# Infinity, A Very Short Introduction, by Ian Stewart

This review was originally written for the London Mathematical Society November 2018 Newsletter. The book can be found here.

The “Very Short Introduction” series by Oxford University Press attempt to take a moderately deep dive into various subjects in a slimline volume. Professor Stuart addresses the apparent paradox of tackling the subject of the infinite in such a small volume right at the start, along with the observation that the topic of infinity has long provided such paradoxes. This particular VSI aims to tackle infinity as found in numbers, geometry, art, theology, philosophy and more, and so it is a tightly packed volume indeed.

Infinity is a concept that is, at least now, embraced in Mathematics, but also reaches into Physics, Philosophy, Theology and Language in significant measure. In this book, Ian Stewart sets off almost immediately into the mathematical interpretations and concepts of infinity, starting with examples that are likely to be accessible to a wide range of readers, but also touching on some that will cause more mathematically advanced readers to consider them carefully and which may be challenging to less mathematically literate readers.

These examples are of the paradoxical issues surrounding infinity; all but one of these is explicitly mathematical, some geometrical, others more algebraic; by the end of the first short chapter we have visited David Hilbert’s famous hotel and explored some of the implications for the arithmetic of the infinite.

The second chapter then moves into a more detailed exploration of the consequences of the infinite in numbers, and in particular explores the infinite, non-repeating decimal representations of irrational numbers and the continuity of the real numbers.

Stewart then explores the history of the infinite in the third chapter, and how it weaved through early Greek philosophy and the classic paradoxes of Zeno, and how for the Greeks issues of infinity were closely tied to their thoughts and theories about motion, and indeed whether motion was in fact possible or an illusion. Some time is spent with Aristotle, and how he dismissed the idea of an “actual” infinity in favour of a “potential” infinity. We then move through with both Locke and Kant to the beginning of more modern philosophical analyses of the infinite. Some time is taken to explore the philosophy of the infinite in Christian theology, particularly through Thomas Aquinas, a philosopher heavily influenced by Aristotle and how he used the infinite in his “proof” of God.

Stewart also explores how, in the modern era, mathematicians take the infinite very much as a normal and integral part of mathematics, with little concern about the distinction of actual and potential infinities that were the great concern of the philosophy of the ancient world.

We dive then, from the infinitely large to the infinitely small in the fourth chapter where the seeds of calculus and analysis are to be seen, and the philosophical objections from Bishop Berkeley to the use of infinitesimals. It is interesting to note that these and other concerns about the theoretical underpinnings of calculus were largely ignored in the face of its obvious utility, until others tried to explore these foundations more deeply. Stewart takes us through this work through Cauchy and eventually to the work of Bolzano and Weierstrass who finally introduced the ? and ? notation that has undoubtedly delighted many undergraduates since and ushered in the start of analysis proper. Stewart than dips into an examination of non-standard analysis, a topic that at least I was never knowingly exposed to in my formal studies; it was intriguing to read of these numbers with “standard” and “infinitesimal” parts.

There follows a chapter on the geometrically infinite, which in particular looks at the role of the infinite in art, but which again after an informal discussion dips into the mathematics of what is going on. The chapter after this focuses on infinities that arise in Physics, particularly in optics, Newtonian and Relativistic gravity, moving on then to discuss the size of the known universe and its curvature. These two chapters are both short and may require some unpacking by readers with less background knowledge.

The final chapter is mostly dedicated to work of Cantor and his systemization of modern mathematical thinking around the concept of the infinite. Here we meet the distinctions between the finite, countably infinite and uncountably infinite, transfinite cardinals and transfinite ordinals. But even here we find the objections of some philosophers, in this case Wittgenstein. This is interesting to read in an era where Cantor’s formulations are considered uncontroversial and part and parcel of the “paradise” of Hilbert’s modern mathematics in the same way that the past controversies of complex numbers are of little interest to modern mathematicians.

The approach taken to infinity in the book, is non-apologetically Pure Mathematical in its spirit, and I suppose this may make the work a little less accessible for some readers, particularly those who are not prepared to think through some of the sections, perhaps with a pen and paper. The Very Short Introduction to Mathematics, from the same series, by Timothy Gowers similarly tackles a cross section of challenging examples from the discipline in a relatively small space.

