Gods as Topological Invariants

Daniel Schoch

(Submitted on 1 Apr 2012)

We show that the number of gods in a universe must equal the Euler characteristics of its underlying manifold. By incorporating the classical cosmological argument for creation, this result builds a bridge between theology and physics and makes theism a testable hypothesis. Theological implications are profound since the theorem gives us new insights in the topological structure of heavens and hells. Recent astronomical observations can not reject theism, but data are slightly in favor of atheism.

Comments:	Please note that the publication date is April 1st 2012
Subjects:	Popular Physics (physics.pop-ph); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph); General Topology (math.GN)
MSC classes:	54-XX, 37C15, 20K45, 14J80
Cite as:	arXiv:1203.6902v1 [physics.pop-ph]

Submission history

From: Daniel Schoch [view email] 
[v1] Sun, 1 Apr 2012 10:11:34 GMT (6kb)

God as  Topological  Invariants

Abstract.

We show that the number of gods in a universe must equal the

Euler characteristics of its underlying manifold. By incorporating the clas-

sical cosmological argument for creation, this result builds a bridge between

theology and physics and makes theism a testable hypothesis.

Theological implications are profound since the theorem gives us new insights

in the topo logical structure of heavens and hells.

Recent astronomical observations can not reject theism, but data are slightly

in favour of atheism.

Motivation

Conventional theology builds on faith and metaphysical assumptions. While

faith is unquestionable, metaphysics is generally considered untestable. This lead to

the widespread assumption that theology and natural science form non-overlapping

magistrates.

However, nothing could be further away from the religious tradition.

Both Arabic and Christian medival thinker were intending a synthesis between the

Aristotelic scientific view and their specific religion, in particular the Abrahamitic

monotheism. Up to the early modern times, Galileis processes and the rejection of

atomism by the Catholic church have revealed possible conflict zones between both

fields.

Although metaphysical speculations always linked theological and mathematical

concepts, the introduction of distinctive mathematical methods to support theism

has long been attributed to Euler. In his famous alledged encounter with

Diderot, Euler presented a mock algebraic proof to embarrass the atheist philosopher

Diderot, which is probably a 19th century legend.

The true core of the legend, however, might be that the possibility of an algebraic

proof of the existence of god was discussed among 18th century intellectuals

[Struik1967, p. 129].

The first successful step in mathematization of theology was taken by Goedel

[Sobel 1987]. His formalization of the ontological argument, originally formulated

in terms of modal logic, has been reconstructed in set-theoretical form [Essler 1993].

It was noticed that the set structure was identical to the topological concept of an

ultrafilter, which led to speculations about the specific relations between theology

and topology [Calude et al. 1995].

It is the aim of this paper to advance these a priori speculations and convert

them to a testable theory, linking theology and cosmology very much in the spirit

of the medival Christain tradition. Although current data is not deceisive in this

matter and is slightly supporting a zero-god universe, the path has been paved for

a fruitful interdisciplinary collaboration of physics, mathematics, philosophy and

theology.

Date: April 1st, 2012.

Key words and phrases.

Topology, Euler characteristics, manifolds, invariants, mathematical theology,

mathematical joke.

1 GODS AS TOPOLOGICAL INVARIANTS 2

2. The Cosmological Argument

The cosmological argument (Platon, Philoponos, Aquinas etc.) states that a

First Cause of the universe must exist, since no causal chains are finite and do not

contain loops. Insofar it is based on the simple fact that any component of an

ordered and circle-free finite graph is a tree and thus must have a first vertex. To

apply this to a pre-Einstein universe one must add the assumptions that matter

does not by itself emerge out of the vacuum, that no motion can occur out of rest,

and that initial or primordial matter is at rest. It follows that a First Mover must

have initiated all cosmic motion at the moment the universe comes into existence.

Since it is not by itself a natural phenomenon, such a mighty cause can only be a

god.

However, this argument does not remain valid if the universe has origined from

a Big Bang. If the universe expands out of a zero-volume point at t = 0, there no

causal relation can connect an event at some time t > 0 of the existing universe by

an event t < 0 before the Big Bang. In the watchmaker analogy, the watch could

not have been wound up.

We nevertheless embrace the result of the argument, as far as it is restricted

to the types of steady-state universes for which it was originally intended. It goes

without saying that anything beyond the three-dimensional Euclidean space was

out of imagination for the medieval scholar. Before Einstein, time was considered

absolute and independent of space and matter.

