Can Intuition be Used in Mathematics to Overcome Limit Laws and Norms?

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A inquiry into the inuition of mathematics, in the sense that a mind-map of mathematics is not a well structured machine with each part serving a purpose, but more as in each function and transform has it's own pitfalls that encourage a need for rigour.
Hi,

I have been studying mathematics for quite some time now, and I have an understand of what each topic constituted up unti and including graduate level mathematics. However my computations in each topic are very instructive, as if each topic carries its own instructions for calculations under the rules of that venue or theoreom.
It is a common occurrence to hear mathematicians talk about pitfalls and incongurences in what I can only interpret as limit laws and norms.
Some example questions to get us on the same page is:
  1. How does a norm of an N dimensional space have it's own weaknesses when comparing it's limit laws? That is to say, what are the limit laws being compared to when considering the accuracy of the solution to the theoreom you are using.
  2. How does the choice of a differential form of a tensor/multi-linear function for a morphism have a weak, dominant or strong topology?
My doubts come from many experts qualitatively explaining rigour as if there are carrying out the instructions of each thereom in contrast to some universal data set, that gauges the quality of the geometry, algorithm or otherwise.

I am looking for what this 'data set' is ideally, but in the real world it is likely there is some form of control between extrema that I am over-looking, this is what I am asking to be explained.
 
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I feel that your survey doesn't really reflect how people learn math. Of course, we all have some intuitive ideas that crop up as we are instructed in it as a kind eureka moment. I remember something like that happening when I learn Vector Analysis and many courses of math came together to make Vector Analysis a truly useful math tool.

Other areas of math like Abstract Topology left me totally confused as it seemed to be more definitions heaped on definitions and I couldn't grasp it visually as I had done with geometric concepts.

I really enjoy Grant Sanderson's math videos on Youtube under the channel name 3blue1brown. They can take many obscure/mysterious problems and frame them into a visual feast for the mind.
 
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Tucker121 said:
My doubts come from many experts qualitatively explaining rigour as if there are carrying out the instructions of each thereom in contrast to some universal data set, that gauges the quality of the geometry, algorithm or otherwise.

I am looking for what this 'data set' is ideally,

You are correct that standards for detail differ in various branches of mathematics. They also have varied over the history of mathematics. In a given branch of mathematics, experts adopt a particular standard of rigor that allows them to settle differences of opinons, avoid wrong deductions, and resolve apparent paradoxes. For example, in the history of calculus, the early development was less rigorous than the current treatment. When differences of opinions about calculus arose that could not be settled from a contemporary standard of rigor, people had to invent more precise definitions and get involved in more detail.

If you experience the usual academic course of education, you are taught the elementary properties of real numbers in secondary school,. In college, you take a course of calculus that assumes you know these properties and does not expect you to prove them. Much later, in a graduate course on analysis, there may a be a chapter in the text that rigorously develops the properties of the real number. If you specialize in "foundations" of mathematics, you may study the development of the real number system in greater detail.

By contrast, if you are self-taught then it can be jarring to read several mathematical textbooks at one time because you do not encounter the material in the order that it is taught in academia. In an academic course, you have time to adjust to the standard of rigor expected in that course.
 
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Aha, yes thanks.
Intuition appears to be more of a working memory based on experience, whereas intelligence is a working memory based on knowledge.
I have looked into real numbers and calculus and how topology is forms for mean 0 variance 1 and onwards.
There are also other quirks like eccentricity which is the accuratey of othogonality etc.
I hope the survey wasn't too disappointing.