In September 2016, the BBC aired an interesting series on Radio 4: “The History of the Infinite” (this is still happily available online for those interested, at least for those in the UK). In this series, Adrian Moore began discussing the original Greek antipathy to the idea in early philosophy, and then how the idea emerged through Aristotelian Philosophy, Christian theology. It was after this that Moore decided to tackle the more serious implications of the infinitely small and big in mathematics, before emerging back through Physics into more philosophical territory.

I suspect this route, sandwiching the more complicated mathematical treatment between philosophy more related to human experience could be more palatable to a general reader.

The Very Short Introduction to Infinity is nevertheless a fascinating and joyful exploration of the topic, accessible to the committed and careful novice, but with enough detail and asides to delight formally mathematically trained readers.

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# Implementing configurable work-flow patterns in Python Django

In my previous article, I discussed some of changes I’ve made to my WAM software to handle assessment and work-flow. I thought I’d have a look at this from the technical side for those interested in doing something similar, this is obviously extensible to general workflow management, where you might want to tweak the workflow later without diving into code.

My challenge was to consider how not to hard code a work-flow, but to have something that would be configurable, in my case in a SQL layer because I’m using Python and Django.

I had an idea about the work-flow I wanted, and it looked a bit like this (carefully sketched on my tablet). These nodes are particular states, so this isn’t really a flow chart, as decisions aren’t shown. What is shown is what states can progress to the next ones. But I wanted to be able to change the pattern of nodes in the future, or rather, I wanted users to be able to do this without altering the code. I also wanted to work out who could do what, and who should know about what.

## Understanding States

The first thing I did was to create a State model class, and I guess in my head I was thinking of Markov Models.

As you can see, I created variables that told me the name of the state, and an opportunity for a more detailed description. I then wanted to be able to specify who could do certain things, and be notified. So, rather than a long series of Booleans, I want for a text field – the work-flow won’t be edited very often, and when it is, it should be by someone who knows what they are doing. So it’s just a Comma Separated text field. For instance.

will indicate that the Module Coordinator and Moderator should be involved (this is an HE example, but the principle is quite extensible).

So the actors field will specify which kinds of people can invoke this state, and the notify field those who should get to hear about it.

I want to draw your attention to this bit:

What on earth does this do? It allows a Django model to have a Many to Many relationship with itself. In other words, for me to associate a number of states with this one. Please also note that presence of

This is most easily explained by comparison to the Facebook and Twitter friendship model. Both of these essentially link a User model in a many to many relationship with itself.

Facebook friends are symmetrical, once the link is established, it is two way. Twitter followers are not symmetrical.

I wanted to establish which successor states could be invoked from any given one. And this should not be symmetrical by default. You can see in my example graph above, I want it to be possible to move from state A to either B or C, but this is not entirely symmetric, it is possible to move from B to A, but it should not be possible to go from C to A. Without symmetric=False, each link will create an implied link back (all arrows in my state diagram would be bi-directional) which would be problematic. By establishing the relationship as asymmetric we can allow a reciprocal link (as is possible in Twitter, and our A and B example), but we don’t enforce it, so that we prevent back tracking in work-flow where it should not be allowed (as in our A and C example).

## Invoking States

I then created another model to keep track of which states were invoked.

This model allows me to work out who (the signed_by field) invoked a particular AssessmentState, when, and with any particular notes.

I also added a field to record when a notification (notified) had been sent. On creation, I leave that field as null. One of the many glorious things about Django is that it’s infrastructure for custom management commands allows you to easily build command line tools for doing cron tasks while your web front end runs without interruption. I found this rather awkward, but not impossible, in PHP, but in Django the whole thing is very organic, and you get access to all your models. If you have pushed plenty of your logic into the Model layer and not the View layer, this can really help.

In my new custom commend I can easily work out which signoffs have not been notified yet:

I can then act upon those, send notifications, and if that’s successful, set the notified field to the time at which I sent them.

In this article I have concentrated on the Model layer, with a few other observations, and in particular the relationship from a State model to itself.

All of the Forms and Views are available within my GitHub repository for the project. They aren’t a work of art, but if you have any questions feel free to look there, or get in touch.

I hope that might be helpful to someone facing the same challenge, and do feel free to suggest how I could have solved the problem more elegantly.

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# Assessment handling and Assessment Workflow in WAM

As is often the way, the scope of the project broadened and I found myself writing in support for handling assessments and the QA processes around them. At some point this necessitates a new name for WAM to something more general (answers on a post card please) but for now, development continues.

Last year I added features to allow Exams, Coursework, and their Moderation and QA documents to be uploaded to WAM. This was generally reasonably successful, but a bit clunky. We gave several External Examiners access to the system and they were able to look in at the modules for which they were an examiner and the feedback was pretty good.

