Here is an excerpt from a piece by Alain Connes in The Princeton Companion to Mathematics:

It might be tempting at first to regard mathematics as a collection of separate branches, such as geometry, algebra, analysis, number theory, etc., where the first is dominated by the attempt to understand the concept of “space,” the second by the art of manipulating symbols, and the third by access to “infinity” and the “continuum,” and so on.

This, however, does not do justice to one of the most important features of the mathematical world, namely that it is virtually impossible to isolate any of the above parts from the others without depriving them of their essence. In this way the corpus of mathematics resembles a biological entity, which can only survive as a whole and which would perish if separated into disjoint pieces.

The scientific life of mathematicians can be pictured as an exploration of the geography of the “mathematical reality” which they unveil gradually in their own private mental frame.

It is this about the nature of mathematics that has led me to wonder about the way it reflects cognition itself. Our nervous system integrates the work of delicate sensory devices, which often overlap in function, bringing discernible form to the flux around us. In this way we find our fundamental orientation in the world. The human development of music, language, art, and science are likely extensions of this. Perhaps mathematics draws on some of these internal mechanisms, ones that extract form from flux, and then somehow formalizes the possibilities for relationship among the things we experience.

Connes later says:

Where things get really interesting is when unexpected bridges emerge between parts of the mathematical world that were remote from each other in the mental picture that had been developed by previous generation of mathematicians. When this happens, one gets the feeling that a sudden wind has blown away the fog that was hiding parts of a beautiful landscape.

It is with mathematics that we find an unexpected link between the imagination and reality. Our confidence that we can rely on the consequences of a particular kind of reasoning leads to startling new possibilities. It makes sense that there was some concern, particularly in the eighteenth century, that we could violate the demands of this reasoning with ill-conceived generalities and non-constructive proofs.

As Donal O’Shea says nicely in The Poincare Conjecture:

Popular accounts of mathematics often stress the discipline’s obsession with certainty, with proof. And mathematicians often tell jokes poking fun at their own insistence on precision. However, the quest for precision is far more than an end in itself. Precision allows one to reason sensibly about objects outside ordinary experience. It is a tool for exploring possibility; about what might be as well as what is.

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