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The language of physics is mathematics. In order to study physics seriously, one needs to learn mathematics that took generations of brilliant people centuries to work out. Algebra, for example, was cutting-edge mathematics when it was being developed in Baghdad in the 9th century. But today it's just the first step along the journey.
With any definition, we must either postulate the existence of what has been defined (that is done in the case of a circle, Postulate 3), or we must prove it. By maintaining the logical separation of the definition of a thing and its existence, mathematics becomes a science in the same way physics is a science.