## What Worked

One of the things that worked best about last year’s experiment was that we put in information about the Programmes (Courses) each Module was on. It’s not at all unusual for many Programmes to have the same Module within them.

This can cause a headache for External Examination since an External Examiner is normally assigned to a Programme. In short, the same Module can end up being looked at by several Examiners. While this is OK, it can be wasteful of work, and creates potential problems when two Examiners have a different perspective on the Module.

So within WAM, I put in code an assumption of what we should be doing in paper based systems – that every Module should have a “Lead Programme”. The examiner for that Programme should be the one that has primacy, and furthermore, where they are presented other Modules on the Programme for which they aren’t the “lead” Examiner, they should know that this is for information, and they may not be required to delve into it in so much detail – unless they choose to.

This aspect worked well, and the External Examiners have a landing screen that shows which Modules they are examining, and which they are the lead Examiner.

## What Didn’t Work

I had written code that was intended to look at what assessment artefacts had been uploaded since a last user’s login, and email them the relevant stuff.

This turned out to be problematic, partly because one had to unpick who should get what, but mostly because I’m using remote authentication with Django (the Python framework in which WAM is written), and it seems that the last login time isn’t always updated properly when you aren’t using Django’s built in authentication.

But the biggest problem was a lack of any workflow. This was a bit deliberate since I didn’t want to hardcode my School or Faculty’s workflow.

You should never design your software product for HE around your own University too tightly. Because your own University will be a different University in two years’ time.

So, I wanted to ponder this a bit. It made visibility of what was going on a little difficult. It looked a bit like this (not exactly, as this is a screenshot from a newer version of an older module):

with items shown from oldest at the bottom to newest at the top. You can kind of infer the workflow state by the top item, and indeed, I used that in the module list.

But staff uploaded files they wanted to delete (and that was previously disallowed for audit reasons) and the workflow wasn’t too clear and that made notifications more difficult.

## What’s New

So, in a beta version of 2.0 of the software I have implemented a workflow model. I did this by:

• defining a model that represented the potential states a Module could be in, each state defines who can trigger it, and what can happen next, and who should be notified;
• defining a model that shows a “sign off” event.

Once it became possible to issue a “sign off” of where we were in the workflow, a lot of things became easier. This screenshot shows how it looks now.

Ok, it’s a bit of a dumb example, since I’m the only user triggering states here (and I can only do that in some cases since I’m a Superuser, otherwise some states can only be triggered by the correct stakeholder – the moderator of examiner).

However, you can see that now we can still have all the assessment resources, but with sign offs at various stages. The sign off could (and likely would) have much more detailed notes in a real implementation.

This in turn has made notification emails much easier to create. Here is the email triggered by the final sign off above.

The detailed notes aren’t shown in the email, in case other eyes are on it and there are sensitive comments.

All of this code is available at GitHub. It’s working now, but I’m probably do a few more bits before an official 2.0 release.

I will be demoing the system at the Royal Academy of Engineering in London next Monday, although that will focus entirely on WAM’s workload features.

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# Migrating Django Migrations to Django 2.x

Django is a Python framework for making web applications, and its impressive in its completeness, flexibility and power for speedy prototyping.

It’s also an impressive project for forward planning, it has a kind of built in “lint” functionality that warns about deprecated code that will be disallowed in future versions.

As a result when Django 2.0 was released I didn’t have to make many changes to my app code base to get it to work successfully. However, today when I tried to update my oldest Django App (started in Django 1.8x) I hit an unexpected snag. The old migrations were sometimes invalid. Curiously I don’t think this problem emerged the last time I tried.

Django uses migrations to move the database schema from one version to the next. Most of the time it’s a wonderful system. In the rare case it goes wrong it can be … tricky. Today’s problem is quite specific, and easier to fix.

Django 2.0 enforces that ForeignKey fields explicitly specify a behaviour to follow on deletion of the object pointed to by the key. In general whether we Cascade the deletion, or set the field to Null, getting the behaviour write can be important, particular on fields where a Null value has a legitimate meaning.

But a bit of a sting in the tail is that an older Django project may have migrations created automatically by Django which don’t obey this. I discovered this today and found I couldn’t proceed with my project unless I went back and modified the old migrations to be 2.0 compliant.

So if this happens to you, here are some suggestions on fixing the problem.

