For an introduction, explanation and links to the entire work, click here.
Having cleared up the matter of the Son being “created” or more informatively “generated” we must turn to the problem of the Son being God and subordinate to the Father at the same time. Like the generation of the Son, this is a problem that only became a serious one with the coming of Arius.
As we have seen, ante-Nicene church fathers had no trouble envisioning the Son and the Spirit as subordinate to the Father. The reason for this is twofold. The first reason is that the Bible clearly taught this subordination. Those who weren’t bogged down in philosophy weren’t concerned with the technicalities of such a declaration; God said it, they believed it, and that settled it. Those who were concerned with philosophical niceties could console themselves with the second reason: the whole universe and everything in it was set up in a hierarchical manner; numerous philosophical schools affirmed this.
Arius, however, mindful of the philosopher’s concept of God such as we have presented earlier, couldn’t see how subordination could exist with all of this primal omni-everything, so he simply rejected the idea of the Son and Spirit being God. Having disposed of this problem, he could happily affirm the subordination of the Son because the Son, being an ordinary creature, would be subordinate to the Father like everyone else. This was one of the appeals of Arianism in its early stages; as long as the Son was important to salvation and subordinate to the Father, many couldn’t see the problem in it.
Trinitarians examined this problem at length and realised that they could agree with the Arians on one point: they couldn’t see how the Son and Spirit could be God and subordinate at the same time either. So Trinitarians, while making the usual allowances for the procession of the persons of the Trinity, basically stated that the Father, Son and Spirit were equal. They found it easier to take this position against the Arians rather than to attempt to work out a way by which the Son and Spirit could be subordinate to the Father. Put another way, faced with the choice of denying the deity of the Son versus his subordination, the Trinitarians chose to jettison subordinationism to preserve the deity of the Son and the Spirit.
This was an entirely sensible and correct choice; recognising that the Son is God is too important to the whole plan of salvation to sacrifice it for the concept of subordinationism. It also made sense in the Greek philosophical system of thought that prevailed at the time. In doing this we lose an important Biblical concept for philosophical reasons that are largely forgotten.
This is one place where Greek philosophy, which furnished Judaism and Christianity (and even Islam in the early years) with some very powerful assistance in understand the nature of God, really fell flat, where the living God of the Bible, who could feel regret and take on the sin of the world, was deprived of these freedoms for the sake of a perfectionistic concept. We need to find another vehicle to examine this subject. We could of course resort to a mystical, subjective approach to the problem, and many have, but since we have worked on an objective plane up to now, we need to stick to it.
Enter the Mathematicians
They who are of the priesthood, or of the clergy, shall not be magicians, enchanters, mathematicians, or astrologers; nor shall they make what are called amulets, which are chains for their own souls. And those who wear such, we command to be cast out of the Church.
It is with a little sense of apprehension that we approach this topic in the way we do, not only because “scientific” explanations of spiritual phenomena frequently fall flat but because of people’s perceptions of math itself. Involving mathematics in a pursuit such as this conjures images of a mad Unabomber blowing up things and people for arcane reasons.
The reason why we have resorted to this, however, is rather simple: mathematics deals more informatively about the concept of infinity better than any other branch of science or art for that matter. Theologians routinely throw out such terms as “omnipotent,” “omnipresent,” “omniscient” and others about God; each of these described a quality (all of which are essentially God’s in any case) that is infinite. While theologians can set these things forth and leave them, mathematicians must actually deal with infinity, sometimes in a theoretical way, sometimes in a practical one. If we can use the concept of infinity to understand the nature of deity then we can make some more realistic assessments of the situation and hopefully understand better what we see in the Scriptures rather than throwing up our hands in confusion.
In employing mathematics to understand these things, we must be careful not to get so far into our analysis that we either lose sight of the main object or lose the comprehension of most of the people we are communicating with. Fortunately the mathematics we employ are relatively simple and, better yet, can be described with pictures and graphs. So we can proceed with some confidence that we will not lose everyone in the process.
