- 4507 -
SQUARE ROOT OF TWO? - The ancient Greeks wanted to believe that
the universe could be described in its entirety using only whole numbers and
the ratios between them. These fractions
are what we now call “rational numbers”. But this aspiration was undermined
when they considered a square with sides of length 1, only to find that the
length of its diagonal couldn’t possibly be written as a fraction.
-
-------------------------------- 4507 - SQUARE ROOT OF TWO?
- The first proof of this (there would be
several) is commonly attributed to Pythagoras, a 6th-century BCE philosopher,
even though none of his writings survive and little is known about him.
Nevertheless, it was the first crisis in what we call the foundations of
mathematics.
-
- That crisis would not be resolved for a long
time. Though the ancient Greeks could establish what √2 was not, they didn’t have a language for
explaining what it was.
-
- For millennia, this sufficed. Renaissance
mathematicians manipulated what they came to call “irrational numbers” while
trying to solve algebraic equations. The modern notation for square roots came
into use in the 16th and 17th centuries. But still, there was something
slippery about them. Does √2 exist in
the same way that 2 does? It wasn’t clear.
-
- Mathematicians continued to live with that
ambiguity. Then, in the mid-1800s, Richard Dedekind realized that calculus,
which had been developed 200 years earlier by Isaac Newton and Gottfried
Leibniz, stood on a shaky foundation. He was a reserved but gifted
mathematician who worked slowly and published relatively little, Dedekind was
preparing to teach his students about continuous functions when he realized
that he couldn’t give a satisfactory explanation of what it meant for a
function to be continuous.
-
- He hadn’t even seen functions properly
defined. And that, he argued, required a good understanding of how numbers
worked — something mathematicians seemed to have taken for granted. How, he
asked, could you know for sure that √2 multiplied by √3 equals √6? He wanted to
provide some answers.
-
- And so he introduced a way to define and
construct the irrational numbers using only the rationals. Here’s how it works:
First, split all the rational numbers into two sets, so that all of the
fractions in one set are smaller than those in the other. For instance, in one
group, collect all rationals that, when squared, are less than 2; in the other,
put all rationals whose squares are greater than 2. Exactly one number plugs
the hole between these two sets. Mathematicians give it the label √2.
-
- For Dedekind, then, an irrational number is
defined by a pair of infinite sets of rational numbers, which create what he
called a “cut'. Dedekind showed that
you can fill in the entire number line this way, rigorously defining for the
first time what are now called “real numbers” (the rationals and the
irrationals combined).
-
- Cantor came up with a different definition
of irrational numbers. He expressed each in terms of sequences of rational
numbers that approached, or “converged” to, a particular irrational value.
Though Cantor’s irrational numbers initially looked different from Dedekind’s,
later work proved that they are mathematically equivalent.
-
- Cantor’s work led him to ask how many
numbers exist. The question might at first seem strange. There are infinitely
many whole numbers. You can always keep
adding one more. Presumably, that’s as big as a set of numbers can get. But
Cantor showed that, paradoxically, though the number of fractions is the same
as the number of integers, there are demonstrably more irrational numbers. He
was the first to realize that infinity comes in many sizes.
-
- The number line was more crowded, and
weirder, than anyone had imagined. But mathematicians were only able to see
that after a change in perspective.
-
- Dedekind’s “cuts” are arguably the
beginning of modern mathematics. It’s
really the first point in the history of mathematics where mathematicians
actually know what they’re talking about.
-
- Dedekind and others used his definition to
prove major theorems in calculus for the first time which allowed them not just
to strengthen the edifice that Leibniz and Newton had built, but to add to it.
-
- Dedekind’s work enabled mathematicians to
better understand sequences and functions. Emmy Noether, a prolific
mathematician who helped shape the field of abstract algebra in the early 20th
century, is said to have told her students that “everything is already in
Dedekind.”
-
- A formal definition of √2 opened new horizons for exploration beyond
the topics in calculus that initially motivated Dedekind. After Dedekind, mathematicians started to
realize that you can invent new concepts altogether. … The whole idea of what
mathematics is about becomes much broader and more flexible.
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-
June 13, 2024 SQUARE
ROOT OF TWO? 4500
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--------------------- --- Saturday, June 22, 2024
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