You will know if you have a problem if when you try to run your test server, or indeed replace runserver by check

you get an error and output like this

I would suggest you try runserver whatever you did before as it will continue to try each time you save a file.

Open your code with your favourite editor, and open your models.py file (you may have several depending on your project), and the migration file that’s broken as above.

Looking in your migration file you’ll find the offending line. In this case it’s the last (non trivial) line below.

To ensure that your migrations will be applied consistently with your final model (well, as long as nobody tries to migrate to an intermediate state) look carefully in the correct model (Activity) in this case, and see what decision you make for deletion there. In my case I want deletion of the ActivitySet to kill all linked Activitiy(s). So replicate the “on_delete” choice from there.

Each time you save your new migration file the runserver terminal window will re-run the check, hopefully moving on to the next migration that needs to be fixed. Work your way through methodically until your code checks clean. Check into source control, and you’re done.

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# Semi Open Book Exams

A few years ago, I switched one of my first year courses to use what I call a semi-open-book approach.

Open-book exams of course allow students to bring whatever materials they wish into them, but they have the disadvantage that students will often bring in materials that they have not studied in detail, or even at all. In such cases, sifting through materials to help them answer a question could be counter productive.

On the other hand, the real world is now an increasingly “open-book” environment, which huge amounts of information available to those in the workplace which is now almost always Internet connected.

So I decided to look at another approach. Students are allowed to bring in a single, personalised, A4 sheet, on which they can write whatever they wish on both sides. There are a few rules:

• the sheet must be written on “by hand”, that is to say, it cannot be printed to from a computer, or typed;
• the sheet must be “original”, that is to say, it cannot be a photocopy of another sheet (though students may of course copy their original for reference);
• the sheet must be the student’s own work, and they must formally declare as much (with a tick box);
• the sheet must be handed in with the exam paper, although it is not marked.

The purpose of these restrictions are to ensure that each student takes a lead in producing an individual sheet, and to inhibit cottage industries of copied sheets.

In terms of what can go on the sheet? Well anything really. It can be sections from notes, important formulae, sample questions or solutions. The main purpose here is to prompt students to work out what they would individually distill down to an A4 page. So they go through all the module notes, tutorial problems and more, and work out the most valuable material that deserves to go on one A4 page. I believe that this process itself is the greatest value of the sheet, its production rather than its existence in the exam. I’m working on some research to test this.

So I email them each an A4 PDF, which they can print out at home, and on whatever colour paper they may desire. The sheet is individual and has their student number on it with a barcode, for automated processing and analysis afterwards for a project I’m working on, but this is anonymised. The student’s name in particular does not appear, since in Ulster University, it does not appear on the exam booklet.

The top of my sheet looks like this:

So, if you would like to do the same, I am enclosing the Python script, and LaTeX that I use to achieve this. You could of course use any other technology, or not individualise the sheet at all.

For convenience the most recent code will also be placed on a GitHub repository here, feel free to clone away.

My script has just been rewritten for Python 3.x, and I’ve added a lot of command line parameters to decouple it from me and Ulster University only use. It opens a CSV file from my University which contains student id numbers, student names, and emails in specific columns. These are the default for the script but can be changed. For each student it uses LaTeXÂ to generate the page. It actually creates inserts for each student of the name and student number, you can then edit open-book.tex to allow the page to be as you wish it. You don’t need to know much LaTeX to achieve this, but ping me if you need help. I am also using a LaTeX package to create the barcodes automatically.

I’ve spent a bit of time adding command line parameters to this script, but you can try using

for information. The script has been rewritten for Python 3. If you run it without parameters it will enter interactive mode and prompt you.

I’d strongly recommend running with the –test-only option at first to make sure all looks good, and opening open-book.pdf will show you the last generated page so you can see it’s what you want.

Anyway, feel free to do your own thing, or mutilate the code. Enjoy!

I use a LaTeX template for the base information, this can be easily edited for taste.

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# The Most Dangerous Idea in History

In the modern world we often throw around the wordÂ meme to mean some comic image, video or idea that has become associated with a concept, but the word has a different origin.

“an element of a culture or system of behaviour passed from one individual to another by imitation or other non-genetic means.”

This usage was coined by Richard Dawkins in his 1976 book “The selfish gene“. Like genes, memes are replicated by one process or another, sometimes with mutations. Like genes, memes are subject to a form of “evolutionary pressure”, a survival of the fittest.