Life in One Dimension
Let us begin by considering the one dimensional co-ordinate system shown below.
We have a line that extends from negative infinity to positive infinity. At the “centre” of this line is the origin, the point “0.” Extending to the right is the positive part of the line, divided up into spaces indicated by the tic marks. These spaces can represent any unit of time, space or whatever you might imagine: inches, feet, meters, seconds, years, etc.. For our present purpose we plan to discuss this as the time line for the entire universe so these units are units of time of whatever length you care to think of.
If we look at this we see that the origin is at a certain spot on the line. However, this is an entirely arbitrary decision. In the course of using co-ordinate systems to describe physical phenomena, people find it convenient to do something called “co-ordinate shifting,” which means moving the origin from one point to another. For example, if they are using the co-ordinate system to describe the motion of an object, and that object starts at a certain point, it makes sense to set the zero point of the co-ordinate system to the place where the object’s motion actually starts. This makes calculations easier later.
If we employ co-ordinate shifting on the time line of the universe, we can place the origin at any point we like. In fact we have been doing this all along. The Jews (and Christians for many years) defined the year “zero” at the point of creation; Christians then took up the idea of having dates from the birth of Jesus Christ, even though they didn’t hit the event accurately when they set up the “co-ordinate system” of the years. Muslims picked Mohammed’s flight from Mecca to Medina in 622 as their origin; to make matters more complicated, they adopted a lunar calendar, which means that the length of the years (the distance between the tic marks mathematically) is different. These are straightforward examples of co-ordinate shifting in time. To change from one to another can be confusing to us with all of the associations of time we have but as we see we can define a point in time in literally an infinite number of ways.
The one thing that doesn’t change, however, is that the origin – wherever we place it – is always at the centre of the co-ordinate system! The reason for this involves the nature of infinity. No matter how far to the left or the right we move the origin, it is still an infinite distance (or strictly speaking a semi-infinite distance) from the origin to infinity. No matter how far the origin goes, it never reaches the infinite point, nor gets any closer, because the remaining distance is infinite. John Newton put it very succinctly in “Amazing Grace:”
When we’ve been there ten thousand years
Bright shining as the sun
We’ve no less days to sing God’s praise
Then when we’ve first begun.
Theologians and philosophers say that God’s existence is limitless in time; mathematicians would say that God exists from negative infinity (-∞) to positive infinity (+∞) (cf. Ps. 90:2) and at all points in between. Thus at once we set very definable “limits” for God’s existence, yet in reality they are not limits at all.
Having set forth both the co-ordinate system and its relationship to the universe, let us make some observations based on both. The first is that there is no real proportion between anything finite and infinity. By “proportion” we generally mean division; any finite quantity divided by infinity is, in reality, zero. On our co-ordinate system, this means that, no matter how long of a finite distance or time period we consider, it is basically nothing in comparison to the infinite time period God exists in. This certainly helps us in conceiving of the real nature of the proportion of God’s existence and his essential attributes as compared to ours; such a comparison is certainly Biblical.
Second, there is certainly nothing impossible about the Son and the Spirit being generated at -∞. It is as valid a point on the time line as any other. It is both this possibility and this necessity that we established in our discussion on the nature of the arche in John 1:1. In fact, we can say that -∞ is in fact the arche of our co-ordinate system; however, we could also say that +∞ is likewise an arche of our system, and that God (and only God) exists at both. Put another way, we could say that α=-∞ and ω=+∞; this is certainly correct since the Scriptures teach that God, the “Alpha and the Omega,” is at both. So we have further illustrated the nature of the arche relative to the whole co-ordinate system, and thus to the entire universe.
Using this to clarify our examination, we have said that the Father is the arche and exists from -∞ to +∞. We have also said that the Son and the Spirit were generated at -∞ and exist to +∞. In doing this we have removed the time proportion of Father, Son or Spirit relative to anything that is finite in nature. This is an essential attribute of deity. There was never a time when the Son and the Spirit did not exist, but on the other hand if we say that the Father generated the Son and the Spirit at -∞ we can maintain at the same time the priority of the Father, as we see in the Scriptures.
