Complex mathematics make sense if the results can be calculated with precision. This applies for the physical sciences like physics and engineering. Economics is more of a social science subject. Does complex mathematics make sense in this context?
In the field of economics complex math is often used for
a) Economic modelling
There are usually tons of factors that influence even the simplest economic event. It is often necessary to resort to complex math to include all the variables that represent those factors in the right way.
b) Analysis of statistical data
It is often very difficult to support economic theories with good data, again because there are so many factors in play. On top of that, many simple mathematical instruments have drawbacks and should not be applied in certain situations. For example, to test relationship between two variables, you can use a simple correlation coefficient, but there are cases where outliers or certain patterns can make it unreasonably high or low. Then you can use simple linear regression, but if you spent months gathering data you may want to spend a few days trying to build a custom regression equation for a better fit. Finally, if you are preparing a paper for publication, you may want to use as many statistics as you can to try to convince reviewers and readers that your findings are actually true.
Drawbacks of using complex math in economics
1) There are not that many readers out there who are proficient in advanced mathematics. Unless you want to limit your target audience to a few math enthusiasts, keep it as simple as possible.
2) If you are preparing a paper for review at a journal, keep in mind that reviewers in economic journals are not necessarily mathematicians. It usually takes longer for math-heavy papers to get reviewed and the quality of those reviews may not be great, unless its a top journal.
I asked in a comment
By "complex mathematics" do you mean "advanced theoretical mathematical methods", or "complicated numerical computations"?
and the OP replied
Both. But "advanced theoretical mathematical methods" is worse.
Well, the use in Economics of "advanced theoretical mathematical methods" has absolutely nothing to do with the accuracy or inaccuracy of numerical calculations/estimations/predictions.
When the mathematization of Economics started back in the 30's and received a critical boost from Paul Samuelson's "Foundations of Economic Analysis" (1947), the scholars may have envisioned an exact social science at the end of the road, social engineering reigning supreme.
We soon learned how implausible such a goal was, while also understanding what this mathematization really offered instead :
A) Sanity-check of economic theorizing to a degree not permitted when this theorizing swims in the moving waters of human language
B) The ability to obtain new economic insights unattainable by human reasoning phrased in human language
c) Greater degree of re-usability of the work of others, thus accelerating research
...and I may think of a few more.
In another comment, the OP seems pre-occupied with the relation between forecasting performance and complex mathematics. Two remarks here: No forecasting method, complex or naive, does well in the face of structural breaks, we are not there yet. Second, the combination "sophisticated forecasting model + experienced forecaster" most of the times performs better than "simple model".
This is not news: 30 years ago, in Granger & Newbold (1986) "Forecasting Economic Time Series" book, in the relevant sub-chapter on evaluating macroeconomic forecasts, the authors reviewed the literature in a very balanced and detached way, and found just that: since this is not social engineering, tools should be handled by humans (like complex models that are handled by their creators), not left alone (as simple models are, by their nature).
In a more recent paper
Dovern, J., & Weisser, J. (2011). Accuracy, unbiasedness and efficiency of professional macroeconomic forecasts: An empirical comparison for the G7. International Journal of Forecasting, 27(2), 452-465.
the overall evaluation of professional forecasters (which certainly use complex models alongside their experience) appears to be improving.