Is Pure Mathematics a Waste of Time?

[Tac-Tics]

What use is a use?

In fact, what does 'useful' even mean?

People keep saying art and music are 'useless' in this thread. Maybe the use is in the joy they bring to people. Or the feeling of accomplishment they bring to the artist. Why be so literal about the meaning of the word? It's not like knowing chemistry is any more useful than knowing how to draw. How often in life do you perform quantitative experiments in your home or work? I'd say there is probably more use to a drawing class than a chemistry lab, because the former you can at least practice on your own.

Just playing a little devil's advocate.

The one aspect of pure math I like is its compressibility. Memorization is hard and time consuming. The more truth we can encode into something, the better. Pure mathematics is very, very good at this. The reason is that one of the unspoken principles in mathematics is simplicity. The axioms are few in number. General theorems trump special cases. Special exceptions to the rules must all be declared ahead of time. Even our books are terse to the point it's often hard to rebuild the main idea the author had when he wrote it. Personally, I find the absolute regularity appealing. It provides a confidence to your results that simply doesn't exist in the real world. (It also happens to remove the need for tedious experiments).

 

[phreak]

I think one point that is not understood here is that the rigorousness of mathematics is necessary for higher-level physics. Let us consider Hamiltonian mechanics, which is a necessary extension of classical mechanics. In Hamiltonian mechanics, symplectic manifolds are indispensable. A symplectic manifold is a 2nd countable Hausdorff topological space with a symplectic structure - that is, a closed, non-degenerate, differential 2-form. Sure, the results of a standard introductory real analysis class could be seen (and "proven") via purely geometric techniques, but how would you construct (or understand) any theorem concerning symplectic manifolds unless you already have a rigorous mathematical setup? The advantage of pure mathematics is precisely that it is so abstract, and so that we can work with and prove the very abstract constructs and theorems that exist in physics, and trust me, there is an abundance of them.

More than likely, you're not to the point in physics where you have noticed just how abstract it can become. In fact, almost all undergraduate classes are very 'physical' in nature - the mathematics behind the theory is almost never employed. However, if you were to become a physicist and take some graduate courses in physics, I think you would understand just how vital pure mathematics is to the state of physics.

Pure mathematics has always been heavily criticized, even after it has, time and time again, proven to be absolutely vital for the progression of physics. It was in the early 1900s when differential geometry was being criticized as a needlessly abstract and useless theory in pure mathematics, and then, Einstein revolutionized physics with his use of differential geometry in the creation of general relativity. As a mathematical physicist, I constantly have to remind myself to look at modern developments in pure mathematics for the answers. For example, even though abstract algebra is widely thought to be rather useless in physics, I wonder if some of the great unanswered questions in physics have solutions there.

 

[Gokul43201]

Topology is way too technical and is quite useless itself. Saying it is used in physics is a stretch.

You can't be serious - not after all that has happened with gauge theories, GR, soft condensed matter systems like liquid crystals, fractional quantum Hall physics, Aharonov-Bohm theory, (and probably dozens of other areas I know absolutely nothing about).

 

[deiki]

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