1.2 The Continents of Mathematics Review

summary

What is mathematics and how does it connect to physical reality?

For centuries, mathematics has been instrumental to the progress of science and technology.
Galileo wrote, “The laws of nature are written in the language of mathematics,” and without mathematics, “we are wandering about in a dark labyrinth.”
Nobel Prize-winning physicist Eugene Wigner remarked on the “unreasonable effectiveness of mathematics in the natural sciences.”
Is math something in nature, or something we create? Why is it so effective in physical reality? In other words, does math exist by itself without physics?

Many important discoveries in physics have been preceded by works in mathematics.

There are many areas of mathematics with no immediately obvious physical connections to the world around us. These areas have been developed within the narrative of mathematics itself, without any reference to physics.
For instance, Einstein’s general theory of relativity was preceded by work by mathematician Bernhard Riemann, who worked on the intrinsic structures of curved geometric shapes.
Riemann looked at these curved surfaces without referring to a “landscape” or ambient space in which they were embedded.
This was contrary to the contemporary description of curved shapes, which fundamentally viewed them as embedded in a flat, Euclidean space.
Einstein’s insight into intrinsically curved spaces is what helped him formalize his theory of relativity.
Mathematics is a hidden “parallel universe,” full of elegance and beauty, intricately intertwined with our world.

Just as in the physical sciences, there is an idea of unification in mathematics.

Mathematicians have also been motivated by this desire to understand our mathematical world from very few, fundamental principles, and to find connections between vastly different areas of mathematics.
We can think of mathematics as a giant jigsaw puzzle—we have all of the pieces, but we don’t have the box, and so we don’t know what the completed puzzle is supposed to look like.
Another way to look at the different areas of mathematics is to imagine them as continents on the same globe, incredibly different and yet all part of the same underlying framework.

What are the different "continents" of mathematics?

Number theory—figuring out what kind of numbers can exist and how we can use them.
Harmonic analysis—deconstructing and reconstructing wave patterns, like different instruments in a symphony.
Geometry—looking at the intrinsic shapes of mathematical objects, even if they defy our extrinsic physical understanding.
These are just three examples of seemingly different areas of mathematics. They are driven by different kinds of problems. They were developed by different people, to different ends. Could it be that they are somehow connected?

The Langlands program looks to unify the different continents of mathematics.

Robert Langlands, a Canadian-born mathematician and professor at the Institute for Advanced Study, proposed the idea in the late 1960s.
In a sense, the Langlands program could be a “grand unified theory.” While it doesn’t connect every area of mathematics, it touches upon many, and has continued to evolve and expand.
This series of conjectures may one day help provide a fundamental basis to many far-reaching branches of mathematics.