First, suppose what's happening when I say I don't understand something (let it be a proof of some mathematical theorem). What does it mean? Probably that I can't see all logical connections between given facts and statements, right?. And then it comes to me that eureka moment, when I finally 'get it'.

What if I shouldn't understand it and I'm just fooling myself this proof or anything similar makes sense? Given some proof we were told is correct, we basically stop thinking about it once we feel we 'understand' it. Maybe we shouldn't think we 'understand' it, because it contains some major logical flaws and our initial intuition with regard to it (that we didn't understand it) was correct?

In other words, when we don't understand something we feel we should, and it makes potential mistakes harder to spot, because we adjust our thinking process to arrive at the conclusion 'yes, this statement makes sense and is correct'. On the other hand, if we attempted to falsify every statement, we could never finish the process, because maybe we would never find any flaw in it. It all comes down to proving axioms, which is impossible (stating that it's impossible to prove an axiom is an axiom - it can't be anything else... quite circular thing though). In the end, we take it as an axiom that if a billion people find no flaw in a statement, then we could accept it as an axiom.

  • 1
    Well, this is why we show our proofs to other people or publish them; there are informal, and formal peer review processes in science and elsewhere. Apr 21 '15 at 21:12
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    How understanding works is a good question (and we've got two good answers to that below), but I'm slightly confused as to whether you are offering a question or an answer here. On SE, you are allowed to provide an answer to your own question, but if you want to do so please separate the question answer bits.
    – virmaior
    Apr 22 '15 at 6:04
  • I just gave my opinion on this problem, nothing else. That's what philosophy is all about - comparing our view on the problem with other people. In fact, there are no ANSWERS in philosophy.
    – user107986
    Apr 22 '15 at 6:21
  • I love how people just put on hold this question BEFORE telling me what's wrong with it. You realize that basically no 'on hold' questions are brought back to normal state.
    – user107986
    Apr 22 '15 at 6:22
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    That's not really true. There are quite some questions that end up in the reopen queue. The on hold reason and Virmaior's comment tell you what you can do to improve your question. As soon as you edit it, it will show up in the reopen queue. You're quite right that there are no answers in philosophy. However, this is not a philosophical site but rather a site about philosophy. Hope this helps to understand.
    – user2953
    Apr 22 '15 at 6:39

Here is a possible set of criteria for understanding something:

  1. You know the concepts and facts, and how they link together
  2. You know the underlying assumptions
  3. You know the implications of the thing you understand
  4. You know the conditions under which your current understanding would be false

These may be either too stringent or too loose, and might be a possibly incomplete list. The term 'understanding' is fairly vague - and as will be clear from what follows, important in clarifying the question.

It is important to note: Whether you judge yourself to have understood something is logically independent of whether you have actually understood it. Consider the following four cases:

  1. You have not understood something, but judge yourself to have.

An example: You believe yourself to have understood why 13 is a prime number, even though you don't know the reasoning behind it, or have a mistaken notion of the reasoning behind it.

  1. You have not understood something, and judge yourself not to have.

You don't get what implicit differentiation is. And you judge yourself not to have gotten it. No problems here

  1. You have understood something, but judge yourself not to have.

This case is probably most problematic. One path I could posit here is that you have met the criteria that others would reasonably expect you to have met - e.g. being able to teach it to someone else, replicate the steps in a proof, but you intuitively believe that there is more that you need to know in order for it to count as understanding, according to your self-imposed definition.

  1. You have understood something, and judge yourself to have.

You understand the principle of addition - 2+3 = 5. And you know that you understand it.

This means you could be wrong about having judged yourself to have understood a thing. You are probably right that there is some complacency involved after we have judged that we have understood something. Maybe we should be more hesitant with our judgments of our own understanding.

On the issue of falsification, it is true that most of what we believe ourselves to understand could be falsified under some imaginable circumstance. And you are right that insisting on complete unfalsifiability is unrealistic. This suggests that in order for the concept of 'understanding' to have any practical import, we have to set a lower benchmark than "complete, unfalsifiable knowledge". Otherwise it would follow that we do not understand anything, in which case 'understanding' would be a useless notion. I think this stems from an unreasonably high benchmark for 'understanding', and that we could moderate it.

I think that criterion 4: "You know the conditions under which your current understanding would be false." - introduces a notion of defeasibility which could be useful in fine-tuning your understanding.

Hope this helps!


“If you can’t say it clearly you don’t understand it yourself” –John Searle.

The way I see it, we could have two construals of “understanding”- a weak and a strong one. The weaker one consists in activity which is essentially intellectual, cognitive, reflective and representational (Ryle’s knowledge-that). The stronger version would be grounded in ‘agency’; namely, you know you have it when there is something you can make happen in the world- predictably and routinely (Ryle’s knowledge- how). Ryle’s regress was supposed to show that “knowledge-that” is merely a species of “knowledge-how”; a ‘cognitive’ version of ‘doing something in the world’. Timothy Williamson begs to differ. He thinks one’s capacity how is always mediated by consulting ‘propositions’ beforehand. Josepha Toribiro has a good paper defending Ryle’s case (against Williamson, et al.) that ‘knowledge-how’ has primacy.

I completely agree that having a capacity to predictably (and routinely) bring about certain outcomes doesn’t necessarily entail “understanding” in the cognitive/reflective sense, but I don’t think it has to matter all that much. If you take the ‘measure’ of agency as simply a capacity for task-completion, then the agent only has to be beholden that. Having an indefinitely deep cognitive account (omniscience?) as to all the pertinent wheres and whyfores of what’s going on is I think redundant (think of Searle’s (non-representationalist) notion of the deep and local background).

The philosopher Stephen Grimm has gone after the issue of understanding in a big way (with a sizable grant of 3 mill!). I think he adopts a position that’s couched in epistemological terms, which comes off as a little circular- explaining a cognitive faculty in terms or more cognitive faculties. I recall in a paper where he cites certain others (?) that suggest understanding fails to bring with it any particular epistemic constraints: the child sees the situation a certain way; the religious fanatic has a certain weltanschauung; the radical skeptic denies climate change; the paranoid believes the government is watching him; the dog can fetch the stick, etc. I think this is correct. If you make (non-cognitive) agency dependent on the (representational) cognitive variety, then it would be hard to see how any organism not blessed with a ‘language capacity’ could ever function properly. Does the dog consider a bunch of propositions before chasing down the stick? (-pace Williamson). Agency in terms of ecologically coupled/productive goal directed activity has to come first. The reflective, representational stuff comes after.

Richard Menary has a good evolutionary account of how and why a capacity for personal level, decoupled (top-down sensory motor emulation/neural reuse) is a ‘Johnny-come-lately’ adaptation in comparison to the sub-personal ‘coupled’ variety. Menary’s ‘cognitive integration’ thesis (and the bulk of Andy Clark’s work (et.al)), detail what I’m essentially trying to say here- cognition subserves/leverages, pragmatic (or ecological) agency. Ryle is right!

Proofs of mathematical theorems are a tricky one, since they are almost exclusively cognitive/theoretical.

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