So memes are not just ideas, but ideas can be seen as memes. I’ll likely use the words a bit interchangeably for convenience, however, in this article. The best ideas or memes, can survive for centuries or millennia, as Dawkins himself noted:

“But if you contribute to the world’s culture, if you have a good idea…it may live on, intact, long after your genes have dissolved in the common pool. Socrates may or may not have a gene or two alive in the world today, as G.C. Williams has remarked, but who cares? The meme-complexes of Socrates, Leonardo, Copernicus and Marconi are still going strong.”

In effect, memes can be more immortal and long-lasting then genes. And the transmission can be more direct as well. You may not descend from Socrates, Newton or Curie, whatever benefit that may or may not give you, but you can easily open a book and have those memes transmitted to you directly (or more likely through one or two intermediaries) very efficiently.

This is a very important feature of humanity, perhaps its most important: the ability of a human to learn from more than just its immediate family or peer group, however valuable that interaction is.

Some people have explored the viral nature of memes, and in this sense, we can easily understand that in terms of the common usage of a word in social media.

Of course, some memes are millennia old, have virally spread and are just plain wrong. The popular meme that humans have five senses is wrong. So memes can survive selection pressure despite error. Of course, the pressure may be more intense and effective if the consequences are more significant.

I contend that the most dangerous idea in history is one of a family of related ideas on this theme:

It is bad / wrong / sinful / wicked to question / doubt.

This is a very widespread meme indeed, and a successful one therefore in terms of its own survival. It exists in various strengths and in various contexts. And it doesn’t seem especially dangerous; it’s an innocuous statement.

So what’s the problem? Well, there are two aspects to this.

Firstly, it knocks out your mental immune system. This idea is almost parasitic because it reinforces itself with circular logic. Once it is in place, it prevents or inhibits its own eviction. After all, one has to challenge the idea to reject it, and the mind the idea resides in has already accepted that this is unacceptable.

People that have been taught this as part of their philosophy, ethics or morality, will of course tend to pass it on as a necessary element in those systems, and one can see why.

Because the second aspect is that, this idea rarely comes on its own. The really big problem with this idea is that explicitly or implicitly it tends to actually be found in this form:

It is bad / wrong / sinful / wicked to question / doubt [X].

And then X is or can be the problem. In other words, this is a mental virus that often comes with an associated payload. Like a two-part drug.

For convenience, let’s call X the “payload“, the idea or collection of ideas that hitches a ride with the “immunosuppressant“, the idea that one must not question the payload.

Maybe the payload is trivial, like somebody learning a martial art who has essentially been told not to question anything in what they are being taught. In such cases the immunosuppressant challenge could cause poor form or technique never to really be corrected, or not to be open to improvement from ideas from others.

Surprisingly one can find the immunosuppressant quite easily in class rooms, where groups of students have been told that they have to accomplish a task by certain means are exhorted not to think about why. Or sometimes they are so frightened out of asking questions that they pick up the immunosuppressant meme all by themselves. This can damage their ability to discern good ideas from bad.

If the payload is something more serious, such as having significant ethical or moral content then it might still be a relatively minor problem. For example ifÂ the payload is ethically benign such as some variation on the Golden Rule, then few issues arise, since the immunosuppressant defeating aspect is reinforcing a behaviour (payload) that is ethically non damaging or even perhaps, life enhancing.

But, if the payload contains many ethically or morally dubious aspects, then you have real problems, because these ideas and behaviours simply cannot be challenged from outside that mind. If the person swallowing the two part pill has accepted the immunosuppressant wholeheartedly then almost nothing can be done to recover that mind’s proper function. It’s trivially easy to see this at work in the world, where people of a given faith can’t even accept that adherents of different strands of that faith are worthy of respect, or in extreme cases, life itself.

In most cases the payload is complex, comprising both good and bad ideas; in these cases the immunosuppressant is the main reason preventing people from discerning which bits to hang on to and which bits to discard. Fortunately, for many the immunosuppressant isn’t full strength, and they quietly, and quite sensibly work out which parts of the payload to discard, but often with no fanfare. They are sometimes still ashamed to state that they do this or don’t even admit it to themselves.

But we shouldn’t be embarrassed to say that parts of a payload are good and parts should be rejected. For instance, most people of faith, from the Abrahamic tradition, quietly reject parts of the payload, let’s take this one:

“If a man has sexual relations with a man as one does with a woman, both of them have done what is detestable. They are to be put to death; their blood will be on their own heads.”

