What does it mean, to actually understand?

[Will Graham]

There are two main topics I want to discuss in relation to 'understanding', the first is a much more abstract discussion. Effectively it' a mk II of Harlequin's conceptualisation thread(which can be found here). Basically, are we able to truly understand anything at all, when we don't know everything? Considering we have no way of viewing or examining the universe in a divorced manner(i.e. as in we cannot view it as if we were not part of it), then is understanding something in its entirety possible?

-------------------------------

Now for the second part of this discussion. We have to assume that understanding is possible, at least in the colloquial sense. As well as some other generalisations and assumptions. I got to thinking about the difference in how I approach exam questions in maths, versus how I approach questions in English. For maths I memorise a sequence of equations, formulae, and computations. I then apply these to relatively standard question formats in order to work out an answer.

In English however, when asked a question on say 'Discuss the themes of love and regret in Derek Walcott's poetry', I use an entirely different approach. I think about what the question means to me, I cross-reference this with what I felt when I read the poems, and what I thought the poet meant. Then I begin to formulate an answer that is unique to me, and what I understood from the poet's work.

Now this made me wonder, what is the difference between understanding something and simply being able to memorise a sequence of facts or operations? Is there even a difference, is understanding simply such a high degree of memorisation it comes across as independent thought and understanding? Or is understanding simply something unique to each person's own perception?

 

[der Astronom]

If you think that math is all about memorizing formulae and facts, then you don't understand math.

In order to truly understand math, you have to be able to understand the context under which the formulae are being used, and in order to understand a theorem or math concept completely, you need to be able to understand how to use it, and under which contexts they are appropriate. And it's not always as easy as it sounds. There are plenty of clever ways in which people can use a theorem or a math idea. That's why I've always abhorred memorizing math theorems and algorithms. It's absolutely useless. If you understand a math theorem or idea, then you can always rederive it. If you forgot it, and can't get it back again, then you probably don't understand it all that well.

If you really want to know if you understand math, you should consider upper level math classes. The majority of them are highly abstract, and in order to do well, you have to be able to understand the theorems to the point where you'll recognize when to use them when you are asked to solve a problem; no amount of memorization will help you then.

I guess this might answer your original question better than expected. It might just be because you might not have considered that math is more than simply memorizing stuff.

 

[Will Graham]

Ah you make some good points regarding the nature of maths, some of which I'll admit I forgot when writing the post, also there are some things I didn't communicate correctly.

In order to truly understand math, you have to be able to understand the context under which the formulae are being used, and in order to understand a theorem or math concept completely, you need to be able to understand how to use it, and under which contexts they are appropriate. And it's not always as easy as it sounds. There are plenty of clever ways in which people can use a theorem or a math idea. That's why I've always abhorred memorizing math theorems and algorithms. It's absolutely useless. If you understand a math theorem or idea, then you can always rederive it. If you forgot it, and can't get it back again, then you probably don't understand it all that well.

All of these things though I feel are the product of memorisation, knowing when to use a theorem, isn't that just recognising a previously seen pattern? Re-deriving the theorem even is simply applying a a set of standard computations, that you have had drilled into you from a young age. Now clearly this could also be down an understanding of maths that allows you to apply information how you wish, but what makes that understanding differnt from memorisation?

Isn't the foundation of that maths something you memorised a long time ago? A set of defined principles to be learned off. Therefore does something first require memorisation before understanding can come? Or is understanding derivative of a natural aptitude for the subject at hand?

I don't mean basic things, for example everyone learns to memorise the order of numbers at young age. When you learned things like maths during youth, practice was required(at least for me, and a lot of people I know). And then once we are capable of solving these exercises by using a set of rules, we built on that knowledge and thus gained a mastery of maths. Even then though are we not simply accessing a data-base(in our minds) and cross-referencing a never before seen problem with patterns we've seen before and stored in our heads?

Something like English(which is only an example) requires something else though I feel, a set of rules can not be crafted for all poetry. There are no real guidelines in understanding a poem, nor is there even a definite answer that can be arrived at.

