note that these triads fall into a patterned set of ranks (marked above by the semi-colons) according to how many 0s or 1s they have. The first rank has only one member, namely the triad consisting entirely of 1s; the three triads in the second rank each have two 1s and only one 0; the three in the third rank have two 0s and only one 1; and the single triad in the lowest rank has nothing but 0s. Using this ranking, we can set the triads out on the page in a balanced way, thus:

 

111

 

110

101

011

100

010

001

 

000

 

This way of setting them out makes it clear that we are not here thinking of the triads as numbers at all. There are many ways of re-interpreting them; I propose that we think of them as states of information. Thus the triad 111 represents the state in which full information is present; the triad 000 represents the state in which no information is present; and the triads in between represent the set of possible states in which partial information is present.

(c) The relation of greater-than-or-equal-to (>=)

We have spoken of the set of triads as a set; but we have not yet established any set-relation between them. We will do this by postulating that the very general notion of "greater-than-or-equal-to" (also called the relation of inclusion) holds between triads of the set. It is this relation which, operating upon an "0-1" system, brings the generalizing power of the Idempotency Laws into play. Even in such an inclusion-system which has only one element, x, x >= x; that is, x is greater-than-or-equal-to itself. So the inclusion-relation can hold whether the two elements concerned in the relation are distinct or not.

(d) Specifications of the inclusion-relations of the system

We can now say that this relation of greater-than-or-equal-to shall so hold between the elements of the set which we are constructing that:

(i) any triad shall include itself.

(ii) triads in which fuller information is present shall include triads in which lesser information is present except where the triad representing the lesser state of information has a 1 where the triad representing the greater state of information has a 0. Thus, the triad 101 does not include the triad 010, but (e.g.) the triad 110 does include the triad 100, because the 1 in the second triad, 100, is in the same position as a corresponding 1 in the triad 110, whereas the 1 in the triad 010 corresponds to an 0, not to a 1, in the triad 101.

(iii) Given any three triads, x, y, z, if x >= y and y >= z, x >= z. (This requires not only that the elements between which the inclusion-relation holds need not be distinct, but also that they need not be in contiguous ranks).

On these three conditions, the inclusion-relations holding between the triads of our system will be:

(From condition (i)): 111 >= 111, 110 >= 110, 101 >= 101, 011 >= 011, 100 >= 100, 010 >= 010, 001 >= 001, 000 >= 000; (From condition (ii)): 111 >= 110, 111 >= 101, 111 >= 011; 110 >= 100, 110 >= 010, 101 >= 100, 101 >= 001, 011>= 010, 011 >= 001; 100 >= 000, 010 >= 000, 001>= 000; (From condition (iii)): 111 >= 100, 111 >= 001; 111 >= 000, 110 >= 000; 101 >= 000, 011 >= 000.

(e) The Hasse Diagram.

We can make a simplified picture (called a Hasse Diagram) of this system of inclusion-relations by representing the eight triads as eight points arranged in four ranks (as in (b) above), and then drawing lines as inclusion-relations between them on criterion (ii) (as in (d) above [see Diagram A]). In other words, we can depict graphically any situation where the inclusion-relation holds between the two triads which are distinct; but we cannot graphically depict any situation in which an inclusion-relation holds between a triad and itself; and if we try to depict graphically the non-contiguous inclusion relations, we find the lines depicting these fall on top of lines already drawn. Such a simplified picture is sufficient, however, to distinguish any finite idempotent system from any other.

In such a picture, the inclusion-relation goes down the page from top to bottom.

(f) Remarks on the mathematical centrality of the system.

The system which we have just constructed can be regarded mathematically in several ways:

(i) As a Boolean algebra.

(ii) As a partially-ordered set. (i.e. as a set of elements interconnected by the >= relation).

(iii) As a particular kind of partially-ordered set called a lattice.

A lattice is a partially-ordered set in which it is true of any two elements, x and y, that there is within the system a unique element, x join y (called the join of x and y) constituting a least upper bound (supremum) of x and y; and also a unique element x meet y, (called the meet of x and y) constituting the greatest lower bound (infimum) of x and y. The idempotencies of the system arise from the fact that, whatever element x may be, both the join of x and x and the meet of x and x is x itself. The fact that the system is a partially-ordered set shows itself in that if, of any two distinct elements, x and y, x join y = x and x meet y = y, then x >= y.

Thus, [see Diagram A] of the two elements (e.g.) 110 and 100, 110 join 100 = 110, and 110 meet 100 = 100, because 110 >= 100. But of the two elements (e.g.) 110 and 011, which do not include one another, 110 join 011 = 111, and 110 meet 011 = 010. (In other words, to find the join of two triads you combine their 1s, whereas to find the meet of two triads you combine their 0s.

