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

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