What follows is not an "answer", but just some ideas or frames of interpretation from discussions with fellow Buddhists with a Western scientific background. So please don't crucify me :-) but tell me if I'm wrong.
The Scientific Method is Compatible with Mahayana Buddhism
Let's first appreciate the success of the western scientific method to explain certain parts of conventional reality that were previously within the religious domain. There is no need to believe in a god of thunder anymore in order to explain certain meteorological phenomena. I believe that this success was just extrapolated to the metaphysical realm somehow unintentionally or without much thinking, so that "scientific materialism" just formed once there was no need for specific divinities anymore. Scientific models just don't talk about metaphysics (for exceptions see below).
Scientific models used to fail completely to deal with "mind" (as the structure of a perceiving subject), at least until a few years/decades ago. Psychology is either considered "unscientific" (early psychology with Freud, Jung etc.) or doesn't actually talk about the mind anymore (focusing on what is measurable). So I would argue that "science" is actually not talking about the mind or any metaphysical concepts like Karma or Sunyata. So I don't see any hard incompatibility here with early Buddhism. This would also hold for most Buddhists schools, if you'd strip of the mythological elements and/or interpret them as psychological images.
Neuroscience, Cognitive Science and Artificial Intelligence
A few decades ago things have started to change, and that is where I believe things get interesting with respect to (in-)compatibility with Buddhism. The disciplines start to talk about the mind and the structure of the perceiving subject. However, current models are very basic and typically limited to specific brain functions. For example, there has been quite some progress to model the processing of images from the eyes in the visual cortex, and artificial intelligence (AI) has become successful performing such tasks automatically, as everybody can see first-hand when using the image processing functions of recent mobile phones. We can expect further progress in this area.
The current results imply that "thoughts" and other cognitive states are represented in the brain as patterns of neuronal activity. This is a pure materialist view of the mind. There are philosophical and religious schools that deny this, but given the success record of the scientific method and the amount of resources invested in this research area, I predict important advances in the upcoming years that will make this view more and more popular.
Areas of Conflict
It may be interesting to examine the specific areas or concepts that might conflict with such material view:
- Karma: Maybe we could live with "Karma in this one life"? Karma could also be seen as related to neuroplasticity and neural pathways that accumulate/aggregate habitual reactions.
- Devas, Nagas and other divinities: I interpret them as personification of states, traits or properties.
- Nihilism: Nihilism gets in the way of Sunyata. There is no answer on this one yet.
- Mind: See below.
- Other areas? Let's discuss.
Materialist View of Mind Preserving Buddhist Properties?
I'm an active researcher in the area of AGI and have published a very technical paper on the subject.
Here is a summary using a simile: The mind relates to the neurons of the brain like a YouTube video on a computer to the silicon logic that only works with 0's and 1's. You can't explain the quality of the video on the level of silicon logic. You will have to talk about applications, libraries, operating systems, programming languages, microprocessors, digital logic and many other "layers" that build on each other in order to produce a video. In a similar way I could imagine that the phenomena of "mind" appear at the highest layer of the "layer cake" of the brain.
Is ... Materialism ... compatible with Mahayana?
To summarize these views: The wealth of mental phenomena in and outside meditation is perfectly compatible with a materialist base of the mind. The physical laws governing the simple neural building blocks don't limit the mind in any way. Nihilism may be a problem. We also may have to "tweak" the interpretation of some concepts in order to avoid conflict. Strict followers of certain Buddhist schools may find this not acceptable.