One is the distinction between art and craft. So the final rigour in a proof is the craftmanship of a mathematician, making sure that everything works as claimed. No patches needed (though they sometimes are).

Another is that proofs are like describing a route, and this is in a certain landscape. How much detail should you give when describing a walk to the station? You do not want to describe all the cracks in the pavement, but you do want to warn of dangerous manholes.

One of the jobs of mathematicians is t build a landscape in which proofs, routes, can be found. I heard a comment of Raoul Bott on Grothendieck in 1958, that:”Grothendieck was prepared to work very hard to make proofs tautological.” There is a good aim to make it clear **why** something is true; that may need new concepts.

I once had a student criticism of may first year analysis course: “Professor Brown gives too many proofs.” So I decided next year there would be no theorems and no proofs; what they will get were “facts” and “explanations”.

However there is a kind of obligation that an “explanation” should actually explain something!

A good test of a future mathematician is not necessarily current level of performance, but do they actually want to know why something is true.

]]>On the other hand, I think I’m good on language. I can do abstract reasoning when interpreting and using linguistic tools like metaphors and metonymy. I think language is underrated as a math tool. It’s as abstract as greek characters, but its mapping to _some_ intuition is more natural, at least to me. Every time I see greek letters in an explanation I translate it to language and then to some intuition, if needed, but I always do the translation step. If some concept from pure math is explained by a textual description, I tend to capture it easier and I can instantly map to a bunch of real-world applications.

I know, someone will say that natural language is ambiguous, but so is the actual academic math. How much papers do you read where there isn’t a single sentence written in plain natural language? In this case, ambiguity is not language’s fault, since the author decides doing a textual explanation because math isn’t enough to cover the entire concept. There isn’t any notation capable of motivating the reader for reading a math paper. Thus natural language is an advertising tool, but also is employed for covering gaps when pure math notation isn’t enough.

In short, I think pure math (i.e., math heavily relied on notation) is overrated as an abstract reasoning tool. You can do similar reasoning with plain natural language, so you could reach a broader audience and allow more intuitions to emerge — even pure math intuitions. Proving my hypothesis is the harder part, since I should use pure math, but I think I could contribute with some empirical results for motivating a mathematician to try something on this path. There is room for all kind of abstract thinking mathematicians.

]]>But being able to read and write the formalism is not very useful if it can’t be tested and applied against our intuitions. Having intuitions or ownership of a model is difficult if you are not the originator and don’t have access to the originator’s intuitions. One of the biggest problems I find with other people’s papers in applied areas (Engineering, Computing, Neuroscience, …), is the tendency to reproduce and even extend formal models without understanding either the assumptions or the intuitions that underlie them. At conferences, authors who are challenged on the appropriateness and applicability of a model often show that they are unaware or unmindful of the assumptions and only understand the model in the formal mechanical sense of being able to manipulate equations, verify derivations and produce proofs within the bounds of the model. This perhaps characterizes the dangers of an applied hybrid of heuristic and rigorous approaches and pinches the square into a figure eight as these are brought together in the same individual, or even the ubiquitous paradigm of an entire field.

I mean that an individual has sufficient rigour to go through the motions at a the model level, but is applying canned heuristics at the application level, without either pre- or post-rigour intuition being in evidence, without addressing satisfaction of assumptions or performing sanity checks on conclusions. And I’m not just talking about students. In some cases, whole fields are operating in a kind of unsound limbo, because a formal but inappropriate model takes precedence over common sense intuition and understanding of the boundary cases and the impact of assumptions.

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