I mean, there’s no getting around it. It’s perfectly clear, in the payload, and it’s equally clear to most 21st century people that this is wrong. Wrong. Illegal. Murder. Ludicrous even. But still many lovely and kind people will try and apologise for this, quoting nicer parts of the payload, rather than just admitting that this is wrong, often because the immunosuppressant part of the pill says we have to not question any aspects of the payload.

And then we are surprised when people kill each other around the world based on the differences in their ideas, even if those differences are trivial, and pose absolutely no threat whatsoever.

But why should we be surprised?Â The answer is all too obvious.

Horrifically, in many cases, they have been explicitly told to do these things. It’s there in writing.Â And the immunosuppressant is strongly in place. It’s no good saying that the payload has lots of nice bits in it too. That’s great. That’s wonderful, but the payload will only become better when people are able to admit that parts of it are just plain wrong, and need to be rejected. For this to happen, the immunosuppressant has to be removed. At this point their natural mental immune response comes back to life. It is then possible for peers to influence people for the best. It is easier and possible to learn from the positive examples of others.

If we want to rid ourselves of some of the worst most horrific memes of our past, we need to admit that this is a possibility, and this is why doubt and questioning isn’t a sin, or an error, but the most basic principle of mental hygiene.

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# Pretty Printing C++ Archives from Emails

I’m just putting this here because I nearly managed to lose it. This is a part of a pretty unvarnished BASH script for a very specific purpose, taking an email file containing a ZIP of submitted C++ code from students. This script produces pretty printed PDFs of the source files named after each author to facilitate marking and annotation. It’s not a thing of beauty. I think I’ll probably write a new cleaner version in future.

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# Installing Android Nougat on a Stock Galaxy Tab 10.1

My daughter uses an Android Samsung tablet (coded GT-P7510) which ended official support on Android 4.0.4. Unfortunately I didn’t pay any attention to this issue until the apps she most wanted to use, namely Netflix and YouTube stopped working on it as the Android version was too low.

I found a ROM to upgrade to Android 7.1 (Nougat) with some cost – for instance, the camera doesn’t work, but Aimee doesn’t care about that. So I decided to try upgrading it since the tablet was otherwise now utterly useless.

To make things more difficult, most of the information on upgrading this tablet on the Internet is outdated or wrong, or pre-supposes that the device was long since updated. I also don’t run Windows, and ran into some problems with the Heimdall alternative.

So this quick article is the result of a couple of evenings running into dead ends. It might help someone else. Certainly if I ever need to do it again it’ll help me.

But as usual, if you break something, you own all the parts. These instructions are completely specific to this particular tablet, and the wifi only version at that. Make sure your device is fully charged before you start.

A new recovery image had to be installed first, and some steps had to be undertaken just to get that far.

## Heimdall

First of all there’s supposed to Windows software called Odin that is used to update the ROM, especially from a stock start. I can’t run that without emulation since I don’t run Windows, and in any case, I suspect it might behave badly in a virtual machine, and probably wouldn’t run correctly on modern Windows.

So I installed a Free and Open Source alternative known as Heimdall. For me, this was simple as it was Debian packaged. I couldn’t get the frontend to be useful, and I couldn’t get the Java version of the frontend to work online or offline. So I defaulted to the command line.

So, as root on Debian GNU/Linux:

This is also ensuring all the command line tools for android debugging are installed (I already had these).

## Receiving TWRP

The device needs to be made ready for Odin / Heimdall upload. Turn the device off, and then hold Power and Volume Down till it appears with two icon choices. You want the one on the right. Use Volume Up to select, and use Volume Up again to bypass the dire warnings.

I had no success in using the Heimdall frontend, your mileage may vary. I got the correct archive for my purposes from here.

I downloaded the archive, and used tar xvf to extract the contents. You will find two .img files, recovery.img and hidden.img. You’ll need both.

Note that the partition target on the device for recovery is not called recovery but is called SOS at least on my device.

Because of the no-reboot option note that the tablet will continue to warn you not to restart it. You’ll need to watch the command line progress carefully to ensure that it is on. Now reboot the machine once again into the Odin / Heimdall mode again. I.e. power it off, and turn it on with Power and Volume Down.

Now flash

For me this successfully got TWRP 3.0.3 loaded. It was a major odyssey of conflicting information to get this far. When you reboot make sure you hold down volume down to get to the recover menu, (and now choose the left hand option). If you don’t do this, the stock ROM overwrites the new one and you’ll need to start again.

## Using TWRP

From here, things were relatively plain sailing. I got the ROM from here. Incidentally, I’d tried other recovery ROMs I got onto the device before when I couldn’t get TWRP onto it, they did not allow the following steps to work.

