Open Journal of Philosophy
2013. Vol.3, No.3, 372-375
Published Online August 2013 in SciRes (
Copyright © 2013 SciRes.
Mathematical Platonism and the Nature of Infinity
Gilbert B. Côté
Sudbuy, Ontario, Canada r
Received April 25th, 2013; revised May 25th, 2013; accepted June 2nd, 2013
Copyright © 2013 Gilbert B. Côté. This is an open access article distributed under the Creative Commons At-
tribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the
original work is properly cited.
An analysis of the counter-intuitive properties of infinity as understood differently in mathematics, clas-
sical physics and quantum physics allows the consideration of various paradoxes under a new light (e.g.
Zeno’s dichotomy, Torricelli’s trumpet, and the weirdness of quantum physics). It provides strong support
for the reality of abstractness and mathematical Platonism, and a plausible reason why there is something
rather than nothing in the concrete universe. The conclusions are far reaching for science and philosophy.
Keywords: Mathematical Platonism; Infinity; Zeno; Torricelli; Abstractness; Quantum Physics;
Dichotomy; Trumpet; Paradox
The ontology of mathematical Platonism remains to this day
an unresolved issue. Most philosophers and mathematicians
concede that numbers exist and are abstract, but there is con-
siderable divergence of opinion on whether they exist inde-
pendently of rational observers (Cole, 2010). The difficulty is
particularly evident when one considers the concept of numeri-
cal infinity and its counter-intuitive properties in the different
fields of mathematics, classical physics and quantum physics.
In the text below, I will endeavour to tease apart the various
meanings of infinity, in an attempt to clear the confusion that
otherwise produces various contradictions and well known
paradoxes. The proposed solutions should illuminate the phi-
losophical debate on the reality of abstractness and the signifi-
cance of mathematical Platonism.
Infinity and Mathematics
Mathematical infinity is clearly defined on the basis of the
original and authoritative work of German mathematician
Georg Cantor (1845-1918) (Dunham, 1990, 1994). The first
type of mathematical infinity that comes to mind is that of the
natural numbers (also called positive integers: 1, 2, 3, 4, etc.).
They are infinite because we can always add 1 to any large
integer previously identified. Any set of numbers that can be
put in a one-to-one correspondence with the positive integers is
also said to be denumerably (countably) infinite because we can
identify all its members one at a time. This turns out to be the
case for the odd integers, the even integers, for the set of nega-
tive and positive integers and 0, the rational numbers (those that
are the ratio of two integers), the prime numbers (those that can
only be divided by themselves and 1) and all algebraic numbers
(the roots of non-zero polynomial equations).
All these sets of numbers have the same infinite size or “car-
dinality”, formally denoted as 0 (aleph-naught).This is very
counter-intuitive because at first sight the number of all positive
integers (odd and even) would appear to be twice as large as the
number of even integers. However, infinite sets of numbers do
not behave like finite sets. To compare the cardinality of the set
of all integers with that of all even integers, we start with the
first member of the set of all integers (i.e. 1) and the first mem-
ber of the set of all even integers (i.e. 2) to establish the unique
correspondence 12. We then continue sequentially with 24,
36, 48, 510, etc. and find that we can carry on indefi-
nitely and establish a unique one-to-one correspondence be-
tween all members of the two sets. For each member of one set,
there is one and only one corresponding member in the other
set. This proof demonstrates that both sets have the same car-
Several infinite sets of numbers have higher cardinality than
those above because a one-to-one correspondence with the
positive integers cannot be established. These sets are said to be
non-denumerable (non-countable, or innumerable) and contain
more members than the denumerable sets do. Their cardinality
is denoted as c. These sets include the set of real numbers con-
tained in the interval between any two numbers, the set of irra-
tionals (like 2), the set of transcendental numbers (i.e. non-
algebraic numbers like π and e), the set of complex numbers
(numbers with a real part and an imaginary part), the number of
points in a square, and the number of points in any n-dimen-
sional space.
In set theory, a power set is defined as the set of all subsets
of a set, including an empty set and the original set. The cardi-
nality of power sets is higher than that of their original sets.
