Number
Contemporary mathematics is an extremely abstract and complex science. But originally, its essential task was to manipulate numbers that represented concrete quantities. There would not have been all subsequent developments without that initial impulse born of the need to harness the power of numbers to solve practical problems. But what are numbers?
A number is essentially an abstraction. It is the concept of a given quantity. It presupposes the cognitive ability to analyze our experience of the world, recognizing similarities and differences of all kinds, from the most evident to the most subtle.
Similarities and differences allow us to catalog experiences and sensations. We learn from an early age to isolate the common traits of what we observe. In this way, we form concepts that transform collections of perceptions into objects of thought: we bite an apple, no longer just a rounded, fragrant thing with a juicy pulp.
Thanks to the capacity to recognize patterns and to catalog them, we soon learn that apples are different from pears, dogs are other from cats, friends are different from enemies and strangers.
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Once the concept of an object has been formed, we can separate it from the context and note the existence of groups of similar objects. And we can create groups of things that share common traits. For example, we can pick apples in a basket, satisfied with the thought that we will have fruit to eat for a few days. Or we can go further and count those apples.
Counting means associating things that we recognize as objects with a label that creates a relationship between them, representing their quantity, that is, their number.
With thousands of years of mathematical history behind us, counting may seem like a trivial operation, but it is not at all. It presupposes the ability to catalog the things we observe as objects of thought. And this, in turn, presupposes the capacity to recognize their common characteristics. We could not say, for example, how many apples there are in a basket that also contains pears if we did not have a well-defined concept of what an apple is and what a pear is. That is, if we had not learned to distinguish the ones from the others.
Therefore, a number is an abstraction that represents the quantity of something sharing common traits. Once we mastered this abstraction, we no longer needed the connection to material objects, from which the experience of counting and the concept of number were born. Thanks to abstraction, numbers have a life of their own, as pure objects of thought, and can be manipulated in complex and sophisticated ways, as happens in algebra.
The surprising and extraordinary thing about mathematics is that those complex manipulations of numbers that human thought has developed over centuries and millennia find a precise correspondence in the physical world, whose laws seem to be widely modeled precisely on mathematics principles, as Galileo already understood four centuries ago.
For example, it would not have been possible to send robotic probes to Mars if the highly complex engineering calculations required to create lengthy automated procedures of related events — from the launch of the rocket to the opening of the parachute in the atmosphere of Mars — were not able to model in advance, with great precision, the behavior of the various elements of a space mission in the changing physical environments in which they operate.
