Hi all,

I have noticed that many of the math helpers in the Nrich forum seem to be mathematical students from Cambridge University colleges. I have a few questions. How is it like to do maths at Cambridge? What are the good things you gain from the math course? Do you do more pure work tham mechanics or visa versa? Does the work sometime go over your own head? If so do you get it cleared with other students and teachers? How often do you take internal tests? How did you get into Cambridge? How did your interviews go? Were they difficult? What kind of questions did they asked you? Were you able to answer them alright? Did you make any mistakes? How did the interviewers aid you to show your potential within the interview time? Does any extra reading help you do better in the interview?

If you have any particular interests in maths can you do your own extra work and research? Is anyone here interested in pursuing a career in research in AI? I would like to know.

I hope no one takes any offence at any of these questions. I'm just querious ;)

HAL2001

Best Regards

Hi, Hal2001

First of all, I have to say that the questions for interview can be found in the thread

Do you think Cambridge (or indeed Oxford) is elitist?

under Open Discussion.

Now for the other questions: For the first year, there seems to be a good balance of pure and applied maths. One can choose to specialise later on, but it obviously does not hurt for learning more before you specialise. You can find out more about Maths in Cambridge in the CU Undergraduate prospectus. Studying Mathematics here in Cambridge is in a rather relax atmosphere, but information will not be spoon-fed to you as in school.

First of all, note that there are some previous discussions about this sort of thing, if you go digging in the archives, ironically the discussion called ''Are Oxford and Cambridge elitist'' might be useful to you.

Now for your questions: doing maths at Cambridge is great if you like to challenge yourself mathematically. You can do as much pure / applied as you like, although in the first couple of years you have to do quite a lot of both, you can specialise in your third / fourth year, I'm only doing pure this year for instance (3rd yr). Almost nobody will be in a position that the work doesn't go over their head at some point (there are exceptions, although I'm not one of them), usually this isn't too much of a problem as you get lots of almost one on one supervision with very clever people. Internal tests? Um, end of year exams, and mock exams in the middle of the first year IIRC. The interviews vary, I found mine very difficult, others I talked to seemed to have had easier ones. My impression (and the impression of lots of other people when we discussed this before) is that they test potential rather well, not relying on your knowing lots of theory before you go there. My interview had lots of matrix questions (exponentials of matrices and complex numbers), and I hadn't even studied matrices except at GCSE level, but I still managed to get in. I would recommend extra reading, as it certainly won't be a bad thing. Basically, you should definitely apply if you want to.

If you're interested in AI, you might also consider applying to do maths+computer science, you get to do some computer stuff in your first year, and you can then change to do either computer science or maths in the subsequent years. You can certainly try and do research or extra work, but you'll probably have enough work to do doing the sheets. There's always the (very long) holidays though.

Thank you Kerwin and Dan for your posts.

Much appreciated!

I had a read of the previous post on the contraversial topic of Oxford and Cam being elitist. I am of the opinion, from the majority of postings at nrich, the people seem to be down to earth ppl with a love for maths. And if thats elitist, let it be so. ;)

Any recommendations on any math books that you've ppl like? I reckon you math CAM circle ppl have read some nice gems.

Signing Out...

Ian Stewart is very good.

There is a reading list on the maths department webpage, try reading list

One good book is ''A Mathematician's Apology'' by G. H. Hardy. Also, ''Ideas and Opinions'' by Albert Einstein is good, although I'm afraid it isn't based primarily around maths.

Thanks Dan, Tom, Marion and Brad for your suggestions!

I shall look up the books through Amazon.co.uk.

They seem to have some good brief reviews of many books. So, I'll check it out.

Has anyone seen the movie called Pi?

Thanks once again.

Hi again,

If anyone know of any other AI related books I would appreciate it if any suggestion could be made. (Dan - thanks for the book suggestion- i'll look it up)

If you know of any teachers/lecturers/students who are interestedin AI, maybe you could badger them for some recomendations! Both technical and non-technical books.

Regards

Two non-technical books on the mathematical and physical aspects of AI are The Emperor's New Mind and the sequel Shadows of the Mind by Roger Penrose.

I wouldn't recommend ''The Uncertain Reasoner's Companion'' to anyone who hasn't done some logic and some probability theory, it's intended as a reference and introduction for higher level study and research (late in a degree course or in postgraduate work).

Thanks for the book suggestions Sean, Dan and Marcus.

I've ordered a copy of Gdel, Escher, Back. The review for the book seem to be impressive. Until I get my copy, anyone care elaborate more? Was'nt Escher the artist?? How is this related to maths, logic and AI? I have also ordered a few by Martin Gardner. I looked at www.Sciam.com (Scientific Americian) but could not find any of his articles there. Anyone know of any online depositaries of his work I could read?

Marcus, how long did it take you to get a doctorate? What was the doctorate specifically in? In a branch of maths I guess? Did you enjoy it? What scope does it bring to your future in terms of career etc. Would you recommend it doing a doctorate?

Regards

Is Deep Blue (the IBM chess playing thing) considered an AI application? By the way, it is the software that they are naming Deep Blue, and not the hardware that it runs on I gather. I would have thought that a chess playing software can't better itself, the essence of what you would want to achieve in pure AI. So implying Deep Blue is just smart coding (mathematically deep!?)and not AI?

Has anyone used the AI bot at www.AliceBot.org, I think that specifically is not AI either??? How can really pure AI be acheieved, if at all. Can a machine really achieve 'self-conscienceness' or a 'soul', like human beings I guess. Or is that not needed to be branded Intelliegent?

This is cool. It is good to see such an interest in maths not only evidenced by threads of conversation like this but by the tremendous response from young people to all sections of NRICH.

If you want to read online the best new popular articles on mathematics then go to The Mathematical Association of America Current Columns.

You will find the latest by Ivars Peterson, Keith Devlin, Alex Bogomolny, Frank Morgan et al. and a rich archive of past columns, great stuff!

Also don't forget Plus, a new edition was published a few days ago.

Enjoy!

Someone I know says his ambition is to live long enough to work with a computer that is proud of him.

It really depebds on what you call AI. Certainly the people working on neural networks, expert systems and the like wold consider themselves as working in AI and this is the sort of stuff we find in the text books. They count as AI because they reproduce artificially some function that we perform with our brains, say making decisions based on inconclusive evidence (such as diagnostics in medicine) and predicting things and larn from mistakes. There are two approachs, expert systems attempt to reproduce results we might have ascribed to intelligence but make no attempt to mimic the way intelligence works, so there is a front that looks intelligent but the working is clearly not, such is the case of Deep Blue. The other, bottom-up attempts to start from modelling the brain, these are neural networks.

