Where are we so far?
We have seen the "MATHeological" part one, the "maTHEOLOGYcal" part two and I hope the "matheoLOGICAL" part three will bring together the final MATHematical model with the THEOLOGYcal doctrine of the Biblical revelation and make sense to everybody's standards of rigor in a LOGICAL way.
So in this part I will tell no more of a math-theological reflection on revelation history, but finally give the last mathematical Trinity object to demonstrate how there can be one object and three truly different projections of it. [And, mathematically, from the three projections and some regularity assumptions, we can recover the knowledge of the original object.]
After we have seen how the duality of circle and rectangle can be united in one cylindrical object, it should not come to anyone's surprise anymore, that even in our very own three-dimensional world, there are objects that incorporate three different projections in one body.
Imagine again a cube of side length 2. If you look at it from above, you see a square, into which we can neatly fit a circle of radius 1. Consider using a rotating cylinder saw of this circle shape, which comes down on the cube in direction of the z-axis and cuts out a circular cylinder of radius 1 and height 2.
Next operation: Look at the cylinder that is left in the direction of the x-axis and cut away in direction of the x-axis everything that is outside of a triangle with base length 2, height two and the other two sides of equal length.
If we project a shadow of this object into the xy-plane (z-direction), we will see a disc shaddow. If we project this object in the x-direction onto the yz-plane we will see a isosceles triangle shaddow and if you project this object in y-direction onto the xz-plane you still see the original 2x2 square shaddow.
[I would really appreciate some computer graphics whizz making me a graphic of this, i.e. the object and its three projections in the three coordinate planes. Imagine the object being completely in the first octant of the coordinate system far enough out that you can see the object and the projections on the planes. I would love to include such a picture file here on the web page version of this article.]
Another aspects of this model: Just as the disc shaddow, the square shaddow and the triangle shaddow are "equally valid" projections of the object into the space of comprehension of a 2-dimensional being, so are the person of Father, the person of the Son and the person of the Holy Spirit equally God - that is what the Bible says - and that is the way God revealed Himself into the realm of human comprehension.
So, this object is undoubtedly one but nevertheless reveals itself as shaddows of three distinct forms.
And don't we all know that we can see only some sides of any 3-d object at one time and never all of them at once (if the object isn't small enough and we don't use mirrors)? Think of a big wonderful villa.
We can walk around it and admire it from all sides, but still only at most 2 sides at a time while we are on ground level. If we could fly or climb a tree or nearby hill, then we might also see the roof and hence 3 sides, but never all 5 theoretically visible sides and never the last side, its bottom side.
This is another analogy of the Trinity. The object of our thinking is too big. We can never at the same time think (see) of all of Him. We can think of God from different sides/angles and we see different aspects of him. And those different sides look maybe so different, that we have a hard time imagining they are the same object. If we were only given the shaddows and not the revelation that it really is one object and the three different shaddows really belong together and are "one" original object we might not even come to the idea that this might be so.
Now, the shadows of the 'object' are dead things. But the 'projections' of God are persons and hence nothing would hinder them of talking to each other (Jesus prays) and about each other to the human beings. And that is exactly what happens. God talks through prophets in the Old Testament to the people. And He talks about Himself as distant and transcendent but also talks about being near and that He himself will come and 'visit' His people.
We can think of the Old Testament as telling us humans who live in one coordinate plane about God's projection onto other coordinate planes. But then God also 'projects himself' into our own world (coordinate plane) as God. The Son and relates to us directly. And while He is visibly among us, He still talks about the other persons of God which are unseen. God, the Father, of whom they have already heard much through the earlier prophets, and He also talks about the third person (projection) who He will send soon.
And if you do not like "projection" since this is something 'outside' of the real object, then you can just as well use 'intersection' with the coordinate planes, and then the 'intersection' is part of the object itself. Then another verse suddenly becomes integrated into the model. Namely, Jesus says, that He has to leave first before the Spirit can come and live inside of the believers. In the model, the (and any material) object can have only have one shape of intersection with the same part of one coordinate plane at a given time.