(iv) The system can also be regarded as an information-system as in (b) above.

(g) Remarks about the system considered as a Boolean Lattice.

Considered as a lattice, the system which we have just constructed is the very well-known Boolean lattice of three minimals, the "cube" lattice. It is exceedingly useful because it can equally well be considered as an algebra, as a partially-ordered set and as a lattice. Not all partially-ordered sets are lattices (consider, for instance, a two-branched tree with, say, seven elements), and not all lattices are Boolean, in that not all represent the maximal number of inclusion-relations between their elements (consider, for instance, the 8-element "lantern" lattice, which has one point at the top, and one at the bottom, and the other six in a single rank in between). The system which we have just constructed is therefore, in the mathematical world of idempotent systems, a cardinal one, about which it is possible to think a number of pure mathematical thoughts of many kinds. It is not the only finite Boolean lattice; the Boolean lattices go up in size, within lattice-theory, by having the numbers of their elements correspond to the sequence of the powers of two: i.e. 2, 4, 8 (our lattice), 16, 32, etc. But, like all Boolean lattices, it is a fully complemented lattice (i.e. every point in it has a corresponding opposite point; contrast it, for instance, with the "limping lattice" of five elements, the Hasse Diagram of which is made by drawing four points in a diamond, adding an extra point on one side, and then joining them all up in a ring). It is a modular and a distributive lattice (modularity and distributiveness both being properties which define differing types of regularity); it is a self-dual lattice (i.e. if you turn Diagram A upside down the same shape re-occurs); and it is the lattice formed by the "centres", or boundary-markers, of the three factors in any three-factor product-lattice, no matter what shape the lattices which constitute the three factors of the product-lattice may be.

Some of these properties, though not all, will be used in what follows.

We have constructed the system, or schema; let us now convert it into an icon -- or rather, into an iconic vehicle [see Diagram B]. That it is not entirely a theological innovation to make an abstract schema of the Trinity is shown by the frequently found mediaeval diagram given in Diagram C, which can be mapped on to the cube-lattice.

To return now to the iconic vehicle given in Diagram B. If we develop this vehicle, we find that we can now think quite a number of mathematical thoughts which are analogous to thoughts which theologians have thought of the Trinity. Among these are the following:

(i) God can be conceived in His unknowable Essence, G, as well as in any of His Personae, F, S, or P. When He is conceived in His Essence, we shall say that F join G = G, and that He is being conceived in the aspect of His supremum. When He is conceived as F, i.e. in His Persona as the Father, we shall say that F meet G = F, and that He is being conceived in the aspect of His infimum. But, since F join G = G and F meet G = F, therefore G >= F; so there are not two Gods, but only two aspects of one God.

(ii) God in His Essence, G, is complementary to God seen in His Energies, E. But (under condition (iii), given earlier) G >= E. Therefore (as St. Gregory Palamas said) there are not two Gods, but only two aspects of the same God.

(iii) From God the Father as conceived in His Essence, F join G, is begotten the Eternal Son, seen as God seen in His infimum, G meet S. But (F join G) meet (G meet S) = S, and (G join S) join (F join G) = G; therefore (F join G) >= (G meet S). So again, here are not two Gods, but one God.

(iv) Similarly, substituting P for S in (iii), we can show that the Paraclete, seen as God in His infimum, proceeds from the Father (but not from the Son, for there is no Filioque inclusion-relation in this icon).

Thus we have shown the Trinity of God, "neither confounding the Persons, not dividing the Substance"; and the Son, "equal to the Father concerning His Godhead, but inferior to the Father as concerning His Humanity".

(v) Consider now the 4-element Boolean sub-lattice G >= S, G >= P, S >= C, P >= C. This shows the Church, C, proceeding from the Son, S, and the Paraclete, P. That the Church is Divine is shown by the fact that C join S = S ("the Church is the Body of Christ") and that the Paraclete works within the Church is shown by the fact that P meet C = C; i.e. the Paraclete, seen in His infimum, is the Church. Moreover, when we say (C join S) join G = G, or, (C join P) join G = G, we show the very humanity of the Church being taken up into God.

Take now the corresponding 4-element Boolean sub-lattice G >= F, G >= S, F >= W, S >= W. This shows how the Father, (in His supremum) through the Son, created i.e. includes the world; and how the Father (in His infimum) together with the Son, includes the world. And, if the two corresponding sub-lattices be compared, it will be seen that the Divine Church is complementary to the created world; and that the created world itself is, in its Essence, Divine.

