Little Mathematical Riding Hood

For (female) mathematicians

GOOD AGAINST EVIL. A classic plot with well-defined characters: a young girl -sweet, innocent and naïve- represents good; and a monstrous, despicable and horrific animal represents evil. And what about justice? Who represents law and order? The hero of the day is the brave hunter who defeats the devil and saves the angelical, defenceless child. The plot, as is only right and proper, comes to a crescendo with a grand finale in which righteousness triumphs over wickedness. That's all very well! But it is not how things really went..

When they wrote the story of Little Red Riding Hood, Perrault, first, and then the Brothers Grimm simply transcribed an old European tale word for word. However, their transcription was not impartial; it was influenced by the social values of their time, the 18th century. Women without men were nothing, and they were expected to be beautiful, submissive and pleasing. In Perrault's version, the girl's naivety is fully deserving of her tragic end and there are no mitigating circumstances. The Brothers Grimm were more magnanimous, and they saved her in extremis, after she had been devoured by the wolf. Salvation arrived, of course, in the form of a male hunter. Nevertheless, Perrault and the Brothers Grimm were the sort of old-fashioned intellectuals who did not appreciate the power of maths. They made the mistake of not recognizing that, when you have to decide between a long and a short path, everything comes down to a mere numerical comparison and, therefore, to a mathematical decision. Let me tell you once and for all what really happened.

The main character of the story was indeed a young woman, just as beautiful as you want her to be. Even so, what really stood out was not her beauty or her figure or whether her coat was red or a light fuchsia pink as dictated by the Parisian fashion of the time, but her eyes, which shone with intense mathematical intelligence. And then there was the big, bad wolf, more conceited than brave, more loud mouthed than intelligent. And if you are wondering about the girl's so-called saviour -that's to say, the hunter- well, you can forget him because our young lady was more than a match for the wolf. She didn't need anyone to do her dirty work for her. If he had realised this, our ravenous wolf would have eaten her alive there and then, and would never have dared to take her on in a face-to-face confrontation.

Our story begins when the young girl was in a corner of the forest in the shade of a majestic ash tree. Let's say that this is point A. She was about to set off for her grandmother's country cottage. Let's say her destination is point C. From point A to point C you could go in a straight line, or follow a different path that went round a midway point marked by a huge centennial birch. We shall call this midway spot point B. This alternative path from A to B was a straight line of exactly 100 steps, and at B it veered 90 degrees to the left and went straight for a further 200 steps to reach C.

-All I asked was where you were going. My head's going to explode with so many points and steps! -the wolf complained in response to the girl's explanations and, in so doing, revealed his intellectual limitations. To help him out, the young girl sketched the situation she had just described on the ground.

In fact, the wolf was only interested in finding out how he could get to grandmother's cottage before the girl. With just a glance at the sketch, the wolf understood that the shortest route was the first one, from the ash tree to the cottage. If he took that path, he would get to the house first and be able to hide and take her by surprise. But what if someone found out that such a magnificent beast as himself had devised such a vile and cowardly strategy to assault the girl? No way! He would make sure that didn't happen by adorning the story, ignoring a couple of minor details and making it quite clear how clever and brilliant he had been once again.

To his surprise, the girl did not complain and made no objection to taking the long path, the one that went past the birch. "She thinks she's so smart but I can twist her round my little finger" -thought the wolf, more conceited than ever. The only condition that the girl insisted on seemed to him to be quite trivial: they both had to promise that they would take full steps or steps that were half the length of the previous step and they could not go past point C.

-That's ridiculous! But, yes, of course I promise! Don't finish off all your grandmother's cake before I arrive -the wolf said, with barely concealed scorn and contempt, as he hurried off to the shortest path, taking care to take full steps just as she had stipulated.

The young girl set off at a leisurely pace, and in one hundred steps she reached B with no difficulties. As soon as she got to the birch, she turned ninety degrees and, after two hundred steps, arrived at point C, her grandmother's country cottage. As she imagined, there was no sign of the wolf.