And it is yet again similar to the Biblical Trinity in the sense, that these three projections/intersections are 'co-existing'. It is not the modalist heresy which says that God came in different roles at different times and transformed from one to the other. No, all three persons are real and alive at the same time.
Now God is not inhabiting 'physical' or 'mathematical' space which are abstractions of the MATERIAL world. He is not so long, this wide, and that high, nor is he " this number of years old" (since we live really in a 4-dimensional space time world). Those are not the "measurements" we can put God into. He is spirit. He is the Creator of all that is and is "above" the created world.
Now, this Creator God in all his unthinkableness of higher "dimensions" did want to communicate with us in some language we would understand. He wanted to reveal of himself so that we can enter in a meaningful relationship with him. And, knowing who He himself really is (while we do not know this without revelation) and also knowing our human limitations, he chose to reveal himself, being one God, as three persons (projections into humanly understandable categories). That does not mean he is three persons and nothing else. That would be Tri-theism, since several persons are always distinct in human experience. But it means he is at least personal, and he is super- (more than) personal, since obviously "one person" is not enough to show all that God is. And on top of it, these three persons are said to be ONE God. Just as the disc shaddow is not all that the above defined object is, but is "part of it".
God being the Trinity means that He is community in Himself. That is also shown in one of his "names" used by the Apostle John who writes: God is Love. Love always needs at least a subject and and object. And true love is always giving and taking. God could love the human beings after He created them, but He could not BE love in himself if He had been "alone" before He started creating anything. Then He would at 'some time' not have been love. And he says He is unchangeable, eternally the same God.
And as we can look at the house only "one side" at a time, so we can speak/think about God only "one person" at a time. Yes, we can in abstract term talk about the whole house as one entity, and abstractly we can talk about the "concept" of God and in this abstract theology, He is One, but in our concrete experience He relates to us threefold. If we want to look at the house "in real life" we have to look at a side a time; similarly in relating to God we can only focus on one person a time. Maybe that is one reason that in Islam God seems so distant, since He is always in "abstract Oneness" and many Muslims would even say "person" is too anthropomorphic a word for God. On the other hand in Christianity and Biblical revelation God draws near to us and meets us in a PERSONAL loving relationship, yet without sacrificing His holiness.
There is so much more that could be said here. But I think, I should close this mathematical-theological exploration at this point.
I hope it gave you some insight and also maybe lets you appreciate some aspects of the revelation of God about Himself as the Trinity.
All of this was a meditation on some 'mathematical' properties. Do not confuse this with the reality of the true God. Though I tried to stay in my explorations near to what God says about Himself, in the end it is the Biblical revelation that counts. Let me just name one aspect of this model which is UNlike the Biblical Trinity. The Bible reveals that God IS Father, Son and Holy Spirit. God does not only "appear" as Father, Son and Holy Spirit. Our 3-dimensional object can not only be viewed with our eyes in direction of one of the coordinate axes, but we can move continuously move around it and see many more different 2-dimensional shapes. But it is to be expected that this or any other material model will not cover all aspects of the reality of the Trinity. God says, He is three in One, and those three persons ("views") are the only ones (at least the only ones which are revealed which is the same as far as we are allowed commanded to think about God). Our first statement of faith is, that God is One, our deeper insight of God is, that God is three in One.
Mankind is created in the image of God, and each human being is a 'trinity' of body, soul and spirit, and the soul is again a trinity of mind, emotions and will. And it is the 'soul-spirit' that makes us a person. If we look at our body only, then it is only a material piece, but it is not person. God says reveals Himself to be 'personal' in each "aspect".
In closing this trilogy off now, I do hope that you can see that the Trinity, the ONE God in Three persons is not illogical at all. Maybe you are curious to learn more about this God and how He wants to enter into this love relationship with you. The Bible is the place to find out about Him. There you will see Him revealing Himself. You will find the different 'shadows' He has made over time as people were privileged to encounter Him. And you will find how He says that those different shaddows fit together in a unity of harmony, love and holiness.
(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C) (C) This original "revelation" and all its future corruptions are (C) (C) the sole property of Jochen Katz in the sense of any copyright (C) (C) law imaginable. But unadulterated copies of it may be distri- (C) (C) buted free of charge for the multiplication of smiles in both (C) (C) the mathematical and the religious community on this planet (C) (C) with its tri-dis-unity of monotheistic faiths. (C) (C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)(C)
P.S.: Ideas for improvement of the model are welcome any time.
