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The symbol for infinity that one sees most often is the lazy eight
curve, technically called the lemniscate. This symbol was first used in
a seventeenth century treatise on conic sections.^{1} It caught
on quickly and was soon used to symbolize infinity or eternity in a
variety of contexts. For instance, in the 1700s the infinity symbol
began appearing on the Tarot card known as the Juggler or the Magus. It
is an interesting coincidence that the Qabbalistic symbol associated
with this particular Tarot card is the Hebrew letter
À
(pronounced alef), for Georg Cantor, the founder of the modern
mathematical theory of the infinite, used the symbol
À_{0}
(pronounced alef-null) to stand for the
first infinite number.

The appropriateness of the symbol for infinity lies in the fact that one can travel endlessly around such a curve ... demolition derby style, if you will. Endlessness is, after all, a principal component of one's concept of infinity. Other notions associated with infinity are indefiniteness and inconceivability.

Infinity commonly inspires feelings of awe, futility, and fear. Who
as a child did not lie in bed filled with a slowly mounting terror while
sinking into the idea of a universe that goes on and on, for ever and
ever? Blaise Pascal puts this feeling very well: "When I consider
the small span of my life absorbed in the eternity of all time, or the
small part of space which I can touch or see engulfed by the infinite
immensity of spaces that I know not and that know me not, I am
frightened and astonished to see myself here instead of there ... now
instead of then."^{2}

It is possible to regard the history of the foundations of
mathematics as a progressive enlarging of the mathematical universe to
include more and more infinities. The Greek word for infinity was
*apeiron,* which literally means unbounded, but can also mean
infinite, indefinite, or undefined. *Apeiron* was a negative, even
pejorative, word. The original chaos out of which the world was formed
was *apeiron.* An arbitrary crooked line was *apeiron.* A
dirty crumpled handkerchief was *apeiron.* Thus, *apeiron*
need not only mean infinitely large, but can also mean totally
disordered, infinitely complex, subject to no finite determination. In
Aristotle's words, "... being infinite is a privation, not a
perfection but the absence of a limit..."^{3}

There was no place for the *apeiron* in the universe of
Pythagoras and Plato. Pythagoras believed that any given aspect of the
world could be represented by a finite arrangement of natural numbers
(where "natural number" means "whole number"). Plato
believed that even his ultimate form, the Good, must be finite and
definite. This was in contradistinction to almost all later
metaphysicians, who assumed that the Absolute is necessarily infinite.
In the next chapter I will discuss the way in which Greek mathematics
was limited by this refusal to accept the *apeiron,* even in the
relatively harmless guise of a real number with an infinite decimal
expansion.

Aristotle recognized that there are many aspects of the world that
seem to point to the actuality of the *apeiron.* For instance, it
seems possible that time will go on forever; and it would seem that
space is infinitely divisible, so that any line segment contains an
infinity of points. In order to avoid these actual infinites that seemed
to threaten the orderliness of his *a priori* finite world,
Aristotle invented the notion of the *potentially infinite* as
opposed to the *actually infinite.* I will describe this
distinction in more detail in the next section, but for now let me
characterize it as follows. Aristotle would say that the set of natural
numbers is potentially infinite, since there is no largest natural
number, but he would deny that the set is actually infinite, since it
does not exist as one finished thing. This is a doubtful distinction,
and I am inclined to agree with Cantor's opinion that "... in truth
the potentially infinite has only a borrowed reality, insofar as a
potentially infinite concept always points towards a logically prior
actually infinite concept whose existence it depends
on."^{4}

Plotinus was the first thinker after Plato to adopt the belief that
at least God, or the One, is infinite, stating of the One that,
"Absolutely One, it has never known measure and stands outside of
number, and so is under no limit either in regard to anything external
or internal; for any such determination would bring something of the
dual into it."^{5}

St. Augustine, who adapted the Platonic philosophy to the Christian
religion, believed not only that God was infinite, but also that God
could think infinite thoughts. St. Augustine argued that, "Such as
say that things infinite are past God's knowledge may just as well leap
headlong into this pit of impiety, and say that God knows not all
numbers. . . . What madman would say so? . . . What are we mean wretches
that dare presume to limit His knowledge?"^{6}

This extremely modern position will be returned to in the last
section of this chapter. Later medieval thinkers did not go as far as
Augustine and, although granting the unlimitedness of God, were
unwilling to grant that any of God's creatures could be infinite. In his
*Summa Theologiae* St. Thomas Aquinas gives a sort of Aristotelian
proof that "although God's power is unlimited, he still cannot make
an absolutely unlimited thing, no more than he can make an unmade thing
(for this involves contradictories being true
together)."^{7}
The arguments are elegant, but suffer from
the flaw of being circular: it is proved that the notion of an unlimited
thing is contradictory by slipping in the premise that a
"thing" is by its very nature limited.

Thus, with the exception of Augustine and a few others, the medieval thinkers were not prepared to deal with the infinitude of any entities other than God, be they physical, psychological, or purely abstract. The famous puzzle of how many angels can dance on the head of a pin can be viewed as a question about the relationship between the infinite Creator and the finite world. The crux of this problem is that, on the one hand, it would seem that since God is infinitely powerful, he should be able to bid an infinite number of angels to dance on the head of a pin; on the other hand, it was believed by the medieval thinkers that no actually infinite collection could ever arise in the created world.

Their proofs that infinity is somehow a self-contradictory notion
were all flawed, but there was at least one interesting paradox
involving infinity that the medieval thinkers were aware of. It would
seem that any line includes infinitely many points. Since the
circumference of a circle with radius two is two times as long as the
circumference of a circle with radius one, then the former should
include a *larger* infinity of points than the latter. But by
drawing radii we can see that each point *P* on the small circle
corresponds to exactly one point *P'* on the large circle, and each
point *Q'* on the large circle corresponds to exactly one point
*Q* on the small circle. Thus we seem to have two infinities that
are simultaneously different and equal.

In the early 1600s Galileo Galilei offered a curious solution to this
problem. Galileo proposed that the smaller length could be turned into
the longer length by adding an infinite number of infinitely small gaps.
He was well aware that such a procedure leads to various difficulties:
"These difficulties are real; and they are not the only ones. But
let us remember that we are dealing with infinites and indivisibles,
both of which transcend our finite understanding, the former on account
of their magnitude, the latter because of their smallness. In spite of
this, men cannot refrain from discussing them, even though it must be
done in a roundabout way."^{8}

He resolved some of his difficulties by asserting that problems arise
only "when we attempt, with our finite minds, to discuss the
infinite, assigning to it those properties which we give to the finite
and limited; but this I think is wrong, for we cannot speak of infinite
quantities as being the one greater or less than or equal to
another."^{9} This last assertion is supported by an
example that is sometimes called Galileo's paradox.

The paradoxical situation arises because, on the one hand, it seems
evident that most natural numbers are not perfect squares, so that the
set of perfect squares is smaller than the set of all natural numbers;
but, on the other hand, since every natural number is the square root of
exactly one perfect square, it would seem that there are just as many
perfect squares as natural numbers. For Galileo the upshot of this
paradox was that, "we can only infer that the totality of all
numbers is infinite, and that the number of squares is infinite . . .;
neither is the number of squares less than the totality of all numbers,
nor the latter greater than the former; and finally, the attributes
'equal,' 'greater,' and 'less,' are not applicable to infinite, but only
to finite quantities."^{10}

I have quoted Galileo at some length, because it is with him that we
have the first signs of the modern attitude toward the actual infinite
in mathematics. If infinite sets do not behave like finite sets, this
does not mean that infinite is an inconsistent notion. It means, rather,
that infinite numbers obey a different "arithmetic" from
finite numbers. If using the ordinary notions of "equal" and
"less than" on infinite sets leads to contradictions, this is
not a sign that infinite sets cannot exist, but, rather, that these
notions do not apply without modification to infinite sets. Galileo
himself did not see how to carry out such a modification of these
notions; *this* was to be the task of Georg Cantor, some 250 years
later.

One of the reasons that Galileo felt it necessary to come to some
sort of terms with the actual infinite was his desire to treat space and
time as continuously varying quantities. Thus, the results of an
experiment on motion can be stated in the form that *x* =
*f(t),* that space position is a certain function of continuously
changing time. But this variable *t* that grows continuously from,
say, zero to ten is *apeiron,* both in the sense that it takes on
arbitrary values, and in the sense that it takes on infinitely many
values.

This view of position as a function of time introduced a problem that
helped lead to the founding of the Calculus in the late 1600s. The
problem was that of finding the instantaneous velocity of a moving body,
whose distance *x* from its starting point is given as a function
*f(t)* of time.

It turns out that to calculate the velocity at some instant
*t*_{o}, one has to imagine measuring the speed over
an infinitely small time interval *dt.* The speed
*f'(t*_{o}*)* at *t*_{o} is
given by the formula (*f(t*_{o} + *dt)* -
*f(t*_{o}*))/dt,* as everyone who has ever
survived a first-year calculus course knows.

