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Mathematics
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| Don't fear the math. |
How Math Works
BY ROBERT LAMB
It's
easy to think of mathematics as a kind of storybook sorcery -- a powerful
secret language known to few, mastered by inhuman agents (such as your
calculator) and underpinning the very fabric of the universe.
Even
if we avoid such hyperbole, the fact remains: Many of us are mathematically
illiterate in a world that runs on math.
When
was the last time you seriously crunched some numbers with only pen and paper?
In
his book "The Geometry of Paradise," Mark A. Peterson described the
people of medieval Europe as a non-mathematical culture in possession of
sophisticated mathematics.
Mathematicians
of the day certainly honed their skills but mostly out of love for mathematical
abstractions.
They
perused few practical applications with it and, according to Peterson, didn't
really grasp what math was.
Today,
the mathematics field is far more vibrant than it was in the Middle Ages, but
it still eludes an alarming number of those who depend on it.
On
one hand, math certainly has a way of solving itself these days through
calculators and hastily keyed-in Google searches.
Yet
for many individuals, mathematical anxiety begins with inadequate teaching from
non-mathematicians who have trouble relaying enthusiasm and practicality.
Factor
in overcrowded classes, and it's little wonder that so many students fail to
latch onto math's logical core.
In
fact, only 40 percent of 4th graders and 34 percent of 8th graders in the U.S. are
proficient in math, according to Arne Duncan, U.S. education secretary
speaking at the National Council of Teachers of Mathematics in April 2011.
The
ramifications of mathematical illiteracy are very real.
In
2005, the United States National Academies identified the country's decline in
mathematics education as having a severe detrimental effect on its scientific,
technological and economic prowess [source: Mullich].
So
let's demystify the world of mathematics.
A
world without math is unimaginable. It's a part of who we are.
It's
the analytical juice of our left brain and, in the words of physicist
Richard Feynman, even a fool can use it.
Here's
a quote from the late great scientist's book "The Pleasure of Finding
Things Out":
“What
we've been able to work out about nature may look abstract and threatening to
someone who hasn't studied it, but it was fools who did it, and in the next
generation, all the fools will understand it. There's a tendency to pomposity
in all this, to make it deep and profound.”
In this article,
we'll take a very wide-angle look at the world of numbers.
Just what are
they, and what does math really do?
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What Are Numbers?
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What Are Numbers?
Mathematics boils down to pattern recognition.
We
identify patterns in the world around us and use them to navigate its
challenges.
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| A boxing referee administers the count. |
What
are numbers?
As
we'll explore more later, that's a deceptively deep question, but you already
know the simple answer.
A
number is a word and a symbol representing a count.
Let's
say you walk outside your home and you see two angry dogs.
Even
if you didn't know the word "two" or know what the corresponding
numeral looks like, your brain would have a good grasp of how a two-dog
encounter compares with a three-, one- or zero-dog situation.
We
owe that innate comprehension to our brain (specifically, the inferior
parietal lobe), which naturally extracts numbers from the surrounding
environment in much the same way it identifies colors [source: Dehaene].
We
call this number sense, and our brains come fully equipped with it
from birth.
Studies
show that while infants have no grasp of human number systems, they can still
identify changes in quantity.
Neuroimaging
research has even discovered that infants possess the ability to engage
in logarithmic counting, or counting based on integral increases in
physical quantity.
While
a baby won't see the difference between five teddy bears and six teddy bears in
a lineup, he or she will notice a difference between five and 10 [source: Miller].
Number
sense plays a vital role in the way animals navigate their environments --
environments where objects are numerous and frequently mobile.
However,
an animal's numerical sense becomes more imprecise with increasingly larger
numbers.
Humans,
for instance, are systematically slower to compute 4 + 5 than 2 + 3 [source: Dehaene].
At
some point in our ancient past, prehistoric humans began to develop a means of
augmenting their number sense. They started counting on their fingers and toes.
This
is why so many numerical systems depend on groups of five, 10 or 20.