However, all these things are only good at one particular area, one skill. The multifunction intelligent robot like those in the Asimov books is not on the cards in the foreseeable future, I don't think anyone has any idea how to even begin designing one. And as Dan said we might not even recognise it when we see it. We don't even understand how our own brain works, we don't understand if consciousness even means anything when applied to people, so you can just imagine how hard it would be to build something...

If you want to have a look at some AI, there's a computer game called Creatures which has little characters whose behaviour is governed by a developing neural network and a genetic code. There's a system of rewards and punishments so that you can train them. Some quite odd things can be made to happen: one of the creators tells a story about two creatures learning how to play catch with each other and I've had one that does everything asleep. There are currently three versions, Creatures, Creatures 2 and Creatures 3, with rising costs, but Creatures can be picked up for about £5 now.

Thanks Mrs Beardon for your input and the links. I'll check them out!

Tom, do you have the link or details of where I can get this AI app that you described?

I now also agree that Deep Blue is not pure AI. But surely the techniques and theory created in devoloping small 'singular' thinking progs that do things better than their creators (ie us- the humans) is a step in the right direction to the eventual pure AI 'thing'. In my view, an AI app needs to be built around thinking processes, maybe something not related to reality (ie chair, face, book, place, guilt etc) but a reality of numbers balancing or an inbalance of numbers. Letting some element of 'randomness' involved in the decision processing task. The final hack should be that allows the AI 'thing' to outthink its creator and lead a independent life, setting its own goals and tasks as we humans do. You can think of them as our future children?!

I hope my thoughts are not too cluttered, but I simply put what came to mind of my feeling towards AI.

Regards

Can I just add that I think when we make the break though in into 'real' AI, it will be by a mathematician! ;)

In regards to How escher is related to AI and maths, he based a lot of his ideas around geometric paradoxes and shapes (I think he made a very famous version of the Mobius Strip). Also, he believed that the route of explaining consiousness lied within self reference(see his [very famous] artwork in which one hand draws the hand that it drawing itself, or his one wiht three spheres), which has much to do with AI.

As far as whether computers can ever emulate human consciousness, AI is far from what I know well, but I don't think so. Given current notions of causualty, the only way to achieve self-awareness is through infinite regression or a paradox. The infinite regresion is because a possible way to achieve conscious is thought A is detected by thought B, which is detected by thought C, ad infinitum. This leads to never achieving consciousness. The example of a paradox conscious is through thought A being detected by thought B, and thought B being detected by thought A. but close examination reveals that this means that A must precede B, and B must precede A: a paradox (one exacly analogous to Escher's drawing). So, until someone comes up wiht another way for consciousness, I don't see how it can be replicated. If anyone else has any thoughts on the matter, I would really enjoy hearing them, as the matter of a conscious mind is something I've been pondering in great detail since the discussion ''An Irrelavent Question'', parts one and two.

Thanks,

Well, certainly programs like Deep Blue are not useless, research into these techniques greatly improves expert systems and the like which have all sorts of uses as described by Sean above. Whether or not they're a step towards true AI is another matter, it might be, as I suspect, that true AI requires a whole new paradigm.

The last point is important, systems like Deep Blue are perhaps more of a red herring than a step in the direction of genuine intelligence because there is no attempt to build something that can works internally as something we might consider intelligent would.

I'm not so sure it would be a mathematician... My bets would be on a neuroscientist, computer scientist or physicist. The first two perhaps for obvious reasons and the third because it may be (and this is what Penrose's books are about) the quantum mechanics (or even quantum gravity!!) is important for a complete understanding of the brain. Mathematicians do not generally make the best scientists, especially when it is a 'messy' science like neuroscience.

What I'm saying though, is that it will be hard to ''write up a program'' with a consciousness or even create a machine with a consciousness because we have no idea how we have a consciousness, and the two most probable ways to explain it result in either a paradox, or a lack of causation. So, perhaps it is right that the consciousness lies in quantum mechanics, but how then would we have control (or even an illusion of control, for that matter) over our own thoughts. Furthermore, wouldn't quantum mechanics neccessarily have to incorporate time travel to allude the paradox of self-reference in my non-infinite explanation for the conscious mind? And, wouldn't this make us in control of quantum mechanics?

Hal, the link to the official site is http://www.creatures.co.uk/. As far as I know, you can't buy the program there, so you've got a walk round your local computer games stores, or a trawl through the internet.

Personally, I'm inclined to believe that a neural network is sufficient for intelligence and conciousness.

Brad, I guess that there are two possibilities; consciousness is a property (in some way) of specific sorts or arrangements of matter OR consciousness is a property of systems with sufficient algorithmic complexity in some way. We really have no way of knowing which is true, or whether it's another possibility entirely. If it's the former, then no piece of software can be conscious, which doesn't rule out the possibility of machine consciousness, it just rules out ''consciousness on your PC''. However, if it's the latter, ''consciousness on your PC'' (perhaps Microsoft ConsciousEntityTM) is a possibility.

Another interesting property of consciousness is that it could be a product of a metaphysical realm, rather than just physical realm. At current, I am very inclined to believe that it exists entirely by physical laws, but there is nothing ruling out the other possiblity...

Hi, Hal (and the rest of you).

This is a great discussion, on a topic which has always interested me hugely. The nature of consciousness is the central theme in ''Godel, Escher, Bach'', and Hofstadter uses the common theme of recusrion and self reference in Godel's proof of his incompleteness theorem, Escher's art (such as the self drawing hands) and Bach's music (notably his canons) to illustrate his ideas.

