3 = 1 ? A MATHeological Meditation
Part One
This is a humoristic semi-serious treatise on the Trinity from a perspective of mathematical theology. :-) !!
And how could it be otherwise, this one matheological revelation will come to you in three parts. :)
Dedicated to all the great Muslim mathematicians ofOne of the recurring ways of ridiculing the Trinity from the Muslim side (and some others too) is to present the equation
all time = the past + the present + the future!
[one time] [another time] [and one more time]
Analogies can be misleading, but they are
the least misleading thing that we have.
Samuel Butler
1 + 1 + 1 = 1
and then smile and think that is enough to expose the utter craziness of the concept and showing the completely illogical content of the Christian faith.
Now, it seems that the people which use this kind of reasoning are apparently stuck in their preschool education and have never heard that there are other operations one can perform on numbers than just addition.
1 x 1 x 1 = 1
wouldn't lend itself quite as well for the effect those people want to achieve. And the Trinity is a powerful revelation of the Almighty God, so "raising to the power of a number" might be an appropriate 'composition' for relating the 'three Ones' mathematically, which is giving us
1
1
1 = 1
so that we have already a two to one victory over those who have not yet grown out of their restricted imagination of only being able to add numbers. And lest you want to even out the score by pointing to subtraction, think if you want to risk me telling you about division. {Not to mention "extracting first roots", identity mappings, inclusion homomorphisms etc, which are all on my side, but are rated "R" and only allowed for grown-up mathematicians.}
Okay. Having established that mathematics might be able to give more helpful insights into theological concepts, let us explore what else is available. Most readers here are part of an academic community, so I think everyone will able to follow when the level of sophistication increases.
Now the Bible speaks of three "persons" in respect to God. It does not say, the relationship between them is "addition". If we want to use mathematics at all, the question now is: What would be the "Biblically apppropriate, mathematical composition" of the three ones to make another ONE?
But before it becomes more difficult, another easy part. We already have seen that multiplication is a very nice operation. When we venture from mathematics into physics, we win another 'dimension' of thinking since now our numbers come equipped with units and are no longer so shamefully "naked".
So we could write for example:
3
(1 m) x (1 m) x (1 m) = (1 m )
I.e. the product of three times one meter in length, width and height gives us one cubic meter. This "analogy" is not a very good one though, since it would make God, the Father, God, the Son, and God, the Holy Spirit sort of 'the same', since one meter is one meter in whatever direction you measure it. But we can not expect that analogies which are necessarily restricted to your created, material world will be able to perfectly illustrate the nature of the uncreated, spirit being (non-material) of God.
Maybe it would be better to take the physical concept of "force":
[And to pay tribute to our science fiction fans here:]
[ "May the force be with you!" ]
2
(1 kg) x (1 m ) / (1 s ) = 1 N [i.e. F = m a ]
or the concept of "energy":
2 2 2
(1 kg) x (1 m ) / (1 s ) = 1 J [i.e. E = m c ]
or "power":
2 3
(1 kg) x (1 m ) / (1 s ) = 1 W
But that may make the "three different ingredients" a little too different in the end. And none of the individual "parts" IS force. Not the mass, not the length, not the time. But the Biblical revelation insists, that the Father is God, the Son is God and the Holy Spirit is God. But nevertheless, there is only one God(head). I think NO analogy from the created world exists that will ever capture it completely. { But don't say yet, "see, it doesn't work, it is illogical, I knew it all along ..." The best part is still ahead. :) }
3 = 1 ? A maTHEOLOGYcal Meditation
Part Two
Back to being more serious, I still have to make good on the initial promise of being "semi-serious", so far I was only "semi".
Muslims say, that God is ONE. And I say, Christians believe this just as well. Christians are strictly monotheistic. There is ONLY one God. Pardon me for my scant knowledge of the Islamic concept of God, but my impression is that Islam is in general not saying very much about what sort of "Oneness" this is. Is Islam saying really more than that "one means that He is NOT two, three or any number of many"? Is Islam giving only such a "negative" description of what God is NOT, instead of trying to give some positive information on what God IS? What is the nature of God's Oneness?
There are people who might say, as soon as someone starts with "God is ..." and fills in something for these three dots, whether it is just one word, one sentence or a long book ..., he is already lying and is plain wrong. And to some degree all religions would agree with this opinion.
God is so much greater than any human being could ever hold together in his brain, much less express in the imperfect categories of human language. But what is the Qur'an saying about God's Oneness? What is a Muslim positively thinking if he says "God is one" other than "He is not 'three'"? Could it be that Muslims are putting God in a category too small for the one true God who is much bigger than any of our thoughts will ever be able to fathom?
[These are not rethorical questions, I really would like to see some articles here on the soc.religion.islam newsgroup who will attempt some answer(s) to this.]
Now, Christians believe God has not stayed silent but has spoken and revealed more about Himself and the nature of His Oneness. And this is that - in some sense - He is a commUNITY without being many God's. The unity is so strong that the image of three SEPERATE gods would be much further from the truth than the other danger of having a onesided picture of making Him a 'onedimensional' oneness.