The quantity *dt* is called an *infinitesimal,* and obeys
many strange rules. If *dt* is added to a regular number, then it
can be ignored, treated like zero. But, on the other hand, *dt* is
regarded as being different enough from zero to be usable as the
denominator of a fraction. So is *dt* zero or not? Adding finitely
many infinitesimals together just gives another infinitesimal. But
adding infinitely many of them together can give either an ordinary
number, or an infinitely large quantity.

Bishop Berkeley found it curious that mathematicians could swallow
the Newton-Leibniz theory of infinitesimals, yet balk at the
peculiarities of orthodox Christian doctrine. He wrote about this in a
1734 work, the full title of which was, *The Analyst, Or A Discourse
Addressed to an Infidel Mathematician. Wherein It is examined whether
the Object, Principles, and Inferences of the modern Analysis are more
distinctly conceived, or more evidently deduced, than Religious
Mysteries and Points of Faith. "First cast out the beam out of
thine own Eye; and then shalt thou see clearly to cast out the mote out
of thy brother's Eye."* ^{11}

The use of infinitely small and infinitely large numbers in calculus
was soon replaced by the limit process. But it is unlikely that the
Calculus could ever have developed so rapidly if mathematicians had not
been willing to think in terms of actual infinities. In the past fifteen
years, Abraham Robinson's non-standard analysis has produced a technique
by which infinitesimals can be used without fear of contradiction.
Robinson's technique involves enlarging the real numbers to the set of
*hyperreal* numbers, which will be discussed in Chapter 2.

After the introduction of the limit process, calculus was able to
advance for a long time without the use of any actually infinite
quantities. But as mathematicians tried to get a precise
*description* of the continuum or real line, it became evident that
infinities in the foundations of mathematics could only be avoided at
the cost of great artificiality. Mathematicians, however, still
hesitated to plunge into the world of the actually infinite, where a set
could be the same size as a subset, a line could have as many points as
a line half as long, and endless processes were treated as finished
things.

It was George Cantor who, in the late 1800s, finally created a theory of the actual infinite which by its apparent consistency, demolished the Aristotelian and scholastic "proofs" that no such theory could be found. Although Cantor was a thoroughgoing scholar who later wrote some very interesting philosophical defenses of the actual infinite, his point of entry was a mathematical problem having to do with the uniqueness of the representation of a function as a trigonometric series.

To give the flavor of the type of construction Cantor was working with, let us consider the construction of the Koch curve shown in Figure 4. The Koch curve is found as the limit of an infinite sequence of approximations. The first approximation is a straight line segment (stage 0). The middle third of this segment is then replaced by two pieces, each as long as the middle third, which are joined like two sides of an equilateral triangle (stage 1). At each succeeding stage, each line segment has its middle third replaced by a spike resembling an equilateral triangle.

Now, if we take infinity as something that can, in some sense, be
attained, then we will regard the limit of this infinite process as
being a curve actually existing, if not in physical space, then at least
as a mathematical object. The Koch curve is discussed at length in
Benoit Mandelbrot's book, *Fractals,* where he explains why there
is reason to think of the Koch curve in its infinite spikiness as being
a better model of a coastline than any of its finitely spiky
approximations.^{12}

Cantor soon obtained a number of interesting results about actually
infinite sets, most notably the result that the set of points on the
real line constitutes a *higher* infinity than the set of all
natural numbers. That is, Cantor was able to show that infinity is not
an all or nothing concept: there are degrees of infinity.

This fact runs counter to the naive concept of infinity: there is
only one infinity, and this infinity is unattainable and not quite real.
Cantor keeps this naive infinity, which he calls the Absolute Infinite,
but he allows for many intermediate levels between the finite and the
Absolute Infinite. These intermediate stages
correspond to his *transfinite numbers* . . . numbers that are
infinite, but none the less conceivable.

In the next section we will discuss the possibility of finding
*physically* existing transfinite sets. We will then look for ways
in which such actual infinities might exist *mentally.* Finally we
will discuss the Absolute, or metaphysical, infinite.

This threefold division is due to Cantor, who, in the following passage, distinguishes between the Absolute Infinite, the physical infinities, and the mathematical infinities:

The actual infinite arises in three contexts:

firstwhen it is realized in the most complete form, in a fully independent other-worldly being,in Deo,where I call it the Absolute Infinite or simply Absolute;secondwhen it occurs in the contingent, created world;thirdwhen the mind grasps itin abstractoas a mathematical magnitude, number, or order type. I wish to make a sharp contrast between the Absolute and what I call the Transfinite, that is, the actual infinities of the last two sorts, which are clearly limited, subject to further increase, and thus related to the finite.^{13}

There are three ways in which our world appears to be unbounded and thus, perhaps, infinite. It seems that time cannot end. It seems that space cannot end. And it seems that any interval of space or time can be divided and subdivided endlessly. We will consider these three apparent physical infinities in three subsections.

Suppose that the human race was never going to die out -- that any given generation would be followed by another generation. Would we not then have to admit that the number of generations of man is actually infinite?

Aristotle argued against this conclusion, asserting that in this situation the number of generations of man would be but potentially infinite; that is, infinite only in the sense of being inexhaustible. He maintained that at any given time there would only have been some finite number of generations, and that it was not permissible to take the entire future as a single whole containing an actual infinitude of generations.

It is my opinion that this sort of distinction rests on a view of time that has been fairly well discredited by modern relativistic physics. In order to agree with Aristotle that, although there will never be a last generation, there is no infinite set of all the generations, we must believe that the future does not exist as a stable, definite thing. For if we have the future existing in a fixed way, then we have all of the infinitely many future generations existing "at once."

But one of the chief consequences of Einstein's Special Theory of Relativity is that it is space-time that is fundamental, not isolated space which evolves as time passes. I will not argue this point in detail here, but let me repeat that on the basis of modern physical theory we have every reason to think of the passage of time as an illusion. Past, present, and future all exist together in space-time.

So the question of the infinitude of time is not one that is to be dodged by denying that time can be treated as a fixed dimension such as space. The question still remains: is time infinite? If we take the entire space-time of our universe, is the time dimension infinitely extended or not?

Fifty, or even twenty, years ago it would have been natural to assert
that our universe has no beginning or end and that time is thus infinite
in both directions. But recently it has become an established fact that
the universe does have a beginning in time known as the Big Bang. The
Big Bang took place approximately 15 billion years ago. At that time our
universe was the size of a point, and it has been expanding ever since.
What happened before the Big Bang? It is at least possible to answer,
"Nothing." The apparent paradox of having a *first*
instant in time is sometimes avoided by saying that the Big Bang did not
occur *in* time . . . that time is open, rather than closed, in the
past.

This is a subtle distinction, but a useful one. If we think of time
as being all the points greater than or equal to zero, then there is a
first instant: zero. But if we think of time as being all the points
strictly greater than zero, then there is no first instant. For any
instant *t* greater than zero, one has an earlier instant
*t*/2 that is also greater than zero.

But in any case, if we think of time as not existing before the Big Bang, then there are certainly not an infinite number of years in our past. And what about the future? There is no real consensus on this. Many cosmologists feel that our universe will eventually stop expanding and collapse to form a single huge black hole called the Big Stop or the Gnab Gib; others feel that the expansion of the universe will continue indefinitely.

If the universe really does start as a point and eventually contract back to a point, is it really reasonable to say that there is no time except for the interval between these points? What comes before the beginning and after the end?

One response is to view the universe as an oscillating system, which repeatedly goes through expansions and contractions. This would reintroduce an infinite time, which could, however, be avoided.

The way in which one would avoid infinite time in an endlessly
oscillating universe would be to adopt a belief in what used to be
called "the eternal return." This is the belief that every so
often the universe must repeat itself. The idea is that a finite
universe must return to the same state every so often, and that once the
same state has arisen, the future evolution of the universe will be the
same as the one already undergone. The doctrine of eternal recurrence
amounts to the assumption that
Figure 11. From R. v.B. Rucker, *Geometry, Relativity, and the
Fourth Dimension.*

time is a vast circle. An oscillating universe with circular time is pictured in Figure 10.

There is a simpler model of an oscillating universe with circular
time, which can be called *toroidal space-time.* In toroidal
space-time we have an oscillating universe that repeats itself after
every cycle. Such a model is obtainable by identifying the two points,
"Big Bang" and "Big Stop," in Figure 11.

Note, however, that if the universe really *expands* forever,
then it cannot ever repeat itself, as the average distance between
galaxies is a continually increasing quantity that never returns to the
same value.