Base-10
or decimal systems stem from the use of both hands, while
base-20 or vigesimal systems are based on the use of fingers
and toes.
So
ancient humans learned to externalize their number sense and, in doing so, they
arguably created humanity's most important scientific achievement: mathematics.
The Tower of Math: Numbers
Numbers pose a difficulty for humans.
Sure,
some of us have more of a gift for math than others, but every one of us
reaches a point in our mathematical education where things become hard.
Learning
your multiplication tables is difficult because the human brain never evolved
to handle such advanced computations as 17 x 32 = 544.
After
a certain point, our mathematical education is largely an exercise in
rejiggering ill-adapted brain circuits [source: Dehaene].
Number
sense may come naturally to us, but mathematical literacy comes only with time.
Likewise,
humanity's use of mathematics has steadily grown over the ages.
Like
science itself, math isn't the product of one mind but rather a steady
accumulation of knowledge throughout human history.
Think
of math as a tower. Natural human height is finite, so if we're to reach higher
into the air and see out farther across the landscape, we'll need to build
something external to ourselves.
Our
mental abilities to understand math are equally finite, so we build a great
tower of number systems and climb upward to the stars.
To
break down the basic structure of this tower, let's first look at the raw
materials. These are the basic types of numbers:
Integers: You
probably know these as whole numbers, and they come in both positive and
negative forms. Integers include the basic counting numbers (1-9), negative
numbers (-1) and zero.
Rational numbers include integers but also encompass simple fractions that
can be expressed as a ratio of two integers. For example, 0.5 is rational
because we can also write it as 1/2.
Irrational numbers: These numbers can't be written as a ratio of two integers. Pi
(the ratio of the circumference of a circle to its diameter) is a classic
example, as it can't be written accurately as a ratio of two integers and has
been calculated to trail off decimal points into the trillions.
Rational
and irrational numbers both fall under the category of real numbers or complex
numbers. And yes, there are also imaginary numbers that
exist outside the real number line, and transcendental numbers,
such as pi.
There
are many other different numbers types as well, and they, too, play a part in
the structure of our tower.
On
the next page, we'll look at some of the core branches of mathematics.
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The Tower of Math: Branches of Mathematics
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The Tower of Math: Branches of Mathematics
Who would you hire to build a tower?
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| Circa 100 B.C., Greek astronomer Hipparchus, inventor of trigonometry, studies the heavens. |
Likewise,
many branches of mathematics play a part in the tower of math. Here are just a
few.
Arithmetic:
This is the oldest and most basic form of mathematics. Arithmetic chiefly
concerns the addition, subtraction, multiplication and division of real numbers
that aren't negative.
Algebra:
The next level of mathematics, algebra, is essentially arithmetic with unknown
or abstract quantities thrown in with the real numbers. We represent the
abstracts with symbols, such as X and Y.
Geometry:
Remember what we said about math helping us navigate a world of numerous and
movable objects? This is where geometry comes into play, dealing chiefly with
the measurements and properties of points, lines, angles, surfaces and solids.
Trigonometry:
Trigonometry concerns the measurements of triangles and the relationships
between their sides and angles. While the historical origins of arithmetic,
algebra and geometry are lost in the fog of ancient history, trigonometry
originates with second century astronomer Hipparchus of Nicaea.
Calculus:
Independently developed by both Isaac Newton and Gottfried Leibniz in the
17th century, calculus deals with the calculation of instantaneous rates of
change (known as differential calculus) and the summation of
infinite small factors to determine some whole (known as integral
calculus). As such, it has proven a vital scientific tool in a number of
disciplines.
The
tower of mathematics has enabled human culture to rise and flourish, to
understand both the inner mysteries of the cells to the outer mysteries of
space.
But
did we truly build this tower out of our own ingenuity? Did we invent
mathematics or merely discover it?
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Math: Human Discovery or Human Invention?
So just what, in essence, is this thing called math?
So just what, in essence, is this thing called math?
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| Does the universe conform to math, or math to the universe? |
Is
mathematics, in the words of Galileo, the language of God?
