I certainly agree that an intuitive understanding of a concept can be helpful as a guide and emotionally satisfying. However, I don't think that it is at all sufficient and it is precisely "mathematical gibberish" which resolves the problem.
An isolated concept is worthless. It is only when you are about to apply it by reasoning with it that becomes valuable. The problem with intuitive explanations is that they don't nail down enough details to allow a person to reason with them.
"I think of points in an n-dimensional space as objects holding n different types of information"
Imagine walking into a room and drawing a straight line on the floor. What is the dimension of that line?
Answer = One dimensional.
Proof: We can describe each point by one type of information. Point = (Distance of that point from the start of the line.)
Answer = Two dimensional.
Proof. We can describe each point by two types of information. Point = (Distance of that point from the East wall, Distance of the point from the North wall.)
Answer = Three dimensional.
Proof: We can describe each point by three types of information. Point = (Distance of that point from the East wall, Distance of the point from the North wall, Distance of that point from the roof.)
Answer = Four dimensional....
So by the intuitive explanation we can make this single line any dimension that we want.
This isn't just a problem within mathematics. For a simple programming example:
Question: How does a computer program work?
Intuitive Answer: You give the computer a list of instructions for it to carry out.
Result: The guy opens up notepad and types in "Make a computer game where I walk around shooting Zombies."
An isolated concept is worthless. It is only when you are about to apply it by reasoning with it that becomes valuable. The problem with intuitive explanations is that they don't nail down enough details to allow a person to reason with them.
"I think of points in an n-dimensional space as objects holding n different types of information"
Imagine walking into a room and drawing a straight line on the floor. What is the dimension of that line?
Answer = One dimensional. Proof: We can describe each point by one type of information. Point = (Distance of that point from the start of the line.)
Answer = Two dimensional. Proof. We can describe each point by two types of information. Point = (Distance of that point from the East wall, Distance of the point from the North wall.)
Answer = Three dimensional. Proof: We can describe each point by three types of information. Point = (Distance of that point from the East wall, Distance of the point from the North wall, Distance of that point from the roof.)
Answer = Four dimensional....
So by the intuitive explanation we can make this single line any dimension that we want.
This isn't just a problem within mathematics. For a simple programming example:
Question: How does a computer program work?
Intuitive Answer: You give the computer a list of instructions for it to carry out.
Result: The guy opens up notepad and types in "Make a computer game where I walk around shooting Zombies."