God is Open
Incarnation is Compactified God
The soul is Open
Reincarnation is Compactified Soul
God and the Soul are Homeomorphic
God is without Boundaries
The soul is with Boundaries
God and the Soul are not Diffeomorphic

God is Open.

God is open. God regrets. God cannot accept the consequences of his own will. If God is omnipotent, and can see perfectly the consequences of his actions, then God would never regret. However, God does regret. God himself is open to the non-deterministic future. This is what the latest theology from the American evangelicals, the theology of the open God, claims.([i])

Let us take a look at the Bible. God regretted creating humanity so much that he sent Noah’s flood. 

“I will wipe from the face of the earth the human race I have created […] for I regret that I have made them.” (Genesis[JDP1]  6:7)

God also tested Abraham by demanding that he sacrifice his only son, Isaac. This is because God could not know what Abraham would do.

“Now I know that you fear God, because you have not withheld from me your son, your only son” (Genesis 22:12)

In addition, God was angered when the Israelites worshipped the golden calf idol they had created, and was about to smite them, but was stayed by a prayer from Moses.

“Then the Lord relented and did not bring on his people the disaster he had threatened.” (Exodus 32:14)

To relent after a prayer from a human seems to mean that the future as God has determined it can be changed by human action. The future is open for both God and Man.

God is open. As a metaphor for this openness of God, we can use the open set of topological space.([ii]) In other words, we can assign God the predicate of being an open set.

Discussing the meaning for using the language of mathematics as a metaphor for God should properly be left to another article, but we can briefly look at it here. In daily language, the metonymic use of metaphor is called poetry.([iii]) It goes without saying that daily language is a metonymic, contiguous combination[JDP2]  and at the same time, has a rich metaphorical, similar selection. However, the language of mathematics is a pure metonymic, contiguous combination and rejects any and all metaphorical, similar selections. We use this language of mathematics, pure metonymy as metaphors. Could we not perhaps call this metaphorical use of metonymy poetry? If we were to do so, it would be an attempt at a mathematical poem.

Incarnation is Compactified God

God was incarnated due to his love of humanity.

“The Word became flesh and made his dwelling among us.” (John 1:14)

“For God so loved the world that he gave his one and only Son.” (John 3:16)

The open God is incarnated. Let us consider compactification in topological space as a metaphor for this incarnation.([iv]) The metaphor of the open God is the open set in topological space. Let us consider the complex plane C as an example. The compactification of the complex plane C is sphere S2={(z,t)∈C×R;|z|2+t2=1}. While the complex plane C is open, the sphere S2 is closed. The spirit of God that is open in Heaven is incarnated as closed human flesh on Earth. This phenomenon can be likened to the compactification and closure of the open set.

The Soul is Open

Let us consider the human soul apart from its flesh. This is the form used in Greek philosophy.

“But [the soul] thinks best when none of these things troubles it, neither hearing nor sight, nor pain nor any pleasure, but it is, so far as possible, alone by itself, and takes leave of the body, and avoiding, so far as it can, all association or contact with the body, reaches out toward the reality.” (Phaedo 65c[JDP3] )([v])

“If, as we are always saying, the beautiful exists, and the good, and every essence of that kind, and if we refer all our sensations to these,” (Phaedo 76d)

“which we find existed previously and are now ours, and compare our sensations with these, is it not a necessary inference that just as these abstractions exist, so our souls existed before we were born?” (Phaedo 76e)([vi])

The soul apart from the flesh reaches out toward the reality. This soul must be open. Human souls are open. We shall once again consider the open set in topological space as a metaphor for this human soul.

Reincarnation is Compactified Soul

The soul is alive – that is, it has flesh in this world. For the soul to abide [JDP4] in flesh in this world is called reincarnation through its repetition. Therefore when the soul leaves the flesh is death.

“Well, then, this is what we call death, is it not, a release and separation from the body?” (Phaedo 67d)([vii])

 The endeavor of philosophy to keep the soul apart from the body to reach out towards reality is…

“the true philosophers practice dying” (Phaedo 67e)([viii])

Let us consider once more the compactification of the open set as a metaphor for this reincarnation of the soul. For example, if we consider the open disc D2={z∈C;|z|<1} as a metaphor for the soul, then its compactification is the sphere S2. The soul that is open in Heaven is closed on Earth, and thus alive. This appears to be a very appropriate metaphor for Platonic philosophy.