Or
is math just a human-created system that happens to correspond with natural
laws and structures?
There
is no definitive answer to this question, but mathematicians tend to side with
one of several compelling theories.
First,
there is the Platonic theory. Greek philosopher Plato argued that
math is a discoverable system that underlines the structure of the universe.
In
other words, the universe is made of math and the more we understand this vast
interplay of numbers, the more we can understand nature itself.
To
put it more bluntly, mathematics exists independent of humans -- and will continue on long after we're extinct.
The
opposing argument, therefore, is that math is a man-made tool -- an abstraction
free of time and space that merely corresponds with the universe.
Just
consider elliptical planetary orbits. While such an elliptical trajectory
provides astronomers with a close approximation of the planet's movement, it's
not a perfect one [source: Dehaene].
Several
theories expand on this idea.
·
The logistic theory,
for instance, holds that math is an extension of human reasoning and logic.
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The intuitionist theory defines
math as a system of purely mental constructs that are internally consistent.
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The formalist theory argues
that mathematics boils down to the manipulation of man-made symbols. In other
words, these theories propose that math is a kind of analogy that draws a line
between concepts and real events.
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The fictionalist theory,
while less popular, even goes so far as to equate mathematics with fairy tales:
scientifically useful fictions. In other words, 1 + 1 = 2 might enable us to
understand how the universe works, but it isn't a "true" statement.
Who's
right? Who's wrong?
There's
ultimately no way to know, but on the next page we'll look at two examples of
what each possibility could mean to our understanding of the universe.
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The Mathematical Universe
The history of mathematics is a history of humanity
seeking to understand the universe.
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| Can math explain it all? |
Math
generally plays a vital role in any theory of everything, but contemporary
cosmologist Max Tegmark even goes so far as to theorize that the universe
itself is made of math.
In
his mathematical universe hypothesis, he proposes that math is
indeed a human discovery and that the universe is essentially one gigantic
mathematical object.
In
other words, mathematics no more describes the universe than atoms describe the
objects they compose; rather math is the universe.
Tegmark
even goes so far as to predict that a mathematical proof for a theory of
everything could eventually fit on a T-shirt.
More
than 60 years earlier, however, Austrian mathematician Kurt Gödel put forth a
theory that argues quite the opposite.
Gödel's
first incompleteness theorem concerns
axioms, logical mathematical statements that we assume to be true but can't be
proven with a mathematical proof.
A
simple example of this would be the axiom of equality (X = X). We assume this
to be a true statement, but we can't actually back it up with a mathematical
proof.
Gödel's
theorem states that any adequate axiomatizable theory is incomplete or inconsistent.
The
implication, according to theoretical physicist and mathematician Freeman
Dyson, is that mathematics is inexhaustible.
No
matter how many problems we solve, we'll inevitably encounter more unsolvable
problems within the existing rules [source: Feferman].
This
would also seem to rule out the potential for a theory of everything, but it
still doesn't relegate the world of numbers to either human invention or human
discovery.
Regardless,
mathematics could stand as humanity's greatest invention.
It
composes a vital part of our neural architecture and continues to empower
us beyond the mental limits we were born with, even as we struggle to fathom
its limits.
About Robert Lamb
As a child, Robert Lamb dreamed of becoming a mad
scientist when he grew up. As this profession proved to be largely fictional,
however, he swallowed his heartbreak and turned his attention to the written
word instead. He earned his bachelor's degree in creative writing from the
University of Tennessee in Knoxville, which launched him on a career path
through high school English classrooms, small-town newsrooms and finally into
the offices of HowStuffWorks.
As a senior writer and podcaster for Stuff
to Blow Your Mind, Robert now spends his days
sifting through all the scientific wonders that make the world so mad and
amazing. He currently lives in Atlanta with his lovely wife and their beautiful
one-eyed cat. When he's not researching the apocalypse or the miracle we call a
space toilet, he enjoys listening to electronic music, painting the odd
miniature and writing fiction.
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