About numbers, episode 4 : infinite numbers

Medium : for amateurs

« Eternity is really long, especially near the end. »

Woody Allen (1935 - )

The lemniscate, symbol of infinity
Source favim.com

What is infinity ? How to grasp such a concept, by definition so immense that nothing can contain it ?
As surprising as it may seem, the answer lies not in philosophy, but really in mathematics.
We will see that it is possible to define infinity in a rigorous and coherent manner.
There even exists numerous different infinities : precisely an infinity of them !
Three separate notions of infinity play a major role in mathematics :
The ordinal infinity, defined as a number greater than any natural number, the cardinal infinity as the number of elements of an infinite set, and the infinity used in calculus and especially limits as an unreachable element.

However, the latter of the three is misleading, being more representative of the concept of unboundedness than that of infinity, as was explained to us by Riemann and Einstein.
Of course it doesn't stop there, and the mathematicians' imagination can give us a lot more definitions of infinity, which shows us how limitless the human mind can be.

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About numbers, episode 3 : rational numbers

Medium : for amateurs

The Vitruvian man, by Leonardo Da Vinci
Source Luc Viatour

What is a fraction ?
It seems easy to answer this question with the concept of proportion, but we will see that rational numbers possess some quite extraordinary properties.
Even though possible, it would not be very enlightening to make the same distinction we made with natural numbers and integers between ordinality and cardinality. But cardinality raises a few questions : what is a fractional quantity ? Worse, what is a negative fractional quantity ?
Once those interrogations answered, we will see a few examples of irrational numbers, such as $\pi$ and $\sqrt 2$.
Then we'll tackle a few structural and topological properties of the set of rational numbers.
Take an aspirin, it is going to be a bit more complicated than the previous posts of the series.

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The myth of Niels Bohr and the barometer question

Neutral : for everyone

Water barometer
Source : Jean-Jacques MILAN
(Wikimedia commons)

In 1959 was published in the journal Pride of the American College Public Relations Association an essay entitled Angels on a Pin, by Alexander Calandra, professor of physics at Washington University in St. Louis, Missouri. The story is about a physics student who surprises his professor on a simple question of physics.

« On almost every level, this essay falls apart on critical analysis. I wonder why it has become such a legend in the physics community ? »
Donald Simanek, emeritus professor of physics at Lock Haven University of Pennsylvania.

The title is supposed to be a reference to medieval scholastics adept of meaningless questions such as How many angels can dance on the point of a pin ?
Personally, I see in this fable an illustration of the importance of creativity in sciences, a skill that is too rarely stimulated in the teaching of sciences, when it isn't consistently ignored...

The student is generally said to be Niels Henrik David Bohr (1885 - 1962), Nobel prize of Physics in 1922 and the referee is supposed to be the chemist Sir Ernest Rutherford (1871-1937), Nobel prize of Chemistry in 1908, even if they didn't meet until Bohr finished his scholarship.

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A mathematician's lament

Neutral : for everyone

« The most desperate are the most beautiful lament »

(Alfred de Musset, La nuit de Mai, 1835)

The keys of success...
Source : Photos Libres

A mathematician's lament (2002) is an article from Paul Lockhart, a first-class research mathematician and teacher at Saint Ann's School in Brooklyn, New York. Not only is it a critique of current K-12 mathematics education in the United States, but it's also a reminder that mathematics are also and above all an art, like music or beaux-arts. I invite you to read at least the two introductory pages.

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About numbers, episode 2 : Integers

Easy : for the curious

Negative numbers
(Source Wikipedia commons)

A quantity of -2 objects, a marathon runner finishing -3rd, numbers smaller than nothing ?! What do negative numbers mean ?

The answer to that question is easy only to those who are used to them. And for most of us, it doesn't matter what they really are or what they mean, because the thing is these numbers are quite useful !

However, for the ancient Greeks as well as for the Arab and Indian mathematicians - who already understood the meaning of the zero and the negative numbers -, the negative roots of an equation are thought of being unnatural, absurd even according to Nicolas Chuquet (~1450-1488) and would not be recognized until the XIXth and XXth century ! So useful, but also very problematic...

We will see that their meaning is more subtle than we would think, and that it is illuminating to base it on the distinction between ordinals and cardinals that we made in the preceding chapter about natural numbers

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