Wanting to understand too much

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Hi, I'm curious about something : Does anyone doing mathematics really understand at 100% in a logical way everything he does in math, or must he take it for granted some things, and that he only needs to do what he has learned without completly understanding these foundations ?( why a thing is the way it is, etc.)

I'm asking this question because it seems that some things that I supposely know, well I can't explain why they are the way they are.
 

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  • #2
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In principle, if you'redoing mathematics, then you should be able to explain everything. Of course, when you're new to learning mathematics, then this is not possible. You will have to take some things for granted. But you should really try to understand these things once.

Maybe you can give examples what is troubling you. What do you think you don't understand well?
 
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Well, a lot of things, not all of them, but it seems that my knowledge is built on shaky foundations.For example, an easy thing like : why a minus times a minus gives +.I'm not sure I could explain you why it's that way, I just know it is. But on the contrary, I could explain you why 2+2=4 because I know what is really happening. I would really like to understand the underlying mecanisms of math.
 
  • #4
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Well, a lot of things, not all of them, but it seems that my knowledge is built on shaky foundations.For example, an easy thing like : why a minus times a minus gives +.I'm not sure I could explain you why it's that way, I just know it is. But on the contrary, I could explain you why 2+2=4 because I know what is really happening. I would really like to understand the underlying mecanisms of math.
What I hate is that High Schools is that high schools teach the math without justification. They just tell you to memorize things like "minus and a minus gives a plus". It really isn't difficult to justify the rule, but somehow they neglect to do it.

I'm not sure what your level of math is. But I recommend picking up "basic mathematics" by Lang. It discusses things like this. Spivak's calculus proves the basic properties of numbers from scratch, but that might be a difficult reference.
 
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What I hate is that High Schools is that high schools teach the math without justification. They just tell you to memorize things like "minus and a minus gives a plus". It really isn't difficult to justify the rule, but somehow they neglect to do it.

I'm not sure what your level of math is. But I recommend picking up "basic mathematics" by Lang. It discusses things like this. Spivak's calculus proves the basic properties of numbers from scratch, but that might be a difficult reference.
That's exactly what I think ! Basically, my math education was remember, don't try to understand. They teach more math like a tool that must be not understood(well, not completelly), but used for other purposes than in way to really understand the mecanisms.
 
  • #6
jbunniii
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It can be counterproductive to spend too much time on foundations at the start, even though logically everything else depends upon the foundations. Far better in my opinion to state the axioms for the system you are working with, and proceed from there.

For example, when learning calculus or real analysis, one can start by listing the axioms of the real number system (a complete, ordered field), mention without proof that there is only one system up to isomorphism which satisfies these axioms, and then proceed with calculus/analysis.

At some point you will be curious enough to want to go back and see how to formally construct the real numbers from the rationals. There are several equivalent ways, and you should eventually know what they are, but once you understand the key ideas in the construction, there is a boring mass of details to verify. It would be horribly demotivating to sit through this before being able to proceed with the fun stuff. This is why Spivak puts the construction into an appendix in his calculus book.
 
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symbolipoint
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What I hate is that High Schools is that high schools teach the math without justification. They just tell you to memorize things like "minus and a minus gives a plus". It really isn't difficult to justify the rule, but somehow they neglect to do it.

.

That must be a huge change from the past (or as so I remember). Almost everything in beginning Algebra was fit to a justification or explanation. Nobody teaches by showing a pattern of symbolic results or refers to a number line anymore?
 
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Attached is an excerpt from the Lang book micromass mentioned, that explains why (-a)(-b)=ab.
 

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  • #9
reenmachine
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Very interesting thread.Due to my situation , I have to learn highschool very fast in the moment yet I always take the time before each exams to "play" with the concepts they are teaching me.I do it because I think it's the best way to digest the concepts long-term instead of just remembering it until the exam is done.

Let's take your exemple of -2 × -2 = 4.I will try to explain it naively.

Suppose you like blackjack and you like to play 2$ a hand.One day you lose all your 2$ hands very quickly and lose all your money in the process.You really want to play a little bit more so you ask your friend for 10$ to play five extra hands.He accepts to lend you the money , you play a couple of hands and you lose that extra 10$.

The day after , you go see your friend with 4$ and you give it to him.You repay 4$ out of a 10$ debt , so it's really -2 × -2$ = 4$.That's because you were already at -10 (-2-2-2-2-2) , so if you pay him 4 , you take out (-2-2) from (-2-2-2-2-2) and are now at -6 (-2-2-2) , which means you are +4 in your debt situation (-10 + 4 = -6).
 
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Hi, I'm curious about something : Does anyone doing mathematics really understand at 100% in a logical way everything he does in math, or must he take it for granted some things, and that he only needs to do what he has learned without completly understanding these foundations ?( why a thing is the way it is, etc.)

I'm asking this question because it seems that some things that I supposely know, well I can't explain why they are the way they are.
Nobody understands everything. There are many seemingly innocent statements that can be very difficult to prove. There are also many things in mathematics that have yet to be formalized all the way down to set theory axioms. There are varying levels of rigor, and the field of mathematics is so broad that no one person could possibly know everything, and nobody has the time or energy to reduce everything to fundamental axioms. What you should do is make every effort to understand something to whatever level of rigor is necessary and acceptable for what you are trying to achieve.
 

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