It's my birthday and I don't want to spend a lot of time responding (and I expect next week to be terribly busy so I probably won't want to continue after that), but there's a handful of things I do want to respond to in here.
I'm saying that it is not possible for multiple systems to be objective. If there is an objective system then all differing systems must be false and therefore not objective. You said “most ethical systems employed by most philosophers, such as deontology and utilitarianism, are both non-theistic and objective.” Calling two differing systems objective makes no sense.
There's two different senses of "objective" being conflated here. One, the sense I meant when I said that there are many different kinds of secular objective morality, is the sense whereby a given point of view or ethical system considers the answer to the question "are there objectively right and wrong moral assertions?" to be "yes". The other, that you're conflating with it, is whether a given point of view or ethical system is the objectively correct one to use.
My initial point was that there are plenty of logically possible, non-contradictory stances one can take where ones believes some things are objectively right and wrong (morally) without believing in God; it doesn't follow from the rejection of God that one must reject morality, unless one also holds onto religious assumptions about morality, which most people who reject God don't. (Some do, and they become nihilists, and even people like Nietzsche argued that that is something to be overcome, a lingering vestige of the religious worldview, and not a good thing that people should strive for).
Which of those stances is the objectively correct one to take epistemically is disputed between them, but that doesn't mean that none of them is correct, any more than the existence of disputes between religions means none of those religions is correct. "What in particular is morally right and wrong, and how can we tell?" is a different question from "is anything actually morally right or wrong, objectively?", and there are lots of secular viewpoints who agree (along with most religious viewpoints) that the answer to the latter is "yes", even though they disagree about answers to the former question (as do different religions).
So if you reject dualism can I deduce that you are a materialist? If all that exists is matter, where does morality come from? Is it matter? How do you account for it? You claim it is universal so it can't be only in my mind. You have rejected that things have moral properties. What is it?
I'm not strictly speaking a materialist, but I'm probably close enough to what you mean by that. I'm a physicalist, which is different from a materialist in that there are physical things besides just matter, and that I reject that there are ontological material substances distinct from their attributes, and I'm also a little unusual (but not alone) amongst physicalists in that I'm also a panpsychist; there are no non-physical mental substances, nor are there even non-physical mental properties of otherwise physical things, but there is a mental way of looking at the same physical properties of physical things.
(This isn't really related to the moral issue to I'll be short about this: basically, what fundamentally exists is a web of interactions, which you can visualize as a graphical web of lines connecting at nodes. The objects that exist in reality are those nodes, which are defined entirely by the lines connecting to them -- there's no such thing as a point connected to no lines. The attributes of a thing are those lines -- the attributes of a thing are the ways that it interacts with other things. Mental experience is then what the lines connecting to the node that is you seem like to you; they are the reciprocal of the attributes you seem to have to the rest of the world, the other half of the equation of how you and the world interact with each other. Every object thus looks to us from the outside, in the third person, like a bundle of attributes, but it, from the inside, in the first person, experiences those very same attributes -- which are really interactions -- as a bundle of experiences of the rest of the universe. That is "phenomenal consciousness", the subject of the so-called "hard problem of consciousness", and is not actually a very interesting thing; rocks have it, electrons have it, but there's not much to say about theirs because the quality of that kind of phenomenal consciousness is defined by the function of the object that has it, and those objects have such simple functions; their behavior is simple, and so is their experience. On the other hand "access consciousness", the subject of the so-called "easy problem of consciousness", is a specific kind of complex function that humans do, which gives us not only complex behavior but also complex experience. But that's a long tangent: back to ethics).
I think that trying to ask about ontology (the kind of stuff that exists or not) to ground ethics is misplaced, because ethical questions are non-descriptive, and so descriptive answers, answers that say something about what or what doesn't exist, have no bearing on ethical questions. You've probably heard a version of this called the "is-ought problem": no "ought" can ever be soundly inferred from an "is". What is or isn't the case has no bearing on what ought or ought not be the case, so it doesn't matter whether the universe is entirely physical or nonphysical or anything like that; moral questions aren't about that at all. But unlike Hume, who coined the is-ought problem, I don't thereby conclude that there just is no objective grounding of "ought" statements, and that they're mere expressions of sentiment. I instead ask: why exactly do we think there are objective answers to "is" questions (even though people disagree about those too), how do we go about finding out what they are, and what would be the analogue of that for "ought" questions?
The scientific method relies on the uniformity of nature. How do you know that the future will be like the past? Do you have an answer for Hume or Russel?
