The n-Category Café

‘Modern mathematics’ is indeed boring and devoid of meaning in most papers. That’s why I stick to reading the works of the greats like von Neumann, Wiener, Einstein, and hosts of others. What made them great? They *explain* things, and then go on to carry out tremendeously complicated calculations. I think the advent of calculators and numerical methods has hurt the advance and understanding of math to a large degree. However, on the flip side, new symbolic methods allow computations which are a great help and lead to new relations not previously seen, so maybe there is hope yet.

I haven’t really thought about it, but I think this is on some level why I tend to write the way I do. My papers tend to read very conversationally, and very often I reverse theorem and proof.

It’s like, we’re just walking along seeing what there is along the road and – hey! – we just proved a theorem!

I also try to write very self-contained papers, not assuming that everyone in the audience is an expert and knows all the folk theorems around, but I think that’s a different effect.

When programming in the language Haskell beginners almost immediately hit the notion of category theoretical monads. Even the simplest “Hello, World!” program requires the use of monads. What I think is interesting about this is that Haskell beginners, many of whom have very few abstract mathematics skills, let alone category theory, suddenly have to leapfrog into completely foreign territory. They can try to read the formal definition of categories, functors and monads, and may even be able to carry out basic proofs with them. But the theory seems completely uninteresting to the beginner, especially when all you want to do is print “Hello, World!”.

So we have a situation where people have been forced to find ways to make monads accessible and interesting, and now there’s a whole industry of people writing ‘narratives’ with florid metaphors in an attempt to get the meaning over clearly. Various writers have used metaphors like containers, spacesuits(!), storage for nuclear waste(!!), types of piping and nesting, Hotel California (you read that correctly someone thinks monads are like the Hotel California!), computations, and ‘unsafe’ functions. It’s incredible how creative technical writers can become when there’s a definite need. It’s also interesting to see people make their private metaphors public like this. I’m sure that all mathematicians have lots of bizarre and interesting metaphors for mathematical concepts that they wouldn’t normally share with other people.

It seems that there may be two pure strategies here: one extreme is to motivate a general audience and the other extreme is to teach a technical audience. Of course, in the real world we often deal with mixed strategies which involve some of each.

Sometimes one can simplify the problem by considering the pure strategies first – in this case, motivate a general audience, or teach a technical audience. Then clearly identify the skills and knowledge of the intended audiences; clearly identify the objective of the paper, talk, book, blog posting, whatever; decide how much space/time we have to achieve the objective. Finally outline solutions and see how the solutions differ, and if appropriate, propose some mixed strategies to cover the in-between cases.

And if you really want a challenge, never use a number bigger than three.

I know, it sounds like a lot of motherhood, but hey, where would we all be without motherhood?

I like John Armstrong’s approach he mentioned above.

Like Einstein says, “The whole of science is nothing more than a refinement of everyday thinking”. This includes mathematics. If we want to contribute to the understanding of something, we should provide enough explanation about things. We should emphasize the purpose of things (why we defined this like that), and review the goals (where we want to get) every now and then in our presentation. We should be able to make distinction between a rigorous proof and its exposition.

Perhaps there would be less confusion in this whole discussion if people illustrated their points with reference to online material. After all, literary criticism with the text in front of you makes much more sense.

Why not have mathematical critics just as you have literary critics, to develop mathematical taste by public criticism? (Lakatos, ‘Proofs and Refutations’, CUP 1976: 98)

Here’s an email I got on this subject, which I was given permission to post:

Dear Professor Baez,

My name is Paolo Bizzarri and I am following your invitation to provide some feedback on the topic “Why is mathematics boring?”

Let me introduce myself: I am 37 years old, got a Degree in Computer Science in 1994. In the last four-five years I have started studying mathematics for fun and passion.

As my job is different, I have a limited time to dedicate to my passion. Even if I find mathematics extremely interesting, I have often found studying maths boring, difficult and hard to pursue. So, I have done some simple reflection on why I find difficult and boring something for which I have anyway passion.

