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When and how did mathematics come to be abstracted away from the physical world?

At first, mathematics would originate in its simplest form of counting and addition as to keep track of certain supplies in ancient civilizations. At some point it came to be used to describe more complex physical situations, such as ancient astronomy or Archimedes' study of force and buoyancy.

At what point did it come to describe inherently non-physical things; doing math for the sake of math? For example, doing things such as an analytic study of functions, the defining of abstract spaces, or even primitive number theory, etc. The earliest example I can think of is Indian and Greek geometry, but even this had a motivation in engineering and design. And yes, you could argue all of math at its core came from a motivation in physics, but today that's not the case.

In the modern day, mathematicians do beautiful work with no thought of how it may effect or be used in physics or any other science for that matter. So when did this switch happen? How did it come to be "math" rather than a tool for physics?

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  • $\begingroup$ Since the earliest we can reliably get back to is the Pythagoreans, one has to suppose that it was earlier than we can reach. $\endgroup$ May 8 at 5:11
  • $\begingroup$ AFAIK, it is not known whether Pythagoras really existed, and what exactly did he do. $\endgroup$
    – markvs
    May 8 at 21:14
  • $\begingroup$ It could be like Bourbaki who never existed, but the followers of Pythagoras certainly did exist. $\endgroup$
    – Somos
    May 9 at 0:31
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    $\begingroup$ Having studied origins of phenomena, it seems to me theory and practice are naturally interdependent and inspire each other. You can't separate them from each other. Practice inspired theory and theory inspired practice. See for example: en.wikipedia.org/wiki/Number_theory#Origins or en.wikipedia.org/wiki/Eratosthenes#Number_theory -- Moreover, Aristotle talked about the concept of epistēmē using the example of mathematics which is universal and eternal; he contrasted this with techne (the intentional making of physical things). $\endgroup$
    – Michael
    May 9 at 11:23

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You are right, this happened in ancient Greece, and is credited to Thales and Pythagoras. Unfortunately, too little of their early work survived (nothing written by Thales or Pythagoras). The main source of our knowledge abut this early epoch is Euclid, where mathematics is already almost completely separated from the "real world". The problems that Euclid considers (constructions with compass and ruler, number theory, etc.) have little to do with the physical world.

This is especially clear in the work of the next generation. Archimedes worked on both mathematics and physics, but made a very clear distinction between the two in his writings.

By the way, Greek mathematical astronomy is of later origin than that. The Hellenistic Greeks made a clear distinction between pure and applied mathematics (though they did not use these terms).

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