> The distinction between the three types of errors can lead to the phenomenon ... of a mathematical argument by a post-rigorous mathematician which locally contains a number of typos and other formal errors, but is globally quite sound, with the local errors propagating for a while before being cancelled out by other local errors
I was initially amazed at this when I was in graduate school, but with enough experience I started to do it myself. Handwaving can be a signal that someone doesn't know what they are doing or that they really know what they are doing and until you are far enough along it is hard to tell the difference.
>Handwaving can be a signal that someone doesn't know what they are doing or that they really know what they are doing and until you are far enough along it is hard to tell the difference.
I found it's very easy to distinguish these two when you have another expert ask questions. But if you don't have someone like that in the audience it might take forever. Or at least until you become an expert yourself.
Good point. Here are some notes on it based on what I've observed happens in Academia and in other environments:
I think handwaving comes in different flavors:
- Handwaving and not knowing what they are doing, when they know they don't know:
This is arrogance and/or fear of people thinking you are a fool. Bad practice. Professionals who do this are status chasers and not fun to be around. Students who do this are mostly insecure, and they might just need some help with their self-esteem. Help them by letting them feel comfortable with being wrong. Foster a good environment so that the arrogance and fear fade away.
- Handwaving and not knowing what they are doing, when they don't know they don't know:
I believe this is a good thing, in particular for Students, if they are within a nurturing environment. It can lead to interesting ideas and to discussions of innovative ways to move forward. I believe this to be a way of actually "training your intuition muscle" both for Students and Professionals. It lets them know not to fear moving on, tackling the thing that captures their attention the most at first, and later on filling some of the gaps, which I feel is common practice for people who have been working on the field for a while. However, if the gaps are left unattended it can lead to bad things... Environment matters.
- Handwaving and knowing what they are doing, when they know they don't know:
For trained Professionals only... :)
This modality kinda kicks in when deep in mathematical work.
It's the path that leads to the Eureka moments...
Pure trained intuition acting almost as a separate entity to oneself. We are facing the unknown and something tells us that certain aspect can be handwaived, we don't fully know why but we feel it is. Later on it becomes clear why we could do the handwave. It works itself out.
- Handwaving and knowing what they are doing, when they don't know they don't know:
For trained Professionals only...
Kind of a stretch, but might be where our intuition either fails us completely, or completely takes us by the hand to turn the unknown unknowns into known unknowns, then it goes back to the previous category.
This isn't set in stone by the way, just some thoughts I had while reading the article...
Any ideas or suggestions for modifications more than welcomed.
I was initially amazed at this when I was in graduate school, but with enough experience I started to do it myself. Handwaving can be a signal that someone doesn't know what they are doing or that they really know what they are doing and until you are far enough along it is hard to tell the difference.