Turning to the creation, since it was created ex nihilo, there was a time when the creation – both material and spiritual – did not exist. Let us consider this event as our zero point, which is fine since the selection of this zero point is arbitrary. The time before this point is infinite, and the time afterwards is likewise infinite. As we said before, if we consider the position of this point relative to infinity, then its position becomes irrelevant, because any point we choose is in the centre of the co-ordinate system. This should help us in answering the question “Why did God create the heavens and the earth when he did?” because the specific time is in reality not a serious consideration relative to God. We can also say that any other finite point of time is the same distance from either infinite point; thus, time in general is not significant relative to God, as we have said before.
We now must consider the course of the universe after this event. The end of the universe and of matter is a debatable point, because we know from physics that matter (or more accurately the matter-energy continuum) cannot be created or destroyed, but only transformed. Since we have undermined the first point (it had to come from somewhere), we could say that at the end of time matter would be destroyed. But the Scriptures do not necessarily teach this; the end of “things” can either be taken to be their annihilation or their transformation. But we know that spiritual beings have an eternal existence from the time of their creation forward. This leaves them, however with at best only half of the existence in time as God has and furthermore they are subjected to other limitations such as limited intelligence, lack of omnipresence and omnipotence, etc.
We have thus seen that the mathematics that we have employed are useful in quantifying (if that term can be intelligently used relative to infinite matters) the relationship between God and his creatures. We have seen that, in drawing the analogy between an infinite God and infinity as a mathematical quantity, we can understand more about what it really means for God to be infinitely anything and everything that he is. We also see that the existence of created beings cannot be compared with God except that, if they have existence at -¥, they can be said to exist in a sense half as long as God has. Our one dimensional graph – as is the case with the Greek philosophers and their Christian students – cannot explain how the Son and the Spirit can be both subordinate to God and God at the same time, so we must expand our view on this subject.
A Broader View
Let us consider the co-ordinate system as shown below.
Instead of the one-dimensional representation we have been used to up to now, we have a two dimensional representation. The following discussion could apply to co-ordinate systems of more than two dimensions but it is simpler to discuss a two-dimensional representation. We also should note that co-ordinate shifting applies to this system as it does to our one dimensional one; moreover, in addition to the translation (linear movement) of the origin we can also rotate the co-ordinate system relative to its original orientation; we also have the option of doing both. Now let is superimpose a circle in the centre of the co-ordinate system of a finite radius as shown below and consider the area contained within.
We know that the area of this circle, as long as the radius is finite, is also finite. We also know from our previous discussion that, if this area is compared with any area with a boundary at infinity, then there is no comparison; the division results in essentially zero. Our use of the term “boundary at infinity” is mathematically sensible but in reality an understatement, since there is no boundary properly at infinity. It makes no difference how far the circle (or any other shape) is extended, as long as it is finite then any comparison with (division by) infinity results in zero.
Now let us look at an area that encompasses the entire co-ordinate system.
We now have an area that is infinite; its “boundary” is at infinity at all points and angles. No matter in what direction from the origin one goes we are still within the area. If we consider the finite area of the previous graph and divide this by the infinite area of the present one, we will still obtain zero.
In this graph we see an area that takes up “half” of the co-ordinate system. It is infinite in all directions to the right of the origin. Its area is likewise infinite because it has a boundary (in this case a 180° boundary) at infinity. Moreover, if we shift the origin in either direction, or rotate the co-ordinate system, the area is still infinite, no more or less so than in the original position.
We previously set forth an area which makes a 180° fan about the origin. We should note that this angle can vary, as we see above. It is important to note, however, that if the angle is greater than zero then the area is still infinite, as the area contains a boundary at infinity. As before any finite area has no meaningful proportion to this infinite area irrespective of how large or small the angle is as long as it is non-zero.
Up to now we have gone through very quickly some very detailed mathematics about areas. It is time to apply this to the matter at hand and come to some definite conclusions.
