What is time? What is space?

Time and space are fundamental concepts that we use as a reference to describe reality, and yet (or rather for this very reason) we are unable to define them unless we start from elementary labels and relationships assumed to be true (axioms).

Let's start from space

The most common notion of space is generally associated with the “Elements” of the Greek mathematician Euclid, who lived over 2300 years ago, between the fourth and third centuries BC. In order to define space and its properties, three elementary entities are used:

1. point: an infinitely small (dimensionless) entity associated with a position in space;
2. straight line: an infinitely long entity (with one dimension), composed by infinitely many points aligned with each other;
3. plane: an entity with infinite flat surface (with two dimensions).

Examples of axioms are the following :

• between any two points there is one and only one straight line;
• a plane is uniquely identified by three points that are not aligned with each other.

In Euclid's original intent, both these labels (point, line, and plane) and their relations were meant to formalize "obvious" circumstances about the space perceived by our senses.

Euclidean space is a great model of physical space in the context of our daily life. Well, the reality described by Euclidean space is that of a static system, made up of physical entities at rest with respect to each other.

However, we know that reality is made of entities and relationships between entities in continuous evolution, including us. In order to describe the evolution of these entities in space, we need to resort to another elementary concept: the instant of time.

The instant of time

Like all elementary concepts, it cannot be defined through other concepts, except in a circular form. However, we can identify some key relationships with the concept of reality:

• in an instant of time the reality we intend to describe does not change, it is frozen;
• between one instant of time and another, reality may change.

Reality can therefore be described through a series of “numbered snapshots”. This corresponds to the creation of a mathematical space with four dimensions: the three of physical space plus a number that reflects the instant in time. This construct is also called spacetime. To each point in spacetime, we can associate an event, that is, what happens in a particular position in physical space and at a certain instant in time.

Now, there is a problem. How to define the measurement unit between two points in time, that is, for the time interval? What we usually do, and often implicitly, is to use an object whose evolution in space is taken as a reference.

Think about our day. The reference system is characterized by the Earth's surface where we live. Using such reference frame, we observe the position of the Sun in space. We say that (the time of) one day has passed every time we observe the Sun in the same position. The same is true for our analog clocks (position of the " clock's hands") or digital (position of an electromagnetic signal), even though they are characterized by greater precision and resolution.

This is a working definition of time: how we measure it in practice.

To define (as precisely as possible) any aspect of reality we need to rely on the notion of elementary entities and axioms.

We cannot escape from that! If you manage to do it, please let me know ;-)

In hindsight, we can say that our assumptions are correct if they work, that is, if they allow us to make correct predictions about what happens in nature by applying logic reasoning (within the uncertainty limits that we can assign to our observations/measurements).