Natasha asked me to write about the topic of studying nature as a religious enterprise in Islam. There's a lot that can be and has been said about this topic and I can only hope to give the briefest of overviews here. There are some websites that have some good information, but the most extensive writings are offline, in books.
There are many websites that contain information on the accomplishments of Islamic science. The best brief overview is Muslim Contributions to Science, Philosophy, and the Arts. However, if you want to go into more depth, I recommend The Oxford History of Islam's chapter on Science, Medicine, and Technology (there's also lots of great stuff in the rest of the book).
The Arabic word "ayah" means "sign" as in "this is a sign of something". The Quran uses the word "ayah" in several different senses. The most broad is to refer to anything in the created world. For example:
We have appointed the night and day as two signs - Surah al-Isra verse 12
And a sign for them is the dead earth, which We brought to life and from which We brought forth grain that they eat - Surah Ya Sin verse 33
And of His signs is the creation of the heavens and the earth, and the variety of your languages and colors - Surah ar-Rum verse 22
Another sense of the word is to refer to the proofs brought by the prophets of God (peace be upon them). For example:
And We sent Moses with Our signs, and a manifest authority, to Pharaoh and his council - Surah Hud verses 96-97
The word "ayah" is also used to refer to verses of the Quran itself. You may notice that the entries in my Quranic Journal blog have titles like "Surah al-Jinn ayah 3" (the most recent entry there at the moment). This is the same as saying "Surah al-Jinn verse 3".
The Quran not only mentions the signs (ayat) of God but also indicates their purpose. Some examples:
Even so does God make clear the signs for you, so that perhaps you may reflect - Surah al-Baqarat verse 266
In the alternation of night and day, and what God has created in the heavens and the earth - surely there are signs for a people who fear God - Surah Yunus verse 6
In general, whenever, the Quran mentions something in the created world, it invites people to think about the meaning. A listing of some of these verses can be found at Quran and Scientific Knowledge.
The Quran also makes mentions of signs within the human soul:
We shall show them Our signs upon the horizons and within their souls until it becomes manifest to them that it is the truth - Surah al-Fusilat verse 53
Thus, Muslims are commanded not only to study the Quran and the signs (verses) contained within it but also to study the created world and to contemplate the workings of their own inner selves. All of these are sources of knowledge to those who seek it.
Seyyed Hossein Nasr explains:
It might be said that from the Islamic point of view creation and revelation are inseparable, and that there are in fact three grand revelations: the cosmos, the human state, and religions - all three of which Islam sees as "books." There is, first of all, the cosmic book to be read and deciphered. Then there is the inner book of the soul, which we carry within ourselves. And finally there are sacred scriptures, which have been sent by God through His mercy to guide humanity throughout the ages and which are the foundations of various religions and keys for reading the other two books, that of the cosmos and that of the soul.The Heart of Islam, p. 15. In An Introduction to Islamic Cosmological Doctrines, Nasr looks at the study of the cosmos by prominent Muslim thinkers. In the prologue he discusses several ways of approaching the study of nature:
In traditional civilizations, the study of nature may be made for the sake of utility as is seen in ancient and medieval technology where aspects of nature were studied with the aim of discovering the qualities which might make them useful to the daily needs of the society. Or such a study may be made with the aim of integrating cosmic existence into a pervasive rational system as with the [philosophers]... Finally, nature may be studied as a book of symbols or as an icon to be contemplated at a certain stage of the spiritual journey... Moreover, all of these different manners of studying nature are sometimes superimposed upon each other in the writings of a single author, especially in the Islamic civilization which forms the subject of our present study.Doctrines, pp. 2-3. In the body of the book, Nasr looks at examples of each of these types of study. The primary example of the utilitarian study is the scientist al-Biruni (973-1030 C.E.). He describes al-Biruni's views about the study of nature:
The study of creation as the handiwork of God is for al-Biruni a natural and noble activity of man... There is for al-Biruni no separation between "sacred" and "profane" learning. Whatever he studies, whether it be historical or physical sciences, takes on a religious character. Particularly the study of the visible world as the "signs"... [is] for al-Biruni characteristic features of the perspective of Islam... Al-Biruni, although not a gnostic, shared that view in his approach to all learning, and considered the pursuit of the traditional sciences as a religious activity... The cosmos in which al-Biruni lived and breathed is the handiwork of God in which all true science leads ultimately to the Creator and possesses truth which is guaranteed only when it is sanctioned by Him.Doctrines, pp. 173-4. The primary example of the rationalistic or philosophical study is Avicenna (ibn Sina; 980-1037 C.E.). Nasr describes ibn Sina's views about the study of nature:
For Ibn Sina the approach toward the study of nature depends completely upon the purpose for which nature was created. Nature is the domain where everything possesses a meaning and an end and where the wisdom of the Creator is everywhere manifest... The study of nature, in which the purpose and wisdom of God is manifested, leads ultimately to the knowledge of the source of all beings. Inasmuch as all creatures by the fact that they exist derive their existence from the Creator who is Pure Being, all sciences of nature have for their purpose the knowledge of the essence of things in relation to their divine origin.Doctrines, pp. 231, 233. The primary example of the mystical study is the group known as the Ikhwan as-Safa (Brothers of Purity; fl. c. 1000 C.E.). Nasr describes the views of the Ikhwan about the study of nature:
The universe described in the [works of the Ikhwan] is a unified whole whose various parts are held together by the analogy which exists between them... The language with which this interrelation is expounded is that of symbolism, particularly numerical symbolism... In this world of symbols the Ikhwan study nature with the purpose of discerning the wisdom of the Maker... Of the many types of symbolism which the Ikhwan use, numbers are the most important because through numbers they are able to relate multiplicity to unity and bring to light the harmony which pervades the universe.Doctrines, pp. 44, 45. From the above, it's clear that al-Biruni, ibn Sina, and the Ikhwan all had the same motivation for studying nature, but they pursued their study in different directions, al-Biruni as a scientist, ibn Sina as a philosopher, and the Ikhwan as mystics. The book also discusses the scientific thought of each of these people so that you can see for yourself what they accomplished in the study of nature. Nasr has also written a shorter essay that explains some of the themes touched on here, Science and Civilization in Islam. Two other essays you may wish to read if you're interested in this perspective are Modern Science and the Eclipse of Tawhid (Unity) and The Antropocosmic Vision in Islamic Thought, both by William Chittick. A more purely religious view on the signs of God is found in The Foundations of Islamic Religious Experience by John Renard.