We need to be clear from the very start that what we are dealing with here is a group of analogies. It is not our purpose to make an exact mathematical representation of the Godhead. Analogies concerning God and the Trinity have been used since the subject first came up. The advantage in using a mathematical analogy is that mathematics can be used to precisely quantify and qualify things that do not have a physical representation, and certainly spiritual things fall into that category. However, we should be aware that it is no more possible to make an exact model of the Godhead using mathematics than with anything else. We are dealing with things that are beyond finite intellectual definition.
So what are we to make of these areas? The areas represent the extent of God’s activity and being, which are both one and the same with God. For such an area to make sense for deity it must be infinite; moreover in being infinite we can draw lines within the area that are infinite, just as the one-dimensional time line is infinite. We should note that these areas are not specifically meant to deal with time progression, although one could pick one of the axes to do this.
If we consider an area, we can consider any number of lines or curves within the area, or even sub-areas within the area. These can be considered to be the various aspects of God’s being, and thus his activity. We should be careful how we delineate these because we are normally used to categorising God’s attributes — power, love, eternity, knowledge, etc. – into a few categories. But these categories reflect how we look at God. Since God is one such categorisation is for our convenience.
The voids we see in the later graphs may be more instructive. In these graphs the area under consideration is still infinite and still has a border on infinity. But there are parts that are missing. What could these parts consist of? What could be missing in God, be it Father, Son or Holy Spirit? It is neither our place nor desire to deprive God of anything, even if it were possible (which it is thankfully not.) The Scriptures, however, speak of things that indicate the existence of these blank areas; some examples are as follows:
- The Son does not know things the Father does. (Matthew 24:36)
- The Father has reserved certain decisions to himself. (Matthew 20:23, Matthew 26:42, John 5:30)
- The Father is greater than the Son. (John 14:28) It is important to note here that the difference here is “greater” and not “better;” it is quantitative, not qualitative.
- The Son was sent to take away the sins of the world, which he did. (John 1:29) How is it possible for the Son to accommodate these sins?
Such things are explained by the voids.
Let us start by stating that the Father – the arche – can be likened to our graph with the entire co-ordinate system part of the shaded area. He is infinite in all respects. There is no comparison to him by any finite creature. Movement by him is meaningless – and philosophically non-existent – because he extends to all infinities completely. Beyond the Father, beyond God, a beyond cannot be said because it does not exist.
Next we may consider the Son as an infinity with a non-zero angle. He is still infinite, but not as great as the Father. Let us assume for simplicity’s sake that that angle is 180°. Strictly speaking this makes the Son “semi-infinite,” but still infinite. Under this assumption the Son is, for practical purposes, half of the Father, but it is important to note that there is in any case some kind of reasonable comparison to the Father. If we turn to comparing the Son to finite creature we discover once again that there is no comparison – no proportion – between the Son and creatures because the Son is infinite.
This then is the key to John 17:3 and indeed where our analysis reaches the critical moment. Jesus, preparing to go to the Cross for our redemption, accurately calls the Father “the only true God.” Why? Because the Father has the quality of “infinity” in all directions, while the Son (and presumably the Spirit as well) have it only in some, albeit having been generated at -∞. Jesus could look at the Father from the garden and, understanding this proportion (or something like it could call the Father the only true God. On the other hand we, finite and created at a certain point in time as we are, have no proportion with either the Father, the Son, or the Spirit; they are all God in the true sense of the word. So we have established the reason why Jesus could on the one hand call the Father the only true God and himself be really God as well.
The one difficult question that comes out of this concerns the proportionality of the Son and the Spirit to the Father. We know that it is less than unity, but how much? This is a question that the Scriptures are loath to answer. Among those who maintain both the deity and the subordination of the Son and the Spirit, there is variance in this. We have seen that many ante-Nicene thinkers such as Origen were prepared to make this proportionality very small; this makes some very nervous. At this point we are in the realm of speculation, something that is very dangerous, but we need to at least hypothesise a bit about it.