Consciousness itself is hard to define, let alone detect in another entity. It is easy for me to convince myself that I am conscious. It is harder for me to say that (for example) any of you are conscious, rather than sophisticated text parsing bots. We tend to assume that people we know to be flesh and blood human beings are conscious, but we really do that by comparison to ourselves - ''you are human, I am human, I am conscious, therefore I infer that you are conscious''. One famous criterion for consciousness (or, as it is usually put, intelligence) is the Turing test. In case you are unaware of it, what you do is converse with two entities via a textual or other indirect interface. The communication is completely free, you can ask whatever you like of either entity. Their objective is to convince you that they are human. There are a couple of problems with the test. Firstly, text parsing bots are becoming increasingly sophisticated - some can fool a decent percentage (30 or 40 last time I looked) of people interacting with them over the Internet (bear in mind one of the central tenets of my life, though - ''People Are Stupid''). These bots are things nobody could classify as intelligent - they are at best ''conversation expert systems''. The second problem is that we are here defining ''intelligent'' as ''like us''. It is not difficult to envisage some alien who is as intelligent as we are, completely conscious with a sense of morality and any other paraphenalia you care to hang on the notion of personhood, and who is also totally incapable of passing the Turing test, since its intelligence is simply different from ours.

On the other hand, it is easy to dismiss some intelligent behaviour as merely ''simulating'' intelligence. An example I recall form GEB is of a (ficticious) argument that women are not conscious, since they do not think, they have internal processes different from real conscious beings' - they ''womansee'' things, ''womanthink'' about them and come up with responses that, although identical to those of people who really see and think, are a product of mere mechanistic processes, not conscious thought. It is actually not possible to prove the falsity of this to a man - and a woman claiming it to be false is producing the effects of ''womanthought'', so this is not proof at all. Now change ''woman'' to ''AI'' and you can see the difficulty of defining ''consciousness'' as having to happen in a meat brain (as opposed to a male brain).

It's a thorny problem, and not one which will be solved until we have an AI which is generally acknowledged as intelligent - which is to say, I guess, that my answer to the question ''What is true intelligence?'' is ''We'll know it when we see it.''

Thanks Tom! for the link, I'll check it out.

Brad, Dan, Sean - thanks for sharing your carefully placed out thoughts. AI does indeed interest me and I think the amount of locked potential that could reap benefits for mankind is huge.

Marcus, thanks for your open enlightening post. Are you considering to further research in the field of AI or reasoning under uncertainty? If so when and with what type of a team? I can see a ?possible? future for me in the field of AI research (in whatever form that may take). I want maths to give me the technical rigor that I feel will be essential in any field of research I might peruse.

The note where we talked about who would the AI breakthrough, my opinion is reformed after reading the above posts, and I feel that a mixture of people with varying expertise in various areas will be the people that possibly makes the AI breakthrough. Team of researchers: mathematicians, neuroscientists, philosophers, computer scientists and an artist.

Marcus, after you take a degree in maths what is the next step towards doing a PhD? What all do you have to accomplish to get your PhD? How flexible is it in terms of the research that you are allowed to do? Are you allowed to do the research for a company and then use it for your PhD? I guess then you can get all three, 1.money/financial stability and 2.the satisfaction that the research brings you, finally, 3.Possible job security at the company for whom you did the research for. What do you think? Also, what type of companies would allow their reseach team double as students doing their PhD in the field that they are doing research for the company as well.?

Yesterday I recived some of the Martin Gardner book I ordered. I look forward to reading them. Thank you everyone who told me about Mr Gardners books! I also should be getting the GEB book later this week.

Regards

I was wondering...

What kind of applications could a real AI 'thing' be used for. How will it help us lead better lives?

Hi again, Hal.

In my case, the route from first degree to PhD was mainly through knowing the right people and being in the right place at the right time. If you get a good degree, it's not too hard to be accepted onto an MSc or PhD course, the tough thing, especially in pure mathematics, is getting funding. I was lucky enough to be in a position where late funding came up and I was in a position to take advantage of it. As for my own future as a researcher, I may develop one or two more ideas from my thesis with my supervisors for publication (one is in the works as we speak), but realistically my full time teaching job takes up the vast majority of my time.

Incidentally, I would tend to agree that a cross disciplinary team is the most likely contender for a big AI breakthrough.

More generally, on the higher degree front, things are really quite flexible. As I said before, getting on a course is relatively easy compared to getting funding. Funding sources include the Government (via the research councils, for most of Maths that'd be the EPSRC, who funded me) either funding you as a student or your project as an entity in its own right; private research grants (few and far between, for the best projects only); and private companies, who would generally expect your research to directly or indirectly benefit them and for you to work for them during your studentship. As far as the actual project is concerned, if you can find someone who will supervise you, it's really wide open. My best advice is to find out more about the people doing the kind of research you're interested in - who is publishing in the topics you like? Find out where these people work and see if they will take you as a student. Don't narrow your search too much - a supervisor only needs to understand and be able to help you with your project, you don't need to be extending his or her central research. Of course, if you're after industrial funding (and the higher paid world of industrial research as opposed to academia after you finish your thesis) you need to coordinate your choice of project with the funding company.

Technically, what you have to accomplish to be awarded a PhD is ''a significant and original contribution to the sum of human knowledge''.

Hal, rather depressingly, the only applications of AI that I've heard about, apart from web bots and gaming, are military. For example, the company that produces Creatures is also working on using neural networks to pilot fighter planes.

Fortunately, there seems to be a happy incompetence in the armed forces; one program was set up to use neural networks to spot camouflaged tanks, they took photos of scenes with tanks in and without and fed them through the network. It seemed to be working, until they tried a new set of photos and discovered that in the initial set of photos, the ones containing tanks had been taken on a different day to the ones without...so the neural network was able to tell whether there was a tank there or not by whether or not the sky was cloudy.

Tom, I am not exactly sure (as you and many of the other in this nrich forum might agree too) that AI should not aid and albit to the progress of war arsenal (like the highly popularised einstien and the atom bomb did for WW2). Science helped the destruction of thousands (or even in the long run millions) of innocent people :( If you were offered money by the military to carry out AI research, you would probably say yes, simple becuase of the money, it'll allow you to carry out your passion. But in the long run your work could/?will? be used to kill other human beings. This is very disturbing. Then again the same can be said for so many other things in life....

Marcus, what do you consider a good degree? Please don't hesitate to answer, you won't offend anyone, because you are stating opinions and everyone is entitled to one, I think, in most parts of the world.

How much in terms of cost are we looking at in funding a PhD? Do ppl often stop half way through a PhD because of the climbing costs? If so what happens to all the work and effort they put in?

Do you know of any ppl wheelchair bound who have succeeded in obtaining funding for PhD or research in the area of Mathematics (or AI related). Do they find any descrimination against them simply becuase of their disability? I would be interested to know.

