Today, I have spent a lot of time on Maths and have very little to show for it. Three double-sided revision cards, to be precise, and three past questions. Wikipedia has confused me with its over-generalisations, and the summary notes don't match up with the lecture notes, even remotely for this particular section. The books that I have from the library have not helped either. OK, so if I understood everything first time then there wouldn't be much point in my doing the course in the first place, but it's still pretty demoralising. I wish that there was someone I could ask for help, but I don't want to take people off their Easter breaks if they're busy enjoying themselves and I know that a lot of my friends on the course understand even less of it than I do. While all failing together promotes team spirit, no doubt, it doesn't stop the fact that we'd all have failed!

 

*sigh*

 

 

The internet hasn't changed a lot either, unsurprisingly given the regularity with which I have been checking it. I liked Kath's idea of doing question-time, though. How about you lot save me from the crazy-Maths-boredom and ask me some questions? Thank you kindly :-)

 

(It occurs to me in retrospect that this will be harder for people who actually know me...)

 

The problem with Maths lecturers is that in order to be in academia in the first place, they are necessarily very clever mathematicians. Especially when you get into Pure Maths, abstract Algebra and the like, this is simply to do with the way that their brains function. Yes, they've had to develop it over the years through study, but they are innately programmed to cope with abstract concepts and that's something that you either have or you haven't got. The result of having this type of mind, however, is they tend not to be very good communicators, cannot organise their material for toffee, and sometimes forget that just because it's obvious to someone who's been working with isomorphism theorems for the past twenty years, it isn't always obvious to their students. You know the stereotypical Maths nerd, socially stunted? John Nash in 'A Beautiful Mind' (only without the schizophrenia)*? That's them.

 

Next factor in** that the two I have for Algebra and Number Theory both have broken English. Their lectures have been incredibly incoherent, they don't always understand questions that are asked for clarification purposes, and between them they have kept forgetting to state important results explicitly, justifying that it's OK because they have covered them by implication. I have been going through the online summary sheets for the first term's work and have just given up because large chunks of the handwritten lecture notes are missing. Does this mean that we don't have to know them? Or if we do, to what level can we rely on the summary sheets? Forget 'revision' - this is 'vision', because most of us spent the majority of lectures completely at sea.

 

And don't get me wrong, I quite like these guys! Particularly in the latter case, they know that they aren't very good lecturers. It's not a case of snobbery, or ivory towers, or even not caring. They just don't know how to lecture better. And that makes it a lot, lot harder to learn, because you've got all of these barriers before you even start on this (or this or this).

 

 

*A wonderful, wonderful film which I saw for the first time on Friday! Wonderful in its own right, and they'd clearly done their Maths homework (, as it were, unlike in 'Good Will Hunting' which annoyed me immensely for that reason). I got terribly excited when he wrote up some results for diffeomorphisms on the board!

**Ha! Sorry...

 

A suggested veto for this blog: the word 'religion' (for a bit, anyway!)

 

 

I have just spent six days away, although it feels like more than that when I consider what I managed to fit in. For the first part of the week I was staying with my family just outside Welshpool. My brother is on a school trip for the week and as he is still too young to stay anywhere independently, it has been a very long time since we went on a family holiday without him - so that was a bit odd*. But it was lovely to be in the countryside for a bit. Shropshire/ Wales is very pretty in places and very good walking country!

 

After four days there, then, I caught various trains down to Surrey to stay with one of my housemates. Her mum had found a deal on lastminute.com which involved dinner at a posh restaurant (with complementary cocktail) and tickets to 'Lord of the Rings: The Musical' for £21 each, which ent bad! 'Lord of the Rings: The Musical!' wasn't bad either, actually - it was visually stunning, musically good, and plot-wise incredibly confusing. Even as someone who has neither read the book nor seen the films, I could tell that it had been vastly, vastly cut down and squashed into the three-hour time gap, which didn't do it any favours. Apparently a complex half-hour battle in the film had been condensed into the line "I was imprisoned... but I escaped!", which while hilarious for those who knew better, just made me feel like I was missing out on an in-joke - but it was a good night out, all things considering! After four hours sleep and a day wandering around Guildford, then, I collapsed into bed exhausted not long after arriving home last night.

