I'm going to give you a short answer first: Muslim states never stagnated in terms of innovation and science. And Neil DeGrasse Tyson needs to stop pretending he's a historian.

While modern Muslim countries economically lag behind the West, and therefore are less able to support modern scientific research, this is the result of a recent history of European colonialism, American neo-colonialism, and the collapse of the traditional order following World War I. It is certainly not the result of any deficiency on the part of Middle Eastern people, nor the result of a "decline" or "stagnation" from an Islamic golden age in the Medieval era. This is a mythology of history, a teleology that has recently come under attack from all quarters of Islamic studies and historical analysis.

To me, the best way to illustrate this myth - and the reality that belies it - is with the story of Al Ghazali and Averroes and the Islamic Golden Age - a story frequently recounted by Mr. Tyson in public forums, but one which I think he gets some really key details about absolutely wrong.

As a figure in intellectual history, ʾAbū l-Walīd Muḥammad Ibn ʾAḥmad Ibn Rushd (1126-1198), known in the West as Averroes, is one of those whose image has been revived over and over and whose overall influence far exceeds that which they exerted during their lifetime. He is best known as a philosopher and commentator on Aristotle, and his legacy includes many translations and commentaries on the Greek masters and defenses of the study of Hellenistic philosophy in the Islamic world. While he died in the beginning of the twelfth century, the Aristotelian corpus was largely revived in Europe as a result of translations from Ibn Rushd's commentaries, giving him a lasting importance. His ideas have been presented as so important to thirteenth century Christian and Jewish scholars (called “Averroists”) that he can be seen as a foundational figure in Western secular thought. He is often remembered in connection with the great Islamic theologian Abū Ḥāmid Muḥammad ibn Muḥammad al-Ghazālī (1058-1111) because of his book Tahāfut al-Tahāfut, or The Incoherence of Incoherence, in which he responds directly to al-Ghazali's criticisms of his understanding of Hellenistic philosophy and those who adhere to it. This link has been of great interest to both historians and students of Islamic philosophy, but unfortunately, it has been often misread, misinterpreted, and even misused in the service of various political, ideological, and intellectual disputes. While this is perhaps par for the course in any academic discipline, the use, misuse, and abuse of Ibn Rushd and al-Ghazali persists to this day and has potentially significant consequences for public discourse about and understanding of Islam and Islamic history.

The “decline” narrative that you’re referring to is intimately related to Ibn Rushd and al—Ghazali, and goes something like this: According to this view, Islamic society enjoyed a brief “Golden Age” following the first few centuries of Arab expansion. From roughly 800-1200 CE, culture, the arts, philosophy, and above all science and technology flourished. It was this period that saw the birth of geography and algebra, which saw the development of Arabic numerals, and which saw intensive study of the classical Hellenistic tradition. However, following the sack of Baghdad, the Muslim world fell into a period of decline, and even when Islamic empires re-emerged, science and rational inquiry never recovered, in part due to the ascendance of anti-rational ideas embodied by theologians like al-Ghazali. Specifically, Ibn Rushd offered a spirited defense of reason and rational inquiry against al-Ghazali, who adopted a view based on the subordination of reason to faith and revelation. As Ibn Rushd's ideas fell out of favor and his book eventually declared heretical, the theology propagated by al-Ghazali became dominant. In short, despite the valiant efforts of Ibn Rushd, al-Ghazali killed philosophy, and with it, the spirit of rational inquiry and the potential for scientific development in the Islamic world. The result today is self-evident: by the 19th century, and perhaps even much earlier, the Middle East “lagged behind” Europe significantly, and today the region is in the grip of a resurgence of religious fundamentalism.

What a great answer u/Xuande88 ! I just want to add a few points that some of the readers of AH might find interesting in terms of changes in the historiography of the history of science over the past decade.

I would take a somewhat different path to answer the original question.The terms stagnation and decline carry a lot of connotations with them. If we assume that there was a decline or stagnation, then there were no major scientific achievements after the al-Ghazali period. u/Xuande88 gave wonderful examples from the Ottoman world after al-Ghazali. Here are a few more examples to add to the list from different regions. During the 13th century an example would be the founding of the Maragha (or Maragheh) Observatory, which was the first large-scale observatory, and was the model for other observatories founded across Asia, the Middle East, and Europe for centuries (Ragep 2008). Besides astronomy, there was also the research of Ibn al-Nafis on pulmonary circulation (predating the research of William Harvey by 400 years) (Fancy 2013). My favourite one though has to be the Darb-e Imam shrine, which includes artwork that is argued to represent ‘nearly perfect quasi-crystalline tiling’ dating almost 400 years before they were studied by Roger Penrose in the 1970s (after whom the Penrose pattern/tiling is named) (Lu & Steinhardt 2007). These are just a few examples, but I think they are a good starting point to get you AskReddit aficionados digging, and also to illustrate the point that science did not stop with al-Ghazali or Ibn Rushd.