Another thing I have been wondering is the following: I have always wanted to do research in a field of mathematics. Though it would be very satisfying to follow your heart and passion, the general message I get is that they are much under paid, and ppl don't employ them when they finally want to get a job (for reason I have no idea). What is myth and what is fact? No one can deny that they would not like to earn more money than they do. So what do ppl who have been trained by the rigor of academia find if they go into industrial research? Is it worth it?? And can they do the transition from academia (driven be the passion of their field) to (the money driven) industry?

Anyone else contemplating doing a PhD?

What are your thoughts?

Regards

Tom, that story is as old as the hills, my Dad told me that one years ago, and I'm sure that he must have heard about it at least 15 years before he told me. However, I agree with your thesis, remember those guided missiles which almost all missed their targets? I think it was either in Serbia recently or in Iraq longer ago.

Hal, a good example of a wheelchair bound guy getting a PhD would be... Stephen Hawking, co-discoverer of Black Holes, world-renowned popular scientist!

Prof. Stephen Hawking got his PhD after he was wheelchair bound, is this true? I think. I also read that his PhD was about Black Holes, is that true? I think I read it in a edition of Readers Digest a few years back, my memory on that reading is not quite 100 theoretical physist/cosmologist, are there many wheelchair bound PhD in mathematics in the UK? I would be interested to know. I have tremendous amount of respect for Proff. Stephen Hawking, he blatently never allows his disability to get in the way of his passion - cosmology. He is an inspiration to us all.

Dan, though I am still a long way off from doing a PhD in anything, my personal view is that is good to think ahead, thats one of the reasons I want hear what other ppl think about doing PhD, so that mistakes can be learned from other ppls hard earned experience.

Also Dan, even though you have not made a decision on what exact area you may do your PhD in, what area of math are you thinking in doing a PhD in?

Is it common or uncommon for ppl to tackle PhD after their degree, what makes ppl choose to go into industry rather than continue academia? Is the pull simply the money? Pls don't flame me for this question, just interested in what other mathematician think.

Does the education background of your parents/family influence on what you acheieve at higher education and your curiousity levels? Is your potential stunted simply becuase your family has not bkground of academia? Or is this complete nonsense?

Regards

Thank you Anon for the Hawking link!

Regards

Hal, not yet, although it'll probably be something in the pure maths line. My two main interests at the moment are in metamathematics (set theory, logic, computability, decidability, etc.) and geometric stuff like topology, complex analysis, etc.

When you choose to do your PhD, do you choose something (a branch of math) that you like/enjoy doing, or something that you scored very high in exams? I guess this is a contradictory statement, as, if you can do it well you normally enjoy it too.

You know of any math jokes?

Regards

I think you choose something you like doing from the set of things you can do well :).

You probably won't get this one for a few years, but here goes:

Let $\epsilon <0$.

Other ones:

Q: What did the constipated mathematician do?

A: He worked it out with a pencil.

Q: What did the very constipated mathematician do?

A: He used logs.

Q: Why did the chicken cross the Möbius strip?

A: To get to the same side.

Some say the pope is the greatest cardinal.

But others insist this cannot be so, as every pope has a successor.

Another favourite:

A mathematician went insane and believed that he was the differentiation operator. His friends had him placed in a mental hospital until he got better. All day he would go around frightening the other patients by staring at them and saying ''I differentiate you!''

One day he met a new patient; and true to form he stared at him and said ''I differentiate you!'', but for once, his victim's expression didn't change. Surprised, the mathematician marshalled his energies, stared fiercely at the new patient and said loudly ''I differentiate you!'', but still the other man had no reaction. Finally, in frustration, the mathematician screamed out ''I DIFFERENTIATE YOU!''

The new patient calmly looked up and said, ''You can differentiate me all you like: I'm e to the x.''

A really bad one:

Two mathematicians are in a bar. The first one says to the second that the average person knows very little about basic mathematics. The second one disagrees, and claims that most people can cope with a reasonable amount of math.

The first mathematician goes off to the washroom, and in his absence the second calls over the waitress. He tells her that in a few minutes, after his friend has returned, he will call her over and ask her a question. All she has to do is answer one third x cubed.

She repeats ''one thir - dex cue''?

He repeats ''one third x cubed''.

Her: `one thir dex cuebd'? Yes, that's right, he says. So she agrees, and goes off mumbling to herself, ''one thir dex cuebd...''.

The first guy returns and the second proposes a bet to prove his point, that most people do know something about basic math. He says he will ask the blonde waitress an integral, and the first laughingly agrees. The second man calls over the waitress and asks ''What is the integral of x squared?''.

lol - good ones Dan! ;)

Regards

What's the difference between an introvert and and extrovert mathematician?

An extrovert mathematician looks at your feet when he's talking to you.

By the way, I think I probably would refuse to work on military applications. Call me ethical if you like.

I don't know that military applications to physics or AI are always evil. Much of the time, fighting a war must be looked at as sorrow leading to happiness. To obtain what is right, it is sometimes neccessary to destroy what is wrong. The way I look at some things, particularly the A-Bomb, is that it was killing thousands to save millions.

Concerning whether parental success affects your appreciation for maths or academia, I would say at times, but not always. While parents educated on how to raise children can almost always make a difference to their child's attitude towards learning (and the opposite is true for parents unconcerned with their child), not all parents have to be involved with maths to be able to cause an interest in maths in their children.

I'm also interested in getting a PhD. but it's still 8 years until graduate school. So, before I commit to anything, I'd like to reiterate Hal's request of how much does a PhD cost? And how much does the average PhD grad. make? I would be very appreciative if anyone could answer either of these.

Thanks for your replies (and the good jokes!:))

What did the English Student say to the Math Student? ''Why does one plus one equal to two?''

Why did the Math student frown? And what did he reply to the English Student?

Sad joke really, I just made it up!

Regards

Well, I'm a pacifist, so I certainly wouldn't do any research which would lead to direct military applications (it would be silly to try not to do any research that might possibly lead to military applications, as almost anything could). I think it would be very difficult to prove that bombing Hiroshima saved millions of lives, unfortunately my knowledge of history isn't good enough for me to make the case against using the A-bomb on Hiroshima. However the book I'm reading at the moment (Lessons from Kosovo by Noam Chomsky) claims that the effect of bombing Serbia was to escalate atrocities, and that moreover the people in command predicted that this would be the likely outcome of the bombing campaign.

