With varying popularity and the tragic loss of numerous writings, Aristotle's basic ideas were preserved and developed throughout antiquity but also continued to compete with other schools of thought. During the Hellenistic Period, the atomism of Epicurus provided one alternative to Aristotelian natural philosophy, and the Stoics developed ethical and scientific views quite distinct from those of Aristotle. During Late Anitiquity, Aristotle's ideas gained more widespread acceptance but were also challenged and expanded mainly by Platonists and early Christian thinkers.
With
the beginning of the Middle Ages, which I am considering to begin around the time of Mohammad (570 - 632 A.D.),
it is in the Arab world that Greek thought, including that of Aristotle, is preserved. While I am not
well-versed in the intellectual history of this period, my impression is that the predominant schools of
scientific thought were founded on Aristotelian ideas but that Platonism was also very influential. During
this time, much attention was devoted to the reconciliation of the Greek tradition with Islam.
The Arab philosophical tradition culminates in the 12th century with Averroes, whose views were
heavily influenced by Aristotle.
After the 12th century, European Christians began to re-examine the Greek tradition and to combine Aristotle's philosophy with Christian beliefs. Aristotelian natural philosophy became the dominant world view in Europe. It is not until the 16th and 17th centuries that Aristotle's view of the physical world was successfully challenged. First Galileo, then Descartes, Newton, and Hobbes all rejected Aristotle's physics when they found that they were better able to explain various observations with atomistic assumptions. Around the same time, Francis Bacon rejected Aristotle's methods in the empirical sciences in favor of a new inductive method which Bacon then developed. It is Galileo's atomism and Bacon's inductive method from which modern physics and scientific method have emerged.
Modern
biology and modern logic, however, have continued to grow from the seeds sown by Aristotle. These sciences
have developed not by breaking with Aristotle but by building upon his work.
In biology, the Linnaean classificational hierarchy
developed in the 18th century marks
an enormous step toward the realization of Aristotle's vision for a classificational framework. Linnaean
taxonomy is one of the pre-requisites without which Darwin could not have developed his theory of evolution.
Finally, Aristotle's system remained the basic framework for the field of logic until the late 19th century. In the interim, several improvements were made upon the scheme outlined in Aristotle's Analytica Priora, but the main ideas remained intact until the late 19th century, when Gottlob Frege introduced a system capable of representing and evaluating statements much more complex in form than those analysed by Aristotle. While Aristotle originated the idea of a calculus through which true conclusions can be derived from true premises, Frege expands this calculus in such a way that Aristotle's original syllogisms can only be viewed as a few special cases in the new system.
It would in my opinion cast serious doubt on the foundations of our sciences had we not developed them over the last two millennia in directions that Aristotle did not foresee. Despite such developments, Aristotle's signature remains clearly recognizable in contemporary science, especially in biology and logic. His Politics had significant influence on the fathers of modern democracy, and in ethical deliberation, much can still be learned from Aristotle's ideas.