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Questions in mathematics are posed as:

Solve for y = yadda yadda yadda

The yadda yadda yadda is some expression involving variables and integrals and so forth. “Solving” for y often leads to another expression, rather than some exact number.

If the result is just another expression, why is this considered a solution? It makes sense to consider it a “simplification” (although I’m not sure what the measure of reduced difficulty is...number of symbols?) but a solution, one might think, should be more than just another way to express the same formula seen in the question.

Further, there are often many different expressions one could come up with as a “solution.” In other words, rarely do 2 people find the same final expression as a solution to the posed mathematical question.

This again begs the question, what exactly is the definition of a mathematical solution given the immense number of possible ways to express it?

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You're right that figuring out a property of a solution or just re-writing the solution in another language may seem useless. But a lot of the great problems in mathematics arise from attempting to build connections between different theories as to enable further inspection of one theory by the results provided by the other.

For instance, consider the number $e$. There are many ways to characterize it which are all equivalent, but looking at a certain definition of this number leads to reasoning that are not possible under other definitions.

You can define $e^x$ as the solution to the differential equation $y' = y$ with initial conditions $y(0) = 1$, and you can define $e \overset{def}= y(1)$. With this definition, the number $e$ is not as interesting as the function $e^x$ it defines, and the number itself doesn't initially look very special.

You can also define $e$ as the limit $\lim_{n \to \infty} \left( 1 + \frac 1n \right)^n$. This gives you the opportunity to approximate the value of $e$ with a concrete expression, something that was not a priori very easy to do with the differential equation.

You can also define $e$ as the value to the series $\sum_{k \ge 0} \frac 1{k!}$. Again, you get a good approximation, but now the $k!$ tells you that maybe the number $e$ has something to do with permutations, since $k!$ is also the order of the symmetric group $S_k$ on $k$ symbols.

It is not directly obvious from the last two definitions that if you look at $\lim_{n \to \infty} \left( 1 + \frac xn \right)^n$ and $\sum_{k \ge 0} \frac{x^k}{k!}$ that these two functions are in fact all equal to $e^x$. If you find a relationship between the symmetric group and $e^x$, because $e^x$ is also the solution to the differential equation $y' = y$, now you can use a property of differential equations to study the symmetric groups.

Maybe my example wasn't the best, but I was trying to indicate that finding connections between fields gives you more tools to study all these fields, and thus proves very useful in practice.

Hope that helps,

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  • $\begingroup$ Excellent answer. Thank you $\endgroup$ – Cybernetic Jan 17 at 20:35

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