'Evil' is an indefinable and pointless word. All the same, although this post is now rather off topic I must dispute your claim that the Atomic bomb 'killed thousands to save millions.' Certainly, for obvious reasons, this was the claim made by the US after the war. However the allies were winning the war. If the Americans had wanted to demonstrate the power of the bomb they could quite easily have dropped it in a less densely populated area. In fact they dropped it twice over extremely populated areas. I find it hard to draw any conclusion other than that they were trying kill people and lots of them.

Can you give any applications of technology to the military that have not lead to more brutal and extensive destruction? Is there any reason why AI should not lead thus?

Hal, the theoretical minimum degree to be accepted onto an MSc course is a 2(ii), for a PhD it's a 2(i). However, in practice, you would almost always need one more than those (i.e. a 2(i) for a Master's, a first for a PhD). The actual cost you should be able to look up exactly on Universities' websites (I expect, I haven't tried). IIRC, tuition costs are around one to two thousand pounds a year (it's more if you go into an experimental science). The main cost is not the tuition but the living costs whilst you do the degree. I, not unusually, did three years fully funded (i.e. the Engineering and Physical Sciences Research Council paid for both my tuition and breadline living expenses) followed by a ''writing up year''. This entails a nominal fee (50 to 100 pounds, I think) and no funding. You're not supposed to be doing any more actual research, although it is not unusual in practice for a little polishing to be done in a writing up year. During that time it is usual to work part time to pay for living expenses - luckily, as a qualified maths teacher, supply teaching work wasn't hard to come by for me. Ideally, of course, you should get the thesis done in the three year funded period, but overruning, if not the norm, is certainly so common that no fuss is made over it.

As far as your own costs for doing a PhD are concerned, then, the only real ones are the reduced income (the postgraduate grant, even tax free as it is, is still less than a top flight maths graduate should be earning in industry), unless you take the unusual step of funding the whole lot yourself, in which case I'd put a ballpark figure of 20 to 30 thousand pounds on the undertaking.

Yet another route is to do the PhD part time whilst holding down a full time job. I have nothing but admiration for people who manage this - I found it hard enough just writing up whilst working on and off.

Just quickly on the subject of the A-bomb: I agree with Brad that the bombing of Hiroshima and Nagasaki was absolutely necessary to end WW2. You have to remember that Japan was then in a fanatical regime, and it is very doubtful they would have surrendered, even though they were rapidly losing lives (and the indications are the figthing was due to get very bloody, as the figthing was moving closer and closer to Japanese territory). The emperor (the dictator) had more or less isolated himself from what was happening outside. In this totalitarian regime individual lives were valued very little (Kamikaze are well documented - they are suicide fighters - it was found to be easier to cause mass destruction if the person causing the destruction didn't have to escape himself, so that's what they did!) So they were not going to surrender for rational reasons, and so the fighting would have continued and very probably many, many more people would have been killed on both sides. Only when the US demostrated they could, and were willing to, use these weapons were the Japanese forced to surrender.

Before President Truman ordered the dropping the bomb, the US dropped leaflets from aircraft informing the Japanese that they had developed a massively destructive weapon (giving near full specifications) warning them that they'd be forced to use it if the Japanese didn't surrender but the Japanese Emperor either never saw this (perhaps people were too frightened to approach him - they might have been executed) or simply didn't respond.

Furthermore there are arguments that the US should have dropped the bomb in the sea or something like that to scare the Japanese government into surrendering. But this was not an option for the very same reason. The Japanese would have been forced to cover up the fact that the US had a new super-weapon. Any other attitude would have been seen as anti- the war effort. Therefore dropping a bomb in the sea would have been a waste of resouces - remember U 235 was very limited in supply and was really the only hope the US had of ending the war quickly.

It goes to show just how fanatical the regime was that even when the US had dropped the bomb on Hiroshima the Japanese government didn't surrender, and it took another bomb at Nagasaki before they surrendered. Does this look like a case in which the war could have been resolved without huge loss of life?

I know it was a pretty terrible thing (about 64,000 Japanese killed at Hiroshima including deaths from radiation sickness later, plus about 30,000? at Nagasaki) but I think the death toll would have been on a different scale otherwise. (There would have been more Japanese deaths as well as American deaths.)

Having said all this, Truman's memoirs on the subject make interesting reading. At one point he seems to imply, if I remember correctly, that he believed that Hiroshima and Nagasaki were not particularly densely populated, and there was in fact a military instillation in these. Whether this was a case of his being misinformed or trying to reconcile his actions falliciously (after all perhaps there was a military instillation in Hiroshima and Nagasaki, but I think he'd be kidding himself if he thought that he was dropping the A-bomb just to destroy these) or plain deception. I suspect not the later, as this deception would not work as there are so many other sources of information.

Hi Marcus,

Thanks for the tips, much appreciated ;)

20-30k is a lot of money indeed! I guess that you really need to love your research to put yourself through a PhD, I guess.

I was wondering, do you use any computer packages to write up you work? ie to type up the math markup? If so, which package do you use. I have heard of Maple, Mathematicia and MathCAD - but never used, yet. Are any of them free or even open source?

Do you enjoy teaching? Just thinking Marcus, what is your full name and title now? ;)

If you get a PhD, what are the chances and how do you go about working for companies like IBM, Merchant Banking, Xerox PARC (if they still exist), MIT (for research)? Do they have specific research departments for people who are well learnered in their field, ie in our case, maths. What is the first step to getting into one of those institutions? Has anyone here attempted this?

Regards

Hal, I think the companies you named would love you to have a PhD in maths. The usual package used for writing up maths is TeX or LaTeX, which can be found at http://www.tex.ac.uk.

Hal, firstly to answer an important question I missed before: I very much doubt that having a disability which does not affect the research would make any difference to the process of getting a PhD whatsoever - in fact, Universities now make big efforts to ensure that people with disabilities have equal access to courses (as well as physical access to resources).

The software used for writing up mathematical theses (and, indeed, many if not most published mathematical papers) is LaTeX - a non-WYSIWYG text processing system. It takes a little getting used to, but it does lots of nice things which are beyond Word et. al. in the area of maths typesetting. I'm sure you can look up more on the web. The software you mention can (depending on your field) be used in actually doing the research, but not in writing it up.

I love teaching. Although we all have our bad days, I don't think you'll find many teachers who don't, on the whole, like teaching. If they didn't like it, they'd leave and do something they also didn't like but which paid a third more, which is most other jobs with the possible exception of nursing.

