How to FULLY understand a definition?

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My differential geomtry professor once told me that with all his 30+ years experience in research in differential geometry he has never actually fully grasped what a tensor truly is.

I believe this is the case for many of us. You can memorize a definition, read many examples of the definition, and then even solve many problems successfully using the definition, and then think you fully understand the definition, but in fact you still may not fully understand what the definition really is. You may think you know what it is, but likely you understand what it really is less than you think.

Going back to my differential geometry professor, I try to give a description of what a tensor is without using mathematical terminology and I cannot, because I cannot feel what a tensor is either. Repeating the definition doesn't really help, and solving more problems won't help either, because I still cannot feel it. I find that solving problems using definitions helps you to improve your solving ability using the definitions, but only helps you to understand the definition a bit more, but not fully.

How do you know when you fully, fully understand a definition? Because solving problems correctly is not enough.
 
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  • #2
Gib Z
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When you can explain it to your grandmother.

Personally that means I don't know anything, she died a long time ago.

I still to this date do not fully grasp The fundamental theorem of calculus. I can repeat it, prove it, but do not know HOW it works. Seems so unexpected the derivative and area are related like that.
 
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  • #3
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Even if you read the entire proof of how something works and even are able to repeat the proof, it's still not always clear what what it really is that you just proved. When I read the proofs of the equivalence of the axiom of choice, well-ordering theorem, and Zorn's lemma, I still could not feel how they were equivalent.
 
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  • #4
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"In mathematics we don't understand things, we just get used to them"

--Von Neumann
 
  • #5
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When you can explain it to your grandmother.

Personally that means I don't know anything, she died a long time ago.

I still to this date do not fully grasp The fundamental theorem of calculus. I can repeat it, prove it, but do not know HOW it works. Seems so unexpected the derivative and area are related like that.
Doesn't the FTC just say that you can approximate areas with rectangles?
 
  • #6
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My differential geomtry professor once told me that with all his 30+ years experience in research in differential geometry he has never actually fully grasped what a tensor truly is.

I believe this is the case for many of us. You can memorize a definition, read many examples of the definition, and then even solve many problems successfully using the definition, and then think you fully understand the definition, but in fact you still may not fully understand what the definition really is. You may think you know what it is, but likely you understand what it really is less than you think.

Going back to my differential geometry professor, I try to give a description of what a tensor is without using mathematical terminology and I cannot, because I cannot feel what a tensor is either. Repeating the definition doesn't really help, and solving more problems won't help either, because I still cannot feel it. I find that solving problems using definitions helps you to improve your solving ability using the definitions, but only helps you to understand the definition a bit more, but not fully.

How do you know when you fully, fully understand a definition? Because solving problems correctly is not enough.
I've always seen Mathematics as the study of mathematical relations among different abstract ideas (i know the defenition is a bit recursive). Most of the time i find it difficult to understand those abstract ideas but get familiazed with the relations among them faster. However, knowing the history of the development of a certain method or definition helps me greatly to grasp more the idea. And sometimes just going along and giving it some time does the job. But maybe Von Neumann said it better.
 
  • #7
Gib Z
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Doesn't the FTC just say that you can approximate areas with rectangles?
>.<" I talking about the version where it says [tex]\int^b_a f(x) dx = F(b) - F(a)[/tex] where F'(x)=f(x)
 
  • #8
arildno
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Doesn't the FTC just say that you can approximate areas with rectangles?
Eeh, no.
What you are talking about, is Riemann integrability, namely that however we choose a sequence of partitions into rectangles and sum together the areas of these rectangles, and let the partition number go to infinity, we'll end up at the same dumb number, which therefore can be assigned the lofty title of being the "area" beneath the curve we're working with.
 
  • #9
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If we assume that the area under a curve is a rectangle in the infintesimal, than we get that dA=fdx. Dividing through by the dx gives us the result that A'(x)=f(x). Of course this is highly unrigorous, but this is how I understand the relationship between area and derivatives.

Arildno is somewhat correct. The FTC is very easy if we accept Riemann's definition of integrals, and then the deep theorem becomes that Riemann integrals actually exist and give sensible answers.
 
  • #10
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The FTC is hardly abstract at all. But can anyone here really, really feel what the covariant derivative, exterior derivative, and darboux derivative are? I'm sure you can state the definitions and solve exercises using them. But even after proving their derivative properties I still could not grasp in my heart what they are really about. Their geometric pictures only tell us superficially what they are.
 
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  • #11
Gib Z
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The FTC is hardly abstract at all.
Are you saying you can intuitively feel why the primitive function gives the area under the curve?
 
  • #12
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andytoh: You may think you know what it is, but likely you understand what it really is less than you think.

This is really a very difficult thing. Now when I taught Beginning Algebra, we always start with, what are called Laws: Commutative, Associative, Distributative. (Sometimes called properties, rules, etc. I guess it is not exactly clear what to call them.)

If you ask the class, "Does anyone have an questions about this?" It is almost certain that everyone understands, thinks its simple, or finds it boring and wants to look at something else. Yet, once the problems start, then specific questions arise, and for a long time to come. Yet, from the teacher's standpoint it does seem that the "Rules" fully explain what to do.

I had a physic's teacher, who after stating Newton's Three Laws of Motion, added, "Well, now that that is explained, no one will have any questions all semester!" (He sounded very sarcastic about this.)

There is that problem with Special Relativity where the light beams, thunder bolts Einstein says, comes at the front and back of the train, and to the trackside observer, the beams proceed at C+V through the front of the train and C-V from the back of the train, or so the equations get written like that. Einstein has no problem with this, he tells us, obviously, that the observer in the middle of the tracks sees one thing when the beams meet, but on the train, why THE MIDDLE OF THE TRAIN WILL HAVE MOVED OFF from the central stationery track position. So it is not in the same position as the track observer when the beams finally meet. Then I wonder, "What did I miss here?" The speed of light is always observed as a constant; it took a long time to get somewhere with that one!