We now turn to a consideration of the possibility of spatial
infinities. The potential versus actual infinity distinction is
sometimes used to try to scotch this question at the outset. Immanuel
Kant, for instance, argues that the world cannot be an infinite whole of
coexisting things because "in order therefore to conceive the
world, which fills all space, as a whole, the successive synthesis of
the parts of an infinite world would have to be looked upon as
completed; that is, an infinite time would have to be looked upon as
elapsed, during the enumeration of all coexisting
things."^{15}

Kant's point is that space is in some sense not already really there -- that things exist together in space only when a mind perceives them to do so. If we accept this, then it is true that an infinite space is something that no finite mind can know of after any finite amount of time. But one feels that the world does exist as a whole, in advance of any efforts on our part to see it as a unity. And if we take all of space-time, it certainly does not seem to be meaningless to ask whether the spatial extent of space-time is infinite or not.

In *De Rerum Natura,* Lucretius first gave the classic argument
for the unboundedness of space: "Suppose for a moment that the
whole of space were bounded and that someone made his way to its
uttermost boundary and threw a flying dart."^{16} It seems
that either the dart must go past the boundary, in which case it is no
boundary of space; or the dart must stop, in which case there is
something just beyond the boundary that stops it, which again means that
the purported boundary is not really the end of the universe.

Dart goes beyond "boundary."

Dart stops at boundary.

So great was their revulsion against the *apeiron* that
Parmenides, Plato, and Aristotle all held that the space of our universe
is bounded and finite, having the form of a vast sphere. When faced with
the question of what lies outside this sphere, Aristotle maintained that
"what is limited, is not limited in reference to something that
surrounds it."^{17}

In modern times we have actually developed a way to make Aristotle's
claim a bit more reasonable. As Lucretius realized, the weak point in
the claim that space is a finite sphere is that such a space has a
definite boundary. But there is a way to construct a three-dimensional
space which is finite and which does *not* have boundary points:
simply take the hypersurface of a hypersphere. Such a space is endless
but not infinite.

To understand how something can be endless but not infinite, think of a circle. A fly can walk around and around the rim of a glass without ever coming to a barrier or stopping point, but none the less he will soon retrace his steps.

Again, the surface of the Earth is a two-dimensional manifold which is finite but unbounded (unbounded in the sense of having no edges). You can travel and travel on the Earth's surface without ever coming to any truly impassible barrier . . . but if you continue long enough, you will begin to recross your steps.

The reason that the two-dimensional surface of the Earth is finite
but unbounded is that it is bent, in three-dimensional space, into the
shape of a sphere. In the same way, it is possible to imagine the
three-dimensional space of our universe as being bent, in some
four-dimensional space, into the shape of a hypersphere. It was Bernhard
Riemann who first realized this possibility in 1854. There is, however,
a traditional belief that anticipates the hypersphere. This tradition,
described in the essay, "The Fearful Sphere of Pascal," by
Jorge Luis Borges, is summarized by the saying (attributed to the
legendary magician Hermes Trismegistus) that "God is an
intelligible sphere, whose center is everywhere and whose circumference
is nowhere."^{18} If the universe is indeed a hypersphere,
then it would be quite accurate to regard it as a sphere whose center is
everywhere and whose circumference is nowhere.

To see why this is so, consider the fact that if space is
hyperspherical, then one can cover all of space by starting at any point
and letting a sphere expand outwards from that point. The curious thing
is that if one lets a sphere expand in a hyperspherical space, there
comes a time when the circumference of the sphere turns into a point and
disappears. This fact can be grasped by considering the analogous
situation of the sequence of circular latitude lines on the spherical
surface of the earth.^{19} This line of thought appears in
Dante's *Paradisio* (1300).^{20}

Aristotle had believed that the world was a series of nine spheres
centered around the Earth. The last of these crystalline spheres was
called the *Primum Mobile* and lay beyond the sphere upon which
were fastened all of the stars (other than the sun, which was attached
to the fourth sphere). In the *Paradisio,* Dante is led out through
space by Beatrice. He passes through each of the nine spheres of the
world: Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn, Fixed Stars,
Primum Mobile. Beyond these nine spheres lie nine spheres of angels,
corresponding to the nine spheres of the world. Beyond the nine spheres
of angels lies a *point* called the Empyrean, which is the abode of
God.

The puzzling thing about Dante's cosmos as it is drawn in Figure 14
is that here the Empyrean appears not to be a point, but rather to be
all of space (except for the interior of the last sphere of angels). But
this can be remedied if we take space to be hyperspherical! In Figure 15
I have drawn the model we obtain if we take the diagram on the last page
and curve it up into a sphere with a point-sized Empyrean. In the same
way, the three-dimensional model depicted by the first picture can be
turned into the finite unbounded space of the second picture if
we bend our three-dimensional space in such a way that all of the
space outside our last angelic sphere is compressed to a
point.^{21} Figure 16 is Doré's
engraving of the Empyrean surrounded by its spheres of
angels.

This whole notion of hyperspherical space was not consciously developed until the mid-nineteenth century. In the Middle Ages there was a general and uncritical acceptance of Aristotle's view of the universe -- without Dante's angelic spheres.

Lucretius, of course, had insisted that space is infinite, and there were many other thinkers, such as Nicolas of Cusa and Giordano Bruno, who believed in the infinitude of space. Some kept to the Aristotelian world system, but suggested that there were many such setups drifting around; others opted for a looser setup under which stars and planets are more or less randomly mixed together in infinite space.

Bruno strongly advocated such viewpoints in his writings, especially
his dialogue of 1584, "On the Infinite Universe and
Worlds."^{22} Bruno travelled freely around Europe during
his lifetime, teaching his doctrine of the infinite universe at many
centers of learning. In 1591, a wealthy Venetian persuaded Bruno to come
from Frankfurt to teach him "the art of memory and invention."
Shortly after Bruno arrived, the trap was sprung. His host had been
working closely with the ecclesiastical
authorities, who considered Bruno a leading heretic or heresiarch.
Bruno was turned over to the Inquisition. For nine years Bruno was
interrogated, tortured, and tried, but he would not give up his beliefs;
early in 1600 he was burned at the stake in the Roman *Piazza*
Campo di Fiori. Bruno's example caused Galileo to express himself a good
deal more cautiously on scientific questions in which the Church had an
interest.

Whether or not our space is actually infinite is a question that could conceivably be resolved in the next few decades. Assuming that Einstein's theory of gravitation is correct, there are basically two types of universe: i) a hyperspherical (closed and unbounded) space that expands and then contracts back to a point; ii) an infinite space that expands forever. It is my guess that case i) will come to be most widely accepted, if only because the notion of an actually infinite space extending out in every direction is so unsettling. The fate of the universe in case i) is certainly more interesting, since such a universe collapses back to an infinitely dense space-time singularity that may serve as the seed for a whole new universe. In case ii), on the other hand, we simply have cooling and dying suns drifting further and further apart in an utterly empty black immensity . . . and in the end there are only ashes and cinders in an absolute and eternal night.

Even though I am basically pro-infinity, my emotions lie with the hyperspherical space. But is there any way of finding a spatial infinity here? Well, what about that four-dimensional space in which our hyperspherical universe is floating? Many would dismiss this space as a mere mathematical fiction . . . as a colorful way of expressing the finite, but unbounded, nature of our universe. This widely held position is really a more sophisticated version of Aristotle's claim that what is limited need not be limited with reference to something outside itself.

But what if one chooses to believe that the four-dimensional space in which our universe curves is real? We might imagine a higher 4-D (four-dimensional) world called, let us say, a duoverse. The duoverse would be 4-D space in which a number of hyperspheres were floating. The hypersurface of each of the hyperspheres would be a finite, unbounded 3-D universe.

Thus, a duoverse would contain a number of 3-D universes, but no inhabitant of any one of these universes could reach any one of the others, unless he could somehow travel through 4-D space. By lowering all the dimensions by one, one can see that this situation is analogous to a universe that is a 3-D space in which a number of spheres are floating. The surface of each sphere or planet is a finite, unbounded 2-D space; and no one can get from one planet surface to another planet surface without travelling through 3-D space.

Following the Hermetic principle, "As above, so below," one is tempted to believe that the duoverse we are in is actually a finite and unbounded 4-D space (the 4-D surface of a 5-D sphere in 5-D space), and that there are a number of such duoverses drifting about in a 5-D triverse. This could be continued indefinitely. One is reminded of those Eastern descriptions of the world as a disk resting on the backs of elephants, who stand upon a turtle, who stands upon a turtle, who stands upon a turtle, who stands upon a turtle, etc.

Note that in that particular sort of cosmos there is only one
universe, one duoverse, one triverse, and so on. But in the kind of
infinitely regressing cosmos that I have drawn in Figure 18, we have
infinitely many objects at each level. Note also that to get from star A
to star B one would have to move through 5-D space to get to a different
duoverse. It is a curious feature of such a cosmos that, although there
are an infinite number of stars, no one *n*-dimensional space has
more than a finite number of them.

