In Ray Monk's biography of Ludwig Wittgenstein, there is a striking account of the weekly discussions on the foundations of mathematics between Wittgenstein and Alan Turing. I found myself in agreement with W. and it seemed to me that Turing had been unable to grasp his own error, because he could not free himself from the mathematical thought-world and its assumptions. Turing could not see, or even acknowledge the reality of, a bigger picture than mathematics.
Wittgenstein was stating that understanding was a simple direct intuitive grasp - when we know that we know; whereas Turing was arguing that we know we understand when the results 'work' - we know mathematics is right because a bridge made using calculations does not fall down - and errors in mathematics will lead to bridges falling down.
The argument is a version of one in which I have participated many times - on both sides at different points. For Wittgenstein, understanding was the most important thing in the world - his whole life was driven by his need to understand. He therefore saw that there was no relationship between understanding and practical results (as we presume we know them) - that science, mathematics, engineering have nothing to do with understanding.
Turing, in a sense, was saying that understanding doesn't matter; what matters is whether we can predict and manipulate the world to achieve our desires. Thus a computer does not understand what it does - but this doesn't matter if it can do things we find useful.
Wittgenstein would realise that the usefulness of computers is beside the point - it just is not relevant to the question of understanding.
To put it another way, Turing regarded 'models' as the furthest we can go in understanding. Each model is a deliberately simplified (hence ultimately false) description of total-reality; but that doesn't matter because (for Turing) all thinking is some kind of model.
Whereas Wittgenstein regarded understanding as the basis of everything; for Turing understanding would just have been a transient psychological state - ultimately a kind of delusion.
Turing's 'proof' that understanding is models only is to point at the (apparent) achievements of modern life - look at that bridge, that computer; look at modern man's capability... That is the result of models, useful models - and there is no truth beyond usefulness, but no further truth is needed...
Whereas Wittgenstein would say that Turing did not understand him; and Turing did not understand his own lack of understanding. Turing displayed a kind of impatience, an inability or unwillingness to see things how Wittgenstein saw them.
Wittgenstein knew what Turing meant, because Wittgenstein had believed the same himself; but further thinking had led to a simple, direct understanding that revealed that Turing's view was false. Wittgenstein had thought the way Turing did, then moved beyond it into a situation so clear it needed no further justification.
The exchange between Wittgenstein and Turing seems to me a microcosm of many arguments and discussions that go on - or rather that fail to happen because one of the parties simply has not thought long, hard, deeply enough to know that they are wrong; and when that party has 'more important things to do' than achieve the clarity and simplicity to which such thinking will eventually lead.
Thus many clever people (and there are few cleverer than Turing!) are wrong about fundamental matters; not least because they are impatient to get on with their main line of work, research, creative endeavour... Yet they do not escape their wrongness; and the wrongness of Turing is not invalidated by the success of and ubiquity of computers - rather, Turing's error is built-into the modern world: baked into it indeed, since our tests of whether something works have themselves become pragmatic.
So now we have, in effect, the situation of non-understanding computers evaluating the truth (i.e. true understanding) of non-understanding computers - because the matter of real (simple direct, intuitive) understanding has become regarded as merely subjective, psychological, contingent. Modern man therefore demands that the need for understanding be eliminated - and he sets-up procedural systems to evaluate the 'truth' (or 'quality') of science, mathematics, engineering and everything else; without any need for (and indeed deliberately excluding) human understanding.
These systems are supposed to be indifferent to whether or not a person has thought deeply and long, whether they understand or merely manipulate. Indeed there is now a suspicion of, hostility to, anything which is not obvious to 'anybody' - including 'anybody' who has never thought about it and never achieved a state of understanding.
A typical modern bureaucrat (and most modern people are bureaucrats, whether professionally or in their private lives) would be saying to Wittgenstein, over and again: That's just your opinion.
(And anyway, do you want us to live in the stone age, or what?)
Wittgenstein would know that he was right, and he would know this for absolute certain - but probably would not be able to convince anybody else. If he couldn't convince Turing that there was more to understanding than models - what chance would have Wittgenstein have with the average modern middle manager, or academic-careerist, or official, or media propagandist?
What you are describing here is surely the fundamental difference between the spiritually oriented person and the materialist, the latter having 'brains' but little imagination. He cannot see that he cannot see because he lacks the organ of vision and so he tries to force reality into his pre-existent limited framework but it can never really fit.
I am minded of the apocryphal Roman custom of having the engineer who designed and supervised construction of a bridge stand in a raft underneath the arch as the first carts rolled across (ancient sources confirm similar customs existing in contemporary societies, but interestingly the only source relevant to Rome laments that it wasn't part of Roman law).
I'm also minded of An Occurrence at Owl Creek Bridge, by Ambrose Bierce, and similar stories which deal with the failure to correctly comprehend sensory perceptions. Turing is correct that we know we understand when the results 'work' - we know mathematics is right because a bridge made using calculations does not fall down - and errors in mathematics will lead to bridges falling down. But he is missing that we will not know the bridge fell down if we refuse to understand when it does...and that this is a real danger.
It is an irrelevancy in the case of a man who is going to die whether or not he accept the current information of his senses telling him that is what is happening. But Wittgenstein is talking of realizing that the model is wrong before it is too late, seeing that the bridge built using it will fall down before it actually happens.
Whether the faculty of understanding the bridge is falling down once it has already begun to fall down is common or rare is of little practical importance. But the faculty of spotting errors in a model before those errors are fatal is of enormous importance. It is one that Turing evidently, and tragically, lacked in his personal life.
Would Wittgenstein's concept of "understanding" be that Platonic truths have a real, independent existence, and that we can grasp these truths by means that are probably not material (e.g., through the operations of the soul)?
@Jonathan - No, I don't think so. What I'm gettng at here is something behind WIttgenstein's 'philosophy' as it was extracted from his work and taught (mostly in a grossly distorted and misleading way) after his death; I'm talking about the basic metaphysical convictions that are prior to his 'philosophy'.
In this regard Wittgenstein wasn't a Platonist, but a religious (Christian) mystic, with a conviction of the possibility a direct (God-given) apprehension of reality.
I do agree that understanding is not necessary in order to realize one's purpose in the material world - for instance, we can command our body competently without any knowledge of neurology. However, I think that there is a necessary connection between true understanding and empirical experience, because I believe in a logically unified world. A disjunction between true knowledge and reality, as revealed by our senses, would imply an imperfect material world existing alongside a perfect mental world. This I see as a kind of metaphysical pessimism, namely the pessimism that the reality outside us is less perfect than the reality inside us.
@RM - This matter was, for me, decisely clarified by Rudolf Steiner's The Philosophy of Freedom (of 1894 - https://en.wikipedia.org/wiki/The_Philosophy_of_Freedom ) this is available free online at Rudolf Steiner Archive, and as an audibook on Rudolf Steiner Audio.
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