> And most mathematicians have been reluctant to invest the time necessary to understand the work because they see no clear reward: it is not obvious how the theoretical machinery that Mochizuki has invented could be used to do calculations.
I find this _deliciously_ ironic since that is the position most students have towards mathematics in general (replace "calculations" with "anything relevant in their lives").
I am a research mathematician. I don't think this quote is an accurate reflection of how 99% of experts feel about Mochizuki's work. Most experts expect that if Mochizuki's work is correct (and even if not) it contains a lot of valuable ideas. Proofs of this kind are almost never mathematical dead ends - they are difficult because they require fresh insights and these insights can always be applied to other areas.
Mathematics is not solely about the proof. Good mathematics is about the communication of the proof and the ideas in it. I think it is fair to say Mochuzuki's work is not being communicated effectively. Though I am not saying the problem lies with Mochizuki alone.
After Perelman's proof, there have been some "filling in of details". [1] Would you say that Perelman's proof is comparable, and that he could also have been more pedagogical? Do you see any parallel at all?
It sounds to me as if you are implying, that it is Mochizuki's responsebility to be pedagogical. If being pedagogical is good (because it is more social?), how is mathematics different from any other discipline? Surely one ought to be social in every regard.
If the proof turn out to be correct, would you still say Mochizuki communicated it wrong?
From what I understand (not exactly my area) Perelman's proof was quite intelligible. Yes there were lots of details to fill in, but for the expert it was clear how this should be done. Perelman's proof also had a long background (it implemented ideas outlined by Hamilton earlier) so for experts it made a lot of sense. I don't think there is a lot of similarity between the two situations.
Research is a social activity. Being a successful researcher means being social. What social means depends on the norms of the relevant field. Yes, we should reflect on those norms and allow innovators to push boundaries but for the science to evolve it has to take everyone with it.
Mochizuki and those around him have a responsibility only if they want take part in the mathematical community - which I think and hope they do.
I am taking the word of experts in the area of arithmetic geometry who say there isn't being enough done to communicate his ideas. Regardless of whether he is right or wrong (really it isn't about this - it is about whether his new ideas have merit - the proof of the abc conjecture would be strong evidence for this) I think the current situation speaks for itself.
Edit: Also, you are entirely correct, we are humans, we should be social in every regard!
I'm not qualified to answer this, but I remember this rather interesting talk where Richard Taylor talks about this somewhat, and there is some discussion of the ABC Conjecture even if somewhat tangentially: https://www.youtube.com/watch?v=eNgUQlpc1m0&t=25m40s
There is a quote from Mochizuki himself in the article: [my proof] "constitutes a sort of faithful miniature model of the status of pure mathematics in human society"
I hadn't followed the original proposal closely, so I can't give a complete list. One feature I know about, that was implementable but very expensive, was the ability to specify multiple ways that a type could match a concept. For instance, a Sortable concept could mean the type has an operator<, or could be passed to the function compare(). I'm sure there was a way to handle cases where a type had both. But then, if you had a type with a method, say lessThan(), you could somehow specify that was an acceptable operator<, and then use anything that relied on the Sortable concept. As far as I know, that isn't in "concepts light."
Hmm, in your opinion do you think this would be a good technique then for digitizing paper maps? And if so, could you point in the direction of a library or textbook you'd recommend?
Is there an e-ink reader with a 8.5"x11"/A4 screen that'll display PDFs? All I want is to be able to bring all my tech ebooks with me to work and be able to look things up/brush up on things on the commute/flight etc.
If you can find an Entourage Edge 10" on ebay or somewhere, you would probably be set. I had the pocket edge version. These were very solid android dual-screen tablets - an e-ink and lcd display. The e-ink was also a digitizer, so you could use a stylus to take notes.
Sadly, being a hardware company is tough and the product line became unprofitable. The android version ended at 2.3, but it was sufficient to put on several readers. If your books were in a standard format, the built in reader worked fine. There was also community effort to either mirror the screens or force other reader apps onto the e-ink panel.
It's actually quite good if you "root" the device and install koreader[1](the original developer is the same guy who wrote the vnc viewer for kindle, that was mentioned in this thread), or its librerator fork[2]. In contrast to Amazon's practically unusable PDF reader, these open source projects support custom zoom levels, 2-column mode for academic articles etc.
Another option is to use k2pdfopt[3].
Horrible. The screen refresh was fast enough, but the actual PDF rendering was grossly underpowered. I was reading technical articles with embedded plots that would take quite literally minutes to render.
I sold my DX (the "current" gen) and bought a 3rd Gen iPad the minute the retina display was released. I still pine for the eInk display though. I've half considered a hobby hack project that involves a Kindle DX brain transplant.
No. That product is years old and has been removed entirely from the market twice now. Amazon deliberately does not market them and only makes them available intermittently.
The PocketBook 912 (and I think a few other models) had a 9.7" (diagonal) display size, with the usual 16 shades of grey, and a reasonable, if not over-powered, CPU.
I have one, and use it for good purpose with a variety of text books. The ability (or constraint?) to not be distracted by colourful networky things is grossly underestimated, but even more so is the ability to stare at the thing for several hours, in a variety of lighting conditions, and not feel any type of eye-strain.
Horribly expensive at the time it came out, and probably rare as rocking horse poo now. Looking at the other responses to your question there is slim pickings for contemporary devices with these specs, sadly.
You probably don't need 8.5x11". There's no reason to display the (usually blank) margins in an e-reader. 6.5x9" should suffice.
I have an old Kindle 3G that I use for reading papers. Its screen is 4.8" in height, which is just about the default column width of LaTeX (= 8.5" - 3.75" margin = 4.75"), meaning I could comfortably read papers in landscape with only scrolling up & down.
PDF reading was okay. It was rather slow, and you couldn't highlight/markup anything, but the rendering was gorgeous.
Formatting -IMHO- depends on context. I have one convention I generally use which is chaining methods together on one line, but when chaining together builder method call I make one line per call. That means the semantics not the syntactics of the code determine my formatting. For this reason I abhor auto formatting code unless it is currently like an unrecoverable mess (ie NO thoguht put into it, not just "not my style"). Honestly, put your curlies wherever you like them, then don't effect readability as much as lacking comments or convoluted flow.
Yes, but it's nice to save my code and have it formatted "the official way". It gets old debating over which standard to use. Who cares what you like. Just pick a standard for every language and get used to it. It's really not that hard.