The question we are concerned with here is whether or not space is
infinitely large. There seem to be three options: i) There is some level
*n* for which *n*-dimensional space is real and infinitely
extended. The situation where our three-dimensional space is infinitely
large falls under this case. ii) There is some *n* such that there
is only one *n*-dimensional space. This space is to be finite and
unbounded, and there is to be no reality to *n* + 1 dimensional
space. The situation where our three-dimensional space is finite and
unbounded, and the reality of four-dimensional space denied, falls under
this case. iii) There are real spaces of every dimension, and each of
these spaces is finite and unbounded. In this case we either have an
infinite number of universes, duoverses, etc., or we reach a level after
which there is only one *n*-verse for each *n.*

So is space infinite? It seems that we can insist that at some dimensional level it is infinite; adopt the Aristotelian stance that space is finite at some level beyond which nothing lies; or accept the view that there is an infinite sequence of dimensional levels. In this last case we already have a qualitative infinity in the dimensionality of space, and we may or may not have a quantitative infinity in terms, say, of the total volume of all the 3-D spaces involved.

In this subsection I will discuss the existence of the infinity in
the *small,* as opposed to the infinity in the *large,* which
has just been discussed. Since a point has no length, no finite number
of points could ever constitute a line segment, which *does* have
length. So it seems evident that every line segment, or, for that
matter, every continuous plane segment or region of space, must consist
of an infinite number of points. By the same token, any interval of time
should consist of an infinite number of instants; and any continuous
region of space-time would consist of an infinite number of events
(*event* being the technical term for a space-time location, i.e.,
point at an instant).

It is undeniable that a continuous region of *mathematical*
space has an infinite number of *mathematical* points. Right now,
however, we are concerned with *physical* space. We should not be
too hasty in assuming that every property of the abstract mathematical
space we use to organize our experiences is an actual property of the
concrete physical space we live in. But what *is* "the space
we live in"? If it is not the space of mathematical physics, is it
the space of material objects? Is it the space of our perceptions? In
terms of material objects or of perceptions, points do not really exist;
for any material or perceptual phenomenon is spread over a certain
finite region of space-time. So when we look for the infinity in the
small in matter, we do not ask whether matter consists of an infinity of
(unobservable) mass-points, but, rather, whether matter is infinitely
divisible.

A commitment to avoiding the formless made it natural for Greek atomists such as Democritus to adopt a theory of matter under which the seemingly irregular bodies of the world are in fact collections of indivisible, perfectly formed atoms. (The four kinds of atoms were shaped, according to Plato, like four of the regular polyhedra. There is one other polyhedron, the twelve-sided dodecahedron, and this was thought somehow to represent the Universe with its twelve signs of the zodiac.) For the atomists, it was as if the world were an immense Lego set, with four kinds of blocks. The diverse substances of the world -- oil, wood, stone, metal, flesh, wine, and so on -- were regarded as being mixtures of the four elemental substances: Earth, Air, Fire, and Water. Thus, gold was regarded by Plato as being a very dense sort of Water, and copper was viewed as gold with a small amount of Earth mixed in.

Figure 20. (A-D). From D. Hilbert and H. Cohn-Vossen, *Geometry and
the Imagination.*

The alchemists and early chemists adopted a similar system, only the number of elemental substances became vastly enlarged to include all homogeneous substances, such as the various ores, salts, and essences. The fundamental unit here was the molecule.

A new stage in man's conception of matter came when it was discovered that if an electric current is passed through water, it can be decomposed into hydrogen and oxygen. Eventually, the vast diversity of existing molecules was brought under control by regarding molecules as collections of atoms. Soon some ninety different types of atoms or chemical elements were known. A new simplification occurred when it was discovered, by bombarding a sheet of foil with alpha rays, that an atom consists of a positive nucleus surrounded by electrons. Shortly after this the neutron was discovered, and the physical properties of the various atoms were accounted for by regarding them as collections of protons, neutrons, and electrons.

Over the last half century it has been learned, by using particle
accelerators, that there are actually many types of "elementary
particles" other than the neutron, electron, and proton. The
situation in high-energy physics today is as follows. A few particles
-- electrons, neutrinos, and muons -- seem to be absolutely
indivisible. These particles are called *leptons.* All others
-- protons, neutrons, mesons, lambdas, etc. -- can be broken
up into smaller units, which then reassemble to form more particles.

The historical pattern in the investigation of matter has been the
explanation of diverse substances as combinations of a few simpler
substances. Diversity of form replaces diversity of substance. So it is
no surprise that it has been proposed that the great variety of
divisible particles that exist can be accounted for by assuming that
these particles are all built up out of *quarks.*

A second element in the historical pattern is that as more powerful
tools of investigation are used, it becomes evident that there are more
types of new building blocks than had been suspected initially. This is
the phase that high-energy physics is currently moving into. First there
were three kinds of quark: up, down, and strange. Now, the charmed quark
has been admitted, and there are two new possible quarks: the top quark
and the bottom quark. It seems likely that the many diverse types of
quark will eventually be accounted for by assuming that each quark is a
combination of a few, let us say, *darks* . . . and that there are
only a very small number of possible kinds of dark. The cycle will then
repeat, with more and more different sorts of dark being indirectly
observed, the new diversity being accounted for by viewing each dark as
a collection of a few smaller particles of which there are a limited
variety, this limited variety beginning to proliferate, and so on.

If this sort of development can indeed continue indefinitely, then we are left with the fact that a stone is a collection of collections of collections of. . . . The stone thus consists of an infinite number of particles, no one of which is indivisible. There is, finally, no matter -- only form. For a stone is mostly empty space with a few molecules in it, a molecule is a cloud of atoms, an atom is a few electrons circling a tiny nucleus. . . . What if any seemingly solid bit of matter proves on closer inspection to be a cloud of smaller bits of matter, which are in turn clouds, and so on? Note that the branching matter tree that I began to draw for the stone has only a finite number of forks or nodes at each level, but that since there are infinitely many levels, there are in all an infinite number of nodes or component particles.

There are various objections to this sort of physical infinity. One is the Aristotelian argument that unless one is actually smashing the stone down to the quark level, the quarks are only potentially (as opposed to actually) there. The point would be that the stone may be indefinitely divisible, but that since no one will ever carry out infinitely many divisions, there are not really infinite numbers of particles in the stone right now.

There is a more practical objection as well. This is that no quark has ever been observed in isolation; the existence of quarks is deduced only indirectly as a way of explaining the symmetries of structure that occur in tables of the elementary particles. This argument is not very strong, however. For one thing, a great number of the things we believe in can be observed only indirectly; and, more practically, if we can continue to increase the energy of our measuring tools, there is no reason to think that quarks cannot be more convincingly detected.

A more fundamental objection to the whole idea of particles,
subparticles, etc., is that the underlying reality of the world may be
field-like, rather than particle-like. By splitting particles
indefinitely we arrived at the conclusion that
there is only form, and no content; many physicists prefer to start with
this viewpoint. For these physicists, the various features of the world
are to be explained in terms of the geometry of space-time. To get a
feeling for this viewpoint, one should look carefully at the surface of
a river or small brook. There are circular ripples, flow bulges,
whirlpools and eddies, bubbles that form, drops that fly up and fall
back, waves that crest into foam. The *geometrodynamic* worldview
regards space-time as a substance like the surface of a brook; the
various fields and particles that seem to exist are explained as
features of the flow.

Does the space-time of geometrodynamics allow an infinity in the small? There is really no answer to this question at present. According to one viewpoint there should be a sort of graininess to space-time, and the grain size would represent a sort of indivisible atom; a different viewpoint suggests that space-time should be as infinitely continuous as mathematical space.

What if there really is nothing smaller than electrons and quarks? Is there then any hope of an infinity in the small? One can argue that a given electron can have infinitely many locations along a given meter stick, so that our space really does have infinitely many points. It is sometimes asserted that the uncertainty principle of quantum mechanics nullifies this argument, but this is not the case.

Quantum mechanics puts no upper limit on the precision with which one can, in principle, determine the position of an electron. It is just that the more precisely the electron's position is known, the less precisely are its speed and direction of motion known. Infinite precision is basically a nonphysical notion, but any desired finite degree of precision is, in principle, obtainable. The precision with which something can be measured is thus a good example of something that is potentially infinite, but never actually infinite.

But this still gives us an actual infinity in the world. For if our electron is located somewhere between zero and one, then each member of the following infinite collection is a possible outcome of a possible measurement:

.2 ± .1, .23 ± .01, .235 ± .001, .2356 ± .0001, . . . , .235608947 ± .000000001, . . .

Although infinite precision is impossible, an electron can be found to occupy any of the infinitely many points between zero and one whose distance from zero is a terminating decimal.

There are, however, some modern physical speculations that regard
"space" and "time" as being abstractions which apply
to our size level, but which become utterly meaningless out past the
thirtieth decimal place. What would be there instead? Our old friend the
*apeiron.* But even if we cannot really speak of infinitely many
space locations, we might hope to find infinitely many sorts of
particle.