I then used TWRP’s wipe option to wipe Cache, Data, and Dalvik Cache.

I used the Advanced button and put the device into sideload mode.

I then, from the Linux command prompt executed

I then did not reboot but went back in TWRP and selected sideload again, this time I was careful to uncheck the wipe data and cache items since I’m loading other items on top of the basic ROM image.

and I repeated the same for the last package

finally I selected to reboot the tablet. It took a pretty long time to boot. Don’t forget it’s a relatively underpowered device.

The device is up and running and now runs the apps my daughter wants again.

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# The Deceptiveness of Coincidence

A friend of mine recently posted about a chain of events – people sharing birthdays – that was so unlikely that a lottery ticket purchase was called for. Most people might make similar comments as the oddity of these events struck them. There followed some discussion about these problems and it made me think of trying to set out the issues more carefully.

I’ve done a lecture, in some variations (for school audiences), almost every year about the deceptiveness of common sense in mathematics. Indeed, it provided the title of this blog.

## The Birthday Problem

One of the problems I examined is the famous birthday problem. Let’s look at two aspects of this.

How many people must be in a room before we are certain that two of them share the same birthday?

This can be answered by using the so called pigeon hole principle. In this case, the pigeon holes are all the days of the calendar in a year, 365, or let’s say 366 to allow for the worst case scenario of a leap year. The people in the room are the pigeons. You can place 366 pigeons into the holes and keep them all separate, but any more ensures that at least two must share one of the holes. So, similarly, you need 367 people to be absolutely sure that one pair in the room shares a birthday.

Let’s look at a more subtle question.

How many people must be in a room so that it is more likely than not that two of them share the same birthday?

It turns out this number is deceptively low. We reach a 50% chance (more likely than not) with just 23 people in the room. Perhaps even more surprisingly we reach a 99.9% chance with 70 people in the room. The calculations for the probabilities are set out in the link to the lecture, and the wikipedia article as mentioned above, but I don’t want to focus on that. I want to focus on why the result is so shocking to most people.

The issue is the collision of relatively rare events (a birthday) with numerous combinations. For instance if you want to consider how many pairs one can form in the room we can see this.

If there are three people in the room, A, B and C, there are three possible pairs: AB, AC, and BC. If there are four people in the room, A, B, C and D, there are six possible pairs: AB, AC, AD, BC, BD, and CD. There’s actually a nice formula for this and you can work this out for more examples.

You don’t need to know the formula, but if you want it is

People in Room Possible Pairs
3 3
4 6
10 45
15 105
20 190
23 253
30 435
50 1225
70 2415

You can see just how rapidly the number of pairs increases. It might seem un-obvious that 23 people is as likely as not to produce a duplicate birthday, but it’s perhaps less so when you realise that we are not predicting which pigeon hole has two pigeons in it to go back to our analogy. But given 253 possible pairings, it seems less unlikely even intuitively that there will be “collisions” within one of those pairs.

## Coin Tosses

Let me talk about a similar problem, flipping a (fair) coin to get a Head (H) or a Tail (T). Let’s consider that we flip the coin three times. The possible outcomes are:

HHH, HHT, HTH, THH, TTH, THT, TTH, TTT.

These are all absolutely equally likely, and even this is sometime unintuitive. The coin has no memory of how it performed, and no desire to conform to some concept such as a “law of averages“. But we tend to categorise these 8 possible events into 4 categories.

• Three Heads: (one way this can happen, so 1/8 probability)
• Two Heads, One Tail: (three ways this can happen, so 3/8 probability)
• One Head, Two Tails: (three ways this can happen, so 3/8 probability)
• Three Tails: (one way this can happen so 1/8 probability).

In doing so, we can see why the HHH and TTT events look particularly exciting and rare. They are no more rare than any other individual outcome, but it’s the groupings of how one can get two heads that makes that collection of events more unexciting.

To take this to more of an extreme. If we flip a coin ten times, then these outcomes are all exactly equally likely.

1. HHHHHHHHHH
2. HTHTHTHTHT
3. HHTHTTHTTH
4. THTTHHHTHT
5. TTTTTTTTTT

But some will be far more remarked upon that others. But even if we have an experiment that flips a coin 10 times over and over again enough times, we will expect to see all of these (more or less, eventually) equally often, but we will probably not consider these exciting except maybe outcomes 1,5 and maybe 2.

