Actual mathematics teaching, as seen in typical University classes typically introduces a new abstract definition, and then immediately follows that up with a number of "worked examples", either in the lecture itself or a follow-up tutorial class.
However, the reference material will typically only have the abstract definitions, but at least there will be a sequence of them building up the formalities piece by piece. In principle, a gifted mathematician could follow the trail and get to the "aha!" moment without even having to do the homework problems.
Encyclopedias like Wikipedia or Mathworld tend to just mention the most abstract, "end product" of a long chain of mathematics. Terse formulas by themselves are utterly useless to anyone who hasn't gone through the step-by-step learning process and already understands the topic completely.
It's like a historian being given a handful of Egyptian hieroglyphic characters before finding the Rosetta Stone. It's just squiggles and shapes!
An example I came across personally was the simplification of Maxwell's Equations using Geometric Algebra. In GA, it is possible to write the equations of the EM fields incredibly tersely, down to as little as:
□²A = J
This is total gibberish to anyone who doesn't already know volumes about Geometric Algebra. It literally states that: Square squared letter A is letter J. What!?
The 5 characters in that equation pack in an awful lot: The Geometric Product. Graded vector spaces. The four dimensional spacetime of special relativity. Vector calculus. Natural physical units. Etc...
Pasting that equation into an article can't possibly cover all of that...
I vaguely recall a presentation from Leslie Lamport on TLA+/formal modelling. I might get some details wrong here, but a member of the audience asked 'why not describe a system by examples' (implying it's easier to see some example interactions to start with), and Leslie's reply was it's not rigorous precise enough (not to mention he was visibly angry at the question). I agree it's not precise enough, but for someone starting, an example is good enough rather than choking on rigor. Sure, rigor does matter, but allow me some toehold first!
Just like 'no taxation without representation', there should be 'no formalization without insight'.
Out of all resources I have found Wikipedia to be almost uniquely useless for teaching yourself a new concept, its like "explanation by committee". On the other hand if I can understand a Wikipedia article I know I am at least intermediate in a area.
Comments
This is a disease of Mathematics in general.
Actual mathematics teaching, as seen in typical University classes typically introduces a new abstract definition, and then immediately follows that up with a number of "worked examples", either in the lecture itself or a follow-up tutorial class.
However, the reference material will typically only have the abstract definitions, but at least there will be a sequence of them building up the formalities piece by piece. In principle, a gifted mathematician could follow the trail and get to the "aha!" moment without even having to do the homework problems.
Encyclopedias like Wikipedia or Mathworld tend to just mention the most abstract, "end product" of a long chain of mathematics. Terse formulas by themselves are utterly useless to anyone who hasn't gone through the step-by-step learning process and already understands the topic completely.
It's like a historian being given a handful of Egyptian hieroglyphic characters before finding the Rosetta Stone. It's just squiggles and shapes!
An example I came across personally was the simplification of Maxwell's Equations using Geometric Algebra. In GA, it is possible to write the equations of the EM fields incredibly tersely, down to as little as:
This is total gibberish to anyone who doesn't already know volumes about Geometric Algebra. It literally states that: Square squared letter A is letter J. What!?The 5 characters in that equation pack in an awful lot: The Geometric Product. Graded vector spaces. The four dimensional spacetime of special relativity. Vector calculus. Natural physical units. Etc...
Pasting that equation into an article can't possibly cover all of that...
I vaguely recall a presentation from Leslie Lamport on TLA+/formal modelling. I might get some details wrong here, but a member of the audience asked 'why not describe a system by examples' (implying it's easier to see some example interactions to start with), and Leslie's reply was it's not rigorous precise enough (not to mention he was visibly angry at the question). I agree it's not precise enough, but for someone starting, an example is good enough rather than choking on rigor. Sure, rigor does matter, but allow me some toehold first!
Just like 'no taxation without representation', there should be 'no formalization without insight'.
Out of all resources I have found Wikipedia to be almost uniquely useless for teaching yourself a new concept, its like "explanation by committee". On the other hand if I can understand a Wikipedia article I know I am at least intermediate in a area.
Yep. Highest recommendation for https://betterexplained.com which is at the other pole.