How important is Rigor in maths? Is it taken to an extreme in academic settings?
How much Rigor would you consider excessive? Can we learn the same things / be as efficient in our applications of math without rigorous proofs & study?
I mean, look at the example image, "remember" is at the bottom and that's as far as medical school will take you really, and doctors are esteemed and successful in their practice.
Perhaps mathematics has indeed practiced excessive rigor without a purpose, you think?
You're conflating academic education with success in medical practice, sir. Medical school alone focuses on remembering and understanding, but learning to practice and practice well requires application, analysis, and evaluation.
I realize that point is tertiary to your argument. Just thought I'd mention.
>Can we learn the same things / be as efficient in our applications of math without rigorous proofs & study?
Yep. Applications don't require proofs.
>Perhaps mathematics has indeed practiced excessive rigor without a purpose, you think?
Nah. My thoughts on this are that mathematicians have to be as rigorous as possible to lay a groundwork that's unquestionable. You can't lay the groundwork for literally all science that isn't pure mathematics on "Meh, looks fine to me".
I find this thread, and the many hundreds before it, particularly amusing, many decades ago, while undergrad, we were taught how to solve calculus geometrically, we were shown that the graphical method is more accurate than the pure mathematical method - which is an approximation.
As an engineer, we have to deal with the accumulation of errors and design them out. Mathematicions simply dont bother with this, if I did not, people would die.
Maths...Accurate? Rigorous? Fuck off.
See the problem with Mathematicians? All with their heads totally up their own arseholes. Follow laid out formulas... sheep.
The whole of cosmology was physics based observations generating mathematics. The only original mathematical based physics is.... string theory. Oh ha ha ha.
First off I cannot see how you can truly understand if you have not analyzed it.
Secondly only hack'n'slash druid doctors only focus on the remembering stage. Analysis is crucial to avoid mis-diagnosis. There are good reasons why Sherlock Holmes was based on a professor in medicine.
Rigor is useful but people outside the sciences tire of it quickly. so you equally quickly learn to do your analysis in private.
an example from my own work: I have a background in research and did a stint in quality assurance where we were measured (KPI and more). I applied full rigor to the analysis and came up with more findings per month than my other colleagues. It was measurable more efficient but I'll leave it to you to guess if others wanted to apply the same level of rigor.
I´m a physicist and you failed to realize that this graphical methods did not just appeard as some sort of hand-wavy method. They have rigour done by proper mathematicians/scientists. Numerical methods is also one of the pillars of pure math and has a well established foundation.
Rigour is important for having a well established theory that we can use.
It depends on what you want to do.
If you want to pursue math, then yes, you will need rigor. But math is not reducible to rigor.
You don't necessarily approach a math problem with pure logic. You might try to make sense of it, get an intuitive feel for it and then see if you can transcribe it with math.
Rigor is there to ensure that everyone speaks a common language