Our universe appears to be
bound by a finite set of laws, yet we often talk about things that go on for an
eternity. "Infinity" is a strange idea. But it's crucial if you want
to understand anything from philosophy to mathematics. Here’s why.
There are three broad
domains where infinity can be applied. It’s used as a conceptual tool to help
us describe the properties and values of objects and processes, it’s an
important notion in philosophy, cosmology and metaphysics, and, of course, it’s
crucial to mathematics. Let’s take a look at each of these in detail.
The Totality of Infinity
We often use the word
‘infinity’ when describing something that goes on forever. But it can also be
used to describe something that doesn’t go on forever, or for which its
value is absolute.
Take, for example, the use
of infinity in sports and gaming. As every chess player knows, each piece is
assigned a numerical value according to its tactical importance and strength.
These values range from one (pawns) to nine (the queen), and are often used to keep
a kind of score as the game progresses. But the king is assigned infinite value
— and for very good reason. Losing the king is fatal. It’s instant game over,
regardless of whatever else might be happening in the match. The king’s worth,
therefore, cannot be bound within a finite set of values.
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Image:
Shutterstock/18percentgrey.
Similarly, an overtime goal
in hockey, or a ‘golden goal’ in soccer, can likewise be ascribed infinite
value. Any situation in which a single goal causes the match to come to an
immediate end, and along with it both instant victory and defeat, hinges on an
absolutist event.
More importantly,
philosophers, religious scholars, legal experts, and ethicists often assign an
infinite value to human life. And indeed, as an enlightened society, we’re
appalled by the idea of attaching a price to such a thing. We simply cannot
buy, sell, or trade each other. Of course, this doesn’t always work in
practice. During war, human lives are sacrificed to protect the larger body of
citizens and to uphold certain values and institutions. And illicit human
trafficking is an ongoing problem. But at a conceptual level, and as many
religious thinkers argue (especially Christians), there is no price high enough
to allow for the ‘buying’ of human life, and there’s no situation grave enough
to warrant killing (i.e. Thou shalt not kill); only God has the power — and the
right — to take away the gift of life.
And therein lies another
kind of infinity: death. Assuming that nothing awaits us in the afterlife, the
termination of our lives represents a kind of eternity. It’s an eternity of
nothingness, but an eternity nonetheless.
Infinite In All Directions
But while infinity can be
used as a helpful conceptual tool, it’s also something that may be physically,
tangibly real — especially when considering some of the freakier aspects of philosophy, cosmology and metaphysics.
Take for example the eerie
findings of string theory. Though controversial, the theory suggests the
presence of 10 or 11 spacetime dimensions. The extra six or seven dimensions
could be compacted at an insanely small scale, or our universe could be located
on a D-brane — a dynamic (3+1)-dimensional object. Columbia University
physicist Brian Greene argues that these branes could support parallel
universe, giving rise to the multiverse hypothesis and the potential for an
infinite set of braneworlds; these worlds aren’t always
parallel and out of reach, resulting in constant collisions that cause an infinite
succession of Big Bangs.
But there are other examples as well. Cosmologist Lee Smolin, in his book The Life of the Cosmos, proposed that universes are nothing more than black hole generators. According to his theory of cosmological natural selection, each black hole spawns a daughter universe. The process is then repeated — likely for an infinity.
And indeed, we’re not
certain if the universe (or multiverse) has an end — or a beginning for that
matter. Perhaps it’s always been here and it always will.
We’re also not sure about
the shape of space-time. If it’s flat, then it could stretch out for an
infinity — but it must start repeating at some point owing to the finite number
of ways particles can be arranged in space and time. If this is the case, then
there are an infinite number of universes.
Quantum mechanics also
suggests an infinite universe. Everett’s Many Worlds Interpretation (MWI)
states that the universe branches off into distinct worlds to accommodate every
single possible outcome. We may live in an infinite web of alternate
timelines.
This raises some interesting
— if not deeply troubling — issues.
If the MWI is true, for
example, every possible outcome that can be observed will be observed — no matter how ludicrous or extreme.
And given an infinite amount
of time, freak disembodied minds, called Boltzman Brains, could eventually pop
into existence owing to the right configurations of matter and energy.
Thankfully, a new interpretation of string theory suggests that humanity will never be overrun by
these spontaneously forming disembodied space brains.
The Math of the Matter
Lastly, infinity is
used in math — though not without controversy. It’s often
mistaken for a number or a singular entity, but it’s more a label that can be
used to describe a variety of mathematical objects and concepts that are larger
than anything that can be physically or conceptually expressed in the real
world. More simply, it’s a term that can be given to any non-finite number or
groups of numbers. And indeed, the whole point of it is that it can’t be
characteristized as there is no end to it.
There are two distinct areas
in which infinity is used in mathematics, namely set theory and topologies
(there are others as well, like limits and algebra, but we’re not going to
discuss those at this time).