I think this time I have explained my self more succinctly on this part of the topic, apologies if it still simply boils down to an ignorance of maths

 

[der Astronom]

All of these things though I feel are the product of memorisation, knowing when to use a theorem, isn't that just recognising a previously seen pattern? Re-deriving the theorem even is simply applying a a set of standard computations, that you have had drilled into you from a young age. Now clearly this could also be down an understanding of maths that allows you to apply information how you wish, but what makes that understanding differnt from memorisation?

Well, not necessarily. The only time where I think memorization adequately comes into play is when we are speaking of convention, which I think you mentioned somewhere near the end of that paragraph--that we have a convention for a base system, and what number constitutes to them. We are naturally more inclined to work in a decimal base system and have a symbol that represents all the numbers from 0 - 9 because we have a symbol that represents every digit on our fingers (although 0 is actually more a symbol that represents nothing; 10 is technically the last digit), but there's absolutely no reason why it has to be that way; we could use binary, octal, hexadecimal or whatever we want to. There's not even a particular reason why we have to use the arabic number system either. It's all just a convention. The same is true for commonly used operation symbols, and sets of symbols for various fields within math, whether it's set theory, number theory, algebra, or any other abstract or advanced field in math. There's absolutely no reason why adding something has to be represented by a plus sign (other than consistency); it could easily be a Greek symbol if people wanted it to be one. And that's the only things we have to remember. Everything else including ideas, theorems, etc. need not be memorized. We are simply being taught as if they are things that need to be memorized, when in fact that's not a necessary condition in understanding something in math.

For a better grasp of this, if you've been following along in the math thread that someone else posted up in the debate section recently, you'll see what I mean when I talked about Gauss. Gauss made lots of significant discoveries in his day, but didn't tell a lot of people about them, and didn't publish prolifically. As a result, other people often made the same discoveries he did, but later. These discoveries are not the result of memorization, but an understanding of a concept, or perhaps even putting several different concepts together. The remarkable thing about this is that you are doing this, regardless of whether or not other people have done the same thing or not; lots of people who made the same discoveries as Gauss weren't even aware that he long had the results already. In fact, you aren't even doing this from anything you remember as being true, in the sense that you aren't starting with the conclusion first. And this works because theorems are not true for the sake of it; they are true if and only if you can prove them. So you can remember a theorem wrong but get it right by finding an adequate proof for it; the only thing you had to remember was the convention everybody knows about and maybe the elementary concepts that lead to the proof. The actual putting together the proof part, however, I think may require a bit of thinking, creativeness and perhaps some cleverness. And because proofs are often lengthy and not always easy to follow, I just find it hard to believe that people who can obtain the proof to a theorem but can't remember what the theorem said originally relied on memory to get the proof.

Another way of looking at this is that there are many different ways of arriving at the same theorem. And the more ways in which you can obtain the theorem, the better you understand it. I don't think memorization can give rise to this kind of creativity.

Isn't the foundation of that maths something you memorised a long time ago? A set of defined principles to be learned off. Therefore does something first require memorisation before understanding can come? Or is understanding derivative of a natural aptitude for the subject at hand?

The only "memorization" I'm aware is the conventions I mentioned earlier, and perhaps basic logic. Because that's really how all math makes sense and how we are able to prove anything. But logic is actually quite a simple building block that they're really not all that hard to remember individually; it's like trying to remember what kinds of Lego blocks exist; it's not hard to remember what they look like, but it's harder to imagine how a complicated spaceship made of Lego blocks might be built unless you designed that model in the first place.

I don't mean basic things, for example everyone learns to memorise the order of numbers at young age. When you learned things like maths during youth, practice was required(at least for me, and a lot of people I know). And then once we are capable of solving these exercises by using a set of rules, we built on that knowledge and thus gained a mastery of maths. Even then though are we not simply accessing a data-base(in our minds) and cross-referencing a never before seen problem with patterns we've seen before and stored in our heads?

Well, that depends. Sometimes, you are using a theorem or math idea in the way you expect it to be used, but there are some problems which might involve you using it in a different way, one which you might not be familiar with. There's lots of things you can do and try with math; people often forget that you are not limited by what you can do to a math problem with any given theorems or math ideas you might happen to know. I know, because I sometimes force myself to solve the same problem a different way (or it just happens by accident), and sometimes, I've never actually done it like that before. If you're given a Lego set, you're not limited to building the exact Lego model that's provided for you in the instructions; you're free to mix and match other Lego blocks and build your own models if you feel like it. It's the same in math.