A physical explanation for a universe emerging out of nothing was unthinkable

and incompatible with the mechanics of their time, may it be Aristotelian, Galileian

or Newtonian. The initial singualarity of an Einstein-Friedman universe is, however,

a distinctive topological feature of the manifold itself. We assume therefore, in

accordance with the cosmological argument, that a finite Aristotelian universe,

which manifold can be desribed by a compact subset of R3 homeomorphic to a

ball (a 3-cell), has one and only one god.

3. The Main Theorem

Let U be a unviverse with underlying manifold MU. By _(U) we denote the

number of gods in the universe. We postulate the following axioms.

From the cosmological argument we obtain

Axiom 1. The number of gods in a 3-cell is one.

Gods are eternal, invariable and do not depend on the evolution of the cosmos

under the laws of physics. Changing the latter would have had no influence on the

existence of gods. Since the laws of physics are continuous, if M0 and M1 are the

underlying manifolds of the spatial universe at some time t0 and t1 (in comoving

coordinates), respectively, they must be homotopy equivalent. We therefore obtain

Axiom 2. The number of gods is a homotopy invariance.

Clearly, each god only belongs to only one universe. In a multiverse consisting

of disjoint unions of universes, the number of gods must therefore satisfy additivity.

Axiom 3. Let U and U′ be separated universes with compact manifolds. Then

the number of gods in the disjoint union is the sum of the numbers of gods of the

parts, (3.1) _(U ⊔ U′) = _(U) + _(U′) .

GODS AS TOPOLOGICAL INVARIANTS 3

Theorem 1. The number of gods in a universe equals the Euler characteristics of

the underlying manifold,

_(U) = _ (MU) .

Proof. By the second axiom, the number of gods only depend on the underlying

manifold. Thus we can write w.l.o.g. _(U) = _(MU). Since the n−cell is null

homotope, by axiom 1 and 2 we obtain

(3.2) _(pt) = 1.

Let further M and N be any compact sets. We construct a homotopy transforming

M ∪ N into separated copies A,B,C of the closure of M \N, M ∩ N, and M \ N,

respectively.

Equation (3.1) together with axiom 2 implies

_(M ∪ N) = _(A) + _(B) + _(C) ,

_(M) = _(A) + _(B) ,

_(N) = _(B) + _(C) ,

_(M ∩ N) = _(B) .

We obtain the inclusion-exclusion principle,

_(M ∪ N) = _(M) + _(N) − _(M ∩ N) .

By a well-known characterization, the latter together with (3.2) implies _(M) =

_ (M) for all compact manifolds M. _

The product theorem for the Euler characteristics implies that for manifolds M

and N

_(M ⊗ N) = _(M) ・ _ (N) .

In particular, if time is itself an interval T ⊂ R, we find that the number of gods

are independent of time,

_(M ⊗ T ) = _(M) .

This is compatible with the scholastic view introduced by Boetius in his Consolations

that god is above time. However, the same formula spells trouble for all

theologies which are based on a cyclic conception of time, which is widespread in

India (Veda) and among native American religions. Since _ 􀀀S1_ = 0 there are no

gods in such a universe, _􀀀M ⊗ S1_ = 0.

4. Universes, Heavens and Hells

The number of gods has come out to be an integral number. This rules out

any demi-gods or lower devas, as they are known from Greek or Indian mythology.

However, the divine cardinal can get negative; with the obvious interpretation of

these gods being devils. By the additivity theorem, components of positive and

negative Euler characteristics could cancel each other out.

We can safely assume, however, (and there is plenty of support from religious texts)

that gods and devils can not have stable coexistence in the same part of the universe.

Thus each component contains only gods or devils, but not both. The absolute value

of the Euler characteristics of the universe therefore equals the number of supreme

and most inferior beings in it, dependent on the sign.

A lot of types of universes are godless. These are all spaces which contain the

1-sphere (circle) as a factor, such as the tori and all products of them with an

GODS AS TOPOLOGICAL INVARIANTS 4

Arbitrary manifold. This also applies if the universe is infinite Euclidean, but has

additional warped dimensions, as suggested in string theory. Also, a spherical

three-dimensional universe would have no gods.

The only non-exotic topology with a positive number of gods are Euclidean spaces,

which all contain exactly one god, which is well in accordance with the Jewish-

Christian and Islamic tradition.

An interesting theoretical question concerns the topological structure of heaven.

The 3-dimensional Eucledian space is suitable, but since souls are immaterial, they

are not confined to three-dimensionality. It is unlikely, however, that heaven is a

bounded manifold, since there should be no limit in heaven.

One possible structure of a monotheistic heaven is the real projective plane.

