Yes I think so. Understanding would involve understanding that it's a short form for adding. Remembering multiplication tables allows you to use math but it doesn't infer understanding.
Training themselves is necessary. All learning is self learning. Teachers can present material in different ways but learning is personal. No one can force you to learn either.
You can check if a model (or a kid) understands multiplication by simply probing them with questions, to explain the concept in their own words, or on concrete examples. If their answers are robust they understand, if their answers are fragile and very input dependent, they don't.
"To understand" is one of those poorly defined concepts, like "consciousness", it is thrown a lot in the face when talking about AI. But what does it mean actually? It means to have a working model of the thing you are understanding, a causal model that adapts to any new configuration of the inputs reliably. Or in other words it means to generalize well around that topic.
The opposite would be to "learn to the test" or "overfit the problem" and only be able to solve very limited cases that follow the training pattern closely. That would make for brittle learning, at surface level, based on shortcuts.
> It means to have a working model of the thing you are understanding, a causal model that adapts to any new configuration of the inputs reliably
The weasel word here is "reliably". What does this actually mean? It obviously cannot be reliable in a sense of always giving the correct result, because this would make understanding something a strict binary, and we definitely don't treat it like that for humans - we say things like "they understand it better than me" all the time, which when you boil it down has to mean "their model of it is more predictive than mine".
But then if that is a quantifiable measure, then we're really talking about "reliable enough". And then the questions are: 1) where do you draw that line, exactly, and 2) even more importantly, why do you draw the line there and not somewhere else.
For me, the only sensible answer to this is to refuse to draw the line at all, and just embrace the fact that understanding is a spectrum. But then it doesn't even make sense to ask questions like "does the model really understands?" - they are meaningless.
(The same goes for concepts like "consciousness" or "intelligence", by the way.)
The reason why I think this isn't universally accepted is because it makes us not special, and humans really, really like to think of themselves as special (just look at our religions).
> It obviously cannot be reliable in a sense of always giving the correct result, because this would make understanding something a strict binary
Our capacity to make mistakes does not necessarily equate to a lack of understanding.
If you’re doing a difficult math problem and get it wrong, that doesn’t necessarily imply that you don’t understand the problem.
It speaks to a limitation of our problem solving machinery and the implements we use to carry out tasks.
e.g. if I’m not paying close enough attention and write down the wrong digit in the middle of solving a problem, that could also just be because I got distracted, or made a mistake. If I did the same problem again from scratch, I would probably get it right if I understand the subject matter.
Limitations of our working memory, how distracted we are that day, mis-keying something on a calculator or writing down the wrong digit, etc. can all lead to a wrong answer.
This is distinct from encountering a problem where one’s understanding was incomplete leading to consistently wrong answers.
There are clearly people who are better and worse comparatively at solving certain problems. But given the complexity of our brains/biology, there are myriad reasons for these differences.
Clearly there are people who have the capacity to understand certain problems more deeply (e.g. Einstein), but all of this was primarily to say that output doesn’t need to be 100% “reliable” to imply a complete understanding.
Indeed, but I wasn't talking about mistakes at all, but specifically about the case when "A understands X better than B does", which is about their mental model of X. I hope you won't dispute that 1) we do say things like that all the time about people, and 2) most of us understand what this means, and it's not just about making fewer mistakes.
Ah, thanks for the clarification; I think I misread you here:
> The weasel word here is "reliably". What does this actually mean? It obviously cannot be reliable in a sense of always giving the correct result, because this would make understanding something a strict binary
What did you mean by "it cannot be reliable in a sense of always giving the correct result", and why would that make understanding something a strict binary?
I do agree that some people understand some topics more deeply than others. I believe this to be true if for no other reason than watching my own understanding of certain topics grow over time. But what threw me off is that someone with "lesser" understanding isn't necessarily less "reliable". The degree of understanding may constrain the possibility space of the person, but I think that's something other than "reliability".
For example, someone who writes software using high level scripting languages can have a good enough understanding of the local context to reason about and produce code reliably. But that person may not understand in the same way that someone who built the language understands. And this is fine, because we're all working with abstractions on top of abstractions on top of abstractions. This does restrict the possibility space, e.g. the systems programmer/language designer can elaborate on lower levels of the abstraction, and some people can understand down to the bare metal/circuit level, and some people can understand down to the movement of atoms and signaling, but this doesn't make the JavaScript programmer less "reliable". It just means that their understanding will only take them so far, which primarily matters if they want to do something outside of the JavaScript domain.
To me, "reliability" is about consistency and accuracy within a problem space. And the ability to formulate novel conclusions about phenomena that emerge from that problem space that are consistent with the model the person has formed.
If we took all of this to the absurdist conclusion, we'd have to accept that none of us really understand anything at all. The smaller we go, and the more granular our world models, we still know nothing of primordial existence or what anything is.
Neither would I. But if someone who can translate a French sentence into English preserving the meaning most of the time, I think it would be reasonable to say that this person understands French. And that is what we're talking about here - models do produce correct output much of the time, even when you give them input that was not present in their training data. The "stochastic parrot" argument is precisely the claim that this still does not prove understanding. The "Chinese room" argument takes it one notch further and claims that even if the translation is 100% correct 100% of the time, that still doesn't prove understanding.
How many people years existed between the time people discovered multiplication and then exponentiation? Did it happen in wall clock time in a single generation?
> How many people years existed between the time people discovered multiplication and then exponentiation?
We don't know, because these concepts were discovered before writing. We do know that far larger jumps have been made by individual mathematicians who never made it past 33 years of age.