Since you ask, it's Marcus Jon Hill BSc(hons) PGCE MSc PhD (though I'm not sure whether a PGCE is a postnominal letters type of qualification).

whoa Michael - I think you account is not historically sound, I understand that there is ample evidence that Japan was in fact about to surrender, that the US knew this, and that the effect of dropping the bomb was aimed more at the USSR than Japan.

The are also cultural problems in your account, for instance the role of suicide in Japan was not something that suddenly sprung up due to a 'totalitarian' regime but dates way back to at least Samurai honour codes.

Dan and others - anything by Chomsky is good. There is something of a parallel in fact between the Serbia bombings and the first A-bomb. The bombing of Serbia was a major conflict is post cold war Europe, the fall of the USSR effectively left NATO (a 'defence' organisation) without it's official role and the bombing had everything to do with NATO redefining itself as an offensive force and also everything to do with the US establishing itself as a now supreme power. This parrallels slightly the A-bomb case because the actual the desired objective is, amongst other things, a demonstration of power.

Totally agree with Sean above; the dropping of the A-bomb was a very dubious action. And I did History GCSE, so I obviously know what I'm talking about. Completely different periods, but still :).

About military funding generally, well, probably depends on the conflict. I don't think anyone here would knock Turing. In any case, I think there is a distinct ethical difference in working on applications during a war, in which you are aware of the causes of that war and uses to which your work is being put and working on military applications during peace-time with no knowledge of or control over the applications to which your work will be put.

Marcus, all you need now is a CBE and an FRS :)

We seem to have entered the strange realm of acronyms; NATO, USSR, UN, CBE, FRS, PhD, MSc, PGCE, GCHQ, DRA, NSA, AI, BSc, GCSE, MIT, PARC, US, IIRC just to name a few... Also, I think HAL might be a TLA as well. The last sentence also allows me to add TLA to the list of acronyms as well.

lol Dan! ;)

May I add:

BOTEC - back-of-the-envelope calculation (a physicist's or mathematician's approximation)

AFA - A friend always.

AFAICS - as far as I can see

AFAIK - as far as I know

AFAIR - as far as I remember

AIUI - as I understand it

aka - also known as

APU - as per usual

BCNU - Be seeing you.

BFN - Bye for now.

BOFH - Bastard Operator From Hell

BRB - Be right back. (Mostly used in chat rooms.)

BTHOM - Beats the hell outta me.

BTW - by the way

Cya - See ya.

dd - dear daughter

dh - dear husband

ds - dear son

dw - dear wife

* ETLA - extended three-letter acronym/abbreviation (Well, some of them are a fair bit longer than three letters.)

FAQ - Frequently Asked (or Answered) Questions

FFPGP - Finger for PGP public key.

FOTFL - Falls on the floor laughing.

FWIW - for what it's worth

FYI - for your information

GTHFY - Good to hear from you.

HAND - Have a nice day.

HTH - hope this helps

IAH - in all honesty

IDTS - I don't think so

IIRC - if I recall correctly

IMAO - in my arrogant opinion

IME - in my experience

IMHO - in my humble (or honest) opinion

IMNSHO - in my not so humble opinion

IMO - in my opinion

IOW - in other words

IRL - in real life (geekspeak for ''not on-line'')

ISTR - I seem to remember

ITWSBT - I think we should be told.

IYSWIM - if you see what I mean

L&P - Love &Peace.

LOL - laughing out loud

MIL - mother-in-law

MOTP - member(s) of the public

ng (also upper case) - newsgroup

OTOH - on the other hand

PMFJI - pardon me for jumping in

PMJI - pardon my jumping in

ppl - people

ROTFL - Rolling on the floor laughing.

ROTFLMHO - Rolling on the floor laughing my head off.

RSN - Real soon now (usually used with a degree of skepticism)

RU... - Are you...?

TTBOMK - to the best of my knowledge

TTFN - Ta-ta for now.

TTYL - Talk to you later.

UCE - unsolicited commerical e-mail

WIH - what in hell

WIHIH - what in hell is happening

wrt - with respect to

WTH - what the hell

YMMV - Your mileage my vary.

YT - Yours Truly

Marcus, do ppl call you Dr, or call you by your name? From your teaching experience, do you find it more interesting to teach the more able students, or the less able student? The interverted student or extroverted student? What about when pupils don't do as you say, like do their hw? Do you forgive them or get angry? Just interested.

Any one read the 2001 odyssey series?

Any one read the book from MIT press ''HAL's Legacy: 2001's Computer as Dream & Reality'.

Regards

I'm also interested to know how different teachers handle difficult situations - if pupils won't do as they say, don't do homework, talk constantly etc.? I suppose the problem varies from school to school, and at different levels of education. I'm not sure if I could handle it if I went into teaching (I too am thinking of eventually doing a PhD). Can't we just be paid all our lives to do research into areas we love? No? Damn.

Of course I can't claim to be an expert on the A-bomb, but I did actually research it pretty thoroughly for my GCSE History coursework. This included reading through many sources, some biased American, some biased Japanese, some biased USSR and some supposedly ''neutral''. I agree Truman was no saint, but personally I don't see what else he could have done to prevent an astronomical loss of life on both sides. I did not find much good evidence that Japan was ready to surrender. The fact that the Japanese were informed about the capability of the bomb, and in fact they didn't surrender even when the first bomb was dropped shows that they weren't about to give in.

I find it extremely difficult to imagine that the US killed 100,000 innocent citizens simply to ''scare'' the Russians (as Kruschev later claimed). Surely dropping the bomb in the sea would have sufficed for this purpose? (Of course this is not ideal either, due to the problems of nuclear waste, but I don't think these had been fully realised at the time and anyway, this must have been better than killing 100,000 people.)

The US actually developed the bomb having been warned by Einstein and Szilard that Hitler's Germany was developing an A-Bomb. On balance I have little doubt it would have been better if nuclear energy had never been discovered, despite the fact that I believe WW2 would have been more devastating if it hadn't been ended quickly. I think the use in WW2 has to be regarded as an exception. Since then it has only helped to make bad situations worse, for instance in the cold war. The problem was that if two countries had a dispute, and one was allied to the US and the other to the USSR, then potentially the two superpowers could come to the aid of their allies. The danger was that one of the super-powers could resort to nuclear energy in the dispute between the two countries, after which the other superpower might well follow suit, and then the crisis could escalate to a full nuclear war. Luckily this never happened.