 

It was really lovely to see people, and also to stay with Pippa. Call it nosiness, but I do like meeting friends' families and home friends and seeing where they live! Pippa's home lifestyle is incredibly different to my own (in the vein of Flix's entry), but I now have much more of an idea of where she's coming from - literally - and that only can help in getting to know her better**. Surrey is very different from how I imagined, although I'm not entirely sure in what way. More affluent, less affluent? More rural, less rural? Prettier? Certainly more densely-populated. It's all part of the travelling experience :-)

 

 

*Although probably more conducive to getting work done, which in my sister's case especially has been important. I was going to say 'that made things a lot quieter', but then I reflected that Cat and Helena came with us!

**Not that I feel that I don't know her, particularly. But the principle goes for any of my university/non-home friends/ the stupidly large proportion of Durham University that comes from Surrey. It's a funny feeling, only ever meeting someone 300 miles away from where they call home.

 

 

 

This is a blog entry that I have been meaning to write for quite some time. It was inspired, albeit somewhat tangentially, by one of Callan's, where he discussed his lack of belief in God. I have already responded to Callan in person about how my own conclusions differ - and while I am quite happy to discuss my beliefs, I'm not entirely comfortable with doing so on here. It feels a bit too personal for that. The second strand to my response, however, was delayed 'til I got round to writing this (only I never did!), and is with regard to the nature of proof.

 

 

The language of proof has to be very specific to convey its meaning. A rigourous, logical argument is one in which no holes can be found and no choices have to be made. Logicians work with true and false statements - there is no grey area, no inbetween ground. To prove something, you have to show that it is true and will always be true however you approach it. The way that proof works is by implication; either you define an abstract concept imposing whatever conditions you like and see what that implies, or you make an assumption and see if you work your way round to any contradictions.

 

Implication can work two ways - 'necessary' and 'sufficient'. Let's say you have two statements, A and B. "A implies B" means that if A is true, then B must also be true; proving A is sufficient work to prove B. Equally, if A being true is necessary for B to be true, then if B is true, so is A - "B implies A"*. Conditions can be necessary or sufficient without being both - if two statements are necessary and sufficient then both "A implies B" and "B implies A" hold; so A and B are logically equivalent to each other. In Maths, this tends to be phrased as 'if and only if' (generally abbreviated to 'iff'), and you've got to be damn careful when you're drawing your implication arrows that you don't jump to any conclusions incorrectly! If you are trying to prove that two statements are logically equivalent then the easiest way to do it is to separately show that A implies B and that B implies A. Sometimes the two-way implication is almost trivial - sometimes it's considerably less so.

 

As a general rule, working from previously proved results makes one's life a lot easier. At some point, everything in Maths that we take for granted will have been proved. Take counting. To us it is obvious that 2 + 4 = 6, irrespective of base notation. Consider apples - six apples is still six apples whether it's written '6 apples' in decimal or '110 apples' in binary. But when we 'count' those apples, a concept which has been instilled in us from a very early age, what we are really doing is constructing a bijection (one-to-one map or correspondance) between the set (collection) of apples and the set of Natural numbers in a very specific way. Think of remote tribal civilisations who only have words for 'one', 'two', and 'many' - for them, the concept of 'six' is by no means intuitive. But hell, it's a concept that we find very useful, and it would be a pain if we had to return to very first principles and reconstruct the bijection every time we went shopping.

 

The danger, of course, is that we assume the truth of too much - working from a result is only meaningful if that result is true in the first place because a single incorrect assumption can lead an otherwise sound proof to a completely false conclusion. To give a simple example, consider the assumption that all siblings have the same colouring. I then tell you that my brother is blond with blue eyes (which as it happens is perfectly true). It would be a completely logical step to deduce that I am also blonde and have blue eyes, but as any of my 'real-life' friends will tell you, this most certainly isn't the case. But given the assumption and given the statement of fact, there is no flaw in the deduction so it would technically have been proved, albeit incorrectly.**

 

There's a whole set of notation for this sort of thing (mathematicians are lazy!***) and a whole load of rigourous implication statements that can be derived. But those are the fundamental ideas behind the concept of proof, and they form the basis for the whole of both Maths and Philosophy.

 

So to return to Callan's blog on religion (another of which went up since I've been typing this entry), this is where you cannot prove anything to do with God, or religion, or faith. This is where faith of any kind, atheism included, rests on assumptions and takes 'logical' steps that cannot be declared true or false by anyone. The only truly logical way to approach religion is to be agnostic - to hold personal beliefs, maybe, but acknowledge that they can never be right.