(vi) The direction of the inclusion-relation in the icon shows that God empties Himself (of information) in creating the world, while yet remaining God; and that, seen in its Essence, everything is transfigured, i.e. carried up the lattice into union with God.

(vii) If we substitute the dual of the lattice for the lattice (i.e. if we turn Diagram B upside down) then, while the whole dynamic patterning remains the same, since the lattice is self-dual, we get an icon of what it looks like to our eyes. For first we see the Energies, the manifestations of God; then, proceeding further we see, on the one hand, the divine springs of natural creation, and, on the other, the divine springs of the Church, i.e. of man "as he shall be". Proceeding further yet, we glimpse the Personae; and beyond that again, apprehension comes to an end in the unknowable Essence of God. And, as we go further and further, the state of information grows greater; i.e. everything becomes not less, but more, real...

Of course it can be objected that, by thinking this way, we only make it possible to think about the interrelations between the different aspects of God, not about the nature of the Personae. It can also be objected, by those who think it wrong to press mathematics into the service of theology, that to mathematicize human thinking about the Doctrine of the Trinity makes this doctrine trivial; (but similar iconoclastic objections can be made, mutatis mutandis, against the use of any icon). What is clear is that whoever in this day and age derides the doctrine of the Christian Trinity, will not be a scientific humanist (he is much more likely to be a Protestant theologian). For I have shown strong reason to think that the Early Greek Fathers were feeling after a very general way of thinking, which, centuries after, other men have mathematicized, and which, as there is coming to be increasing reason to believe, may be fundamental for thinking about the foundations of the Universe itself.


Comments:

In a later passage, in Theoria to Theory vol 1 issue 4 p 343, Masterman makes a brief but telling comment:

Thus, in my Boolean lattice of the Trinity, in Theism III, it could be inferred that the Holy Spirit P in his supremum, (P join G) included the Father: i.e. (P join G) >= F; but I have purposely not made this inference...

It is clear, in other words, that not all the theological inferences drawn from this mapping are in accord with Trinitarian theology, and this alone would be enough to give one pause before making the assertion that the mathematics employed here "demonstrates" the theological rightness of the exclusion of the (theologically disputed) Filioque clause. For theologically inclined readers, this is an important concession...

On p 401 of the same issue, Masterman writes:

Apologia.

The idea of making a Trinitarian interpretation of an 8 element Boolean lattice has produced strong reactions, both of an approving and a disapproving kind. Those who disapprove are mainly of two kinds: those who accuse me of prostituting computer-science in order to bolster up Trinitarian religious obscurantism; and those who accuse me of taking all consolation and warmth out of mystical experience -- in fact, of radically misunderstanding the nature of contemplation. With regard to the first accusation, I am not trying to bolster up anything; I am only trying to see what it would be like to make a theistic informational model (of which the lattice was probably an over-simplified version); not to support such a model if the inferences from it turn out to be false, as may well occur.

With regard to the second accusation, this has a verbal variant which consists in saying, "I thought you were so spiritual, until I met you." This attack leaves me in the predicament of refusing to admit an evident fact, namely, that they are perfectly right, I am in no way spiritual, through exasperation at their truncated, quietist, self-cossetting conception of spirituality.


Footnotes:

11. St. Gregory Palamas spoke of the Divine Energies as God expressed in creation. (See St Gregoire Palamas et la mystere orthodoxe -- Jean Meyendorp. Editions du Seuil).

12. He wrote a sonnet to the Number Three which echoes the sentiments of St. Gregory Nazianzen on the Trinity: "the ineffable radiance common to the Three". (Cf. Lossky, Mystical Theology of the Eastern Church, p.43.)

13. See Dedication of English Churches: Ecclesiastical Symbolism, Saints and Emblems, by Francis Bond. Oxford University Press, 1914, p. 25 (two variants on the mediaeval geometrical emblem of the Trinity).

The existence of this schema, and the possibility of the mapping, was pointed out to me by Mother Geraldine Mary, sometime Superior of the Society of St. Margaret, East Grinstead.


Endnote regarding errata and typographic problems:

Please note that in Masterman's original printed texts, "join" is shown by the sign:

and "meet" by:

both of which caused her considerable problems with the typesetter -- and would cause even greater problems if we attempted to reproduce them in HTML. We have accordingly represented them by "join" and "meet" respectively.

Similarly, since the usual printer's sign for "greater than or equal to" is not available in the HTML special character set (as far as we know), we have substituted ">="

I have also attempted to incorporate the errata which Masterman published on p 401 of T&T vol 1 issue 4 into the body of her text, though I have made some further adjustments where it seemed clear that there were errata in the errata.


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