Platón is a Greek philosopher famous for his theory of ideas. I always had hard time to envision what exactly it is supposed to mean in the real world (knowing mostly the cave and shadows story). However Petr Vopěnka managed to explain the ideas quite nicely on the example of Platón's geometry.
Ideas and Geometry
According to Platón, there is a world of ideas which are independent on the real and geometric worlds (yes, the real world is different than the geometric one; at least in Platón's geometry). What we can do is to seek for these ideas to appear on object of real/geometric world.
Envision an idea of being round. That is an idea that lives in the world of ideas.
Now apply the idea to the world of geometric objects. Imagine circle and ellipsis. Certainly both these objects are round - the idea of roundness is present in both of them. Moreover one can argue that circle is more round than ellipsis, as its roundness is the same at every point of its edge (while ellipsis is deformed). From the above it is just a step to conclude that circle is more beautiful (with respect to roundness) than ellipsis.
Apply the idea of roundness to apple and pear. Again apple is clearly more round (thus more ideal) than a pear. However compared to a circle, a normal apple is just a poor imitation of the roundness of a circle. This is a common observation when comparing objects from real world to objects of a geometric world. The conclusion to draw from the observation - when seeking for beauty, do it in the geometric world. As the geometric world is much likely place for ideal beauty than the real one!
Platonic vs. Pragmatic
The relationship between the world of ideas and the geometric world nicely illustrates what Platonic approach to science means. The ideas are eternal. Unchangeable. Often (if not exclusively) they represent something positive. Beautiful. The task of a science is to analyse its field (its world) and seek for the ideas reflected in objects of such world. By shining more light on such world (e.g. enlarging one's horizon), by discovering more objects, more relationships between such objects and connecting that with ideas one discovers beauty in such world.
The heritage of such approach can be traced in rationalism and in some form remains in all of us. On the other hand, often we need results and as such it seems pragmatism is often the measure of success of R&D we use in day to day comparison.