Of course the obvious example is the Cuban missiles crisis, where it was only really because Kruschev, the successor to Stalin, was sensible enough to step down at the vital moment that a world-wide disaster was prevented. A more fanatical leader would not have given in, and the consequences could have been devastating. The fact he did give in was actually one of the main causes of Kruschev's downfall in the end, but it was an extremely good decision.

The US were unlikely to give in as otherwise the Russians would gain a nuclear base in Cuba which would jeopardise US security (apparently it would only take a few hours for a bomb launched in Cuba to arrive in Califournia but I'm not sure about this detail; certainly it is unlikely the US could have responded in time). Of course if the US hadn't behaved so irresponsibly towards Cuba in the first place (the Bay of Pigs etc) this could all have been avoided, but the presence of nuclear bombs made it all the more frightening.

You ever seen the movie ''War Games''?

Sometimes there is no winner.

War is something like that.

Getting back to the path of AI, suppose that we had a AI 'thing'. If you feed it in the variables and situations for all the elements in about the time of WW2, what would it have deicded to end the war as quick as possible. Remember the AI's only task is to figure out the quickest way to end the war, and would have no bias towards any side. What would it suggest? Would it have made the same choice to drop the Bomb?, even twice? What would have been its reasoning. Is there a name for this type of reasoning?

Regards

Just a small point, but the argument that dropping a bomb as a warning would clearly have been ineffective as the Japanese did not immediately surrender after the Hiroshima bomb does not take into account the fact that the Japanese actually surrendered on August 14 1945, 5 days after the Nagasaki bomb. It should also be pointed out that a Soviet invasion of Dongbei on August 9th (the USSR only declared war on Japan on August 8th) may also have been a factor in the Japanese surrender given the imperialistic fear of Communism.

In view of these factors, it seems not unreasonable to condemn the bombing of Hiroshima and Nagasaki, not because it was unnecessary, but because it may have been unnecessary. If the USA had previously dropped a bomb in an uninhabited but observable region in or near Japan and the Japanese had failed to surrender, there would be a much better case for the dropping of the bomb as necessary in order to end the war.

Michael, another way of seeing the fact that the Japenese didn't surrender after the first bomb is that the bomb made no difference to their surrendering. Also, I don't find it hard to believe that the US would drop the bomb to scare the Russians, they were (and many still are) fanatically anti-Communist.

1) the dropping the bomb into the sea argument is a red herring because it is manifestly obvious that however big a wave you might create it is not going to have as much of an impact on world conscious (or conscious of political leaders) as the well known pictures of a devastated city.

2) I think it is a little ironic when people talk about nuclear or biological weapons being in the hands of 'lunatics' who might use them 'irresponsibly' when the only nation to have used a nuclear weapon at all is the US and the US also used large quantities of biological weapons in Vietnam. Which doesn't leave in much of a position to speak.

3) The situation is more subtle regarding 'scaring' the USSR. In particular, when the bombs were dropped, Russian troops were advancing across the then Japanese occupied Manchuria. So by making Japan surrender EARLIER than it would have otherwise (i.e. before Russian troops reached Japan), the US was able to ensure the USSR had no say in what happenned to post-war Japan.

4) There were three days between the bombings... not particularly much time. If the Japanese were so careless about the lives of people why should they have cared whether it was 100,000 or 200,000 that died? There is no logic in saying that the fact that they surrended after the second and not the first shows that they were not about to surrender. One bomb would have been quite enough for the purpose. Three days is not enough for a proper assesement of damage, evaluation of situation etc.

5) since early 1945 the US had been conducting firebombing raids on Japanese cities, specifically designed to kill as many civilians as possible. So there is no grounds to credit the US with a care for life.

6) By the spring of 1945 it was clear that Japan was militarily exhausted. Its air force and navy had been virtually destroyed and in April the war government fell and was replaced by a cabinet under Admiral Suzuki who was keen to look for ways to make peace. There were divisions in the Japanese leadership over this issue.

7) Did the US know? Well, they had cracked Japanese codes and intercepted messages which were, in the words of US secretary of the navy James Forrester, 'real evidence of a Japanese desire to get out of the war'. In the week before the Potsdam Conference, cables revealed the formal decision of a Japanese Imperial Conference to try to stop the fighting.

On 2 August the Americans intercepted a cable from the Japanese government to the Japanese ambassador in Russia: 'The battle situation has become acute, there are only a few days left in which to make arrangements to end the war... it is requested that you immediately have a talk with Molotov.' On 6 August the atomic bomb was dropped on Hiroshima.

What else is there to say?

Another reason why Japan surrendered was that America threatened to bomb Tokyo. We still do not know whether US had another A-bomb at their disposal at that stage, some sources said it was a bluff, and some said they would have bombed Tokyo on the 14th August.

To address Sean's points.

1) Yes, true. But I would like to think that the factor of saving 100,000 lives would have played some influence.

2) I am not speaking on behalf of the US (at least that is not my intention). I have mentioned other instances above where their international policy has been less than responsible. I don't know enough about Vietnam to comment, but on the subject of Hiroshima/Nagasaki my view was that the US dropped the bomb to end the war. A few of your later points have started to shift my position.

3) Even if Japan was attacked on two fronts by Russia and the US, this doesn't necessarily mean Japan would have immediately surrendered. They would probably have fought to the very end, and there would have been a lot of bloodshed.

4) I would have thought that there could be little doubt how destructive the atomic bomb was. OK, they won't have realised that that bombing was going to result in 80,000 deaths, but they must have noticed how much damage was caused with minimal effort. I don't think they knew that U 235 was so short in supply, so for all they knew the US might have gone on and on dropping A-bombs for months. I think if I was in their position I would have surrendered immediately after the 1st bomb to prevent it happening again. And if I was already thinking of surrendering, my response would have been instantaneous.

5) Well it was war. My guess would have been that the US wanted to make the Japanese surrender without having to resort to atomic energy. After a while it became clear that the Japanese were going to hang on for as long as possible, so the US decided to end it. As I say I am less sure now.

6-7) These are the two crucial points. I was aware that militarily the Japanese were exhausted, but again I think they could have hung on for a lot longer.

But what really concerns me are the two quotations you mentioned. If I had been aware of these my view would have been very different. Out of all the sources I looked through (both in textbooks and outside) I found very little to suggest Japan was on the verge of surrender. Even my history teacher (who incidentally was personally against the use of atomic energy in WW2) never mentioned once that the US had intercepted Japanese messages saying they were just days away from surrender.

