This won't be hard.

Taking 'materialism' to indicate any philosophy that views existence as purely composed of, or reducible to matter, we're going to show how such a view is at the core, incoherent.

We can observe the various processes of evolution on every level of biological organisation -- we can first-hand observe the processes of speciation, natural selection and genetic drift whether in cell cultures or entire ecosystems -- this discovery initially provided tremendous support for materialist viewpoints, but in turn actually provides a stark demonstration of materialism's incoherence.

The modern boogeyman in science and philosophy is the hard problem of consciousness: How and why do we have subjective experience? The so-called easy problems can explain human actions and behaviours in terms of physical systems and their functions, for example the structure of the eye and how that consequently creates our vision, how a variety of receptors and their neural correlates create physical sensations etc -- in essence, explaining human actions as the functioning of biological organisms and their processes, principally so the processes of evolution -- the hard problem is that none of these explanations (which we are still coming up with) can adequately explain why we are conscious, that is, if through entirely natural and mechanistic or random processes animal life and consciousness developed, why is it that we can individually observe and take notice of the fact that we are aware of all of this, if everything is a mechanistic, material process, why is it that we can experience these processes, when there would be no need or function to the experiencing of at all.

The core principle of materialism, then, relies on the key assumption that eventually, with enough Science-ing, we'll have a complete enough web of information to explain all of subjective experience as the result of purely physical processes, thus resolving the hard problem as another process of evolution. There are basically two ways to argue against this point.

The first is to say -- Any statement that reduces consciousness to matter is making the assumption that consciousness can be explained by matter eventually, because right now we cannot explain consciousness by matter. We currently do not know how to explain consciousness, and until we know, we don't know, ergo, we're forced to simply say 'we don't know'. Until we have the content of every physical process mapped out and understood, a complete theory of everything so to speak, we would not be able to confirm if something is or isn't material, and similarly, until we have consciousness fully understood, we cannot confirm what the entirety of it is. Basically, until we know, we're guessing either way.

The counter-argument to that leads into the second way to make this argument: Yes, materialism is assuming that eventually science can explain everything, but that is a better assumption than any sort of religious or spiritual framework which is, often by nature, unverifiable and therefore by structure, un-scientific. Look at how much Science has done! And given that we can verify a scientific explanation in every country and by anyone who verifies it, whereas religious and spiritual experiences are, often by nature, culturally and geographically bound (indicating biological factors as the primary mechanism) and can't be replicated or observed under a microscope. Balance of probabilities, and it seems like a better assumption that our subjective notions are essentially unreal whereas the only thing that is real is material processes that science will soon fully understand. An argument that, if true, irrevocably ends any sort of spiritual notion of the world, and an argument that has implicitly become accepted within the scientific community.

So here's my preferred way to argue the issue, which is to show that no amount of science will ever prove materialism:

it just feels right. Haha.

No seriously though no materialist account of reality will never be complete because materialist viewpoints themselves stem from a tragic misunderstanding of what science actually is. Ideas, philosophies, statements of belief, information itself, is, by definition, nonmaterial. Here's the tragic misunderstanding: if everything is material or connected to what is material, it doesn't say anything about why anything is the way it is or what to do with it or what it means, these are all, by definition, non-material ways of encoding information regardless of if that information is connected to a material process (with any sort of directionality) at all.

So even if we can reduce an explanation of the world and consciousness to physical processes, that doesn't actually say anything about the nature of moral experience as in the form of experience in which you consider what you should do (if anything, it is still a form of questioning you are compelled to).

Going a bit further with that.

Positing consciousness as a mechanism of evolution necessarily ties that into the larger materialist framework of explaining reality and existence as a purely material one; if reality itself is defined by matter, consciousness would exist as a byproduct of physical processes, such as in the form of evolution. If everything is a physical process it can be logically understood as the relationships and properties of that process, therefore evolution itself would be a physical process governed essentially by logical relationships, therefore consciousness itself would be a physical process governed essentially by logical relationships.

If consciousness is a physical process governed by logical relationships, as the observed phenomena of evolution is, if that's what fully defines us, we should not be able to do anything counter to that, but we constantly do, and in fact, whatever we the value most, runs counter to our evolutionary instincts. This is the fundamental paradox at the core of the human condition: That we live in a physical world governed by physical rules and laws that we are just beginning to understand, but that there is something more to us. Grieving for a dead person has no evolutionary advantage. Art can encode information, but there has always been better ways to encode information, the way we do art itself has no evolutionary advantage. Going through, the entire nature and scope of subjective experience itself, has no evolutionary advantage, it shouldn't be happening as a result of purely physical process but it is ergo there is something more than just the physical.

Some form of materialism could be argued here in the notion that consciousness could still emerge in a distinct and disparate way from the physical whilst still remaining fundamentally physical.

That still puts us right back to the hard problem: how can subjective phenomena itself emerge and exist, especially in counter to the nature of the physical processes that it supposedly emerged from? More to the point, how would it be possible for us to pass on traits that directly code for behaviours that actually increase our own chance of death? Even if that trait increases overall survival, an individual organism itself is attempting to survive and reproduce, so it would not be able to do anything but what would increase chances of survival, therefore it would not be possible for us to have traits that run fundamentally counter to our survival without something else. Therefore, consciousness itself cannot be a purely physical process nor can it be an emergent characteristic of a physical process.

Consciousness itself, and therefore elements of reality itself, are necessarily nonmaterial. 

The works of Gödel and Tarski are elegant demonstrations of this.

From Gödel, we know that no mathematical system is every fully complete, meaning that in every mathematical system there are statements that can't be proven, we also know from him that the proof of consistency for any mathematical system cannot itself come from that system. From Tarski, building off Gödel, we know that no formal language or system can fully define itself but must itself be evaluated through a metalanguage, which itself can only be evaluated by a meta-metalanguage.