Power sets of non-denumerable sets thus have cardinality
higher than 0
and c. If we take the power sets of power sets
and repeat the operation again and again, we can build an infin-
ity of infinite sets, each set having higher cardinality than the
precedent. This never-ending process looks like mathematical
infinity feeding upon itself to produce ever greater degrees of
infinity. This is the archetype of self-referral and is inherent to
Infinity and Classical Physics
If we try to apply the notion of mathematical infinity to con-
crete reality, we run into endless conceptual difficulties and
paradoxes. The best examples are more than 2400 years old and
were originally presented by the Greek philosopher Zeno of
Elea. His dichotomy paradox, for example, presents the appar-
ent difficulty of travelling from point A to point B, due to the
fact that one must first reach the midpoint C between A and B.
Once at point C, one must then reach a new midpoint between
point C and point B. New midpoints thus appear ad infinitum,
preventing the tired traveller from ever reaching point B.
Two main solutions have been proposed along the ages.
Practically, one can simply brush off the paradox by walking
from A to B and beyond, apparently demonstrating that Zeno
was wrong. The solution is pragmatic but ignores rather than
answers Zeno’s question about what happens when one travels
an infinite number of small stretches along a Euclidian line.
A more sophisticated mathematical reply consists of using
the infinite series
to show that, at the limit, the sum of this infinite sequence of
diminishing distances converges to the whole A-B distance.
This is a counter-intuitive but solid demonstration that the sum
of an infinite number of values can add up to a finite value. It is
often claimed to solve Zeno’s dichotomy paradox but on the
contrary, it simply illustrates it by showing that the sum tends
to the limit without ever reaching it. Strictly speaking, mathe-
matical infinity has no end, so point B gets closer and closer
along the series but is never reached (unless we truncate the
process and abandon infinity).
The essential problem with the dichotomy paradox is that it
follows Euclid instead of Democritus and Planck. The appro-
priate answer to Zeno is that in practice, we cannot divide our
travel in an infinite number of diminishing halves because dis-
tances are not continuous. We now know that roads are not
endlessly divisible, because the concrete world is made of indi-
visible atoms (or sub-atomic particles). We must not confuse
Euclidian geometry with the physical world. An infinitely long
Euclidian line can be divided ad infinitum, but a road or any
piece of concrete matter cannot. Concrete distances can be very
small but not infinitely so. In modern physics the smallest pos-
sible distance is known as Planck length and is equal to 1.616 ×
1035 meters. Similarly, the shortest period of time is known as
Planck time and is equal to 5.39 × 1044 seconds. If parts of
space-time cannot be infinitely small, instants of time cannot
Mathematically speaking, anything that cannot be divided in-
finitely cannot be infinitely large. Accordingly, our concrete
universe is definitely not infinitely large and will not last for-
ever. It is astronomically large and amazingly old, but not infi-
nitely so (especially if it started at the Big Bang). We do not
have the capacity to count its galaxies but their number is not
mathematically infinite. Our universe will last for eons, but not
forever. In other words, mathematical infinity does not exist in
our concrete, finite universe.
Infinity and Torricelli’s Trumpet
Torricelli’s trumpet is another renowned paradox dealing
with infinity (Clegg, 2003; Weisstein, 2013). Its careful consid-
eration will help us firmly establish the difference between the
abstract realm of mathematics and the concrete world. The
trumpet was first described by the Italian mathematician Evan-
gelista Torricelli (1608-1647). First consider the graph of the
equation y = 1/x for all values of x 1.0, as shown in Figure 1.
Then rotate the graph around the x axis to obtain the trumpet
shown in Figure 2. Note that the trumpet has a flared bell of
diameter 2.0 and is infinitely long.
Torricelli’s trumpet is also known as Gabriel’s horn, in ref-
erence to the biblical Archangel Gabriel who is said to use a
horn to announce God’s news; it seems fitting to equip an ar-
changel with an infinitely long horn. The paradox stems from
the fact that the horn apparently has an infinite surface but a
finite volume. Indeed, it has been clearly shown mathematically
that as x approaches infinity, the horn’s surface area also tends
to infinity, while the volume enclosed by the curve tends to-
wards the value of π cubic units (where π = 3.1416··· is con-
ventionally considered to be a finite value between 3 and 4).
The existence of a finite volume bound by an infinite area is the
paradox that is still considered unresolved today.
Gabriel’s horn cannot concretely exist. It is longer than the
diameter of the universe, and no one can blow in it since it goes
on endlessly without ever reaching a mouthpiece. In practice,
we could not fill it up completely with a quantity π of real paint,
or cover its interior with an infinite amount of real paint since
the paint molecules would be too large to squeeze into the infi-
nitely thin extremity.