Thanks Dan for the TeX link.

If the bomb stopped the war, then it was for the better good? Not though for the people of Nagasaki & Hiroshima :(

I personnaly think that a nuclear bomb will never be used again on civillians, within my lifetime anyway, but then again I am an optimist.

I'm staying right out of the bomb debate...

Hal, to answer your teaching questions, people call me by name, or if they are being formal Dr. Hill. Depending on the institution, formality towards teachers may be required (usually in schools). I teach in a sixth form college, so formality is not important.

I find it most interesting mathematically to teach more able students, since they not infrequently solve a problem in an elegant way which had not occurred to me. However, I find it most gratifying to teach students of any ability when they make progress, and have what Martin Gardner calls that ''aha!'' moment - that's what teaching is about, whether it's a student who is a better mathematician than I am coming to grips with a complex theorem or a kid with special needs seeing how to work out the mean of a set of numbers.

There are always systems in an institution to maintain discipline and work quality by students, and I tend to stick to those. I don't get angry at students for not doing homework, merely disappointed - they are only harming their own progress. I never let people off when they don't do homework, that would encourage people to not do homework!

The point about the US launching the bomb twice to scare the USSR is somewhat illogical when saying that it would only take 1 bomb to scare the Japenese. If it was known that only 1 bomb would be needed to ''scare'' the world, then why would two have been dropped to scare the communists. When it's not your own people, ''quality matters, not quantity''. So why would the U.S. have dropped the second bomb just to scare the USSR. It was either dropped twice to defeat the Japenese, or because of military ignorance. Given some of the arguments already on the page perhaps it was simple ignorance (something all too common in American Politics). But, I still think that the bomb accelerated the end of the war.

Finally, the people designing the bomb were certainly not doing evil, as if the bomb had been dropped as early as possible many lives would have been saved. It has been estimated that half a million lives were lost every month in the War. That would mean that if the team could have gotten the bomb ready 2 months earlier, we would be granting 1 million lives in exchange for .2 million. This, while hurting the individual will help so many more individuals. But, I am as willing as anyone to admit that there was a bit of a double motive in the US government's use of the bomb (to use the bomb for intimidation). I am not willing to admit that the US merely used the bomb as a political weapon though.

Given recent threads on the US's anti-communist agenda, I've decided to post these CIA plots actually considered to stop Castro and Communist Cuba.

1.Use agents in Cuba to spread rumors that the second coming of Christ is imminent and that Casro is the anti-Christ.

2.Suprise him at the beach with an explodinbg conch shell.

3.Put thalium salts in his shoes so that his beard and hair would fall out during the ''David Susskand Show''.

4.Put ithching powder in his bathing suit and LSD on his mouthpiece so that he would be driven crazy and drown.

5.Offer him exploding cigars designed to blow off his head.

6.Shoot him with a TV camerz that has a machine gun inside.

7.Spray his broadcasting studio with hallucinogens

8.Actually send a plane over Cuba dropping pamphlets explaing that due to A-bomb tests, the Cuban island was now bombarded with radiation causing impotency, and the only way to get rid of radiation is to shave off your beard.

Read once again: these are not jokes!(but I've intended them to be in case there was question)

fair point about the two bombs Brad, my guess, but it just a guess, would be that it was always planned to drop two, seing as it was the first time one was had even be really used, there might have been quite a high probability of something going wrong, hence the need for two to be sure one would work. Of course, as you said there may not be a rational explanation.

I agree the bombs accelerated the end of the war, but only by the few weeks that were necessary for the war to end before Russian troops reached Japan.

The conjectured intimidation of Japanese would have been for a very different purpose than the conjectured intimidation of the USSR. I don't see the rejection of the former as necessarily contradicting the latter.

If the idea were to force the Japanese to surrender than I cannot see how more than one bomb would be necessary. If on the other hand the aim was to demonstrate American military strength then there is every reason to destroy as much as possible.

Marcus, if you spot unusually high potenial in a student what w ould you do? a) continue as normal, b) tell the student and set harder work, c) somthing I haven't put as an option?

I would be interested to know how you would nurture potential in a student so that it does not go to waste.

I was reading the book ''Uncle Petros and Goldbach's Conjecture'' (not finished it yet though). Has anyone proved is disproved it yet? How far have they got?

Regards

Marcus, Thanks for your view. Its very interesting, the way you put it.

Do you know anything about the history of the Indian mathematician Ranuajuman? How did he derive so much maths that were already mainstream in the western world independently? What basic education did he have. What books did he initially learn from. What kind of new mathematics did he derive in his days with Hardy and that Long guy (can't remember his name) at Trinity. I would be interested to know. Thanks in advance.

Thanks

Hi Anon, do you know of any of the titles to which you talk about. Much appreciated.

Emma, thanks for the Bio link.

Marcus, I was thinking about what you said, ''intuition''. I was reading some material about Deep Blue. Some of the moves that Deep Blue could come up with was sometimes counter inuitive to a human chess player, and the reasons for the move only becomes apparent to the human later on in the game, if at all. So is this thought ''intuition'' by computer or human standards. The sheer number of paths the computer analysied brought foward, some moves that would be dismissed by a human counterpart. But since it is not thinking like a human but simply going through as many possible moves at it can in a space of finite time, why is this type of thinking not considered human or ''intuitive''. Maybe, Ramamujan ''intuitive'' thoughts that were counterintuitive by others standards were simply working in the same mathod of deep blue, but not fully even aware to the knowledge to even the mathematician himself. Hope I make sence, this is a bit rushed.

Hal-

I found a book about Ramanujan which I started reading a while ago. It's called ''The Man Who Knew Infinity'' by Robert Kanigel. I haven't gotten that far in it, but it seems quite a good book so far (it includes the story of his life as well as some Mathematical notes).

Thanks Olof, I'll check out the book.

Michael - I've finally tracked down a reference for some of the quotes I used above. Have a look in the book ''The Decision to use the Atomic Bomb'', by Gar Alperovitz, New York Vintage Books, 1995. There are quotes there from American military and government people that really leave little doubt.

Thanks Sean - I'll have a look for that in the library later. I'm a bit surprised that I didn't come across anything like this in any of the sources I looked at, and that my teacher (strongly against the A-Bomb) didn't mention it. Maybe it has only been uncovered recently. Anyway thanks for the reference,