A (very) brief history of infinity

Once con­sidered as a sac­red concept (only God is infin­ite) or a meta­phys­ic­al one (the human mind will nev­er be able to entirely con­ceive it), infin­ity has since entered the domain of sci­ence and tech­no­logy. Today, we meas­ure it, com­pare it, study it, and use it almost like a nor­mal number.

But what is infin­ity? And how did we learn to tame it?

It seems that ques­tions about infin­ity are almost as old as human­ity. Unfor­tu­nately, the writ­ings that attest to this are all the rarer because they are old. We know that the philo­soph­ers of the first mil­len­ni­um B.C. were already won­der­ing about the amaz­ing prop­er­ties of infinity.

But is infin­ity only a concept? A weird idea that math­em­aticians play with? Or does it have a con­nec­tion with the world around us? Is any­thing really infin­ite?

For Anax­i­m­ander, infin­ity is the found­ing prin­ciple of real­ity. From it are born an infin­ite num­ber of worlds that fill the volume of the Uni­verse. For Her­ac­litus, on the oth­er hand, it is time that is infin­ite. It has always been and will always be. It is through infin­ity that we per­ceive our own existence.

Of course, none of these state­ments are sup­por­ted by an “exper­i­ment” or a “meas­ure­ment” by cur­rent sci­entif­ic stand­ards. It is more a philo­soph­ic­al pos­i­tion that dif­fer­en­ti­ates one school of thought from another.

Of course, infin­ity is first and fore­most a math­em­at­ic­al concept. And it is math­em­aticians who took it upon them­selves to observe it a little closer. Zeno of Elia (~450 BC) tried to show the “phys­ic­al” impossib­il­ity of infin­ity, not by meas­ur­ing it but by using it to divide things into smal­ler and smal­ler elements.

The res­ult is his fam­ous “arrow para­dox” that, accord­ing to him, should nev­er be able to hit its tar­get. Indeed, one can always divide its remain­ing path by two and there will always remain a por­tion of path to cov­er (1/2, 1/4, 1/8, 1/16, 1/32), ad infin­itum.

Since it takes an infin­ite num­ber of steps to cross the dis­tance between the bow and the tar­get, Zeno con­cluded that it was impossible for the arrow to reach its des­tin­a­tion in a finite time.

How­ever, this para­dox was solved much later by one of the branches of math­em­at­ics that stud­ies infin­ite sums of num­bers: the series.

Adding 1/2+ 1/4+ 1/8+ 1/16+… is like adding 1/2+ 1/2²+ 1/2³+ 1/2⁴+…

This series is called a geo­met­ric series. It is writ­ten in the form: 

Its res­ol­u­tion is very simple. When n tends to infin­ity, the value of this sum tends nat­ur­ally to 1.

A schem­at­ic res­ol­u­tion is even sim­pler. In the fig­ure below, we can intu­it­ively see that, to fill a square with side 1, we must add the ele­ments whose area cor­res­ponds exactly to the series above it.

What Zeno was miss­ing was the coun­ter­in­tu­it­ive res­ult that the sum of an infin­ite num­ber of num­bers does not always give an infin­ite result.

In oth­er words, it is not because the tra­ject­ory of the arrow can be decom­posed into an infin­ite num­ber that the time it will take to cov­er them will be infin­ite. Para­dox solved. The arrows can now reach their tar­get with com­plete peace of mind.

Later, Isaac New­ton per­fec­ted the art of meas­ur­ing arbit­rar­ily small val­ues by devel­op­ing infin­ites­im­al cal­cu­lus. This led to the fam­ous deriv­at­ives (and integers) which math­em­at­ics, but also mod­ern phys­ics, could not do without today to describe and under­stand the world.

There is no need to under­stand or visu­al­ise infin­ity to use it. In the end, infin­ity is just a tool among many oth­ers that math­em­at­ics puts at our dis­pos­al to meas­ure, cal­cu­late and under­stand our environment.

But a Ger­man math­em­atician from the end of the 19^th Cen­tury went much fur­ther than any­one else at the time to manip­u­late infin­ity, or more pre­cisely infin­ite sets.

Georg Can­tor asked him­self a simple ques­tion: can we com­pare two infin­ite sets? Can one be “big­ger” than the other?

His answer lies in the way he com­pares two sets: instead of count­ing the num­ber of ele­ments of the lat­ter and com­par­ing them (which can­not be done with an infin­ite set), the meth­od con­sists in try­ing to match an ele­ment of the first set with an ele­ment of the second. If each ele­ment finds its part­ner and none remains alone (we call this a bijec­tion), we can then say that the two sets are equal. And this meth­od applies to both finite and infin­ite sets.

This is how we can prove that the size of the (infin­ite) set of pos­it­ive integers is strictly equal to that of the set of integers (pos­it­ive and negative).

What’s more, we can also show that, although there are infin­itely many frac­tions between two integers, the size of the set of integers is strictly equal to the size of the set of num­bers that are writ­ten as frac­tions. How­ever, it has also been proven that the set of real num­bers (all num­bers writ­ten with a decim­al point and a finite or infin­ite num­ber of decim­al places) is strictly lar­ger than the set of integers.

As counter-intu­it­ive as it may seem, two dif­fer­ent infin­it­ies can have the same size, but, con­versely, not all infin­it­ies are equal.

Can we draw fig­ures with infin­ite parameters?

Besides the circle (which can be con­sidered as a poly­gon with an infin­ite num­ber of sides), oth­er strange fig­ures star­ted to emerge dur­ing the second half of the 20^th Cen­tury: fractals.

One way to cre­ate them is to build them by iter­a­tion, step by step. After an infin­ite num­ber of steps, the fig­ure is fin­ished, and we can study its prop­er­ties. The Von Koch flake, for example, is an extraordin­ary fig­ure: although its sur­face is finite, its peri­met­er is infinite.

This kind of geo­metry has been suc­cess­fully applied in the field of tele­com­mu­nic­a­tions. Since the end of the 80’s, fractal anten­nas have been developed, whose length, if not infin­ite, is very large, but whose volume remains small, which allows to obtain com­pact and effi­cient systems.