It seems to me like you're claiming that there are only two points of view (or maybe a single one-dimensional axis of points of view) in situations where I see no reason to expect that. For example:
Some people are born with major disadvantages and we need to be sympathetic to them. At the same time, people can act in a way which makes their decisions better or worse and we need to encourage personal responsibility. If we’re too harsh, we don’t give them the help that they need, if we’re too sympathetic, we simply enable people to ruin their own lives. We need to find a balance between the two
There seem to me to be a bunch of implicit claims in this paragraph I don't agree with, like "sympathy doesn't help people," "personal responsibility is the only way to change people's behavior," and also this thing about balance.
My own point of view here is neither of these two points of view, and it's not a balance between them either. I want to help people, but I basically don't want to use the concept of responsibility at all to do it. Generally I try to help people by updating their beliefs about me, them, and/or the world (I mean this in a pretty broad sense, e.g. the update might be "I like you and don't want to hurt you" and I might tell them this using my body language, not words), and teaching them important skills like how to access their actual feelings. I don't want people to do things because they feel responsible to themselves, I want them to do things because they want to do them. (If they want to feel responsibility then that's their business, but I don't want to impose it on them.)
I also have reservations about your use of the word "duality," but based on my experiences from how it's used in mathematics as opposed to mysticism.
Just adding an additional comment. The mathematical definition is not well-known enough for me to avoid using this word in general conversation.
"Also, undoubtedly there are times when you want to synthesise more than two ideas. I don't have a word for this, but it is much less common." - It's very common to have one main axis with which you are concerned, but in any case, two ideas is sufficient to demonstrate this in general.
In general though you can always pick a single dimension along which to work. It may not, as you get at, describe reality in all its richness, and it will definitely confound ideas, but it will also always create a dialectic where one can find balance between one side and the other. I think maybe this is more how this is useful, as a way of exploring an idea, rather than as an ontology to depend on.
You are clearly talking about dialectic as you suggest yourself. I think working to increase awareness of this idea when you want to use it makes more sense than trying to reframe an existing word used for something else to mean what you want.
Duality also has a pretty precise meaning in optimization: determining prices is dual to determining allocations; finding a counterexample is dual to finding a proof. This is very closely related to the broader usage in mathematics.
You need to pick a word that isn't already closely used. Think of this like mathematical variables. If I call. It E and I'm talking about motivation equation and expectancy, that might be okay. Equations about motivation don't usually also talk about energy in the E=mc2 sense. but if I'm talk about energy I probably don't want it reuse E.
I’ve been finding the word duality useful quite a bit recently. I find it very useful for describing situations where there are two very valuable perspectives (or lens) through which we can look at a situation and any attempt to answer the question needs to grapple with and account for both of these. The way I use the word, I’m not claiming that two logically contradictory viewpoints are simultaneously true, but rather that aspects of both can be synthesised together to reach the truth.
Here’s a few examples. Maybe you agree or disagree with these specific examples, but I think they should suffice for illustrative purposes:
Dualities allow you to avoid a particular dysfunctional pattern of thought. Most people will have one side of the duality stick out for them more than the other. Since both sides contradict (or seem to contradict), they conclude that they must reject the other side of the duality. This resolves the contradiction, but it doesn’t give them the truth. Finding the truth would require considering both sides, but now that they’ve settled for consistency instead, they’ve prevented themselves from making further progress. There is value in having a part of the truth-finding process where you can identify ideas as containing a lot of truth without having to worry about whether they contradict, at least temporarily. Thinking in terms of dualities encourages this.
I really value consistency and I encourage others to value it too, but perhaps I should value it a bit less as overvaluing consistency seems to be what leads to these kinds of mistakes. If your model of the world contains an unsynthesised duality, then this won’t allow you act consistently, but it may still give you a more accurate model of the world than picking one side or the other. Of course, attempting to eventually synthesise these dualities should be the eventual goal, but we should also recognise that this may simply be beyond our own personal abilities and that we may never be able to completely remove the duality.
It is also very useful for explaining ideas that you haven't completely worked out in your head. Quite often you are aware that there are two main sides of the issue, both of which make good points and you want to establish that you will be drawing on both sides. Once you've set that up, you can then move towards synthesise. Even if you've already figure out a synthesis, there can be value in taking the listener on the same intellectual journey so that they can understand your conclusions. Also, undoubtedly there are times when you want to synthesise more than two ideas. I don't have a word for this, but it is much less common.
Why use the word "duality"?
The word duality communicates this concept well and if you want to be understood, you need to use language that people understand. Undoubtedly, many people don’t want to use this word as it is usually associated with forms of mysticism and the belief that something can be two things that logically contradict at the same time. This is not a viewpoint that I embrace at all, but I’m not convinced that using the word “duality” encourages this to the extent where I would want to stop. After all, while it may encourage more people to use the word duality and some of these uses may be incoherent, using it properly may also led some people to replace incoherent uses with coherent uses. So I see the overall effect as something of a wash. I would actually prefer the word dialectic, as it connotes the idea of two opposites which can then be synthesised together, but I don’t feel it’s meaning is known well-enough.