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Would mathematics exist if people didn't?
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Since ancient times,
mankind has hotly debated
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whether mathematics
was discovered or invented.
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Did we create mathematical concepts to
help us understand the universe around us,
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or is math the native language of
the universe itself,
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existing whether we find
its truths or not?
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Are numbers, polygons
and equations truly real,
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or merely ethereal representations
of some theoretical ideal?
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The independent reality of math has
some ancient advocates.
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The Pythagoreans of 5th Century Greece
believed numbers were both
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living entities and universal principles.
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They called the number one, "the monad,"
the generator of all other numbers
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and source of all creation.
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Numbers were active agents in nature.
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Plato argued mathematical
concepts were concrete
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and as real as the universe itself,
regardless of our knowledge of them.
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Euclid, the father of geometry, believed
nature itself
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was the physical manifestation
of mathematical laws.
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Others argue that while numbers may
or may not exist physically,
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mathematical statements definitely don't.
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Their truth values are based on rules
that humans created.
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Mathematics is thus an invented
logic exercise,
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with no existence outside mankind's
conscious thought,
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a language of abstract relationships
based on patterns discerned by brains,
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built to use those patterns to invent
useful but artificial order from chaos.
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One proponent of this sort of idea
was Leopold Kronecker,
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a professor of mathematics in
19th century Germany.
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His belief is summed up in
his famous statement:
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"God created the natural numbers,
all else is the work of man."
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During mathematician
David Hilbert's lifetime,
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there was a push to establish mathematics
as a logical construct.
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Hilbert attempted to axiomatize all
of mathematics,
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as Euclid had done with geometry.
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He and others who attempted this saw
mathematics as a deeply philosophical game
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but a game nonetheless.
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Henri Poincaré, one of the father's of
non-Euclidean geometry,
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believed that the existence of
non-Euclidean geometry,
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dealing with the non-flat surfaces of
hyperbolic and elliptical curvatures,
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proved that Euclidean geometry, the
long standing geometry of flat surfaces,
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was not a universal truth,
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but rather one outcome of using one
particular set of game rules.
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But in 1960, Nobel Physics laureate
Eugene Wigner
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coined the phrase, "the unreasonable
effectiveness of mathematics,"
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pushing strongly for the idea that
mathematics is real
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and discovered by people.
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Wigner pointed out that many purely
mathematical theories
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developed in a vacuum, often with no view
towards describing any physical phenomena,
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have proven decades
or even centuries later,
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to be the framework necessary to explain
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how the universe
has been working all along.
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For instance, the number theory of British
mathematician Gottfried Hardy,
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who had boasted that none of his work
would ever be found useful
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in describing any phenomena
in the real world,
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helped establish cryptography.
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Another piece of his purely
theoretical work
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became known as the Hardy-Weinberg
law in genetics,
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and won a Nobel prize.
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And Fibonacci stumbled
upon his famous sequence
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while looking at the growth of an
idealized rabbit population.
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Mankind later found the sequence
everywhere in nature,
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from sunflower seeds
and flower petal arrangements,
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to the structure of a pineapple,
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even the branching of bronchi
in the lungs.
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Or there's the non-Euclidean work of
Bernhard Riemann in the 1850s,
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which Einstein used in the model for
general relativity a century later.
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Here's an even bigger jump:
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mathematical knot theory, first developed
around 1771
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to describe the geometry of position,
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was used in the late 20th century
to explain how DNA unravels itself
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during the replication process.
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It may even provide key explanations
for string theory.
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Some of the most influential
mathematicians and scientists
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of all of human history
have chimed in on the issue as well,
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often in surprising ways.
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So, is mathematics an
invention or a discovery?
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Artificial construct or
universal truth?
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Human product or
natural, possibly divine, creation?
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These questions are so deep the debate
often becomes spiritual in nature.
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The answer might depend on the specific
concept being looked at,
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but it can all feel like a
distorted zen koan.
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If there's a number of trees in a forest,
but no one's there to count them,
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does that number exist?