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As a math major, it caused me great pain when my family thought I was just learning how to do addition really well...

Generally, pure math (i.e., excluding applied math) can be thought of as having twotwo three main branches (although, this is likely an over-simplification):

  1. Algebra - how to use operations on sets of elements to combine two elements into another (potentially different, potentially not) element
  2. Geometry - deals with the distance between points and the things that fall out from that
  3. Fundamentals - logic and set theory, which serve as the basis for the rest of mathematics.

Certainly, counting is used for examples in bothexamples all fields, but in proper mathematics one usually operates in a verymore abstract setting (and so. That is to say, one often does not work with numbers), arithmetic or counting directly, but rather considers things that follow the same rules, and reasons about things in that abstract setting.  

For example, take the set of functions with certain technical limitations (e.g., measurable, or integrable, or differentaible... any "well-behaved" set of functions). You can define operations on them to combine them in different ways (algebra). You can define a metric that puts a distance on the elements of this set (geometry). But the idea of "counting" in this set is very unnatural.

As a math major, it caused me great pain when my family thought I was just learning how to do addition really well...

Generally, pure math can be thought of as having two branches:

  1. Algebra - how to use operations on sets of elements to combine two elements into another (potentially different, potentially not) element
  2. Geometry - deals with the distance between points and the things that fall out from that

Certainly, counting is used for examples in both, but in proper mathematics one usually operates in a very abstract setting (and so, not with numbers).  

For example, take the set of functions with certain technical limitations (e.g., measurable, or integrable, or differentaible... any "well-behaved" set of functions). You can define operations on them to combine them in different ways (algebra). You can define a metric that puts a distance on the elements of this set (geometry). But the idea of "counting" in this set is very unnatural.

As a math major, it caused me great pain when my family thought I was just learning how to do addition really well...

Generally, pure math (i.e., excluding applied math) can be thought of as having two three main branches (although, this is likely an over-simplification):

  1. Algebra - how to use operations on sets of elements to combine two elements into another (potentially different, potentially not) element
  2. Geometry - deals with the distance between points and the things that fall out from that
  3. Fundamentals - logic and set theory, which serve as the basis for the rest of mathematics.

Certainly, counting is used for examples all fields, but in proper mathematics one usually operates in a more abstract setting. That is to say, one often does not work with numbers, arithmetic or counting directly, but rather considers things that follow the same rules, and reasons about things in that abstract setting.

For example, take the set of functions with certain technical limitations (e.g., measurable, or integrable, or differentaible... any "well-behaved" set of functions). You can define operations on them to combine them in different ways (algebra). You can define a metric that puts a distance on the elements of this set (geometry). But the idea of "counting" in this set is very unnatural.

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As a math major, it caused me great pain when my family thought I was just learning how to do addition really well...

Generally, pure math can be thought of as having two branches:

  1. Algebra - how to use operations on sets of elements to combine two elements into another (potentially different, potentially not) element
  2. Geometry - deals with the distance between points and the things that fall out from that

Certainly, counting is used for examples in both, but in proper mathematics one usually operates in a very abstract setting (and so, not with numbers).

For example, take the set of functions with certain technical limitations (e.g., measurable, or integrable, or differentaible... any "well-behaved" set of functions). You can define operations on them to combine them in different ways (algebra). You can define a metric that puts a distance on the elements of this set (geometry). But the idea of "counting" in this set is very unnatural.