It is sometimes thought that quantum mechanics *proves* that
there is a smallest size of particle that could exist. This is not true.
Quantum mechanics insists only that in order to "see" very
small particles, we must use very energetic processes to look for
them.

It is illuminating, after all this, to learn how the high-energy physicists actually go about finding new particles. The process is a little like finding stations on the radio by inching the dial back and forth until you hear music instead of static. One uses a particle accelerator in which collisions (between electrons and positrons) are continually taking place. The energy of the collision processes is varied by turning the voltage on the accelerator up and down. There is number R that measures the "particleness" of the reaction taking place. R can be thought of as being a little like the information parameter that enables you to tell whether you have found a station, even though the sound of music is no louder than the sound of the static. When an energy is found at which the graph of R versus energy has a sudden peak, then it is assumed that the energy in question is characteristic of the rest-mass of a new particle. This process is called "bump-hunting." It is interesting to note that the sharper and narrower the peak, the more long-lived, and, thus, more "real" the particle is.

The question of whether or not matter is infinitely divisible may never be decided. For whenever an allegedly minimal particle is exhibited, there will be those who claim that if a high enough energy were available, the particle could be decomposed; and whenever someone wishes to claim that matter is infinitely divisible, there will be some smallest known particle which cannot be split. One is almost tempted to doubt if the question of the infinite divisibility of matter has any real meaning at all, particularly in view of the fact that such concepts as "matter" and "space" have no real meaning in the micro-world of quantum mechanics.

To return to something a little more concrete, let us consider the
divisibility of our perceptual field. There is a limit to the
subdivisions that this field can undergo. If two clicks happen close
enough together in time, they cannot be distinguished; if a spot of ink
is small enough, we can no longer see it. Hume makes much of this fact
in his *Treatise of Human Nature* of 1739:

Put a spot of ink upon paper, fix your eye upon that spot, and retire to such a distance, that at last you lose sight of it; 'tis plain, that the moment before it vanish'd the image or impression was perfectly indivisible.

^{23}

The best way to
understand Hume's view of the world is to regard our space-time as being
supplemented by an additional dimension of *scale.* To represent
what I have in mind, let us forget about time and drop all the space
dimensions but one. In Figure 24 I have drawn the space-scale continuum
for a one-dimensional world. An individual's perceptual field has a
certain fixed size, as drawn; the field is made up of a certain finite
number of slots or tiles -- minimal perceptual units. In this
model, the one-dimensional creature has two dimensions in which he can
move his perceptual field. He can move to the left and right in space,
and he can enlarge and contract his perceptual field. Rather than
thinking of the field as enlarging and contracting, we think of the
field moving up and down on the scale axis.

If the labelled objects (mountain, stone, speck of rock dust) occupy the appropriate regions of the space-scale continuum, then we can think of the ordinary perceptual level as being when the field is placed somewhere in the middle of the picture. At this perceptual level stones are visible, but one has neither enlarged one's field of vision enough to see the mountain as a single object, nor contracted one's attention enough to see the specks of dust on the rock. Notice that changing the size of one's perceptual field amounts just to moving this field about in the space-scale continuum.

Hume takes perceptions as primary. Although he is often thought of as an empiricist, his is actually an extremely idealistic viewpoint. The perceptions are "out there"; one's consciousness seems to move among them like a butterfly flitting from flower to flower.

One's perceptual field has minimal elements, yet these minimal elements can be resolved into smaller elements by altering one's field (by paying closer attention, using a telescope, or moving closer to the object in question). The only way to reconcile these two apparently contradictory aspects of our perceptual world is to view the world as a five-dimensional, space-time-scale continuum.

The question of the existence of an infinity in the small now becomes
the question of whether or not the space-scale continuum drawn in Figure
24 extends *downward* indefinitely; similarly, the question of the
existence of infinity in the large is the question of whether or not the
continuum extends *upward* indefinitely.

I have long been interested in a curious trick that eliminates the
infinity in the large and the infinity in the small without introducing
any *absolute* perceptual minimum or maximum. This is simply the
trick of bending the space-scale diagram into a tube, by turning the
scale axis into a circle. Here the universe could consist of many
galaxies, which consist of many star systems, which consist of many
planets, which consist of many rocks, which consist of many molecules,
which consist of many atoms, which consist of many elementary particles,
which consist of many quarks and leptons, which consist of many darks,
which could consist of many universes.^{24}

A problem with the circular scale model is that if our universe is broken down far enough, one gets many universes, each of which will break down into many more universes. Are all of these universes the same? Perhaps, but then it would be hard to see how there could really be more than one object in the world. Another difficulty is that if there are many universes, each of which breaks up into many more universes, how can each of the component universes be one of the starting universes?

There is no problem if we have infinitely many universes. To
illustrate this, I have drawn a picture of the simplest case: the case
in which each universe is made up of two universes. We can see that 1
splits into 1 and 2, 2 splits into 3 and 4, 3 splits into 5 and 6, and
in general *n* splits into 2*n* - 1 and 2*n*. We can
continue splitting any given universe indefinitely, thus obtaining an
infinite number of components in any bit of matter.

What is gained here is freedom from the belief that any size scale is intrinsically more basic or important or complex than any other size scale. Why waste time on the six o'clock news when you are no more nor less important than a galaxy or an atom? The point of this question is that one is often pressured to feel that the concerns of society or the world are more significant than one's own immediate personal concerns. But this is based on the assumption that some sizes are in an absolute sense bigger than others, and it is this assumption that circular scale undermines.

In conclusion, note that it is entirely possible that our universe is in every sense finite. A toroidal space-time of the sort mentioned in the section on temporal infinities eliminates all infinities in the large; and if circular scale is introduced as in the section on infinities in the small, then there are no discrete infinities in the small. These finitizations can be accomplished smoothly: there need be no end of time, edge of space, or smallest particle.

But it is hard to believe that there would be only *one* of
these totally finite universes. First, it is difficult to see how to
apply circular scale unproblematically unless there are infinitely many
universes; second, the principle of sufficient reason is violated if
only *this* particular finite universe exists; and third, there is
the feeling that the "space" in which our space-time is curved
should be real.

In the section on spatial infinities it was pointed out that if, on
the one hand, one repeatedly finitizes by replacing lines with circles,
and if, on the other hand, one never accepts some particular finite
*n*-verse as the end of the line -- if, in other words, one
thinks along the lines sketched in the last two paragraphs --then
one is forced to conclude that space is infinite dimensional and that
there are infinitely many objects in this cosmic space.

In the last section I discussed some of the ways in which an actual infinity could physically arise. But there are things that are not physical. There are minds, thoughts, ideas, and forms. In this section we will see if any of these familiar nonphysical entities are actually infinite.

In order to appreciate the section at hand, it is necessary to keep
an open mind on the question of whether or not mind equals brain, for if
one assumes *a priori* that a thought is nothing more than a
certain biochemical configuration in a certain finite region of matter,
then (unless one has infinite divisibility of matter) it seems to follow
automatically that infinite thoughts are impossible.

To cast a few preliminary doubts on the hypothesis that brain equals mind, let me quickly raise a few questions. Is what you thought yesterday still part of your mind? If you own and use an encyclopedia, are the facts in that encyclopedia part of your mind? Does a dream which you never remember really exist? How can you grasp a book as a whole, even though you only read it a word at a time? Would the truths of mathematics still exist if the universe disappeared? Did the Pythagorean theorem exist before Pythagoras? If three people see the same animal, we say the animal is real; what if three people see the same idea?

I think of consciousness as a point, an "eye," that moves about in a sort of mental space. All thoughts are already there in this multi-dimensional space, which we might as well call the Mindscape. Our bodies move about in the physical space called the Universe; our consciousnesses move about in the mental space called the Mindscape.

Just as we all share the same Universe, we all share the same Mindscape. For just as you can physically occupy the same position in the Universe that anyone else does, you can, in principle, mentally occupy the same state of mind or position in the Mindscape that anyone else does. It is, of course, difficult to show someone exactly how to see things your way, but all of mankind's cultural heritage attests that this is not impossible.

Just as a rock is already in the Universe, whether or not someone is handling it, an idea is already in the Mindscape, whether or not someone is thinking it. A person who does mathematical research, writes stories, or meditates is an explorer of the Mindscape in much the same way that Armstrong, Livingstone, or Cousteau are explorers of the physical features of our Universe. The rocks on the Moon were there before the lunar module landed; and all the possible thoughts are already out there in the Mindscape.

The mind of an individual would seem to be analogous to the room or to the neighborhood in which that person lives. One is never in touch with the whole Universe through one's physical perceptions, and it is doubtful whether one's mind is ever able to fill the entire Mindscape.

One last analogy. Note that there is always a certain region of physical space that only I can ordinarily know of -- barring surgery, no one but me is in a position to assess the physical conditions obtaining within my stomach. In the same way, there is a certain part of the Mindscape that only I can ordinarily know of -- unless I am to be greatly favored by the Muse, the feelings that pass over me when I think of my childhood will always remain private and inexpressible. Nevertheless, these almost ineffable feelings are part of the common Mindscape -- they are simply difficult for anyone else to get to.

