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I was an applied math major and took several calculus-based courses, including differential equations and real analysis, as well as an introductory course sequence in probability and statistics. Both calculus and statistics are extremely useful and broadly applicable to the natural and social sciences, but it seems like their tools and methods are completely different. Are the "statistical" and "calculus-based" approaches to solving problems fundamentally different, or is there actually significant overlap between these fields that is not evident to someone who only learned about these fields at the undergraduate and introductory levels?

It seems like one of the major goals of a calculus-based approach to understanding a system is to come up with a differential equation to model the data. And it seems like the statistical approach seeks to obtain causal and inferential relationships between variables. So it seems as if one of the primary goals of both fields is to come up with quantitative relationships between variables being observed in a system. So why are their methods and tools so different?

Also, certain sciences use calculus more than statistics (like physics) whereas others use statistics more than calculus (biology), and others use both extensively (economics). What about a scientific discipline makes either analytical or statistical methods more useful?

Jay Youn
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    Calculus is heavily used in statistics. Studying distributions, especially continuous distributions, requires differential and integral calculus on a routine basis. At a more foundational level, probability theory employs measure theory, which is an advanced topic in real analysis. So no there is a lot of overlap between these fields. – balddraz Dec 26 '20 at 01:54
  • At some universities, there is a need to teach some introductory statistics and/or probability to students with weak mathematical backgrounds, hence, no calculus is used. If you go past these to the more advanced courses, calculus comes in with a vengeance. – Gerry Myerson Dec 26 '20 at 02:06
  • So little overlap?!?!?! optimization, infinite series, and improper integration are ALL OVER THE PLACE in stats!!! Just to name a very few – Matthew H. Dec 26 '20 at 03:23

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It depends on the quantities that you measure or calculate. Most of the measurements are discrete and are processed using statistical methods. This is field independent. Then, when you model the underlying process, one usually has continuous variables (for example some time evolution). Those are described by calculus.

In the undergraduate physics, most of the time one looks at the models. The experiments were done long time ago. Everyone knows that the gravity next to the surface of the Earth is described by a downward constant acceleration. Then you use calculus to describe projectile motion.

In biology, there is much less detailed knowledge about some underlying mechanism. To try to tease out the correlations, one will therefore use some statistical methods first. Once you have a model, you write the differential equations, solve them to predict other outcomes. But this is also true in the current physics research.

Andrei
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As far as I know, all mathematics fields are deeply connected to each other. For example, Riemann hypothesis can be "reduced" to a "simple" arithmetic problem: M(n)=O(sqrt(n)) where M(n) is the arithmetic mean of mu(1), mu(2), ... , mu(n), O is just big-O and sqrt(n) is a square root of n.

Here's a discussion in Russian: Habr.com

And here's the original article.

Vasily Dolgov
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  • The Mobius function $\mu(n)$ belongs to number theory, so saying RH can be described in terms of the growth of the summatory function of $\mu(n)$ is saying number theory can be described in terms of number theory. It is not really an example of connections between different areas of math. – KCd Dec 26 '20 at 02:22
  • @KCd Of course you are right, and understand mathematics on much higher level than me. What I had in mind is that Riemann zeta function can not be studied without deep understanding of complex analisys. – Vasily Dolgov Dec 27 '20 at 12:35