And if it were that easy, then any computer could solve all our math problems. They can't. They lack the creativity required to prove and use theorems, or find more efficient/interesting solutions to a problem. Blindly going through all the possible theorems you know just to solve a problem is memorization. Making educated guesses at which theorems you know are relevant requires you to know why you're trying them, and an adequate understanding of math.

Something like English(which is only an example) requires something else though I feel, a set of rules can not be crafted for all poetry. There are no real guidelines in understanding a poem, nor is there even a definite answer that can be arrived at.

So is there any way at all to tell if someone is just bullshitting their understanding of a particular poem though? Because people do get graded on this at school, and if there really were no guidelines, then it would be impossible to tell the difference between someone pretending they knew what a poem was talking about and someone simply having a different interpretation of the same poem.

Which brings me to my next point; you'd actually have to remember what a poem is saying or what happened in a story to even be able to talk about it. If you can't remember it, anything you say about it amounts to bullshitting. So I guess it sort of does have a building block at the beginning as well.

I think this time I have explained my self more succinctly on this part of the topic, apologies if it still simply boils down to an ignorance of maths

If you haven't seen this a lot, don't be too worried; most people don't get to see this side of math unless you've taken any upper level math classes in university. And personally, I think that part of math makes it more exciting than what most people think of it.

 

[Harlequin]

Basically, are we able to truly understand anything at all, when we don't know everything? Considering we have no way of viewing or examining the universe in a divorced manner(i.e. as in we cannot view it as if we were not part of it), then is understanding something in its entirety possible?

The understanding I feel is different to the knowledge of the understanding. For example, I could understand something correctly or incorrectly, but how would I know to distinguish between the two? Which one is the correct understanding?

I think the comparison Licky was making in regards to the precision of maths and the ambiguity of English is a good way of describing the various dynamics in which we choose to interpret and understand the world around us. It's not a comment so much on maths as a concept, but rather how it's applied when in the pursuit of evidence to better understand the world around us. If I ask someone to prove the sky is blue, they'll tell me about light refraction, if gravity exists, they'll tell me about mass, etc.

Similarly, social abstrations like Pride, Honour and Morality aren't subject to maths so are relegated to opinion rather than fact. Fact becomes suspended in the realm of mathematics. What we get as a consequence is a slight arrogance on our perceptions as a collective of the world and indeed our existence. We have no definative perspective from which to make an infallible assertion from, yet we choose to believe that we do merely because the numbers 'add up', as it were.

Sorry if it just sounds like waffling my mind is full of fuck at the moment.

 

[der Astronom]

Similarly, social abstrations like Pride, Honour and Morality aren't subject to maths so are relegated to opinion rather than fact. Fact becomes suspended in the realm of mathematics. What we get as a consequence is a slight arrogance on our perceptions as a collective of the world and indeed our existence. We have no definative perspective from which to make an infallible assertion from, yet we choose to believe that we do merely because the numbers 'add up', as it were.

Sorry if it just sounds like waffling my mind is full of fuck at the moment.

Actually, that's where science and math differ in that math doesn't necessarily have to be talking about something that's real; it just needs to be logically consistent. So while it may follow particular rules, they don't have to say anything about reality at all. They just happen to agree with reality occasionally. I don't think the concept of imaginary numbers, for example, was born out of a math that was based on reality. If it had any use, it's because physicists happened to find a use for it. But we can talk about them anyways, regardless of if anyone ever finds a use for imaginary numbers in reality, or even if infinity exists at all; it wouldn't matter in the slightest if it did or didn't exist.

I'm not entirely sure what you're getting at there though, but science is not born out of arrogance; it is born out of observations of the things that we can sense and are tangible. The question of whether or not what we actually see exists is not relevant to science, and that we even know anything in science is always subject to error. Yet we call it empirical because it is based off of things that are real, and they operate independently of any one person's individual perceptions (that is, we're less likely to discover something that's the result of one person's hallucinations because then that would mean everyone else would also have to be hallucinating about the same thing as well). We are always improving and expanding on our knowledge of science, and it is the closest thing we have to understanding what is real and what isn't. It's not a complete knowledge set, which is also why it's different from math in that it's inductive, and not deductive (math). If we believed that we were right about everything discovered from science without taking falsifiability, peer review and induction into consideration, then maybe that might be considered arrogant, but that's not the way science works. Science just happens to be the best method in which unreliable minds like ours are able to grasp at reality.