Figure 1.
Graphical representation of the equation y = 1/x for x 1.0.
Copyright © 2013 SciRes. 373
Figure 2.
Torricelli’s trumpet (Image from
However, the horn evidently exists in the abstract, mathe-
matical realm where the infinitely large and infinitely small
readily co-exist. Without finite molecules to worry about, we
can see that the horn’s internal volume is also infinitely long as
it extends into the endless and infinitely narrow tube. Obvi-
ously, a volume with an infinite dimension is definitely not
finite. This presents no difficulty because π is not a finite inte-
ger. It is a transcendental number and has an infinite decimal
expansion with cardinality 0.Using Cantor’s results, we can
see that there is a one-to-one correspondence between the
points in the horn’s volume and the points on its surface. In fact,
we already mentioned earlier how the numbers of points on any
surface and in any volume are infinite, with the same cardinal-
ity c.
This may seem counter-intuitive, but it is in keeping with the
even integers having the same cardinality 0 as all integers
(odd and even) together, and with the real numbers in the in-
terval (0,1) having the same cardinality c as the real numbers in
any other interval (smaller or larger). A comparison can be
made with the points on the surface of an abstract pea having a
one-to-one correspondence with the points on the surface of an
abstract star, both with cardinality c. This does not apply to
concrete stars and concrete peas but it is mathematically sound.
Let us note here that a drop of abstract paint is all we need to
paint an infinite object. It suffices to apply an infinitely thin
layer of paint (i.e. a trick we cannot do with concrete molecules
of paint).
The horn’s paradox thus disappears. It only exists when we
mistakenly attempt to apply the properties of abstract infinity to
finite matter. In abstractness, where there is no physical dis-
tance, infinity takes no room; there is no physical difference
between infinitely large and infinitely small abstractions. Per-
haps the most interesting aspect of an abstract Gabriel’s horn is
that when one uses infinitely small units to build it, the horn is
both infinitely small and infinitely long.
Infinity and Quantum Physics
The rules of quantum physics are also counter-intuitive and
paradoxical in their own way. They radically differ from those
of classical physics and include the apparent possibility of be-
ing at two places at once, the wave-particle duality problem, the
quantum entanglement of sub-atomic particles, the popping up
of virtual particles from the vacuum, quantum tunnelling, and
the fundamental importance of infinite probabilities and com-
plex numbers (with non-zero imaginary parts). This is very
different from what occurs in the case of concrete matter. None
of these phenomena make sense in classical physics, but all are
crucial and fundamental in quantum mechanics. The wave-
particle riddle has particularly struck the popular imagination
because it immediately caused disagreement and controversy
amongst the leading physicists of the time when it was first
discovered. Eight decades later, physics textbooks, scientific
journals and popular science magazines still present this duality
as weird and defying common sense although scientifically
proved beyond doubt: depending on the type of measuring ap-
paratus one choses, photons and other quantum elements turn
out to behave either like waves or like particles, and their
strange, fundamental nature remains elusive.
For the purpose of this article, the importance of infinity and
imaginary numbers is especially relevant. In classical physics,
infinities are always truncated, and imaginary numbers are
simply used as mathematical tricks to solve complex problems
without altering our view of the world. In quantum physics, on
the contrary, infinities and imaginary values are inescapable
and absolutely essential for comprehension and ontology.
It is important at this point to consider the following argu-
ment. When we throw a single die, we expect one of six results.
If presented with a new game that produces twelve possible
results, we know that two throws (or two dice) must be in-
volved. If we randomly pick up numbered balls from a bag and
only ever get numbers from 1 to 100, we can reasonably con-
clude that the bag must contain a minimum of 100 balls num-
bered from 1 to 100. Similarly, the fundamental importance of
infinite probabilities in quantum physics implies the back-
ground presence of an infinite and therefore abstract source.
This means that the existence of quantum particles depends on
the existence of abstract infinity.
If one accepts this, one must logically conclude that we live
in a Universe where quantum particles lie at the interface be-
tween the abstract and concrete aspects of reality.
