Mathematical Platonism and the Nature of Infinity

doi:10.4236/ojpp.2013.33056

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Open Journal of Philosophy

2013. Vol.3, No.3, 372-375

Published Online August 2013 in SciRes (http://www.scirp.org/journal/ojpp) http://dx.doi.org/10.4236/ojpp.2013.33056

372

Mathematical Platonism and the Nature of Infinity

Gilbert B. Côté

Sudbuy, Ontario, Canada r

Email: [email protected]

Received April 25th, 2013; revised May 25th, 2013; accepted June 2nd, 2013

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

Introduction

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 1↔2. We then continue sequentially with 2↔4,

3↔6, 4↔8, 5↔10, 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-

dinality.



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.

G. B. CÔTÉ

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 ×

10−35 meters. Similarly, the shortest period of time is known as

Planck time and is equal to 5.39 × 10−44 seconds. If parts of

space-time cannot be infinitely small, instants of time cannot

either.

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.

G. B. CÔTÉ

Figure 2.

Torricelli’s trumpet (Image from http://en.wikipedia.org/wiki/File:GabrielHorn.png).

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-

servers.

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

374

G. B. CÔTÉ

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-

tence.

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

number:





ie

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.

Conclusion

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.

Acknowledgements

The constructive criticism and encouragement of Prof.

Lucien Pelletier of the Dept. of Philosophy, University of Sud-

bury (Ontario) is gratefully acknowledged.

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