The point of all this is that just as the finiteness of our physical bodies does not imply that every physical object is finite, the finiteness of the number of cells in our brains does not mean that every mental object is finite.

Well . . . *are* there any infinite minds, thoughts, ideas, or
forms or what have you in the Mindscape?

The most familiar candidate is the set *N* of all natural
numbers. If I try to exhibit *N,* all I can really do is show you
something like this: *N* = {1, 2, 3, . . .]. What the ". .
." stands for is something that is evident, yet basically
inexpressible. The idea, of course, is that all of the natural numbers
are to be collected together into a whole. Each of them would seem to
exist individually in the Mindscape, and one would suppose that the set
consisting of exactly the natural numbers would be in the Mindscape as
well -- one almost feels as if one can *see* it.

We might try to avoid the use of the ". . ." by saying
something like this: "*N* is the set that has the following
property: one is in *N,* and for any number *x* that is in
*N, x* plus one is in *N* as well." The trouble with this
definition is that it does not uniquely single out one particular set.
If, for instance, there were some infinitely large number *I,* and
if *N** were the set consisting of all the numbers in *N* and
all the numbers of the form *I* + *n* for some *n* in
*N,* then *N** would satisfy the property that for every
*x* in *N**, *x* plus one is in *N** as well . . .
but *N** would be different from *N.*

We might try to get around this difficulty by saying that *N* is
the *smallest set in the Mindscape* that has one in it, and that
has *x* plus one whenever it has *x.* But, for reasons that I
will begin to explain in the next section, the word
"Mindscape" cannot be meaningfully used in a definition. The
concept of "Mindscape" is too vast to be represented by any
word or symbol.

If we try to avoid *this* difficulty by substituting some sort
of finite description of the mental universe for the word
"Mindscape," then we get the same problem as before. By the
classic work of the logician Thoralf Skolem, we know that for *any*
finite description of *N* one might come up with, there will be a
different set *N** that also satisfies the description. So it is
quite literally true that what is really meant by the ". . ."
is inexpressible.

Some thinkers have taken this to mean that
there is, after all, no unique *N* in the Mindscape. This could be
true. But one need not take this to mean that there are no infinite sets
in the Mindscape: if there are many, many versions of the set of natural
numbers, then there are many, many infinite sets. However, it is
normally more desirable to assume that there is a simple unique *N*
in the Mindscape, just as it is simpler to assume that there is only one
universe instead of a whole slew of "parallel worlds."

I might note here that if time is indeed infinite, then just as we
can indicate Earth by saying, "this planet," we could indicate
our *N* by saying, "the number of seconds left in this
time." This is, in fact, what people do when they attempt to define
*N* by saying, "*N* is what you get if you start with one
and keep adding ones forever."

If infinite forms are actually out there in the Mindscape, then maybe
we can, by some strange trick of mental perspective, *see* some of
these forms. The philosopher Josiah Royce maintained that a person's
mental image of his own mind must be infinite.^{25} His reason
is that one's image of one's own mind is itself an item present in the
mind. So the image includes an image that includes an image, and so on.
This infinite regress can be nicely visualized by imagining a United
States in which a vast and fanatically accurate scale model of the
country occupies most of the Midwest. The scale model, being absolutely
accurate, includes a copy of the scale model, etc. This regress is
occasionally used to make a striking label for a commercial product. The
old can of Pet Milk, for instance, bore a picture of a can of Pet Milk,
which bore a picture of a can of Pet Milk, etc.

In a physical situation we would probably never actually be able to
finish making such a label in all its infinite detail. But this is not
to say that no such label or country-plus-scale-model could
*exist.* There would be no problem, if matter were infinitely
divisible. (If scale is indeed circular, then everything is, in a sense,
already an object of this nature!)

There is certainly no reason why a nonphysical mind should not be
infinite; and Royce's point is that if you believe that one of the
things present in your mind is a perfect image of this mind
and its contents, then your
mind *is* infinite. One might try to avoid this conclusion by
adopting a circular scale attitude and insisting that there is no
difference between the mind and the mind's image of itself, so that the
allegedly infinite set of thoughts {image of the mind, image of the
image, image of the image of the image, . . .} is really the same as the
set {mind, mind, mind, . . .}, which is just a set with one member:
{mind}.

I would like to discuss this a bit more, but first let me formally
introduce some of the apparatus of set theory. In Cantor's words,
"A set is a Many that allows itself to be thought of as a
One."^{26} A set is usually given as a pair of curly
brackets enclosing some description of the contents of the set. It is
easiest to think of the curly brackets as a thought balloon. Thus the
set {1, 2} is the unity obtained by taking the multiplicity consisting
of the numbers 1 and 2 and treating this multiplicity as a unity. That
is, we can think of the set {1, 2} as being represented by a thought
balloon that has 1 in it and 2 in it.

Of particular interest in set theory is the *empty set,* . is the One obtained
by taking together . . . nothing. If we write out in the ordinary way we get { }, which I have
drawn as an empty thought balloon.

More and more complicated sets can be built up using only the brackets in various arrangements. Thus we have the set {{ }} depicted in Figure 28B, and we could equally well form {{ }, {{ }}, {{ }, {{ }}}} which is how the number 3 is usually represented in terms of pure sets. (See Figure 29.)

Now let's get back to the question of whether or not a mind that has
a perfect self-image is infinite. Really to get down to the bare bones,
say that we have a mind or label or set *M* such that the only
member of *M* is *M.* That is, *M* = {*M*}. Now, if
we change this equation by replacing the *M* on the right by
{*M*} then we get *M* = {{*M*}}. If we could continue
replacing *M* by {*M*} forever, we would wind up with *M*
= {{{{{. . . . . .}}}}}. This could actually be a definition of an
*M* whose only member is itself, for note that placing another pair
of brackets around {{{{{. . . . . .}}}}} changes nothing. In plain
English, *M* is the set whose only member is the set whose only
member is . . .

Figure 30. Based on a drawing from Robert Crumb, *Your Hytone
Comix* (San Francisco: The Print Mint, 1976).

But if the only member of *M* is indeed *M* itself (rather
than a copy of *M*), then *M* really only has one element. It
is just that if we try to describe this element by using brackets we get
an infinite description. We call thoughts like *M
self-representative.* Whether or not such an *M* is to be
regarded as infinite depends on whether you experience the *M*
subjectively (in the way you experience your own mind), or objectively
(as a feature of the Mindscape that is to be precisely described in the
language of set theory).

Set theory is, indeed, the science of the Mindscape. *A set is the
form of a possible thought.* Set theory enables us to put various
facts about the Mindscape into one framework in the same way that the
atomic theory of matter provides a framework in which the diverse
physical and chemical qualities of matter can be simultaneously
accommodated.

Before the atomic theory of matter, such phenomena as melting and burning, rusting and freezing were regarded as qualitatively different. Once a good atomic theory was developed, however, all of these phenomena could be thought of in more or less the same way. The notion of set was consciously introduced only at the turn of the century. Before long, it became evident that all of the objects that mathematicians discuss -- functions, graphs, integrals, groups, spaces, relations, sequences --all can be represented as sets. One can go so far as to say that mathematics is the study of certain features of the universe of set theory.

The universe of set theory is closely bound up with the Mindscape -- one can, perhaps, think of the former as a sort of blueprint of the latter. A set is obtained when we take a thought and abstract from it all the emotive content, keeping only the abstract relational structure. A set is the form of a possible thought. So the question of whether or not there are any infinite entities in the Mindscape is really equivalent to the question of whether or not there are any infinite sets.

According to set theorists, there certainly *are* infinite sets.
Indeed, there is to be an endless hierarchy of infinities: the set of
natural numbers, the set of all sets of natural numbers, the set of all
sets of sets of natural numbers, etc. Each member of this sequence can
be shown to be of an infinity greater than that of the earlier members.
In modern set theory there is a whole field of study called large
cardinals, whose specialists study a dizzying array of higher and higher
infinities.

But many mathematicians and philosophers do not go along with the set
theorists. The traditional *finitist* viewpoint is still with us.
According to the finitists, there is nothing that is infinite, in heaven
or on earth.

Those who assert that infinite sets of every size have a secure
existence in the Mindscape are usually called *Platonists.* This
name is a bit inapt, since Plato did not believe in infinity; but he
*did* believe in the existence of ideas independent of thinkers,
and it is for this aspect of his thought that the Platonists are
named.

It is not likely that the finitist vs. Platonist debate will ever be
concluded. On the one hand, it is probably impossible to meet the
demands of a finitist who says that he will believe in infinity only if
he is *shown* an infinite set right now; on the other hand, the
notion of infinite sets appears to be logically consistent, so the
finitist can never prove that infinite sets do not exist.