Fun With Sets
Set theory, in which numbers
with corresponding items can be grouped into sets, shows that there are
multiple types of infinities and that some are bigger than others — a
surprising result to say the least. Back in the 19th century, Georg Cantor used
this insight to describe two different kind of infinities, countable and
uncountable.
Countability describes
anything that can be lined up such that it can be numbered; all sets that can
be put into a one-to-one correspondence with natural numbers can be considered
countable.
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Image: Rasmus Holmboe
Dahl/Shutterstock.
In set theory, this would
include a group like {1,2,3,4} and {banana, cake, fork, napkin) — a set with a
cardinality of 4 (i.e. the size of a set). Some of these countable sets are
infinite (called enumerable infinities), like the set of all integers (yes,
members of a countably infinite set can be counted, even though it would take
an infinity to do so). But because a single number cannot describe the size of
an infinite set, Cantor used the term aleph-0, or aleph-null (aleph being the
first letter of the Hebrew alphabet) to signify the cardinality of an infinite
set.
Indeed, aleph-0 is a
cardinal number despite the fact that infinity should not be considered a
number; but numeration can in fact be extended to infinite quantities so long
as consistent definitions are maintained. Needless to say, infinite cardinal
numbers don’t follow the same mathematical rules as finite number — you can’t
just throw them into an equation and hope for the best.
Aleph-0 can include such
sets as the set of all prime numbers, all rational numbers, all algebraic
numbers, the set of binary strings of all finite lengths, and the set of all
finite subsets of any countably infinite set.
Interestingly, the number 2
is a countable infinity. It’s a number that can be physically realized, but it
consists of an infinite number of fractions.
Uncountable infinities, on
the other hand, happens when you get into irrational numbers, like the square
root of 2, or transcendental numbers like pi and e. The reason why these sets
are considered uncountable is because they cannot be numbered.
Cantor proved this by
offering a thought experiment. If we had a numbering scheme that was supposed
to count all the real numbers, we could go down that list, and for the nth
term, read off the nth digit. But we could then change it to something
different and use it on our new number. But since this “newly constructed
number” is different from every other number on the list (at the nth digit at
the very least), then it must have been missed in the enumeration. The list
cannot be enumerated in any way. It’s simply too big, and thus uncountable.
Numbering Nonsense
As an aside, number systems
cannot accommodate the concept of infinity. No “infinity” concept exists in the
context of any number system, if by number system we mean any collection of
concepts that have operations like addition and multiplication — operations
which obey the usual properties of arithmetic.
For example, what would
infinity minus 1 be? It couldn’t be a finite number, since no finite number
plus 1 equals infinity. This would violate the rules of arithmetic, leading to
such absurd equations like -1=0, which isn’t true. So infinity doesn’t exist,
if by “exist” we’re talking about the context of a number system.
Topological Spaces
Topological spaces describe
the properties of surfaces outside of angles and distances. So, if two surfaces
can be mapped together (i.e. made the same) by stretching and pulling them,
rather than cutting and pasting them, they’re considered identical from a
topological perspective.
Interestingly, the real
number system is considered a topological space. We can come up with a
definition of what it means for a sequences of numbers, rather than surfaces,
to converge. For example, the sequence {1.1, 1.01, 1.001, 1.0001...} converges
to the number 1, while the sequence {1, 2, 1, 2, 1, 2, 1, 2...} doesn’t
converge to anything.
In calculus, it’s said that
sequences like {1,2,3,4...} converge to infinity. But can it truly be said that
there’s an actual object called “infinity” that this sequence can be converged
to? Is there some topological space — i.e. a set of objects plus a definition
of what convergence means — that includes real numbers and also an infinity
concept to which some sequences of real numbers converge?
Perhaps surprisingly, the
answer is yes.
When topologists work with
the real number line (i.e. the set of real numbers together with the usual
notion of distance which induces topological structure), they sometimes
introduce a “point at infinity”. This point, denoted \infty can be thought of
as the point that you would always be heading towards if you started at 0 and
traveled in either direction at any speed for as long as you liked. Strangely,
when this infinite point is added to the real number line, it makes it
topologically equivalent to a circle (think about the two ends of the number
line both joining up to this single infinite point, which closes a loop of
sorts). This same procedure can also be carried out for the plane (which is the
two dimensional surface consisting of points (x,y) where x and y are any real
numbers). By adding a point at infinity we compactify the plane, turning it
into something topologically equivalent to a sphere (imagine, if you can, the
edges of the infinite plane being folded up until they all join together at a
single infinity point).
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Image: AAM.
As a concluding note, and as
Ask a Mathematician points out, it’s important to remember that the
question of infinity in mathematics cannot be answered indisputably. We can
ask, “how do infinite things arise in math”, but we can only answer that they
arise in many, very important ways.
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