Is the concept of Aporia the same as the zen concept of A Beginners Mind, or related to it, or a completely different concept?
Welcome, Sam Wheel.
In philosophy aporia retains, at least standardly, its Aristotelian sense. This has no commonality with zen so far as I can see if by 'beginner's mind' we mean what Suzuki does:
So the most difficult thing is always to keep your beginner's mind... Even though you read much Zen literature, you must read each sentence with a fresh mind. You should not say, "I know what Zen is," or "I have attained enlightenment." This is also the real secret of the arts: always be a beginner. Be very very careful about this point. If you start to practice [Zen], you will begin to appreciate your beginner's mind. It is the secret of Zen practice. (Suzuki, S. (1973). Zen mind, beginner's mind, New York: Weatherhill: http://www.cuke.com/bibliography/ZMBM/prologue.html.)
See now what Aristotle says about aporia and the contrast will be appparent.
Aporia and Aristotle
Here is a useful account by Gareth Matthews of how the notion occurs in Aristotle:
Often Aristotle, when he begins the discussion of a new subject matter, first identifies the aporiai (perplexities, difficulties, antinomies or just problems) that belong to that subject and the sets as a constraint on his own inquiry the resolution (euporia) of each of the perplexing problems he has identified. ...
We must, with a view to the science which we are seeking, first recount the subjects that should be discussed. These include both the other opinions that some have held on certain points ... For those who wish to get clear of the difficulties it is advantageous to state the difficulties well; for the subsequent free play of thought implies the solution of the previous difficulties, and it is not possible to untie a knot which one does not know. But the difficulty [perplexity, aporia] in our thinking points to a knot in the thing [peri tou pragmatou]; for in so far as our thought is in perplexity [aporia], it is like those who are tied up; in both cases it is impossible to go forward. Therefore one should have surveyed all the difficulties beforehand, both for the reasons we have stated and because people who inquire without first stating the difficulties are like those who do not know where they have to go; besides, we do not otherwise know even whether we have found what we are looking for; the end is not clear to such a person, while to one who has first discussed the difficulties it is clear ...
The first problem [perplexity, aporia prote] concerns ... (Aristotle, Metaphysics, IIIB.995a24-b4, J. Barnes, The Complete Works of Aristotle, Princeton: Princeton University Press, II, 1984 : 1572-3)
The philosophical methodology this passage suggests goes something like this. Suppose one wants to know what it is for something to be a place, or a location (topos). One begins by collecting some of the relevant things people say about places, things that might help us determine what a place is. One of the things people say, it seems, is that the place of a thing contains it, but is not part of it, since the thing can leave its place without growing smaller. Then we collect the puzzles (aporiai) about place. For example, there is a puzzle about whether a place is itself in a place. If every- thing in the physical world had a place, then, it seems, a place must have one, too. But then there would be an infinite regress of places, which is absurd.
Now the point of the inquiry is to see if we can somehow respect most of what people are preanalytically inclined to say about places and, at the same time, resolve the relevant aporai. Suppose some- one then says, how can you inquire into what a place is? Either you know already, and so the inquiry is phony, or else you don't know, and so you won't be able to recognize a satisfactory answer. Aristotle replies, first, that what people are inclined to say about places provides some general parameters for a satisfactory account. As for being able to recognize what we are looking for, should we find it, that consists mainly in being able to tell whether the puzzles that our object of inquiry gives rise to have been solved in a way that respects what people are inclined to say. (Gareth B. Matthews, 'Perplexity in Plato, Aristotle, and Tarski', Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition, Vol. 85, No. 2/3, Papers Presented at the American Philosophical Association Pacific Division Meeting 1996 (Mar., 1997), pp. 213-228: 225; Aristotle, Physics.IV.)
Plato had been here before, of course.
Plato has Meno introduce the puzzle about how there can be such a thing as inquiry, for example, trying to find out what virtue is. Either one knows already, and there can, for that reason, be no real inquiry. Or one doesn't know, and so won't even know what to look for (80d-e). (Matthews: 223.)
While there are many examples of aporiai in the Platonic dialogues, particularly in the earlier, Plato shows little interest in the notion of aporia as such. In Meno, for example, he is more interested in the question, 'What is virtue*, than in the paradox of inquiry just cited from 80d-e.
A sharp difference from Aristotle, as any reader of the Republic will know, is that Plato sees no need to respect 'what people are preanalytically inclined to say'.
The two things are unconnected. A dictionary will make this clear. Beginner's mind is an attitude and approach, a recognition of our own ignorance and an opening of our mind. It has nothing to do with aporia, defined as 'an irresolvable internal contradiction or logical disjunction in a text, argument, or theory.'
Not on the face of it. Aporia means, in the clearest form, that two common citizen's opinions, commonly held by the general population, are in contradiction or in some way conflicting. A political difficulty. This leads to the need to think them through and thus to prima philosophia, ergo: political philosophy. More in the manner of an inner teaching it means that a difficulty, an impasse, is stumbled on by an adept. So that one, Socrates being the main example, discovers a strange difficulty which turns out to be a kind of "wonder" (thaumizein). However, the implication is not that he has now become a beginner, who is always "setting out" anew, rather, he has come to an exact difficulty which exercises him (Socrates never, strictly speaking, "doesn't know what he knows," rather this is always in play with his famous "irony" and with other factors, most especially: HISTORY). Thus, the impasse itself is a gateway to the wonder or sheer bliss of knowing being. Unlike in "beginner's mind" where the presupposed difficulties themselves are "problems" only for the ones who have advanced too far and become mired in a sylabus of bigoted narrow bespoke difficulties.
Aporia is less radical from the Eastern perspective in that it presupposes being. And thus it views there to be problems that can become questions, that which makes one wrestle and lose sleep. Form the Western perspective "Beginner's mind" lacks content, and so is mere "New Age bullshit." Ergo, it is only pure experience, even denying the concept of experience, it is without reality and so "against reason."
I provide some links to primers on the subject matters involved: https://www.youtube.com/watch?v=g_F_cxM9d5Q
As the other answers have indicated, aporia and beginner's mind are not equivalent concepts. However, I don't think it would be a grossly implausible stretch to make a connection, perhaps, between the pedagogical force of Plato's aporetic dialogues and the pedagogical force of the beginner's mind concept. For Socrates the worst epistemic state was thinking that one knows what one does not know, or failing to understand the limits of one's own knowledge. Thus in the aporetic dialogues, Socrates uses philosophy in a purgative dimension, purging the faulty thoughts and paving the way for better ones. One is prepared to learn only when one knows one does not know.
The connection here is the emphasis on the purgative aspect of pedagogy: not placing positive doctrines in the head, but rather emptying. I think the two approaches (Socratic and Beginner's Mind) have their differences, but there might be a story to tell here that isn't hugely implausible.