Good question. The uniformity of nature must be taken as an axiom for the same reason that realism in general must be: to do otherwise is simply to give up on even trying to answer questions about what is real, which in turn would guarantee that we will never find any answers, even if we could had we tried. (Causes, like substances, numbers, universals, space, and time, are part of the structure we must assume -- what Kant would call "categories" -- in order to divide up the mish-mash of our experience into comprehensible patterns, and not directly a part of that experienced reality itself). Even in the face of apparent non-uniformity of some class of phenomena, the scientific method is to assume that that's something we merely don't understand yet, not that it's something that cannot be understood, and to then continue trying to understand it despite any lack of success so far. That's why science gets the results it does: it tries, always, doggedly and determinedly, and so wherever success is possible, it actually has a chance of finding it, unlike alternatives that either assume success or failure is inevitable, and so don't try, and so guarantee failure.
Can you clarify this? Perhaps with an example?
This is the real reason I wanted to reply; I kinda got carried away with littler stuff above.
With regards to descriptive questions, that is to say, questions about reality, the relevant aspect of our experience is what we would call sensation. Certain experiences, like of sight and touch etc, make certain things just seem true or false; "seeing is believing" as they say. Except it's actually not, not directly; sensation per se, as the term is used in psychology, is the unmediated, uninterpreted stream of data from our senses; the raw colors of light, pitches of sound, etc. Perception is then the automatic, first-order interpretation of that sensation into descriptive propositions: rather than just seeing a pattern of colors, I see that my phone is on the desk next to me, for example. Belief is then assent to that proposition; sometimes, you don't believe your perceptions, and sometimes you shouldn't. But when trying to figure out what to believe, we should look past those perceptions and attend to the raw sensations -- which, in discussing science, is more generally called observation. And more than just our own sensations or observations, we should listen to the reports of others, and then try to replicate those reports by placing ourselves in the same circumstances and seeing if we sense or observe the same things. If we don't, we should try as hard as possible to figure out what could possibly be different about us two different people or the circumstances of our observations to see if there's any way we can account for that difference, and some to agreement on what seems to be true or false to the senses of everyone. Importantly, we are not just asking people what they perceive, and we're especially not asking them what they believe. We're asking what their senses say, what, on the most fundamental level, just can't help but seem true or false to everyone, and then we can reason about possible higher-level models that accord with that raw data, testing those models against further data collected in the same way, which never directly tells us that any particular model is the ultimate absolute indisputable truth, but continually and unendingly narrows the range of models that might possible be true to a smaller and smaller set.
With regard to prescriptive questions, that is to say, questions about morality, the relevant aspect of our experience is what I call "appetites", for lack of a better-recognized English word for this category of experiences. Things like hunger, thirst, pain, pleasure, etc, that make certain things just seem good or bad. To get my moral analogue of science, just substitute these in place of sensations or observations, and likewise substitute desires in place of perceptions, and intentions in place of beliefs: desires are an automatic first-order interpretation of appetites into prescriptive propositions -- rather than just a pattern of feelings in my skin, I feel like I ought not to touch that hot pan -- and intentions are then assent to those propositions, which we don't always and shouldn't always give; just because I feel like I ought to eat that cake, doesn't mean that I agree with that feeling and intend to eat that cake (though since it's my birthday, in this case I do intend just that). When trying to objectively answer "ought" questions, then, just like science we don't ask people what they intend or even what they desire, but we attend closely to our appetites, paying attention to what brings us flourishing or suffering, what, on an undeniable fundamental level, just can't help but seem good or bad. And like with science, we don't just pay attention to our own appetites, especially not just our own appetites at the instantaneous moment but at the very least the pattern of our appetites over time in different circumstances (just like we account for our changing sensations over time and don't just believe in a static two-dimensional world that appears to us at any given moment). But also, importantly, to each other's appetites, confirmed by placing ourselves in the same circumstances as them -- "walking a mile in their shoes" if you will -- and seeing if those things really do just seem good or bad, on an appetitive level, to us in those circumstances; and, just like with science, trying really hard to account for any differences that appear when we do try that, seeing what are the differences between the two people with differing reports of how they experience the same circumstance, and if it really is exactly the same circumstance and didn't differ somehow, until we come to agreement on what seems to be good or bad to the appetites of everyone, and then building higher-level models that accord with that raw data, and testing them against further data collected in the same way, never positively affirming one specific model to be the absolute indisputably correct one, but continually and forever narrowing in on a smaller and smaller set of models that might be correct.