My conclusions are as follows:

• mathematics is taught in an unnatural way;
• mathematics is a practical discipline, but it is taught as an abstract one;
• there is lots of implicit knowledge that is not made available through books.

I will try to expand each of these sentences in the following.

Mathematics is taught in an unnatural way.

My point is here referred mainly to textbooks, and their typical structure of definition/lemma/theorem/definition.

Where is the problem? The problem is that this approach is unnatural. It is not in that way that mathematicians reason and produce their work.

If you see a demonstration, it is as terse and essential as it has to be. Each passage is perfectly connected with the previous passages. Each hypothesis is done exactly when it is was needed, and it is absolutely minimal.

The question (my question) was: yes, everything works perfectly, but this is not science. This seems more an Hollywood film, where everything happens for a precise reason.

But mathematicians do not work in that way (or, at least, this is my understanding). The real problem in mathematics is often not to demonstrate a theorem: it is to find a good object to study. The definition comes AFTER a theorem has been demonstrated. The demonstration itself is refined numerous times, in order to obtain the “perfect”, textbook demonstration. Which is the only demonstration you see, and I found them quite unnatural exactly because they were perfect.

The real point should be that mathematics text book should provide the context, the reason WHY they are studying something, and what they are trying to study. Which bring us to my second point.

Mathematics is a practical discipline, but it is taught as an abstract one.

Again, this is based on my limited experience, and can be pretty typically Italian. But, anyway, it is the only experience I can provide.

One of the main points about mathematics is that it is “abstract”, “pure”, not tied to any practical problem.

Which is false, from an hystorical point of view. But it is false also on a more concrete, day-by-day, practical matter.

Maths is boring for non full-time mathematicians because they don’t know the “tools of the trade”. They are not used to manipulate mental objects like grups or vector spaces. When I have read for the first time about groups, I have found difficult to understand lots of things about them and their importance. When I have started to see them used, they became much more clearer to me.

Perhaps mathematicians do not feel this problem strongly, because they are used to work with abstract objects. It is the same problem that a non-IT professional has when he has to use a computer program done for an IT professional.

This separation is strong also because you are not supposed to use the tools you learn by yourself: the demonstrations are already provided, and you have not to improve them (you would not be able anyway…). You have to use them in some cooked up situations, but again there is little understanding that this is done for a specific reason. Which bring us to the third, and possible final argument.

There is lots of implicit knowledge that is not made available through books.

It is one of the most striking things I have seen: there is lots of thing about mathematics that are not explicitly expressed in mathematics textbooks.

One is what all mathematics call “elegance”. It is a fuzzy concept, but it is fundamental. I have not seen a single, explicit reference to it (except, perhaps, in Topics in Algebra). But this is a fundamental criterion to create and judge mathematical theories, and a strong guide about which structure you expect.

The second is about the “styles” of demonstration you adopt. Given a certain domain, it is quite common to see demonstrations that use a limited set of methods to be carried out, but these methods are never expressely nominated or indicated.

But, without a name, it is difficult to effectively teach these things. We cannot even speak properly about them.

Conclusions.

Well. If you didn’t find boring these writings of mine, I owe you a pizza in Pisa, if you ever come to my city.

Best regards.