Is the cube of side length 1 meter that I described in Part One more like a "One meter line segment" or more like "Three seperate line segments"? Well, not really like any of those two suggestions. It has more complexity, that is, more 'dimensions' to it.
After thousands of years of philosophy and by now already a hundred years of psychology, sociology etc "man" is still a mystery in many aspects. It is hard for us to even understand ourselves. We can much easier understand things which are 'below us' in terms of complexity.
Therefore, for the sake of argument, let us assume we were 2-dimensional beings living in a plane [x,y-coordinate plane for example] and have the same problem. We can recognize each other and understand quite about about our 2-d life, but truly understanding and grasping we only can 1-dimensional 'objects'. That is what our thinking is comfortable with.
For those 2-dim beings, 'saying "Cube is One"' is true, but if you [as a low-dimensional being] then think of it as "one line segment" then you are very far from the truth since you disregard in your imagination nearly ALL of the cube. [Mathematically: The piece of the cube you are thinking of has volume or "measure" zero and is negligable.]
But if you say the "Cube is three" because it has three independent line segments in the x-, y-, and z-axis direction then you are also thinking only about a tiny part of the cube and disregard most every- thing of it. Furthermore, if you think of those three line segments, just as "unrelated" because they are independent (linearly independent in the language of vector spaces) then you commit another blunder, because you think [as a 2-dim being and only really able to think comfortably about 1-dim entities] of them like three line segment on the same line with distances in between them and those can never be one. That makes the crazy impression of "3 = 1". And you have introduced the blasphemy of "poly-segmentation" and might be called a "poly-objectist".
But everyone should see, how the "Cube is one" and "Cube is three" are both correct when you look at it the right way. We just should not impose our dimension of experience on it. In the higher reality [dimension?] of God, they make perfect sense.
The cube is more than a 'two-dimensional being' could think. But if I am forced to chose among the two given alternatives of "one piece" and "three pieces" then I would have to go for the "one piece" since the cube is surely a unity and not a multiplicity of seperate things.
And that might be one reason that God in His progressive revelation chose to first stress the fact that He is ONE God and not many, because the idolatry of polytheism was the greater danger of going astray. And He left the parts of revelation concerning the nature of this Oneness for later revelation to make plain. So, it really is important to first stress "Cube is One" and then slowly lead the people to more insight and add the other dimension and tell them that the Cube is a trinity, it is
So, understood, that human being can not even understand themselves completely, will we restrict the reality of God, by what is true in our 'lower dimensional' reality? And if we would concede that God is indeed living in a higher reality [for example outside of space and time, on which Muslims and Christians agree], would a Muslim have to insist that in God's reality it is impossible that there is
and would he have to call the Trinity blasphemy if understood in analog to this example of the cube?
Christians are monotheists in the strongest form! No true Christian believes in three gods. Three gods would be blasphemy and every Christian would agree. But this is not what we believe. And could it be that Muslims are in danger to think too 'low' of God's Oneness, just as 2-dim beings would (automatically) think of 'Cube' (which is something they can't really imagine) as a "1-dim thing" when hearing the emphasized expression that "Cube is One"? Mono-segmentism is one 'low' truth of the Cube's oneness, Tri-segmentism is an even worse 'fragmented' view. But the 'segmential trinity' seen in the right relationship of all of its dimensions is the right view of the cube, even if it might never lose its mystery for the 2-dim beings. In the same way, I believe that the Trinity is the only model that makes sense of all the Biblical data and it is logical, when seen in the right light. But also, it still will never lose all its mysteriousness to us, because God's reality is ultimately beyond the grasp of human thinking.
Now, let us slowly work up to the main example I want to present. Let us start with two in one before we go to the more complicated three in one.
Imagine to live in the two-dimensional world (of the xy-plane). In two dimensions, a rectangle (not just the boundary, but filled in) and a circle or disk are two very different shapes. And if we were to live in just two dimensions and that is all we know and then a rectangle and a disc come along and claim to be one and the same "object" and you should believe in the doctrine of Duality, you as most others of those two dimensional beings would just laugh at them and declare them to be crazy.
But do you see that the doctrine of this "Duality" does make sense if you allow just one more dimension? There might even be some religious group of those two dimensional beings who happen to believe in the Duality although they have a hard time making that intelligible to their fellow 2-d'ers ...
Okay, here is the solution, which the 2-d'ers might never completely understand, but you and I are luckily living in 3 dimensions and can only smile at the limited brain power of 2-dimensional beings (as God is maybe sometimes smiling at us in our petty fights).
And for the sake of argument, let us assume, that all the 2-d beings are of rectangular shape. And like in the real world, there are small and large ones, fat ones and tall ones etc. I think you get the gist.
The solution of the mysterious unity of circle and rectangle is the cylinder. The equations for the 'standard' cylinder in 3 dimensions centered at the origin of the coordinate system [I hope I don't lose you now] is [r = radius = 1, h = height = 2 in our example]:
2 2 2
1 = r = x + y,
2 2
or, if we want to have the filled in cylinder: x + y =< 1
and -1 =< z =< 1 for the height, where "=<" means "less or equal".