I had a logic instructor who said the best kind of proof was so well set up that a computer could grind out all the theorems. YES! At that point ALL THE ASSUMPTIONS MUST HAVE BEEEN WRITTEN INTO THE COMPUTER.

But, the question that comes up in my mind is, are there assumptions we are really learning to use that are not written down in the rules, axioms, properties, etc? Because, why, why if everything is right there in the "Rules," why, why is it not immediately clear what the consequences are?
 
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  • #13
mathwonk
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as to FULLY understanding, do you remember the shot of John Belushi at the back of the reverend theopus church in the blues brothers? When he saw the light? thats it. thats understanding. "The BAND, the BAND!"

(this is for you, complex philosophy.)
 
  • #14
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Sigh. It happens almost everyday. Just now, I finished the easy exercise of proving that a submersion is an open map. But to be honest, I am not one step closer to understanding what a submersion truly, truly is.
 
  • #15
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Because, why, why if everything is right there in the "Rules," why, why is it not immediately clear what the consequences are?
I don't know why, but my lack of imagination is not proof that we are using some rules implicitly. Consider cellular automata, they operate according to dead simple rules and yet I have not met a human for whom it is immediately clear what the consequences are.

However, I do believe it is likely that all the math we do will be redone more rigorously by future generations, just as Peano revised Euclid. As a side note, I always find it amusing when philosophers wax on about the perfection of Euclid's geometry, when in fact Euclid used an abundance of implicit assumptions.
 
  • #16
arildno
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If we assume that the area under a curve is a rectangle in the infintesimal, than we get that dA=fdx. Dividing through by the dx gives us the result that A'(x)=f(x). Of course this is highly unrigorous, but this is how I understand the relationship between area and derivatives.

Arildno is somewhat correct. The FTC is very easy if we accept Riemann's definition of integrals, and then the deep theorem becomes that Riemann integrals actually exist and give sensible answers.
I am not just SOMEWHAT correct, I am totally correct, you are DEAD WRONG.
That you are unable to distinguish between the condition of integrability, and the condition for when the derivative of the area function equals the integrand, does not mean they are the same condition.
 
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  • #17
Gib Z
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Statement made by adrildno=True.

And you still haven't answered me andytoh. Can you intuitively feel FTC? If so, you're a better mathematician than Newton and Liebniz were.
 
  • #18
arildno
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But, the question that comes up in my mind is, are there assumptions we are really learning to use that are not written down in the rules, axioms, properties, etc? Because, why, why if everything is right there in the "Rules," why, why is it not immediately clear what the consequences are?
Why should these consequences be immediately clear to us?
If, for example, you actually don't formulate a particular question, the "answer" to it will be as meaningless to us as the number 42 is. And, even more poignant, that answer won't pop into your head unless you pose the question in the first place.

Maths evolves not just by answering old questions, but also by posing new questions.
 
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  • #19
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Can you intuitively feel FTC? If so, you're a better mathematician than Newton and Liebniz were.
Read the proof of FTC. It's just a bunch of rectangles and I don't think anyone feels obscure about the area of rectangles, and neither did the people from the past.

If you want to look at geometry more mysteriously, can someone tell me how is differential geometry the study of a connection on a principle bundle?
 
  • #20
Read the proof of FTC. It's just a bunch of rectangles and I don't think anyone feels obscure about the area of rectangles, and neither did the people from the past.
neither do i feel obscure about the area of the rectangles. what i feel obscure about is how area and derivative can be related. that is, how
[itex]\int^b_a f(x) dx = F(b) - F(a)[/itex] where [itex]F'(x) = f(x)[/itex]

i understand the proof but i still can't "feel" it.
 
  • #21
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When Zermelo shocked the world by proving that every uncountable set can be well-ordered, no mathematician could "feel" the correctness of the proof, and many rejected the axiom of choice as a result. A hundred years later, still no one has been able to find a well-ordering of the real numbers, so I doubt anyone can "feel" it to this day.
 
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  • #22
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neither do i feel obscure about the area of the rectangles. what i feel obscure about is how area and derivative can be related. that is, how
[itex]\int^b_a f(x) dx = F(b) - F(a)[/itex] where [itex]F'(x) = f(x)[/itex]

i understand the proof but i still can't "feel" it.
Feeling? It has been a very long time since mathematicians do not rely on "feeling" anymore.
 
  • #23
arildno
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Feeling? It has been a very long time since mathematicians do not rely on "feeling" anymore.
Of course mathematicians rely on their feelings of what is true!
In fact, the ability to "feel" what is true, or false, is probably a distinguishing mark between a good mathematician and a mediocre calculationist:

It is due to their feelings that mathematicians are able to formulate conjectures in the first place.
After that initial stage, they must use logical analysis to either prove or disprove the conjecture.
 
  • #24
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andytoh: When Zermelo shocked the world by proving that every uncountable set can be well-ordered, no mathematician could "feel" the correctness of the proof, and many rejected the axiom of choice as a result.

Proved what? The Axiom of Choice is equivalent to the well-ordering of every set. So the proof is merely the assumption.
 
  • #25
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andytoh: When Zermelo shocked the world by proving that every uncountable set can be well-ordered, no mathematician could "feel" the correctness of the proof, and many rejected the axiom of choice as a result.

Proved what? The Axiom of Choice is equivalent to the well-ordering of every set. So the proof is merely the assumption.
When he proved the Well-Ordering Theorem was equivalent to the Axiom of Choice.
 

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