Another is the 2-sphere, but it requires a pair of gods. The 2-spehere would have

been a preferred choice of Greek philosophers, since its imbedding in the R3 is

in perfect alignment with the Pythagorean-Platonic idea of a perfect body, which was

influential in the early Scholastics.

A suitable pair of gods entrenched in the Abrahamitic tradition is

Jahwe and Asherah, but apparently the couple broke up some time after the late

Bronze age [Finkelstein and Silberman 2001].

Constraints by traditional religion are more relaxed insofar there is no dogma

imposing an upper boundary for the number of devils. As in the case of heavens,

souls are not restricted to exist only in spaces with dimesion at least three. If hells

are two-dimensional closed surfaces with topological genus g > 1, then the Euler

characteristics is 2 − 2g, which would correspond to a hell with an even number

of 2 (g − 1) devils. Such hells can best be envisioned as multiple tori.

The double torus in the form of the figure 8 has Euler characteristics -2.

Each torus attached decrease the Euler characteristics by a further -2.

This suggest that the rings of hell are not concentric, as Dante speculated, but that

they are lined up such that the soul transgresses through a complete half ring of hell

before entering the next level. This is a more hellish scenario than Dante’s concentric

model and thus more realistic.

5. Evidences

The topology of the universe could in principle be observed if it is finite and

small (respectively old) enough to have light travelled through it. In this case one

would observe multiple images of the same constellations of a matter, which for

each point source of light takes the form of a circle. However, this method is not

discriminative for a large or young universe, where it only yields a lower bound

for the size of the univese [Bielewicz and Banday 2011].

A test for infinity of the universe in one or several dimensions can be based on

statistical analysis of the temperature fluctiations of the background radiation,

which is a remainder of the Big Bang. If one or more dimensions of the space

are topological circles, space remains homogenuous, but isotropy is violated.

Unfortunately, topology is just constrained, but not determined by local curvature.

Data from the Wilkinson microwave anisotropy probe suggest that the universe

is flat with only 0.5% margin of error [Cornish et al. 2004]. A flat universe can

have vanishing total energy consistent with an origin from nothing.

An infinite Euclidean space fits the data.

Some exotic topologies such as the Poincaré homology sphere and the Picard

horn have been claimed consistent with the findings, but for the former this has

been challenged [Key et al. 2007].

A recent statistical analysis on the number of infinite dimensions compared the

Euclidean space R3, the 3-torus

GODS AS TOPOLOGICAL INVARIANTS 5

T 3 and the manifolds T 2 ⊗ R and S1 ⊗ R2 [Aslanyan and Manohar 2011].

Only the Euclidean space has a non-vanishing Euler characteristics.

The most probable topology of the universe was found to be T 2 ⊗ R, which

would support the atheist view brought forward by many leading cosmologists.

References

• [Aslanyan and Manohar 2011] Aslanyan, G. and Manohar, A. 2011. The Topology and the Size of the Universe. Arxiv Preprint. arXiv: 1104.0015.
• [Bielewicz and Banday 2011] P. Bielewicz, P. and A.J. Banday. 2011. Constraints on the topology of the Universe derived from the 7-yr WMAP data. Monthly Notices of the Royal Astronomical Society Vol 412 Issue 3, 2104-2110.
• [Calude et al. 1995] Calude, C., Marcus, S. and Stefanescu, D. 1995. The Creator versus its Cre- ator. A Mathematical Exercise. Preprint.
• [Cornish et al. 2004] Cornish, N.J., Spergel, D., Starkman, G. and Komatsu, E. 2004. Constrain-ing the Topology of the Universe. Phys Rev Letters Vol 92 No 20.
• [Essler 1993] Essler, W. 1993. Grundzüge der Logik, Vol.2, Klassen, Relationen, Zahlen. 4th ed. Klostermann.
• [Finkelstein and Silberman 2001] Finkelstein, I. and N. Silberman. 2001. The Bible Unearthed: Archaeology’s New Vision of Ancient Israel and the Origin of Its Sacred Texts. Free Press, New York.
• [Key et al. 2007] Key, J.S., Cornish, N.J., Spergel, D.N. and G.D. Starkman. 2007. Extending the WMAP bound on the size of the Universe. Physical Review D, Vol 75 Issue 8, DOI: 10.1103/PhysRevD.75.084034.
• [Sobel 1987] Sobel, J. 1987. Goedel’s ontological proof, in: J.J. Thomson (eds.)., On being and saying, MIT Press.
• [Struik1967] Struik, D. 1967. A Concise History of Mathematics, Third Revised Edition, Dover.

Autor: DANIEL SCHOCH

Source: arXiv:1203.6902v1 physics.pop-ph

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