 

 

*You can see the real potential for problems when lecturers have broken English?

**As a side-note, I've often wondered what it would like to be blonde. Not in a I'm-going-to-dye-my-hair-tomorrow way, but in a what-if-every-time-I-looked-in-the-mirror-there-was-someone-standing-there-with-pale-hair-and-blue-eyes? way. I wonder whether it would affect my personality? I digress.

***Though to be fair, why write out "For all positive numbers delta, there exists a positive number epsilon such that if the distance between x and a point c is less than delta, then the distance between the function evaluated at x and the function evaluated at c is less than epsilon", when you could have "δ>0, ε>0 s.t. |x-c|<δ => |f(x) - f(c)| < ε" ? (- proof for continuity of a function, in case anyone's interested...)

 

In the past three days I have spent £67.95 on thetrainline.com. That sounds like a lot until you realise that it covers my entire mainline-Britain travel needs for the next five weeks (and three lots of postage because I was disorganised and didn't book everything at once) -

 

Durham to Birmingham

Welshpool to Guildford

Guildford to Birmingham

Birmingham to Leeds

Leeds to Birmingham

Birmingham to High Wycombe

Gatwick to Birmingham

 

And most of those cover me out to my local train station too. Something tells me that not enough revision is going to get done this holiday. It's gonna be awesome!

 

Today, just for the hell of it, I went to an English lecture.

 

Basically I wanted to find out what it was like! One of the things that I really liked about A-Level was the variety, both in terms of the material and in terms of the variety of the way it was taught. I suppose that my combination – Maths + Further Maths, Music, Classical Civilisation, French AS – promoted that more than some, but the potential’s there for anyone. By the time you get to degree level, however, the expectation is that you specialise a bit more, and that’s fair enough. (In fact one of the things that Durham prides itself on is the flexibility of its Natural Sciences/Combined Social Sciences/Combined Arts programmes – you can basically pick up to three subjects and so long as they can timetable it, that’s your degree. There are big downsides to this, especially in Science, but this isn’t a prospectus so I won’t go into them now.)

 

Through doing ‘elective’ (free-choice) modules last year, I have had direct experience of three departments – Maths, obviously, Music, and Computer Science. Apart from any subject knowledge gained, the differences between the three different Depts. and their approach to undergraduate learning were fascinating. The level of support and indeed input in Music, for example, was absolutely minimal, which gave me much more respect for arts degrees than I might otherwise have had; the lectures in Computer Science were much more practical-based (possibly to do with the nature of the module) but also relied much more heavily on PowerPoint slides and print-outs; Maths set much more regular homework but this in turn meant that we received more practice at exam-style questions etc, etc.. Of course to an extent the precise lecture style will vary from course to course, from lecturer to lecturer even within a subject. But there is generally more cohesion, an accepted style of teaching.

 

Maths, for instance, works with pretty abstract material in the main. It’s proof-heavy*, rigourous, and logical. Long gone are the days of GCSE and A-Level where they taught you a method and then made you practise applying it. Nowadays we work with a brief section where the lecturer outlines a need for a particular result (‘Motivation:’), followed by an outline of the first principles or previously proved results from which we will work (‘Recall:’) and probably a definition (‘Defⁿ:’). Next (depending on the lecturer/the material), there will either follow a long section of reasoning which cumulates in a result, or they will progress through ‘Lemma:’s, ‘Theorem:’s, ‘Corollary:’s, each of which has a subtly different implication and each of which will be followed by a ‘Proof:’. Occasionally there are ‘Remark:’s, or even ‘Fact:’s (when the lecturer can’t be bothered to do a proof or it isn’t in the course for some reason). Then there’ll hopefully be an example (‘e.g.:’) or two. Some lecturers sub-divide their notes precisely, everything given a reference number down to ‘Remark 13.5:’. Some just use section headings and expect us to use our common sense – but only the really bad lecturers don’t structure their notes at all. Maths is incredibly cumulative, so if you weren’t reasonably organised about notes you wouldn’t be in with a chance in hell.

 

More to the point, everything is hand-written. The lecturer writes everything up on the whiteboard/blackboard and we copy down. Yes, it makes it hell to catch up if you miss something, but at least we’re actively processing the information – I hate, hate, hate lecturers who just read off handouts or stick everything on PowerPoint slides with the excuse that it’s on DUO (Durham University Online). And then we have problems classes which are similar to lectures but where they just go through a load of examples, and tutorials where we try questions ourselves and can ask for help if needs be. And then we get set homeworks.

