But if so, one ends up, so the argument concludes, with P(1) part of the explanation for P(k) and P(k) part of the explanation for P(1) and this contradicts the non-circularity of mathematical explanations (which is one of the premises about mathematical explanations defended by Lange).
In both occasions, diagrams of the given word problem were not utilized to support mathematical reasoning and explanations, which distinguishes the semi-symbol precedence view from the verbal precedence view.
We will discuss it, in section 6, in the context of the treatment of mathematical explanations within mathematics to which we now turn.
Dr. Hogan admitted that this undermined some of the traditional aspirations of physics, writing, "at least some properties of the world might not have an elegant mathematical explanation, and we can try to guess which ones these are".
Related to inductivism is Cellucci 2008, which emphasizes the connection between mathematical explanation and discovery.
See also Pincock 2011b for a detailed working out of the mathematical explanation of the rainbow that extends this approach and Lange 2009 for dimensional explanations (which are also mentioned by Batterman as the simplest forms of asymptotic explanations).
Steiner's theory, as we have mentioned, requires an account of mathematical explanations of mathematical facts, which he provided in Steiner 1978a.
Indeed, this was also the major intuition behind Steiner's account of mathematical explanation in science offered in 1978b, whose central idea was that a mathematical explanation of a physical fact is one in which a mathematical explanation of a mathematical fact results when we remove the physics.
On the surface, there is a simple mathematical explanation for such speculation.
In 1927 Nyquist provided a mathematical explanation of the unexpectedly strong thermal noise studied by J.B. Johnson.
