Number as Matching
What is the number 4? In many ways, this is similar to asking what the color red is. There is a reality behind the description, but fundamentally, humans are comparing two aspects of that reality and giving it a name.
The fundamental meaning of 4 is that we can take a collection of objects and say that there are 4 objects in the collection if we can match each object in that group with a distinct object in another group of 4 objects.
The mapping from the four squares to the four circles is good as each square has exactly one corresponding circle. Each circle has exactly one matching square. These groups have the same number of objects.
While the squares and dots have the same number of objects, the mapping indiciated in the picture is a bad mapping as there are two squares that map to a single dot.
We also have the mapping from circles to triangles. Each circle maps to exactly one triangle, but there is a triangle that is not mapped to. This not only means that these two groups of objects are not equal in number, but that the number of circles is smaller than the number of triangles. In this instance, we have four circles and five triangles. This is the sense in which five is larger than four.