Mathematical Platonism
Infinity is an abstract concept not concretised in physical
matter, but it does exist. All abstract ideas do exist even if we
cannot touch them. In fact, existence out of space-time is con-
fidently assumed by those mathematicians who believe in
mathematical Platonism, the idea that mathematical statements
literally exist in an abstract realm independent of rational ob-
It is standard practice to accept at least a watered-down ver-
sion of mathematical Platonism restricted to the first two items
of the definition: 1) infinity exists and 2) it is abstract. Whether
it exists independently of rational observers is still an object of
mathematical and philosophical debate, but if infinite probabili-
ties have governed the production and existence of quantum
particles since the Big Bang, their presence at the beginning of
Copyright © 2013 SciRes.
the concrete Universe definitely did not depend on rational
observers born more than 13 billion years later. Viewed in this
light, the full version of mathematical Platonism becomes a
logical necessity and an essential prerequisite for our very exis-
Believing in the consequential reality of abstractness is a
mental process comparable to the widespread belief in a Hea-
ven that exists out of space-time or in a Nirvana where one can
escape from space-time. Philosophically, the full version of ma-
thematical Platonism means that mathematicians do not invent
theorems, but discover them. In quantum physics, it gives an
altogether new significance to the equations that precisely de-
scribe quantum elements, by making us realise that quanta are
neither waves nor particles but are mathematically virtual, in-
tangible entities. Photons, for instance, are well known to be dis-
crete packets of pure energy without concrete substance. Ever
since they were discovered, they have been fully described as
mathematical functions, i.e. intangible entities. It is only when
they are detected that they lose their virtual character and inte-
grate concrete space-time.
For mathematical Platonists who consider the relationship
between infinity and quantum mechanics, it is natural to accept
the view that the quantum particles making up the universe first
pop up into the vacuum out of an abstract and infinite source.
Such a mathematical interpretation is indeed much more logical
(or at least more palatable) than the currently popular view
among physicists that our universe sprang out of “nothing”
(Hawking & Mlodinow, 2010; Maxwell, 2011; Krauss, 2012;
Holt, 2012; Set & Rêve, 2012; Côté, 2012). An origin from
abstractness also has some commonality with religious creation
dogmas developed throughout human history by people who
knew little about the properties of infinity and, for lack of a
better explanation, resorted to imagining various anthropomor-
phic gods. In fact, an origin from abstractness is supported by
the elegant equation first discovered by the Swiss mathemati-
cian and physicist Leonhard Euler (1707-1783) who showed
that the imaginary power of an imaginary number can be a real
In summary, we have now established the following logical
sequence: the sheer existence of abstract infinity implies end-
less self-reference, the existence of further power sets, greater
infinity, the production of virtual quantum particles and, from
the known combinations of these particles, the consequent for-
mation of space-time. In other words, the existence of abstract
infinity is the basic reason why, in the concrete universe, there
is something rather than nothing.
The solutions proposed above, often counter-intuitive, are far
reaching and can hopefully encourage further research and
discussion until mathematicians and philosophers make sense
of this most logical topic and quench our inborn thirst for the
absolute and the Infinite.
The constructive criticism and encouragement of Prof.
Lucien Pelletier of the Dept. of Philosophy, University of Sud-
bury (Ontario) is gratefully acknowledged.
Clegg, B. (2003). A brief history of infinity. The quest to think the un-
thinkable. London: Robinson.
Cole, J. C. (2010). Mathematical platonism. Internet Encyclopedia of
Côté, G. B. (2012). Latest quantum experiments point to the reality of
abstractness and timeless ne ss .
Dunham, W. (1990). Journey through genius. The great theorems of
mathematics. New York: Penguin Books, John Wiley & Sons.
Dunham, W. (1994). The mathematical universe. An alphabetical jour-
ney through the great proofs, problems, and personalities. New York:
John Wiley & Sons.
Hawking, S., & Mlodinow, L. (2010). The grand design. New York:
Bantam Books, Random House.
Holt, J. (2012). Why does the world exist? An existential detective stor y.
New York: Liveright Publishing, W. W. Norton.
Krauss, L. M. (2012). A universe from nothing. Why there is something
rather than nothing. New York: Free Press.
Maxwell, N. (2011). Three philosophical problems about consciousness
and their possible resolution. Open Journal of Philosophy, 1, 1-10.
Set, J., & Rêve, J. (2012). Our theory of everything. Raleigh, NC: Lulu.
Weisstein, E. W. (2013). Gabriel’s horn. Math World—A Wolfram
Web Resource.
Wikipedia, Gabriel’s horn.’s_Horn
Copyright © 2013 SciRes. 375