God and the Soul are Homeomorphic

If we consider the complex plane C as a metaphor of God, and the open disc D2 as a metaphor of the soul, then we have considered the sphere S2 that is the compactification of the two open sets of the complex plane C and the open disc D2. The mathematical term “open” is the dual of the mathematical term “compact.” Therefore the metaphors for “open” and “compact” are combined contiguously and metonymically. In other words, the metaphors of open and compact are metaphorical uses for metonymy.

Moving on, the complex plane C that is the metaphor of God and the open disc D2 that is the metaphor of the soul are the same in terms of topological space: in other words, they are homeomorphic.([ix]) This proposition is to be demonstrated, or in other words, the metonymic contiguous combination of the complex plane C and the open disc D2 itself. Using the complex plane C and the open disc D2 as metaphors for God and the soul is, in this sense, simply a metaphorical use of metonymy.

However, are there no problems with God and the soul being the same? Even if the spirit of God and the human soul, appear the same when given flesh on this earth, can they be the same when open in Heaven? We are asked what the transcendence from God to Man is.

God is without Boundaries

As a manifold that reifies topological space, the complex plane C that is the metaphor of God has no boundaries.([x]) The metaphor that God is unbounded is appropriate if we think of the infinity of God.

The Soul is with Boundaries

The open disc D2 that is the metaphor of the soul has the circumference S1={z∈C;|z|=1} that is its boundary as its manifold. The metaphor that the soul is bounded is also appropriate if we consider that the human souls are individual souls.

God and the Soul are not Diffeomorphic

God and the soul are homeomorphic as topological space. This proposition stands regardless of whether there are boundaries or not. However, if we view God and the soul as manifolds, then the existence of a boundary becomes crucial. The complex plane C that is the metaphor of God and the open disc D2 that is the metaphor of the soul are not the same as manifolds, or in other words, they are not diffeomorphic.([xi]) With the exception of constant mapping, there is no differentiable mapping from the complex plane C which has no boundaries to the open disc D2 which has the circumference S1 as its boundary, and there cannot be any differentiable inverse mapping from the open disc D2 to the complex plane C.([xii])

Therefore God and the soul are the same in terms of topological space but not the same in terms of manifolds. God and the soul both exist in Heaven, but God is absolutely transcendent over the soul. This is because God is both open and without boundaries, but the soul is both open and with boundaries. Without boundaries as a metaphor for God and with boundaries as a metaphor for the soul are also dual as mathematical terms, and combine metonymically and contiguously. Furthermore, the fact that the unbounded complex plane C that is a metaphor for God and the bounded open disc D2 that is a metaphor for the soul are not diffeomorphic is a proposition to be demonstrated, and is simply a metonymic, contiguous combination. Here too the metaphorical use of mathematics as a pure metonymy is applied.

Apeiron Centre, 2014

[i] Rice, Richard, The openness of God, Downers Grove, IL. InterVarsity Press, 1994,  pp15-16.

[ii] Bourbaki, Nicolas, Topologie générale, chapitre 1, Berlin, Springer, 2007, p1.

[iii] Jakobson, Roman, Language in Literature, Cambridge, MA. Harvard UP, 1987, p71.

[iv] Bourbaki, p59.

[v] Plato, Phaedo, Oxford, Oxford UP, 2009, p11.

[vi] Plato, p27.

[vii] Plato, p13.

[viii] Plato, p14.

[ix] Bourbaki, p2.

[x] Morita, Shigeyuki, Geometry of Differential forms, Providence, RI. American Mathematical Society, 2001, p15, p45.

[xi] Morita, p5.

[xii] Kodaira, Kunihiko, Complex Analysis, Cambridge, Cambridge UP, 2007, p59.


Ochiai Hitoshi

Ochiai Hitoshi is a professor of mathematical theology at Doshisha University, Kyoto. He has published extensively in Japanese. All books are written in Japanese, but English translations of the most recent two books Kantoru—Shingakuteki sūgaku no genkei カントル 神学的数学の原型 [Cantor: Archetype of theological mathematics], Gendai Sūgakusha, 2011; and Sūri shingaku o manabu hito no tame ni 数理神学を学ぶ人のために [Those Learning Mathematical Theology], Sekai Shisōsha, 2009) are available from the author at hochiai@mail.doshisha.ac.jp.


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