 

[coffeecup]

When it comes to ideas that are given man-made explanations, these explanations never do justice for the underlying idea. To understand is to essentially be able to get around the language barrier (be it mathematics or English itself) to tap into the true nature of what is trying to be represented. Words are just words. They're used to convey something by the speaker. And since this is the only vehicle of communication, a lot can be lost in translation when it comes to conveying even a simple idea. This is generally why I dislike the debating style of dissecting each post sentence by sentence. In order to understand what the member is trying to say you need to be able to go behind his arguments to the essence of his message. Who cares if he gets a logical argument here or there wrong (unless the argument itself is what is being conveyed), what matters is his actual expression of an idea. That is the important focus, not the frills.

Naturally, understanding begins with reading what was said or the mathematical notations that were used. But you never want to stop there because words are not ideas. Formulas aren't either. Unless what is being expressed are the words or formulas themselves (which is sadly possible in our defunct education system), you're missing the entire message.

This is the problem with modern education relying so heavily on test scores to gauge a student's understanding of a subject. Normally if students properly learn the material, a test may be able to tell (statistically) whether certain students are grasping the subject better than others. But today the test is no longer a metric but the means. You always hear "I need to study for an exam." It's no longer reliable because understanding takes a back seat to the exam itself. And to make it worse, many teachers tend to produce standardized exams that follow the questions of the book. This makes studying for an exam even more effective and more debilitating towards the entire purpose of education. Studying for an exam and acing it does not necessarily mean that you championed the subject matter anymore.

You'll see what I'm saying once you get proper teachers who are able to formulate questions that rely very little on the superficial aspects of the subject. Many of my exams in upper division mathematics are open book, open notes, and it's basically a 4 hour exam with 5 or 6 questions. Yet the mean exam score always hovers around 60% (fyi this refers to the actual exam grade, the course GPA is naturally curved). And this is at a reputable state university, so it's not as if half of us are total morons. You see the professor is able to readily test the underlying idea. He doesn't care if you bring in a book or some other piece of material that gives a static and rigid definition of the concept. He doesn't care if you have examples of questions displaying a certain approach to a problem. An able professor will always be able to get around all that flimsy wrapping paper and test the core of the subject.

 

[der Astronom]

You'll see what I'm saying once you get proper teachers who are able to formulate questions that rely very little on the superficial aspects of the subject. Many of my exams in upper division mathematics are open book, open notes, and it's basically a 4 hour exam with 5 or 6 questions. Yet the mean exam score always hovers around 60% (fyi this refers to the actual exam grade, the course GPA is naturally curved). And this is at a reputable state university, so it's not as if half of us are total morons. You see the professor is able to readily test the underlying idea. He doesn't care if you bring in a book or some other piece of material that gives a static and rigid definition of the concept. He doesn't care if you have examples of questions displaying a certain approach to a problem. An able professor will always be able to get around all that flimsy wrapping paper and test the core of the subject.

This is actually the typical approach to many upper division math classes, and basically what I've been talking about all along about math problems at that level. As soon as you see problems of a nature which involve "prove this" or "show that", no amount of textbooks you have in front of you are going to help unless you understand what you're supposed to do with them. Now I've actually had math classes like yours, except they were never open book, which probably made it harder to tell that simply remembering the theorem won't save you on an exam. And unfortunately, I think this is the kind of thing very few people are exposed to because even people working in more scientific fields rarely ever bother with math classes at this level because they don't need it. If every other math class were taught this way, and you would be asked to show or prove something on a test, math would be considerably harder. It would be more interesting in my opinion, but I'm afraid there probably aren't nearly enough teachers devoted enough to understand the math at this level in order to teach it or encourage students to develop this kind of understanding and thinking.

I took a similar math class where the test was open book. However, it was that way because most of the algorithms that you had to learn about where very lengthy and extremely difficult to remember. I think the whole point of that course was to see if you understood how to use the algorithms; not whether or not you have the memory of a supercomputer.

