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I noticed that both terms referred to the study of mathematics. Is there a
difference in usage between the two terms?
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word-choice words
What is the difference between 数学 and 数理?
数学 is (pure) mathematics.

数理 refers to [mathematical
sciences](https://en.wikipedia.org/wiki/Mathematical_sciences), "group of
areas of study that includes, in addition to mathematics, those academic
disciplines that are primarily mathematical in nature but may not be
universally considered subfields of mathematics proper." In other words,
数理系の研究者 use difficult mathematics to solve actual problems that matter in
human life, such as cryptography or machine learning.
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2018-08-01T14:34:25.947
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I'll illustrate using examples.

The [Kyoto University Research Institute of Mathematical
Sciences](http://www.kurims.kyoto-u.ac.jp/ja/index.html) (KURIMS, 京都大学数理解析研究所)
is an example of a 数理研究所, and you see [their
research](http://www.kurims.kyoto-u.ac.jp/en/about-06-01.html) span many
areas. They have pure math research (like algebra and analysis), but they also
work in mathematical physics, fluid dynamics, algorithms, and theoretical
computer sciences. Similarly, [名古屋大学多元数理研究科](https://www.math.nagoya-u.ac.jp/)
conducts research in pure mathematics, applied mathematics, and "mathematical
sciences" (i.e., computational fluid dynamics, mathematical physics,
theoretical computer science, etc).

OTOH, the [Hokkaido University Department of
Mathematics](http://www.math.sci.hokudai.ac.jp/) is "just" a math department,
and their research is mainly pure mathematics. They do have a 数理科学 section,
but all applied mathematics/"mathematics sciences" are confined to this
section only.

So, the conclusion is: the practical difference is not that great. 数学 is a
true subset of 数理科学. 数学 is more or less the pure math part, but 数理科学 includes
what you'd call "computational methods" and "applied mathematics":
specifically, this usually includes the fields of operations research,
computational engineering methods, mathematical physics, and theoretical
computer science.

Practically, though, most Japanese universities have departments of 数理科学 (at
least at the graduate level), not (just) departments of 数学. The difference is
not notable. Even if you work in mathematical physics or TCS, non-
professionals would not care if you pass around as a mathematician (数学者)
instead of a mathematical scientist (数理科学者) - they likely don't understand
your work anyways. If you're communicating with professionals, you'd likely be
telling them your specific sub-field as well (e.g. 計算機科学 "theoretical computer
science", 数理物理学 "mathematical physics", 複雑系科学 "complex systems"). Whether
_they_ will consider you colleagues in mathematics is a totally different
question :-)
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