If we stick to the “pie shaped” diagrams that we presented above, we said that, as long as the “angle” (which establishes the proportionality) is between 0° and 360°, then we have something reasonable, although small angles, as we have said before, make some uneasy. We should like to present an interesting alternative for consideration.
This depicts an area that encompasses the entire co-ordinate system except for a line directly to the left of the origin (the negative x-axis in Cartesian co-ordinates.) The angle is of course 360°, but we leave out the line where the angle starts and ends. This line of course has zero width as is the case with all lines; the line is depicted with additional thickness for visual purposes. Since the line has zero width it has zero area; thus, this area is the same as the one for the entire co-ordinate system. We see, however, that this area is not continuous as the previous one but has a break, a break that extends back to infinity. Such a break is referred to as a “branch cut” and is important in complex analysis.
Where We Stand
We have completed a very unusual analysis of the concept of the subordination of the Son and the Spirit within the Godhead. Our purpose is not to denigrate the Son and the Spirit but to show that their subordination, which is taught in the Scriptures, is not contradictory to their divinity. We have employed mathematics to accomplish this because it is a convenient language to do so; it contains the concept of infinity while at the same time enabling us to look at such infinity in a multi-faceted way.
The main weakness of Greek philosophy in this regard is that it looks at the unique existence of divinity in a one-dimensional way; divinity according to this model has characteristics that do not really describe the relationship amongst the Father, Son and Spirit. To make a Trinitarian concept work in this framework either involves the denial of the deity of the Son or the denial of his subordination to the Father. From a strictly Biblical view this is unacceptable, but its upholding of the divinity of the Son has outweighed this problem for many years, and certainly still does as opposed to Arianism.
The evident question now is this: what use is all of this, other than making the Arian’s life miserable? We want to turn to this subject now, while at the same time investigating the basic reason why Arianism failed in the first place and why it is not a viable system of belief now.
Canon 36, Synod of Laodicea. In this case “mathematicians” are those “who hold the opinion that the celestial bodies rule the universe, and that all earthly things are ruled by their influence.” (Balsamon) Such activity presupposes that the stars, planets, etc. are living beings, a very common belief in antiquity.
We should note that co-ordinate shifting in more than one dimension can involve both rotation of the co-ordinates as well as translation of the origin. However, in one dimension it should be obvious that only translation can be done.
This is a simplification of how one would state this mathematically. Strictly speaking, this should be stated as
assuming x is a finite quantity.
Strictly speaking, the correct term for this is “semi-infinite.” This means that something is infinite in “one direction.” With a line, it proceeds from a point to infinity in one way; with a plane, it is everything from one side of a bisecting line onward; in a three dimensional space, it is everything on one side of a bisecting plane, etc.
This is a complicated point because even people who admit that the universe was created at one point in time deny the occurrence of creative miracles because they contend that matter cannot be created further. But it should be evident that, if God had the power and intelligence to create the universe in the first place, he could create other beings or matter at a later time. It should be clear, though, both that matter cannot be created by natural means and that God is capable of using pre-existent matter for his own purposes, in addition to retaining the option of a fully creative miracle.
We are aware that our friends in the Watchtower teach that many beings are annihilated as opposed to receiving eternal punishment. But this does not affect our argument because at least some created beings according to their own teachings remain forever.
We have elected to use a polar co-ordinate system as opposed to a Cartesian one. The reasons for our choice are rather involved but it will make some of the following discussion simpler. As was the case with the “zero” point on the one dimensional system a choice of co-ordinate systems can be made to make the solution of a given problem simpler.
Three-dimensional systems are the first ones to come to mind, although mathematically any number of dimensions can be represented. The physical representation of more than three dimensions, however, becomes difficult.
We choose this word very carefully; it is the same word Jesus used to compare the kingdom of Heaven to various things.
This proportion manifested itself in a number of ways which we have already seen, i.e., the fact that Jesus did not know the hour of his return, that the Father’s will prevailed in the garden and on the cross, etc. The most important aspect of this concerns how the Son could take on the sins of the world; this will be dealt with later.
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