## Genealogy

Another example, closer to home. Let’s assume that there have been 7,500 generations of Homo Sapiens since its emergence as a species.

The probability that in each of those 7,500 generations in both your paternal and maternal line that all the pairings took place between the right people at the right time in the right circumstances to give rise to you is astronomical. You have two parents, four grand parents, 8 great grand parents and so on. Just for that collection of people. If we add you into the mix, then the number of pairings involved over these 7500 generations is

That number has 2259 digits, you can see them here on Wolfram Alpha.

That’s the number of people involved to get to you, including you. They all had to survive and meet at the right time, and um, do everything else at exactly the right time in the right way. What are the odds? And yet you are here… You might find that exceptional, and of course in a way it is, but the point is, you are not more likely or unlikely than most other human beings that have been born, or countless potential humans who might have been but were not born because their parents never even met. The fact that you specifically exist is unlikely, but the fact that if you did not someone else would exist instead is highly likely and predictable.

Naturally, however, we tend to think of our own existence rather more than some random person on the other side of the planet. But each one of us is an example of the coin flipping perfectly as needed every single time. Sometimes a Head, sometimes a Tail, but that exact perfect sequence essential to produce you. Naturally we are each quite taken with the outcome that produces us and also our nearest and dearest. But if it had not happened, we wouldn’t be here to wonder about it.

## Availability and Psychology

As you can see there is more to this than mathematics and probability, there is also perception and psychology, specifically a concept called the Availability Heuristic. Some things stand out to us as being significant when they are no more significant than other events we ignore.

You might live in a city of a million people. You might personally know only a hundred. You might pass a thousand on your commute to work each day. You do that hundreds of times a year. The number of possible combinations of people you will see over a year is enormous, but you won’t remember it as a special event when you bump into all the people who aren’t dear to you.

## The Anthropic Principle

A nice example of this thinking is the question that many people have asked which is essentially: why is the Universe, and our Earth, set up so nicely to allow humans to exist? If certain physical constants were just a bit different then matter as we know it couldn’t form. If we were just a bit closer to the Sun we couldn’t survive and so on.

These things are all true. But in a Universe with inappropriate physical constants there will be no matter by definition, and certainly no sentient matter like us to notice.

If the Earth was closer to the Sun, homo sapiens would not exist as it currently does, but like as not some other species would, and it would also think how perfectly the world was formed for them. This concept, that we tend to think the Universe is so special to have created us, despite the fact that if not we wouldn’t be here to think about it, is often called the Anthropic Principle.

We believe that the improbability of life itself or us as a species is an example of the collision between billions of individually improbable events with the billions of events than could have occurred, and the billions of years in which they might have done so.

We end up having to think about such things with great care because our evolution on the African savannah didn’t really equip us to consider them. So we shouldn’t be surprised that we are surprised, but we still are.

It’s the old cliché: you are unique and special; just like everyone else.

There are always dangers in imagining our special place as a species.

“This is rather as if you imagine a puddle waking up one morning and thinking, ‘This is an interesting world I find myself in — an interesting hole I find myself in — fits me rather neatly, doesn’t it? In fact it fits me staggeringly well, must have been made to have me in it!’ This is such a powerful idea that as the sun rises in the sky and the air heats up and as, gradually, the puddle gets smaller and smaller, frantically hanging on to the notion that everything’s going to be alright, because this world was meant to have him in it, was built to have him in it; so the moment he disappears catches him rather by surprise. I think this may be something we need to be on the watch out for.”

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# Python Script To Copy a Playlist and Linked Files

Most tools for copying files onto MP3 players (often actually a phone these days) work on the basis that you copy your entire music catalogue and then make playlists linking to various files on it.

It can be a trickier process to copy your favourite selected audio files – the ones specifically used in your playlist, that usually exist in a deep structure of their own. There seem to be some proprietary tools for it, but here’s a Free (in all senses) and Open Source solution.

I adapted my previous Python scripts (that allow for files to be appended to a playlist on the command line, and shuffle a playlist from the command line) to make a – well – bit of a Frankenstein’s monster that will take a playlist as input, and copy the playlist and all its referenced file to another directory. It will create the required directory structure, but will only copy the actual music files on the playlist.

I have used this successfully to move a playlist and the selected directory structure onto more modest sized music players.

Usage looks something like this.

Have fun, use with care, and I’ll try to clean it up at some stage. If you break something, you own all the pieces, but feel free to write me a (nice) comment below.

At some point I’ll move these scripts to Python 3 and off the deprecated option handling code module.

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