 

[M1ghty Mous3]

To understand ANYTHING, you need to know it's past, future, and how it is in the present. To understand a person, for example, you must know EVERYTHING about them. heir medical history, how they think, why they thing the way they do, how they apply their thoughts to reality, ect.

Literally, to understand anything, you must KNOW it completely. This means having no doubts at all and being sure your thoughts are 100% correct on the subject.

No one knows anything.

The way it will be and the way it has been.

 

[Harlequin]

Actually, that's where science and math differ in that math doesn't necessarily have to be talking about something that's real; it just needs to be logically consistent. So while it may follow particular rules, they don't have to say anything about reality at all.

You've misunderstood me. Wasn't saying maths is restricted to reality, I was saying that what we define us 'fact' is subject to maths. It's not a comment on maths, it's a comment on how we perceive fact.

I'm not entirely sure what you're getting at there though, but science is not born out of arrogance; it is born out of observations of the things that we can sense and are tangible.

I didn't say anything about Science. I was talking about people's tendency to be lazy when it comes to thought. You tell someone something - they're skeptical, you give them numbers and they believe absolutely anything you tell them. These days it's even enough to say a couple of scientists said it and people will believe it (not a comment on scientists but on people in general, I don't have a problem with Science).

 

[moogling]

I didn't say anything about Science. I was talking about people's tendency to be lazy when it comes to thought. You tell someone something - they're skeptical, you give them numbers and they believe absolutely anything you tell them. These days it's even enough to say a couple of scientists said it and people will believe it (not a comment on scientists but on people in general, I don't have a problem with Science).

So you're suggesting that science in itself has become a faith (albeit to those who perhaps are too "thick" to understand complex calculations and formulae aka quite a lot of people)? They don't understand so they just accept it much like religious people perhaps. However science can provide evidence, it can show us its calculations, it can explain things in terms we understand (or attempt to at least). The difference being that so far the entire world is yet to hear from god, allah or any of the other deities.

Granted this isn't a debate about science vs religion but I think it's related to some of your previous posts in similar threads about how we know what we know - or rather that we CAN'T know what we know. It's all very abstract tbh and while that's essentially what this debate is about I don't think its helpful at all to discuss things this way. It's more helpful to develop an understanding of a particular subject than an understanding of understanding...surely?

 

[Harlequin]

So you're suggesting that science in itself has become a faith (albeit to those who perhaps are too "thick" to understand complex calculations and formulae aka quite a lot of people)? They don't understand so they just accept it much like religious people perhaps. However science can provide evidence, it can show us its calculations, it can explain things in terms we understand (or attempt to at least). The difference being that so far the entire world is yet to hear from god, allah or any of the other deities.

Granted this isn't a debate about science vs religion but I think it's related to some of your previous posts in similar threads about how we know what we know - or rather that we CAN'T know what we know. It's all very abstract tbh and while that's essentially what this debate is about I don't think its helpful at all to discuss things this way. It's more helpful to develop an understanding of a particular subject than an understanding of understanding...surely?

I didn't suggest it was science vs religion, I believe in many scientific theories. But the problem is how many seem to think science in limitless. It's not, it's limited to the natural world.

I'm suggesting that science has always, is and will always be a faith. That everything is faith based, seeing as we have no means from which to removed ourselves form 'reality' (as it were) to assess everything in it's entirety.

The understanding of understanding is the building blocks if you like. One thing is understanding something, another is understanding why you understand it, how you've come to understand it. People these days are too transfixed on the former they ignore the latter completely.

Whether or not we have the ability to understand is a matter I would go on to, but I'm tired and my brain needs rest from debate

 

[Sum1sgruj]

I think there is a supreme difference between 'understanding' and 'knowing'.

For example, it is easy to understand the nature of something after confirmation by mathematics and observation. To actually know something, however, is only based on it's understanding.

In other words, you can understand anything, but cannot know anything.

 

[Rydrum2112]

I think there is a supreme difference between 'understanding' and 'knowing'.

For example, it is easy to understand the nature of something after confirmation by mathematics and observation. To actually know something, however, is only based on it's understanding.

In other words, you can understand anything, but cannot know anything.

What?

I know my name. I know where I live. I know how to drive. That statement makes no sense.