I incline towards Platonism; but if you are stubborn enough, how can I possibly convince you that infinite things are real? All I can do, after all, is to make a finite number of marks on a finite number of sheets of paper. If you are truly committed to disbelief in the infinite, then you will not be satisfied by anything less than my simultaneously exhibiting each member of some infinite set . . . and whenever I claim that I have done so, you will triumphantly point at the finiteness of the number of marks on paper which I have really shown you.

In pre-Cantorian times finitists sometimes thought that they had
proved the impossibility of actually infinite sets. These proofs,
however, were always fallacious. Such proofs usually deal with some
particular property *P* of numbers that each natural number happens
to enjoy. *P* might be the property of being odd or even, having an
immediate predecessor, being the sum of finitely many units, or being
greater than any predecessor. The false proof that no infinite numbers
exist then takes the form: "Every number has property *P.* If
*x* is an infinite number, then *x* cannot have property
*P.* Therefore no infinite numbers can exist." The fallacy in
such a circular proof is that when it is asserted that "every
number has property *P,*" it is being quietly assumed that
anything that fails to have property *P* does not exist.

But, of course, one cannot assume that the infinite sets must have
certain properties before one has ever looked at them! Galileo's
paradox, for example, showed that an infinite set can be put into a
one-to-one correspondence with a proper subset of itself. Had we assumed
in advance that no set could be put into a one-to-one correspondence
with a proper subset of itself, then we would have had a proof that no
infinite set can exist. But such an assumption is *totally
unwarranted;* indeed, to make such an assumption is essentially to
assume in advance that every set is finite . . . which does not make for
a very productive debate.

But are we quite sure that the finitists will never come up with some
*valid* proof that the notion of infinite sets is incoherent and
fundamentally meaningless? A Platonist would answer that yes, he is sure
that there is no inconsistency in the theory of infinite sets. He is
sure of this because the theory in question is a description of certain
features of the Mindscape that "anyone can see."

But the finitist can still hope. There is a curious proof, discovered by Kurt Gödel in 1930, that the consistency of set theory cannot be finitely proved. The time will never come when the finitist is absolutely forced to admit that it is safe to talk about infinite sets.

In mathematics no other subject has led to more polemics than the issue of the existence or nonexistence of mathematical infinities. We will return to some of these polemics in the last chapter. For now, let us reprint Cantor's opening salvo in the modern phase of this age-old debate:

The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.

^{27}

Strong words! But what does Cantor mean when he says that the highest form of infinity created us? Read on!

There is a certain type of non-physical entity that was not discussed in the last section. God, the Cosmos, the Mindscape, and the class V of all sets -- all of these are versions of what philosophers call the Absolute. The word "Absolute" is used here in the sense of "non-relative, non-subjective." An Absolute exists by itself, and in the highest possible degree of completeness.

As I mentioned earlier, Plotinus held that the One could not be
limited in any sense. As Aquinas, the quintessential theologian, says:
"The notion of form is most fully realized in existence itself. And
in God existence is not acquired by anything, but God is existence
itself subsistent. It is clear, then, that God himself is both limitless
and perfect."^{28}

The limitlessness of God is expressed in a form closer to the
mathematical infinite by St. Gregory: "No matter how far our mind
may have progressed in the contemplation of God, it does not attain to
what He is, but to what is beneath Him."^{29} We have here
the rudiments of the infinite dialectic process that takes place if we
systematically try to build up an image of the whole Mindscape.

Suppose that I want to add thought after thought to my mind until my
mind fills the whole Mindscape. Whenever I make an attempt at this, I am
collecting together a group of thoughts into a single thought *T.*
Now, when I become conscious of my state of mind *T,* I realize
that this is a new thought that I had not yet
accounted for . . . so I improve my image of the Mindscape by passing
to the thought that includes all the elements of *T* plus *T
itself,* viewed objectively.

This is a dialectic process in the sense that the thetic component is
one's instantaneous unconscious image of the Absolute, the antithetic
component is the conscious formalization of this image, and the
synthetic component is the formation of a new unconscious image of the
Absolute that incorporates one's earlier images and the awareness that
they are inadequate.^{30}

This process is most clearly understood if we start with nothing at
all, as in the cartoon strip of Wheelie Willie in Figure 32. (Wheelie
Willie is a character whose adventures I occasionally used to draw for
the *Rutgers Daily Targum* when I was in graduate school there.)
Notice that in each of the shifts, what takes place is that Wheelie
Willie forms a thought that has as its members the members of the last
thought plus the last thought itself. Looked at another way, the thought
at each stage has all of the previous stages as components.

If we call the *n*th thought *T _{n}* we can
define

Some readers may have asked themselves if the
thought *T* plus "*T*" really has to be different
from the thought *T.* And the answer is, not always. In the last
section we were looking at a mind, *M,* which has *M* as one
of its components. Such an *M* is already fully self-aware, and
*M* plus "*M*" is no different from *M.* In
terms of sets, *M*
{*M*} = *M.*

It would seem, in particular, that God should be able to form a
precise mental image of Himself. Insofar as the Mindscape is God's mind,
what I am saying is that one of the objects in the Mindscape should be
the Mindscape itself. That is, the Mindscape is an *M* that has
*M* as one of its members. Now, any object in the Mindscape is, in
principle, something that one can perceive through one's consciousness.
So it would seem to be possible for our minds actually to attain a
vision of God or of the whole Mindscape.

Now this seems to contradict St. Gregory's dictum and the general
feeling that the Absolute is unknowable. But there are two *kinds*
of knowing: the rational and the mystical.

If I know something rationally, then I have some thought that is built up from simpler thoughts, which are in turn built up from still simpler thoughts. This regress is not infinite, but goes only through some finite number of stages before certain simple and unanalyzable perceptions and ideas are reached. My idea of "house" consists of a collection of ideas, each one of which represents a certain type of house (e.g., my house, brick house, hovel). Each idea of a type of house consists of ideas of various components and functions (doors, windows, shelter), which can in turn be explicated in terms of certain simple ideas (walking, vision, warmth).

When I communicate a rational thought, what I do first is to show what the components of my thought are, and then to show how the components fit together. If one of the components of the final thought were to be the final thought itself, then this rational communication would be blocked by an infinite regress. To explain the thought, I would first have to explain the thought. I could not finish unless I had already finished.

In terms of rational thoughts, the Absolute is unthinkable. There is no non-circular way to reach it from below. Any real knowledge of the Absolute must be mystical, if indeed such a thing as mystical knowledge is possible.

Mathematics and philosophy do not normally have a great deal to say
about the mystical way of knowing things. Mystically speaking, it is possible to
experience a direct vision of the whole Mindscape. This vision cannot be
rationally communicated for the reasons just outlined. Of course, it is
possible to communicate mystical knowledge in an indirect way, for
example, by advocating that a person prepare his or her mind through
carrying out some physical or spiritual exercises. But, ultimately,
mystical knowledge is attained all at once or not at all. There is no
gradual path by which to build up an *M* that has *M* as one
of its elements.

Even if full knowledge of the Absolute is only possible through
mysticism, it is still possible and worthwhile to discuss *partial*
knowledge of the Absolute rationally. A significant thing about the
Mindscape and the other Absolutes is that they are actually infinite.
Indeed, in 1887 Cantor's friend, Richard Dedekind, published a proof
that the Mindscape is infinite, where Dedekind's word for Mindscape was
*Gedankenwelt,* meaning thought-world.^{31}

Dedekind's argument for the infinitude of the Mindscape was that if
*s* is a thought, then so is "*s* is a possible
thought," so that if *s* is some rational
non-self-representative thought, then each member of the infinite
sequence {*s, s* is a possible thought, *s* is a possible
thought is a possible thought, . . .} will be in the Mindscape, which
must, therefore, be infinite.

A very similar argument proves that the class of all sets is infinite. The class of all sets is normally called V, or Cantor's Absolute. We can use the Wheelie Willie sequence of sets to see that there are infinitely many different sets in V.

Dedekind modelled his argument after an argument that appears in
Bernard Bolzano's *Paradoxes of the Infinite* (ca. 1840):

"The class of all true propositions is easily seen to be infinite. For if we fix our attention upon any truth taken at random . . ., and label it

A,we find that the proposition conveyed by the words 'Ais true' is distinct from the propositionAitself . . ."^{32}

So we can see that the Mindscape, the class of all sets, and the
class of all true propositions are all infinite. Does this guarantee
that infinite objects exist? Not really. For a case can be made for the
pluralist claim that the Mindscape, the class of all sets, and the class
of all true propositions do *not* exist as objects, as unities, as
finished things.

In more familiar terms, it is not hard to prove that God is infinite . . . but what if you don't believe that God exists? It may seem hard to doubt that the more impersonal Absolutes -- such as "everything," or the Mindscape -- exist, but there are those who do doubt this. The issue under consideration is a version of the old philosophical problem of the One and the Many. What is being asked is whether the cosmos exists as an organic One, or merely as a Many with no essential coherence. It is certainly true that the Mindscape, for instance, does not exist as a single rational thought. For if the Mindscape is a One, then it is a member of itself, and thus can only be known through a flash of mystical vision. No rational thought is a member of itself, so no rational thought could tie the Mindscape into a One.

