One of those is the concept of universals. It's the idea that there's another world 'out there', some metaphysical plane of existence, in which categories-of-things-in-themselves literally exist.
For example, we agree that chairs exists. But what defines a chair? Is a bean bag a chair? What about an ergonomic kneeling chair? What about a tire suspended from a tree branch? Plato would have said that there is an inherent property of chairness that permeates all things that fit within the category chair. Aristotle might have called it the essence of that thing — that which makes a chair a chair.
Color is another commonly cited example of a universal. Grass is green, and so are leaves. According to Platonic thought, grass and leaves are particulars. They are individual objects that are not repeatable — that is, they are discrete. Examples of particulars are generally material objects, but some philosophers take 'immaterial' objects to be particulars as well — sensory data, God, etc. A universal is a property that exists independently of these particulars, but inhabits them. If I burn a patch of grass away, the essence or property of greenness still apparently exists in many other things — and so, Platonic thought teaches, it must be metaphysically real. How, precisely, these universals exist and interact with particulars is a mystery yet to be solved, yet even today philosophers write lengthy books arguing that universals must indeed exist.
For the purposes of this post I'm not going to get into nominalism, realism, and conceptualism — all attempts to account for universals. Are they really 'out there'? Or are they illusory, just part of the mind? Are they just artifacts of language?
If Plato were a psychologist...
My background in studying these kinds of questions is psychology — cognitive psychology in particular. You generally don't hear cognitive psychologists talking about universals or debating nominalism versus conceptualism. That's because cogntive psychologists have been able to study, and answer, the question of whether categories-of-things-in-themselves exist independently of human brains. The answer is no.
Let's return to the chair. We all agree that chairs exist. But "chair" is a fluid, and even disputed, concept. From kneeling chairs to tires-on-trees to conventional wooden chairs, what we understand a chair to be is dependent on our experience.
|Is this a chair? Is it art?|
Worse for universals is the fact that categories are often radial. We can, for example, think of a general type of car. We can also think of more specific categories, like particular types of cars. We can do the same for boats, planes, and trains. But we don't have a general categorical image of "vehicle", even though it subsumes all those other categories. Similarly, while we can think of a general concept of a chair, categories of specific types of chairs, as well as other categories of furniture, we can't conceive a representation of the general category of 'furniture'.
The simple answer that cognitive psychology gives us is that while the objects we know as chairs indeed exist outside of our minds (we're not endorsing relativism here), the category of chairs does not. Rather, the category itself is imposed upon the physical world and is inherently a fluid, social construct.
The idea of greenness faces a similar problem. We know from physics that particles of light do not have color. Color cannot be a surface property of objects because some things that are said to have color, like the sky, do not have a surface at all. We know still that a study of the human eye has revealed that we can see only a narrow range of the electromagnetic spectrum, which we call "visible light" — not because it's inherently visible, but simply because it's visible to us. What we call green cannot then be an inherent property of a thing, but rather is a multiplace interactional property that crucially depends on our biology — the interaction of our eyes and visual cortex with the environment. We say that "green" is "real" because we experience it as real, and because others appear to share that experience. But there's no basis to establish the existence of the category of greenness outside of the human experience.
Folk theories versus scientific theories
What's often overlooked in these discussions — particularly since they're steeped in literally centuries of philosophical tradition being pitted against a relatively nascent field of cognitive neuroscience — is that Plato's theory, like Aristotle's theory of essences, is not a scientific theory but a folk theory. Aristotle believed, for example, that the continuity of the self was sustained by an essence that persisted even as our bodies and minds changed. But we now know that the continuity of the self is a product of biology and social construction, and that it can be radically disrupted — as in the case of physical injury to brain that radically alters a person's memory and/or disposition.
Plato may have thought that his theory was scientific, at least in the sense that it proposed hypotheses that could reliably explain the world around us and our relationship to it. But Plato didn't have access to the structure of his own process of reasoning. He couldn't have possibly known about scientific discoveries in cognition like primary metaphors or overlapping conceptual hierarchies, because those things required empirical study of the mind and brain. The "solutions" of nominalism, realism, and conceptualism are fundamentally solutions to a problem that arose precisely because the concept of universals is a folk theory steeped in metaphorical conceptual systems, and not a scientific theory.
Unfortunately, ideas like essences and universals persist to this day despite the advances in cognitive psychology that severely undermine them. My guess is that it's going to take some time for centuries (millennia, actually) old ideas still deeply studied in academic institutions to succumb to a competing field. But the reality is that despite all those centuries of debate, philosophers are no closer to resolving these questions than Plato himself was. Instead, we've needed a radical paradigm shift in how we frame and investigate the questions in the first place, and that's why a study of cognitive psychology is critical to empirically responsible metaphysics.