What makes people irrational thinkers?

[Belacqua]

The allegory is supposed by academics to be a reference to Plato's tripartite soul. […] Plato’s well-developed theory of the tripartite soul.

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While Plato is wrong ultimately about our minds having three discrete parts […] But I don't think Plato was really trying to understand how the brain works (like a neuroscientist does). He was referring to what he saw as internal "forces" of the mind that we all have to deal with in life.

I’ve added emphasis in these quotes to point out a concern. 

The allegory of the chariot, and the theory of the three parts, is about SOUL. ψυχή.

Mind is different, and it's not correct, in Plato, to conflate the two. Mind is usually said by Plato to have two parts: nous, νοῦς, which is something like the modern intellection or understanding. The other part is Dianoia, διάνοια, Latin: ratio. This is more like discursive reasoning, as carried out when we do math or figure out problems. 

Nous is one of those big words with lots of nuances, but in general people here are not going to like it, because it allows the possibility of gaining knowledge without conscious reasoning. For Plato, this includes direct apprehension of the Forms, which is what the chariot allegory is mainly about. 

I’m picky about this because in the past people on this forum have just declared, by fiat, that mind is the same as soul. I guess if you don’t believe in soul you can just define it any way you want, but in philosophy they are different. Soul is more generally about the structure of our animate lives — it is our animation and how our animation functions. The Latin is anima. This includes but is not limited to the actions of the mind. It also includes the activities that the rest of your body is up to right now, including whatever your liver and your gall bladder are doing.

I have some concern as to whether the three parts of the chariot analogy map exactly onto the tripartite soul as described in the Republic. 

Here is a quote from the Phaedrus:

Now, while the horses and charioteers of gods are always thoroughly good, those of everyone else are a mixture. Although our inner ruler drives a pair of horses, only one of his horses is thoroughly noble and good, while the other is thoroughly the opposite. 

So one horse is all good and the other is all bad. But the tripartite soul is less black and white. 

The “top” level of the soul is logos, λογιστικόν, which does the reasoning and regulating. 

Next is the thymos, θυμοειδές, related to spirit.

And at the “bottom” is eros, ἐπιθυμητικόν, which is desire. 

Now the logos level does seem to map onto the charioteer, because he’s trying to regulate the horses. But Spirit, thymos, is certainly not seen as all good or all bad. For example, bad anger is from this part, but so is righteous anger, which is a good thing. Spirit has all kinds of positive properties and serves to motivate us. Likewise eros, obviously, can have good ends and bad ends (cf. Symposium). The whole point of becoming wise is to learn to direct one’s eros to the best ends (or strictly speaking, to remove the misunderstandings which prevent our eros from aiming to those best ends which it naturally desires). 

Similarly, if the three levels of the soul are meant to map onto the social classes of the polis, then even an elitist like Plato wouldn’t say that one social class is all good and one all bad. Each class performs its function for the community.

I’ve seen articles in which people claim the three parts of the chariot map onto the tripartite soul cleanly, but I don’t think that’s right.

He was referring to what he saw as internal "forces" of the mind that we all have to deal with in life. He saw three of them as distinct and important (reason, passion, and appetite). They aren't really as distinct as he portrays them, but they're "distinct enough" from the first-person perspective (ie. how we experience them).

If we say soul instead of mind then I think that’s right. These are something like forces within us. In the chariot allegory he even implies that the horses are doing some reasoning of their own, in the service of their goals, so it's not entirely reason vs. desire. So the allegory is a simplification — exaggerated for clarity — about how we might feel about what’s going on inside our souls. 

I think Plato was correct about a number of things: first, he was correct that we are more desirous and spirited than we are reasonable. It isn't just that horses are bigger than the charioteer. He goes into this in the Republic, arguing that our reasonable nature is the "smallest part." Most people in the city are "producers." The producers represent the desirous part of the soul. Fewer in number are the noble auxiliaries... this class represents the spirited part of the soul. Fewest in number are those fit to be philosopher kings: the guardians. They are the least populous class and correspond to the smallest part of the soul, the reasonable part, (the tiny charioteer if you like).

There are people on this forum who insist that every sentence written in ancient times absolutely must be read literally. Somehow they think that ancient people weren’t smart enough to do allegory. That’s just foolish, of course. Plato is usually quite good about specifying. So he calls the Myth of Er a myth, and he calls an allegory an allegory. He says of the chariot that this is “what the soul is like,” not that this is a true ontological claim about soul. 

And I think that each non-literal explanation is crafted for a specific end, and that it would be dangerous to apply it to every other case as well. So here is where our materialist friends on the forum will say that the chariot allegory is a terrible thing to think about: 

The purpose of the allegory is to explain anamnesis. The chariot is meant to illustrate the condition of the human soul in the upper world, before it is born into a material body. The chariot goes around in this upper world, and occasionally, if properly controlled, can get a glimpse over the top edge into the world of Forms. This direct glimpse of the Forms is what allows us to have knowledge of them -- through memory -- even after the soul is joined with the body. 

I’m not sure that it’s good to take the part of the allegory we like and ignore the purpose for which it was designed.