So if there are examples to a flourishing “Muslim innovation and science” after al-Ghazali, how come we don’t hear more about them? I am going to mention two possible explanations for that, but there are many historians working within the field of history of science who I’m sure would give you different answers.

One explanation goes at the very root of the problem, which is the definition of science. To quote Kapil Raj’s excellent essay on the matter, historians tend to view science as ‘universal knowledge, ideally founded on mathematical formalization and experimental verification’ (Raj 2013:337). However, more recent scholarship has approached science not as ‘logical step-by-step reasoning’, but rather as ‘pragmatic judgement’ (Raj 2013:341). Such a reframing of science shifts the emphasis from big-picture accounts of a system of knowledge, to smaller case studies that explain how judgement was exercised to make decisions about specific cases. (As a side-note, this shift in approach is partly the reason why you sometimes end up seeing strange sounding case studies as in this video parodying Vice article titles https://youtu.be/Ia7fUQXskvA) Historians and philosophers will chastise me for making the following comparison, but a useful way to think about this shift is like a move from science as “knowledge” (episteme) towards science as “craft” (techne). The other major reconceptualisation targets how we think about the diffusion of science (or knowledge in transit) (Secord 2004). Traditional accounts tend to look at unidirectional “dissemination” of knowledge, thereby creating a centre-periphery distinction and dismissing the ability of local individuals to transform or adopt new knowledge. By contrast, more recent approaches focus on the “circulation” of knowledge that allow for the mutation and reconfiguration of knowledge within local contexts as well as for the return of the mutations and reconfigurations to the points of origin (Raj 2013:342-344). Through these reconceptualisations, we can create a theoretical framework that allow historians of science to move away from discussions about the "essence" of “modern science” to more detailed analysis of how science was actually put into practice. Kapil Raj’s article is open-access and includes many references to these case studies that you can explore yourself.

The other explanation for why we don’t hear more about “Muslim innovation and science” has to do more with history, society, and everyday life. Downplaying the significance of Arabic science has been an influential part of culture for decades if not centuries. As u/Xuande88 also pointed out, it’s an old trope, and Renan’s characterisation of Arabic science is one example to this. During the early 20th century the trope was sometimes based on racial accounts, as in the case of the French physicists and intellectual, Pierre Duhem. Despite the fact that Duhem was familiar with the work of a few Arabic authors, he argued that Arabic people were incapable of abstract thought, which type of thinking was a requirement for western science or Christian positivism (Ragep 1990). However, the seed of that idea dates further back to the decoupling of Greek ideas from their Arabic circulation (Chakrabarti 2004). Unfortunately, these ideas survived well into the second half of the twentieth century. Although there were major historians of science like George Sarton and Marshall Clagett who even learnt Arabic, their acts were the exceptions rather than the norm. Within a more contemporary setting there is also the problem of the connotations attached to discussing Islam. Jamil Ragep summarises the impact of this political issue clearly and concisely: “For if a single individual [al-Ghazali] could stop Islamic science in its tracks, then the problem must ultimately be somehow inherent in Islam itself. An alternative view would hold that Islamic science, like all scientific traditions, made its accommodations with the social, political, and religious contexts in which it found itself, and continued on long after Ghazali.” (Ragep 2008:3)

So I’d say that the takeaway message is that talking about “stagnation” and “decline” is misguided. The terms are also loaded with preconceptions. So is how we define science. However, by the end of the day, historians endlessly find new ways to look at things, and debates are still ongoing. It’s always worth re-reading the works of historians and following threads that interest you - there is always the potential to see things in new light and to spot errors that have gone unnoticed. As a result, in the sources below, I included open-access (free) sources at the top of the list so you can read the more detailed explanations offered by the historians themselves. They have seemingly endless stories to offer.

Sources:

Raj, Kapil 2013 - Beyond Postcolonialism… and Postpositivism: Circulation and the Global History of Science - https://www.journals.uchicago.edu/doi/pdf/10.1086/670951

Ragep, Jamil 2008 - When did Islamic science die (and who cares)? https://islamsci.mcgill.ca/Viewpoint_ragep.pdf (This is a wonderful concise summary of Ragep’s thought; the magazine Viewpoint is the public facing outlet of the prestigious British Society for the History of Science; the same Society held its first open-to-all week-long conference last year online, and you can still watch all the replays of the talks and panel discussions via this link: https://www.crowdcast.io/bshs)

Fancy, Nahyan 2013 - Science and Religion in Mamluk Egypt: Ibn al-Nafis, Pulmonary Transit and Bodily Resurrection (Fancy’s book is partly based on his dissertation, which is available to download for free: https://curate.nd.edu/show/cz30pr78k14)

Chakrabarti 2004 - Western Science in Modern India: Metropolitan Methods, Colonial Practices (The first chapter of this book, which talks about the decoupling of is available for preview on Google Books)

Secord 2004 - Knowledge in transit

Sabra 1984 - The Andalusian revolt against Ptolemaic astronomy

Lu & Steinhardt 2007 - Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture

Ragep 1990 - Duhem, the Arabs, and the History of Cosmology