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Does this form of reasoning have a name. I often see it but am a little confused on how to read/understand it and wanted to look more into it but don't know what to call it

Ex. Let's call the force law between the objects $F(m_1,m_2,r)$. We know that if we put the body $m_1$ in free fall, the acceleration doesn't depend on the mass, so

$$F(m_1,m_2,r)=m_1G(m_2,r)$$ So that the mass will cancel in Newton's law to give a universal acceleration. This gives you the relation

$$F(\lambda m_1,m_2,r)=\lambda F(m_1,m_2,r).$$

Source: Combining Proportions to get Newton's Law of Universal Gravitation

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  • $\begingroup$ Are you asking whether there is a name for mathematical reasoning in general (based on the title of the question), or specifically a name for the type of scaling argument you give in your example? $\endgroup$
    – Andrew
    Sep 28, 2021 at 12:41
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    $\begingroup$ A function with this property is called homogeneous. In this case homogeneous of degree 1. If that's what your asking $\endgroup$
    – tomtom1-4
    Sep 28, 2021 at 12:50
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    $\begingroup$ This is not exactly the case, but dimensional analysis helps you to check the soundness of mathematical relations between physical quantities. $\endgroup$ Sep 28, 2021 at 12:57
  • $\begingroup$ What you've learned in mathematics classes applies to physics classes as well. I am not familiar with any new version of math named for use in physics or engineering $\endgroup$
    – Steeven
    Sep 28, 2021 at 12:58
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    $\begingroup$ There's no use in the inference$$F(m_1,\,m_2,\,r)=m_1G(m_2,\,r)\implies F(\lambda m_1,\,m_2,\,r)=\lambda F(m_1,\,m_2,\,r)$$alone. The linked source uses the second mass in a similar way to show$$F(m_1,\,m_2,\,r)=m_1m_2F(1,\,1,\,r).$$It's probably worth asking about the whole argument. $\endgroup$
    – J.G.
    Sep 28, 2021 at 13:32

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