 

So it was fascinating to go to an English lecture where for the hour (their entire week’s input for that module), a woman just stood and talked at them for an hour. They had a couple of handouts with extracts from several books on and a long back-up reading list, but she didn’t use them much. What really struck me was how abstract it all was – she was less discussing the literature itself than the concepts, ‘civilisation’ as it was viewed in the 1930s, the role of literature in the progression of ‘thought’. Even if you could write at the speed of light (and believe me, some of the people in there weren’t far off), you could never have got down half of what she said. I think I’d purchase a Dictaphone if I was on an arts degree. And yes, it was interesting, parts of it particularly so, but it all felt rather directionless.

 

I do not think that I have missed my academic vocation in not taking English. But it was a good thing to go to. It gives you more idea of what your fellow students are dealing with. Next on the list, I reckon, would be Geography or possibly Psychology...

 

 

*, which reminds me, actually, of an entry I was going to write a couple of weeks ago or so. One for the to-do list!

 

Eight or nine of us students from Quaker Meeting got together this evening for an 'Epilogue'. We sat in a circle in a small-ish room in St John's, candles in the middle, and held what was effectively silent worship in our normal form - only this time there was a speaking stone, and the emphasis was on looking back over the past term, from whatever point of view we chose. We ended by each saying what we were letting go - be it stress, tiredness, relationship issues, whatever.

 

I haven't had a great few weeks recently for one reason or another. This was just what was needed.

 

 

I love gathering with the other Quaker students. There are few enough of us that we all know each other reasonably well, although enough to provide new faces once in a while and some good variety of discussion. We are of all ages, of all colleges, of all degree disciplines - we come from all over the place, and have all different life stories. We do not spend day in, day out with each other in the most part, and mutual friends tend to be symptomatic of how small a world Durham University is, not through direct connections. But what we share is a belief system and a set of core values. That's a pretty powerful thing.

 

It's funny. You don't notice an absence of pain until it's gone - that is, until it hurts, it hurts, it hurts all the time for days on end.

 

Today has not been a good day.

 

 

I would like to defer this entry to Callan's most recent (if he doesn't mind!). Although I had heard of the concept and the Hemingway quote before, I am still having to think quite hard about my own response, and as such am not commenting yet. But particularly in the context of my last post, it'd be interesting to see what you'd all put!

 

(Credit to you, Callan, for managing to distract me from the long, incoherent and ultimately futile rage-against-the-Maths-Dept that was going to be posted here instead... Oh, and I'm pretty sure I know which one yours is :-) )

 

I need a rant. I really need a rant. Preferably, I need a rant to someone who I have never met, am never likely to meet, and who will just listen and say one or two words in comfort or in advice. I find it ironic that when it comes to certain things in life then the better I know someone, the harder I find it open up to them - and for my friends reading this, don't worry, it is relatively few things, but it is a problem nonetheless!

 

I sometimes use this blog as a release - to discuss an issue bugging me or to ask for advice on something I'm not sure about, which is great. But I have a policy of censoring somewhat what I talk about. I won't discuss family issues, or things that are hugely personal to me. I won't talk about something if it feels like a personal attack on someone, and as a general rule (although there are exceptions), I try hard not to talk about friends as any more than names to give context because I know that potentially they could read this and feel like I was talking about them behind their backs. Eighteen months into university, an increasing number of people who I spend my days with are also on my MSN from which this site links. There is a link to here from my Facebook profile which I know from the Stats page gets used semi-regularly (- and although the chances are that it's just Kat or Dickie or somebody who I know reads regularly clicking on the link as a convenience, it still could be any of my Facebook friends or anyone from Durham). Yes, the internet is a place to publish our thoughts and to express creativity in whatever way we choose, and yes, if people don't want to read what's written on here then nobody's forcing them to. But just because we say something in cyberspace, it still has the potential to give as much impact as if it were said out loud or written down with pen and paper.

 

And as commented on an entry of Hannah's not 24 hours ago (I find the timing somewhat ironic), you can only block out strangers with privacy settings. You can't block out people you know without preventing everyone and anyone from viewing it, which right now really doesn't help!