Here are the implications of those two simple proofs:

This is not a function unique to our current logical and mathematical systems, but to all formal systems, it is inherent to the structure of the system and it can't be any other way.

Every formal system that we can conceptualise, will require unproven but assumed to be true axioms and deductive rules off which we can make inferences and conclusions, the very nature of the system is such that a formal system can be tremendously useful in deriving conclusions from a starting set of rules, observations and data, but that's it. The tragic misstep in modern philosophy is to assume that we can derive anything causative at all from logic and science, when we can't.

How does that show that we can't or won't eventually reduce all of nature to a set of physical explanations?

Let's say as Science keeps developing we eventually come up with a full explanation of consciousness, of matter, of the universe, and we can encapsulate all of that information into a language or a system of formal languages. We would have defined everything, except we wouldn't have yet defined the actual language we used to define all of our explanations, therefore there is something we haven't defined, therefore our system is incomplete, and can never be complete. Therefore, no set of physical explanations will be able to define everything, there will always be some statement that may be true that is unprovable in all of our current languages, necessitating a new language, which itself requires an explanation, and so on.

This is an easily forgotten fact: That no language can fully encapsulate itself. This renders materialism dead-in-the-water; a scientific method can very accurately test for verifiable hypotheses, and a formal language can accurately provide deductions from a starting set of axioms, that is what they do by their very nature, but within that same nature, there are by definition and scope things they can never investigate, claim, or know.

We actually have two seperate things to notice here: Firstly, that even if everything was solely material, we could never confirm that, the very nature of our apparatus prohibits it, but secondly, that the very nature in which we use science and logic is itself immaterial.

The first claim we know from Gödel and Tarski, the second claim is a bit harder to make, but is a direct consequence.

If the world was defined solely by logical processes, there would be no way to know that, yes? So, there would be no way to know anything, despite the fact that our knowledge works. Meaning, from Tarski's proof it's clear that any formal language is defined by something outside of it that is itself undefined, so the very nature of us using logic precludes logic. Again necessitating the hard problem -- if everything can exist as logical consequences, why is there an ability to interpret that logic in the first place? -- Any explanation we come up with will fall into the problem of infinite self-regression, needing to provide an explanation for that explanation -- therefore, it would be impossible to know anything logically, however it is that we know, it must necessarily be immaterial.

What this very clearly shows us is that some element of how we know is itself always logically undefined. Similar to how the very nature of what we construe as valuable goes directly against the processes of evolution that affects all living organisms, similarly the very nature in which we know things itself precludes logic, cannot ever fully be defined by logic and therefore is always outside of logic.

That also means that all of our scientific explanations, at the root, began with non-logical presuppositions -- unproven axioms, an undefined set of assumptions, and so on. So the very nature by which we know things, as well as some element of reality, as well as some element of consciousness itself, is always necessarily undefined logically.

The structure and function of the proposition: 'everything is material' is actually illogical; it is an assumption that, even though we can never know or confirm it, there is some universal truth of the reality of all of physical existence. We have no proof that such an explanation of truth exists, but we know that the function and structure of our logical and scientific systems can never define such an explanation, therefore, all it is, is an assumption.

More than that, by nature, the statement 'everything is material' is a form of interpreting what is and defining some sort of causative explanation from it, therefore 'everything is material' is a metaphysical speculation that simply assumes from the progress of science that science will eventually explain everything not realising that the very structure of the formal systems that science relies on precludes such an explanation.

Materialism as a form of metaphysics therefore becomes self-refuting, it collapses on itself.

The very form of the speculation, the assumption, that such an explanation of reality as matter exists, is itself a statement that is undefined logically and precludes logic, and therefore is itself not logical, but rather, metaphysical. So to claim that nothing exists outside the physical, necessitates that you use a language that is itself outside the physical, therefore disproving the very claim you seek to make. Therefore, materialism as a philosophy can never be coherent.

The practical consequences of this are tremendous. The subtle attitude of attempting to verify or disprove subjective, spiritual, or religious experiences, statements and beliefs is itself illogical, though nobody seems to realise it. By its very nature, science can model a given set of natural phenomena, and logic can expound a series of connections between propositions and statements, neither of them can make any meaningful claim as to the reality of your own subjective experience, or to the reality of your metaphysical speculations, or to the reality of any metaphysical speculations.

To claim that we should only rely on scientific explanations is itself by structure not a scientific but moral statement and therefore again self-refuting. So what we've cleverly done is, kill religion and metaphysics by philosophically assuming that science can explain everything, not realising that we've subtly turned science and logic itself into a sort of faith by using a form of belief that is itself non-scientific and non-logical. It's tremendously silly, actually.

Part of the confusion comes from the fact that religions will simultaneously make metaphysical and scientific claims, so when one claim collapses we naturally lose trust in the other claim, not realising that the reality of a God or a spirit or a soul is not itself contingent upon the scientific veracity of various religious beliefs, the fact that religious people made scientific claims that were wrong does not itself say anything about the nature of the religious claims necessarily.

So we could certainly say that a given metaphysical system shouldn't be considered seriously because they've been wrong on the science, but that itself does not say anything about the very nature of the subject matter (reality, existence), science itself cannot encapsulate those matters, so we're always going to be using some form of philosophy - a nonmaterial speculation --  itself to generate explanations for any of those subjects matters, therefore any speculation that everything is physical refutes itself, and therefore can't be coherent.

So this isn't me arguing for or against any specific metaphysical claim except for the claim that everything is reducible to matter, which as I've shown, can't be, as the statement itself is self-refuting and incoherent.

The end.