"But where was the blessed wolf?" -you must be wondering. He had taken the shortest path, hadn't he? Indeed he had, but the young girl's condition had been very clear: he could only take full steps or steps that were half the length of the previous step. The first 223 steps did not pose a problem for the wolf, who walked triumphantly on, delighted to see how fast he was moving away from point A towards point C. But it was when he was about to take step 224 that his problems began: he realised that if he took the whole step he would go past point C, leave it behind him, and he would no longer be on the segment AC. If he had taken this extra step, if he had gone past point C, he would have fallen into sidereal space and been lost forever more. The fact is that the universe of mathematics is a very serious, even dangerous, thing and should not be taken lightly. Even a stupid wolf like himself knew that!

"Fortunately," -thought the wolf- "I can take a half step." This half step allowed him to move forward a distance of 0.5 steps, so he was now 223.5 steps from point A. But he had not yet reached point C. The next step was also complicated. If he had taken a new step, it would have taken him past point C. He tried to take a step that was half the length of the previous half step: that's to say, a quarter of a step or, what amounts to the same thing, a step of 0.25. But that didn't do the trick either! The wolf went on trying to divide steps until he found the next little step that he could take without going past C. It was extremely small, a step of just 0.0625 in length, which took him to a point that was 223.5625 steps from point A. Every step was taking him further from point A, but he still hadn't got to point C. And that is what the wolf was doing, calculating and recalculating the size of the next little step he could take without going past C. He was getting further and further from point A, but never quite reaching reach C without going past. And if you ask me, I expect that he is still there, calculating and recalculating, going out of his mind, becoming a little more irrational with every infinitely small step forward that he takes. As irrational as the irrational distance √50,000 which he was attempting, in vain, to cover by using regular intervals.

You can leave it up to Perrault and the Brothers Grimm to discuss the girl's fate at the end of the tale. You now know what really happened, and that she decided her own future freely and independently. So now that the wolf has been defeated, and the young mathematician is quietly having tea with her grandmother, the tale has come to an end, she has won and she will live happily ever after.

This version of the tale, the true account of what happened, also has a lesson to teach: don't follow the example of the brainless wolf who agreed to a mathematical deal even though he didn't understand maths. You would do better to look to the example of the young girl who did understand THE POWER OF MATHEMATICS.

Mathematical note

About 2,500 years ago, Pythagoras founded a mathematical society known as the Pythagorean School. Intuitively, Pythagoreans assumed that if you took two numbers (for example, the numbers 5 and √5) the interval between them could be exactly divided using a common interval. The discovery that this was not always true was the origin of irrational numbers. Thus, √5 was recognized as an irrational number. Irrational numbers raised doubts about all the mathematical knowledge of the age, and sparked an existential crisis among the Pythagoreans. Although he was not aware of it, the wolf in our tale was trying to find this interval that does not exist, which is why I can say that it will take him all eternity to complete the task he was set by the young girl.

Once they had got over their initial confusion, the Pythagoreans were able to draw an important conclusion: mathematicians cannot rely on their intuition; they can only state that something is true if it has been proved irrefutably. This attitude and the concept of mathematical proof are the two essential legacies of the Pythagoreans, which are still valid in mathematics today.

The Pythagorean School accepted both men and women. Its members lived a community life and made vows to keep all their mathematical findings in strict secrecy. In the following centuries, since nobody really knew who had discovered what, all the school's findings were attributed to its founder, Pythagoras. The school was so prestigious that it was attributed with more discoveries than it actually made. For example, nowadays it is known that, 1,500 years before Pythagoras was born, the Babylonians used a numerical formula to draw perpendiculars that was comparable to Pythagoras' famous theorem. What the Pythagoreans did was to formally prove the theorem for the first time. However, it is more likely that the theorem was put forward by a member of the school and not Pythagoras himself. In fact, there is a 50% chance that it was put forward by one of the female mathematicians.

Urbano Lorenzo Seva, Reus 2019

Translation by John F. Bates and Urbano Lorenzo Seva