Now if this ONE object "passes through" (incarnates into?) the two- dimensional world of the xy-plane in the direction of the z-axis, i.e. we add the parameter t = time and get -1 =< t ="<">
{And for for other times, the cylinder/circle would disappear completely and be unseen, which is somehow similar to our situation today, it seems to me.}
In this setting, at all times apart from some finite time interval of 'circular theophany', or better 'circular cylindrophany' the object would be invisible. [theophany = the visibility of God, apparition of God; so 'circular cylindrophany' which is a word I took the liberty to invent, would be the appearance of a cylinder in a circular shape.]
Now, no doubt you have already seen, that the cylinder might decide to rotate 90 degrees at some time where it is outside the xy-plane and come back at the same place and pass through the xy-plane as a rectangle (of varying size). Each intersection of a cylinder that is perpendicular to its base circle is a rectangle.
Alternatively, the 'object' might decide to only do the circular revelation by the method of projection, i.e. the light of revelation in parallel beams is going to reveal a circular shaddow in the xy-plane. And the beings only used to rectangular shapes are very much awed of it. When this object then also promises that it will come and live among them, this is all too mysterious for them and not well understood by them. How can a circular being live in a rectangular coordinate world? This is absolutely impossible. And then, when this object passes through in this rectangular shape, looking so very much like them and does all sorts of miracles and heals some badly dented rectangles and even some which had been squeezed dead into a purely onedimensional piece of line are resurrected into a two-dimensional rectangular life again, this awesome event does cause quite some uproar in xy-world.
All this has very much confused our 2-dimensional population to the greatest extent and created or changed different religions. Those who believe in many 'objects of worship' (poly-objectists), and have already a whole garden variety of shapes they worship, will have not trouble to incorporate also the circle and rectangles of any form into their "pantheon" of adorable shapes.
And then there are different groups of mono-objectists who agree that the 'polies' have an abominable theory and are idolators. Among these 'monos' are those who believe in the circle only and are stuck in the first revelation completely blind to the events that happened among them. The second group has seen the many miracles of this "rectangle just like them" and has decided to believe ALL that is revealed, and trust the 'object' when the 'object' was saying that it is ONE although it seems to look different at different times.
And then, yet another group comes on the scene quite a bit later after all of this had long happened and they decide to not completely ignore the second revelation. After all, something quite momenteous has happened. And they do believe that the object has done miracles through the hand of this extraordinary rectangle, but they declare the records of all this rectangle has said to be corrupted and it in reality was not the object itself but just some ordinary rectangle who was only a messenger of the object. And they side with the first group who says, that it has been clear from the beginning in the projection revelation that the object is a disc and that is it. So, the third group of 'mono-objectists' just declare that all true worshippers of the true object have always and at all times believed only in the disc and nothing else. In all this, they seemed to have no problem in declaring hundreds of years later without having any evidence and although the eye witnesses had meticulously written down their observations of the rectangular revelation of the 'object' and all it has done and said which was clearly more than any "purely 2-dimensional being" could do or say.
When God reveals His name to Moses, He gave a name that can be translated as "I am who I am" or "I will be who I will (show myself to) be".
The Jews, when confronted with God's new revelation about Himself in Jesus, seem to have changed this name into a different one and seem to translate it now as if He had said: "I will always be ONLY what you thought yesterday that I was". To them we have to say: Do not restrict God to the past and to your understanding of His earlier revelation. Do NOT accept CONTRADICTION, because God will never contradict Himself. He is true to Himself and changes not. But maybe you have not yet seen all that He wanted to show about Himself. Come and see that the new revelation fits in with the old in complete harmony if you take the correct "view point". And even the old prophets announced time and again, that God says: "See, I am about to do something NEW."
God calls us to trust in Him and to accept all that He has revealed and grapple with the parts that look like "difficult, mysterious and illogical" until we understand them. And since He is a God, who created our mind and reasoning capabilities, He expects us to use them and not give up too easily. Thinking about God (= theology) will most likely be the most mind-stretching exercise you ever undertook. Woe to him who thinks he can box God into an "easy concept" and because of it belittles his Creator. Do not think, it should be easier to "understand God" than it is to understand "man", and after thousands of years of philosophy and in more recent time sociology and psychology, "man", the created being, is still a mystery to us. Do you really expect God, the Creator, to not be mysterious?
Jews reject anything that is more than what they had from old times, while Muhammad/Muslims agree God has sent more revelation, but they seem to pick and choose, taking snippets from the old revelation, pieces of the new revelation and fill it up with with a number of ideas of their own. And though there are many "authentic" elements in it, by differently putting them together, they create a very different (image of) God. They base too much on their own limited (human) logic capabilities and say "God can not be what I can not think". To them God says: Submit to all I have revealed about myself and do not limit me by your preconceived understanding of "how God has to be" to make sense to you.
3 = 1 ? A matheoLOGICAL Meditation
Part Three
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.
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