If you do not want to lose all sense of proportion,
keep your distance!
The Latin word distare means "to be far away." For a long time, the word was used in geometry without ever being properly defined. The distance between two points meant the length of the line segment joining them, expressed in the unit of measurement used at the time or in the country concerned. Simple tools and techniques were used to measure it. Surveying was a profession.
But some distances were out of reach. Tools became more sophisticated, chiefly by incorporating mathematical methods that were more ingenious but still elementary.
Yet the need to generalize the notion of distance became apparent. The notion becomes more ambiguous when we assess the distance between two cities: do we mean the distance as the crow flies, the length of the journey by road—or even by highway—or the time it takes to travel from one to the other?
It was not until the early 20th century that the concept was clearly defined in mathematics and then generalized, following Georg Cantor’s work at the end of the previous century laying the foundations of set theory. Beyond geometric problems, this new conceptual framework made it possible to describe, and even understand, problems arising in new contexts. In analysis, for example, clarifying the notions of neighborhood and limit made it possible to study properties that are preserved on taking limits in certain function spaces.
The notion has continued to spread into a host of fields, making it perhaps one of the mathematical concepts with the greatest number of applications in everyday life. Without realizing it, whenever you give your social security number or use your car’s GPS, you are "working with" distances.
It is time, dear reader, for you to discover all this for yourself…
