In all the previous Islamic responses to Greek astronomy one can detect a consistent tendency to solve the problems of that astronomy one problem at a time. From the prosneusis point, to the equant, and finally to the harmony between observations and predictive models, what the astronomers seemed to be doing was developing theorems and techniques that allowed them to reconstruct Ptolemaic astronomy along lines that would make that astronomy consistent with its own physical and cosmological presuppositions. What the predecessors of Khafrī seemed to be doing was trying to cleanse Ptolemy's astronomy from its faults, by using new mathematical techniques and theorems that were either unknown to Ptolemy or unnoticed by him. No one though seemed to think of the very act of mathematical theorizing itself and its relationship to the physical phenomena that were being described until Khafrī.
With Khafrī, Islamic astronomy moved to still newer territory. He was the first to begin thinking about the role of mathematical representation itself, the functionality of predictive models, and the relationship of all that to the actual physical phenomena. We had already noted the beginnings of this kind of thinking when we described Quṭb al-Dīn al-Shīrāzī's attempt at producing nine different models for the planet Mercury as a new step that marked the search for mathematical alternatives to the ones that were inherited from Ptolemy. But we also noted Shīrāzī's failure to pursue this line of thought when he seemed to have been still mired in the process of finding a unique solution, or say a unique representation, of the physical phenomenon that could be summed up in one true mathematical model. He spoke of such truths himself, for he was the one to alert his reader that of all the nine models he proposed for the planet Mercury in two of his books, seven were faulty by his own admission, while the eighth was left to the student to figure out its failings, and only the ninth was the true solution. So despite the fact that he was beginning to think that there were alternate mathematical techniques to describe the same physical phenomenon, he still thought that those techniques must climax in a unique solution that represented the truth of the matter.
With Qushjī (d. 1474) we can begin to see this trend pushed slightly forward. For although he must have known that Shīrāzī's model for Mercury solved the problems of the Ptolemaic model quite adequately, he still felt he could produce one more model that could do the same, which he did with his own new model. Was he thinking that his model, which relied entirely on 'Urḍī's Lemma to solve Mercury's equant problem, was only an alternative to that of Shīrāzī, which used only the Ṭūsī Couple, in the sense that it was a deployment of an alternative theorem to solve the same problem, or was he in fact thinking that the problem itself admitted multiple solutions? At this stage, we do not yet know. But from the fact that Qushjī's treatise on the Mercury model is a very short treatise devoted to this model only, one can presume that he was thinking of it as an alternative to Shīrāzī's and thus as just another way of thinking of mathematics.
With Khafrī the issue becomes very clear. In one full swoop he produced four different models for Mercury's motions, which he called wujūh (approaches), all of them accounting for the observations in exactly the same fashion, and none of them similar to the others in terms of its mathematical constructions. It is as if Khafrī had finally realized that there was a difference between two ways of thinking about Apollonius's theorem. On the one hand it could be thought of as representing two different cosmological solutions for the conflict between Aristotelian presuppositions and observations, and it can be thought of as two different mathematical ways of speaking about the variation of the solar speed with respect to an observer on Earth. It is the latter understanding that was finally realized by Khafrī's wujūh, for all his four models were mathematically equivalent in the same way the eccentric representation was mathematically equivalent to the epicyclic one. And although it is not stated in quite those terms, one could almost hear Khafrī say that the mathematical models he was devising were only different linguistic phrases used to describe the same phenomenon.
Seen as a tool, mathematics in the hands of Khafrī would become just another language of science, a tool to describe physical phenomena, and nowhere required to embody the truth or the correct representation, as was apparently thought by Shīrāzī before.[264] Mathematics became just as simple as describing a phenomenon with poetic language, with prose, or with mathematical figures, and as such the language itself can then be isolated from the phenomenon itself.
By focusing on the major shifts in astronomical thought that characterized the Islamic responses to Greek astronomy, it is now easy to see in hindsight the major features of these shifts. We can see how important it was to explore the full technical details of the most sophisticated Greek astronomical texts (the works of Ptolemy), not only to correct their mistakes, observational and otherwise, but also to investigate their presuppositions and the manner in which they related the observed phenomena to the methods of representation that allowed for the prediction of those phenomena. This close look at the foundations of those texts gave rise, as we have seen, to a series of Arabic texts written specifically for the purpose of critiquing the shortcomings of this imported Greek tradition. From istidrāk to shukūk, to straight forward rejection, all this full exposure left the Greek astronomical tradition in desperate need of reform.
The most important transformation that took place during this time was the shift from Ptolemy's instrumental approach to astronomy (which satisfied itself with the pragmatic success of the predictive features of the mathematical models) to a more theoretical approach which required that predictive results be consistent not only with the observations but also with the cosmological presuppositions of the observations themselves. In other words, in Islamic astronomy, it was no longer sufficient to say that a specific predictive mathematical model, such as that of Ptolemy, gave good results about the positions of the planets for a specific time. The new requirement was that the model itself should also be a consistent representation of the cosmological presuppositions of the universe, in addition to its accounting for the observations. If the universe was composed of combinations of spheres, and if those spheres were, by their very nature, supposed to move in uniform circular motions, then it was no longer acceptable to represent those spheres with mathematical models that deprived those spheres of their essential properties of sphericities and satisfy one's self by saying that they yielded good predictive results.