Normally the word "set" is restricted by definition to apply only to collections that are not members of themselves. Under this use of the word, the class V of all sets cannot be a set, for if it were, we would have a set V such that V is a member of itself. So V becomes a collection that can never be formed into a One.

Suppose that we do not believe in circular scale and assume that any physical thing is not a part or component of itself. Is the Cosmos, the collection of all physical things, a thing? If it is, then it has to be a component of itself, which we do not allow. So the Cosmos is not a thing, but only a Many that can never be a One.

There is a highly relevant passage in a letter Cantor wrote to Dedekind in 1905:

"A multiplicity can be such that the assumption that

allits elements 'are together' leads to a contradiction, so that it is impossible to conceive of the multiplicity as a unity, as 'one finished thing.' Such multiplicities I callabsolutely infiniteorinconsistent multiplicities.As we can readily see, the 'totality of everything thinkable,' for example, is such a multiplicity . . ."^{33}

Again, the reason that it would be a contradiction if
the collection of all rational thoughts were a rational thought *T*
is that then *T* would be a member of itself, violating the
rationality of *T* (where "rational" means
non-self-representative). The upshot of all this is that God, the
Mindscape, the class of all sets, and the class of all true propositions
all seem to be infinite, but it is at least possible to question whether
any of these Absolutes exists as a single entity. Certainly they do not
exist as entities that can be fully grasped by the rational mind.

In this section I would like to explore some of the connections
between the various sorts of infinities that have been
discussed.^{34} In his 1887 essay, "Contributions to the
Study of the Transfinite," Cantor quotes a passage from Aquinas's
*Summa* and states repeatedly that in this passage appear the only
two really significant objections that have ever been raised against the
actual infinite.^{35} Let us examine this quote from Aquinas
here, reproducing Cantor's italics:

The existence of an actually infinite multitude is impossible. 1) For any set of things one considers must be a

specificset.And sets of things are specified by the number of things in them. Now no number is infinite,for number results from counting through a set in units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited. 2) Again, every set of things existing in the world has been created, and anything created issubject to some definite purposeof its creator,for causes never act to no purpose.All created things must be subject therefore to definite enumeration. Thus even a number of things that happens to be unlimited cannot actually exist.^{36}

It seems clear
that Aquinas's first point is that an infinite set can occur only if
infinite numbers exist, and he does not believe that infinite numbers
exist. Cantor's theory of transfinite numbers stands as the only
adequate response to this objection. For many years, it was believed
that the notion of actually infinite numbers was fundamentally
incoherent. It was only with the birth of Cantor's theory in the late
1800s that a consistent and reasonable theory of infinite, or
transfinite, numbers was developed. As Cantor remarks in his discussion
of Aquinas's objection, this objection against the existence of actually
infinite collections is to be met *positively* by exhibiting a
theory of infinite numbers.

It is not so obvious what Aquinas's second point might be. It might be taken to be simply a variation on the first point. Under this reading, the first point says that any set must have a number of cardinality, but all numbers are finite; and the second point says that any set must have a purpose or significance, but any definite purpose is finite. If this is indeed Aquinas's meaning, then we can say that once again the Cantorian theory of infinite sets provides a positive rebuttal.

Aquinas's whole view of the infinite is not really tenable, for he
held that God is infinite, but that no created thing is infinite. This
contradicts a widely accepted principle known as the *Reflection
Principle.* The Reflection Principle as formulated in set theory goes
as follows: every conceivable property that is enjoyed by *V* is also
enjoyed by some set. (Recall here that *V* is Cantor's Absolute, the class
of all sets.) Philosophically it would run: every conceivable property
of the Absolute is shared by some lesser entity; or, every conceivable
property of the Mindscape is also a property of some possible
thought.

The motivation behind the Reflection Principle is that the Absolute
should be totally inconceivable. Now, if there is some conceivable
property *P* such that the Absolute is the only thing having
property *P,* then I can conceive of the Absolute as "the only
thing with property *P.*" The Reflection Principle prevents
this from happening by asserting that whenever I conceive of some very
powerful property *P,* then the first thing I come up with that
satisfies *P* will *not* be the Absolute, but will instead be
some smallish rational thought that just happens to reflect the facet of
the Absolute that is expressed by saying it has property *P.*

Let me give an example of a Reflection Principle argument. *For
every thought S in the Mindscape, the thought "S is a possible
thought" is also a thought in the Mindscape.* By Reflection
there must, therefore, be some thought *W* such that For *every
thought S in W, the thought " S is a possible thought" is also
in W.* This *W* reflects, or shares, the italicized property of
the Mindscape. But note now that this *W* must be infinite. So an
infinite thought exists.

Again, it is true that each of the Wheelie Willie sets
*T _{n}* is a member of

The point I wish to make is that if one accepts the existence of any of the various infinite Absolutes, then one is fairly well committed to accepting the existence of infinite thoughts and sets. For to deny the Reflection Principle is practically to assert that the Absolute can be finitely described, which is most unreasonable.

The passage from St. Augustine that I referred to earlier contains a
kind of Reflection Principle argument for the reality of the set
*N* of all natural numbers. In that passage Augustine argues that
God must already know each and every natural number and that he even
knows "infiniteness" in the form of all the natural numbers
taken at once -- for otherwise the set of natural numbers would
exhaust his abilities. God, according to Augustine, must lie
*beyond* the set of natural numbers.

To summarize the points in this chapter:

- The infinite normally inspires such feelings of helplessness, futility, and despair that the natural human impulse is to reject it out of hand.
- There are, however, no conclusive proofs that everything is finite; and the question of whether or not anything infinite exists remains as an open, almost empirical problem.
- There are various sorts of physical infinites that could actually exist: infinite time, infinitely large space, infinite dimensional space, infinitely continuous space, and infinitely divisible matter. Each of these infinites is, in principle, avoidable; whether or not our Cosmos actually does avoid infinities remains to be seen.
- In Cantor's set theory we have a great number of infinite sets. This simple and coherent theory of the infinite provides a logical framework in which to discuss infinities. Moreover, if we feel that the things that mathematicians discuss are real, then we can conclude that actually infinite things exist.
- Attempts to analyze the phenomenon of consciousness and self-awareness rationally appear to lead to infinite regresses. This seems to indicate that consciousness is essentially infinite.
- The Absolute is certainly infinite. So one must either deny the reality of the Absolute or accept the existence of at least one infinity.
- According to the Reflection Principle, once one has an infinite Absolute, one must also have many conceivable infinities as well.

- It is sometimes said that if infinitely many planets existed, then every possible planet would have to exist, including, for instance, a planet exactly like Earth, except with unicorns. Is this necessarily true?
- Consider a very durable ceiling lamp that has an on-off pull string. Say that the string is to be pulled at noon every day, for the rest of time. If the lamp starts out off, will it be on or off after an infinite number of days have passed?
- For each observer
*O,*there is some fixed upper bound N_{o}to the number of stars that*O*can physically see. Therefore, for each observer the universe is finite. Does this imply that the universe is finite? - "I have five fingers on my left hand" means the same thing
as "When I count up all the fingers on my left hand, the last
number I say is
*five.*" What might "I have fingers on my left hand" mean? - Suppose that we find an infinite number
*I*that is the largest possible number. But now, what about*I*+ 1? - In the little-known field of "enumerative geometry," it is
said that there are points on a line and
^{2}points in a plane. There are said to be^{2}*lines*in the plane as well: "To get the correct number^{2}of straight lines in the plane, we must divide the number^{4}of pairs of points in the plane by the number^{2}of pairs of points on a straight line."^{37}How many circles should there be in the plane? How many ellipses? - Can you prove, without circularity, that seven is a finite number?
- The universe has lasted about 10
^{10}years since the Big Bang. There are about 3 ¥ 10^{7}seconds in a year. According to quantum mechanics, the usual conception of continuous time does not extend to intervals shorter than 5 ¥ 10^{-44}seconds, so we might think of this unit as being a kind of "instant," faster than which nothing can happen. How many "instants" of time does that come to so far? Is it reasonable to argue that larger numbers, such as 10^{100}, do not yet exist? - Say that the space we live in is infinitely large. Consider an
infinite line
*L*contained in our space.*L*is infinity yards long, and*L*is infinity feet long. But since each yard is three feet,*L*is also three-times-infinity feet long. How can infinity equal three times infinity?^{38} - Here is an example of an infinite regress. Suppose that some person wishes to prepare a text in which every appearance of the letters "man" is replaced by the letters "woman." If this is rigidly adhered to, then "man and woman" becomes "woman and wowoman," then "wowoman and wowowoman," and so on. What do you reach in the limit?