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I am occasionally asked what is meant by abstract math and what the point is. My usual canned answer is that abstract is the opposite of concrete or specific. Concept X is more abstract than concept Y if Y is an example of X. The point is that knowing some fact Z about X is more useful than knowing the same thing about Y since its truth with respect to Y follows from that of X. Furthermore, if we encounter other examples of X, then fact Z holds there as well.

Some examples I usually end up giving:

A right triangle is less abstract than a triangle. If you notice that the interior angles of a right triangle sum to 180 degrees, you're noticing a special instance of some more "abstract" truth about triangles.

In this example, general feels like a more appropriate word, but is it? Are abstract and general essentially synonyms?

Another example that seems more appropriate:

The number 2 is more abstract than 2 birds or 2 meters.

If there is a difference between abstract and general, this example seems to hint at it because it doesn't seem right to say 2 is more "general" than 2 birds, but perhaps it is.

H_R
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    Generalization is a type of abstraction – Rushabh Mehta Jan 03 '22 at 16:51
  • Well that settles it. Can you give an example of abstraction that doesn't count as generalization? – H_R Jan 03 '22 at 17:35
  • Art increasingly is becoming more abstract, as it refers not so much to the real world, as older art used to, but rather to the context established by the art before it. The more time goes by, the less understandable art will be to those lacking context. This is abstraction, but not generalization. – Rushabh Mehta Jan 03 '22 at 19:27
  • I don't think either of these two words have a formal definition, so this seems to be a question about word usage which might be very subjective. – Torsten Schoeneberg Jan 03 '22 at 19:32
  • @TorstenSchoeneberg I agree, but the general notions of abstraction as the process of filtering information, and generalization as filtering information for the purpose of application to more general situations is well-established even outside of mathematics. – Rushabh Mehta Jan 03 '22 at 19:35
  • @RushabhMehta Thank you for the art example. That's illuminating. So how abstract something is how decoupled from the physical world it is? – H_R Jan 04 '22 at 00:07

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