I understand and know how statistics work.

 

[Donald Trump]

In other words, you can understand anything, but cannot know anything.

I entirely disagree with that.

Knowing is to be aware of facts, or truths. Understanding is to attach a meaning to something. And understanding can be subjective. Literature and History are entirely subjective. You have the things you know; the author, the genre, the time, the text etc etc, and you use them to develop an understanding. And my understand may be different from yours, and both of our understandings different again from the understanding of the author.
To kind of answer Finn's question you cannot have one 'true' understanding of something that requires interpretation. The validity of one's point of view comes from the ability to support it with what we know/evidence. But different views can be equally valid.

 

[Sum1sgruj]

I entirely disagree with that.

Knowing is to be aware of facts, or truths. Understanding is to attach a meaning to something. And understanding can be subjective. Literature and History are entirely subjective. You have the things you know; the author, the genre, the time, the text etc etc, and you use them to develop an understanding. And my understand may be different from yours, and both of our understandings different again from the understanding of the author.
To kind of answer Finn's question you cannot have one 'true' understanding of something that requires interpretation. The validity of one's point of view comes from the ability to support it with what we know/evidence. But different views can be equally valid.

You can assume what you observe by understanding it's properties.

The best fundamental example would be an elementary particle, the building block of everything you observe. You can understand their properties and give them a name, such as a quark or photon, but you cannot know what they are.

Do you know if you existed five minutes ago? You can only assume, despite evidence and memory.
Is the sky blue when you are not looking at it? Only one can assume.

'Knowing' is really just a word for convenience, if you think about it.

 

[Donald Trump]

Do you know if you existed five minutes ago? You can only assume, despite evidence and memory.
Is the sky blue when you are not looking at it? Only one can assume.

Ah yeds, Solipsism. In which case I'd be discussing this with myself.
Solipsism doesn't prove anything. It can neither be proved nor disproved.

The best fundamental example would be an elementary particle, the building block of everything you observe. You can understand their properties and give them a name, such as a quark or photon, but you cannot know what they are.

You can know what their properties, but not understand. You can test what you know, but that isn't the same as understanding. To understand is to know its meaning and its significance.

 

[Sum1sgruj]

Ah yeds, Solipsism. In which case I'd be discussing this with myself.
Solipsism doesn't prove anything. It can neither be proved nor disproved.


You can know what their properties, but not understand. You can test what you know, but that isn't the same as understanding. To understand is to know its meaning and its significance.

Never was a fan of Solipsism either, because it is practically meaningless

It's not really important to 'know' anything. I was just pointing out the semantic value of 'knowing'

 

[der Astronom]

I didn't suggest it was science vs religion, I believe in many scientific theories. But the problem is how many seem to think science in limitless. It's not, it's limited to the natural world.

I'm suggesting that science has always, is and will always be a faith. That everything is faith based, seeing as we have no means from which to removed ourselves form 'reality' (as it were) to assess everything in it's entirety.

Well, the problem is that there's no reason to assume that anything I see and can confirm with someone else who has a different perception of reality from me does not actually exist as I see it, or as anyone else can see it universally. Maybe "red" looks different to me than it does to you, but no one can deny that it can be described by the frequency of its light waves. Maybe in a different reality, it has a different frequency, but there's no reason to assume it does, quite simply because you have no evidence to support such an assertion. Likewise with anything else in reality; maybe there's an alternate version of reality we have no access to out there, but there's no reason to assume it exists. Assuming that the reality that we know it is different from what it appears to be would be faith. I don't claim to know what reality actually looks like either. I just have no reason not to believe that the reality as I see it in front of me doesn't exist, or is really something else. And if it is, then I'll accept it if it can be demonstrated. But until then, I'm just going to hold a default position that what I see exists. It's not any more unreasonable than not believing in god or fairies at the bottom of the garden because it's based on the exact same logic. And if you don't agree, think of it this way. You'd be telling everyone who doesn't believe in Santa Claus that they're not believing out of faith (and think about all the gods you don't believe in). And that's logically impossible because that would be like having faith in a negative. I hope you're not confusing people who believe in a reality just because and people who accept a temporary model of reality because we don't have any evidence that that reality is different from what it appears to be.