And for those who urgently need the mathematical equations for my object of mathematical trinity, here they come:
Starting with the "unit cube" : -1 =< x, y, z =< 1
x^2 + y^2 =< 1 for the disc in the xy-plane
z + 2y =< 1
z - 2y =< 1 for the triangle in the yz-plane
z >= -1
and unchanged by the process of cutting away
-1 =< x =< 1,
-1 =< z =< 1 for the square in the xz-plane
All equations together give the intersection of cube,
cylinder and prism by satisfying simultaneously
-1 =< x, y, z =< 1
x^2 + y^2 =< 1
z <= min{1 + 2y, 1 - 2y}
is automatically contained in the first four equations putting y = 0, as was to be expected from the constructive algorithm of wood cutting.
[If some talented person in wood carving or so, would like to carve me personal model of this "mathematical trinity". That would be wonderful. Taking all units to be inches might be a good choice.]
APPENDIX:
Another mathematical model of different character that didn't fit in anywhere else in the article, which may be a bit too mathematical for the taste of some, but others might enjoy it:
For beings living on the x-axis and thinking "God" to be the Graph of f(x)=x-1 or f(x)=(x-1)^3 is nearly the same in most repects. This graph has ONE intersection point with he x-axis at the point x=1 [so 'rightfully' the people on the x-axis could say, "Graph is 1", since that is the only part of 'Graph' that is showing up in their world, and both graphs are going to infinity in both directions (+ and -) [that is the unseen but believed transcendent part of faith]. The 'static' picture is the same. So, what is the difference between them? The difference becomes clear when "Graph" is "moving". And God is not 'static' but is 'dynamic', He is active. Well, what I am talking about is 'perturbations', i.e. slight changes in the numbers involved. If we perturb f(x)= x - 1 we might get "3/2 x -.5" which is still basically the same, i.e. a straight line, infinite in both directions, and one intersection point with the x-axis. So, this activity does not change the appearance or character of the graph in any substantial way.
If now such dynamics is introduced to f(x) = (x-1)^3 then something different can happen. f(x)= (x-1)^3 = x^3 - 3x^2 + 3x - 1, so with the coefficient change dynamics, I can change it into for example into x^3 - 2x^2 + x = x(x-1)^2 and suddenly this Graph, which is still a cubic (polynomical of degree three), still infinite in both directions, but it has now two intersection points with the x-axis. Though it is still just a cubic polynomial, it 'touches down' and the x-axis inhabitants see a new feature and realize that Graph can not be linear but must be of higher order since it intersects more than once.
Yes, many features in (x-1) are the same as in (x-1)^3 but one most important feature is not visible as long as it is 'static'. And when it gets moving, then the difference in nature is revealed. Mathematicians call (x-1) the first approximation or linearization of (x-1)^3. It is a 'lower level' approximation of the real thing, good enough for some purposes, but it doesn't catch all the properties of it. Both functions mean "x = 1" for the x-axis inhabitants, but what sort of a '1' is it? What is the 'multiplicity of the root? f(x) = (x-1)^3 has ONE root x=1 of multiplicity three. They are 'the same 1' as long as it doesn't move. But they become distinct when the dynamic is considered and that reveals that there have been three all along, even though they were overlapping and the 'threeness' of the '1' could not be seen at this time. And even in the dynamic picture it is one graph, one function, which is 'generically' intersecting the x-axis in three points. So, the 'involvement' or interaction of the cubic function with the x-axis is higher.
In a similar way, God is one, but that is in some sense only a first approximation. His involvement with us is higher, He has more dynamic structure. His love for us touches down to us. It seems to me that in Islam the concept of God is relatively 'static', unchanging and Allah is in some sense a 'distant' God. The Bible speaks about God in much more 'personal' terms. God reveals himself in the Bible as highly interested and involved in the lives of his creation "man". God draws near and touches down in love and his threefold nature becomes visible.
Matheology [[Part 1], [Part 2], [Part Three]]
The Trinity
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