Paolo Bizzarri

there are a couple of things that i think make mathematics dull:
1. difficulty in rapidly developing an overall sense of what is going on
2. the slackening of the ties between math and physics.
the first is, to some extent, unavoidable. there is probably something biologically tiresome about any activity in which the actor is forced to labor fairly strenuously before a reward is forthcoming (indeed, before he knows if there will ever be much of a reward). to make matters worse, some people have a habit of obscuring central ideas with a lot of distracting trivia. outside of mathematics, people commonly employ this tactic in order to conceal a lack of substance in the product they are promoting. it’s called marketing. there’s nothing wrong with marketing in mathematics, but the variety practiced by mathematicians who actually believe that inpenetrability => depth is unhelpful and unfortunate. as for the 2nd factor contributing to the dullness of mathematics: i have no idea why anyone would study mathematics unless they were interested in physics. explaining such a phenomenon to anyone without much experience in mathematics is a task i am certainly not up to. perhaps some people are. however, it’s very easy to convey the importance and interest of mathematical research if you are able to connect it with black holes, oceanic dynamics, the fundamental structure of matter, cryptography, communication networks, and so on. this connection was very deeply appreciated at the university where i went to grad school, but i’m not sure that the same can be said of most schools (the bourbaki shadow is rather long). hopefully this is not the case. in my opinion, it’s fairly easy to make mathematics quite engaging if these issues are addressed, editorial ‘we’ or no.

Rolfe Schmit has blogged about making math less boring — check it out!

Great post John, and great comments too. In my opinion, Mr. Bizzarri is dead on. Too many mathematicians seek to obscure the path of discovery rather than illuminate it.

At the mention of mathematics,3/4 of the class will start groaning..Why?Simple!
Mathematics is equal to only one thing..
Formulaes..
Formulaes,is equal to only one thing..Or two..
Lots of thinking..
And remembering..
Thinking and remembering is equal to only one thing..
BOREDOM..
Agree,anyone?

Hey everyone.

This is slightly off-topic, but I hope you don’t mind if I use this thread to advertise my new blog squarkonium.blogspot.com, where I posted a few ideas of my own about school education (more abstract than those discussed here, though).

Everyone is welcome!

Is Thales and Friends still going? In 2008, I was asked to blog by the online incarnation of a famous computer magazine that started during the hobby-computer era of the 70’s. I thought this would be a nice opportunity to explain category theory to my readers, most of whom are professional programmers, and to show what it can or might do for computing. But this needs more care than, e.g. explaining how to use PHP or why spreadsheets are risky, and since I’m no longer at a university, I don’t have the paid thinking time that most n-Category-Café readers do. So I mailed Thales and Friends to ask whether they could suggest how I might fund such writing. An organisation devoted to popularising maths: surely an ideal advisor. But despite several messages, both via the contact form on Thales’s site, and directly to their email address, I never got a reply.

I’ve had the same non-reaction when asking advice from academics on how to fund my Web-based category-theory demonstrations and related work. Some of the non-replies were even from regulars on this blog, which was a real enthusiasm killer. (Some people I mailed did give helpful replies though.) So such non-response isn’t unique to Thales. But perhaps in their case, problems in the Greek economy have killed the organisation? Does anyone know?

I think that the style in Wikipedia/nlab which replaces Definition/Lemma/Theorem with Motivation/Definition/Lemma/Theorem/Example is very helpful. All the best mathematical writers do this (even if the omit the Motivation subtitle). First they provide a sketch of the informal idea that they are trying to formalise might be (provide a story) Then they provide the formalisation. Finally the show how the formalisation gives new insight into the original idea that motivated the whole exercise. (Some authors have examples scattered throughought the exposition, which is even better).

Of course, depending on the writer, non mathematical content can help or hinder regardless of structure. I found Russell’s writings very difficult to study because of the condescending nature of his exposition. Nearly every paragraph seemed to have the subtext “I am so much more clever than you that I don’t know why I am bothering to talk to you at all”. Just because the implicit statement is true doesn’t mean that it is helpful to rub my nose in it frequently.

I also want to add that Mathematics is used for lots of things besides physics. Although the recent discussion of Arrow’s theorem (and related issues) skirted the original motivation, this is an example of interesting new mathematical thinking being generated in social science. Personally, I work in Transport Planning where graph theory interacts with econometrics to generate interesting maths. Game Theory is another example of maths forming because of the needs of social science.

The boredom starts early claims Michael Green.