AI and Maths

 As an old mathematician I have been trying to follow the use of AI in maths and trying to understand what is going on. There are two aspects which are confusing me. The first is how a system like an LLM can write proofs given that it is a stochastic system that tries to work out what the next 'token' will be.  The second issue relates to how it would know that it gas a proof? There a re formal languages which have been developed, like LEAN, which can formally check that a proof is valid, at least as I understand it.  There has been a claim from Anthropic;  https://www.anthropic.com/research/formalizing-fermats-last-theorem. This turned the proof into LEAN and then used that to check. Kevin Buzzard at Imperial College has been working on this for sometime; https://profiles.imperial.ac.uk/k.buzzard. Somehow this I can understand so that AI can take a proof and turn it into code in LEAN which can then be read makes sense. 

I have tried to read the statement that Open AI published; https://openai.com/index/navier-stokes-solution/. To those who might read this who do not know me I should make clear that I am an algebraist who has worked mainly in Group Theory and have no expertise in partial differential equations of which Navier-Stokes is an example. As a consequence what I was trying to understand how AI tries to put together a proof. Progress does seem to have been quite dramatic, a recent article by Marcus du Sautoy; https://www.ft.com/content/05a7292e-4931-4631-8f77-164fb727c203, expresses this quite well.

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https://www.ft.com/content/05a7292e-4931-4631-8f77-164fb727c203

Three months ago I tried using OpenAI’s ChatGPT to assist me in my mathematical research to discover non-polynomial behaviour in the zeta functions of free nilpotent groups. It was useless. All it did was regurgitate statements in papers that I’d already published. Two weeks ago I decided to try again. The result was phenomenal. I suddenly had an insightful collaborator that knew how to code and do experiments in the structures I was investigating. It made mistakes but corrected them often on reflection. It responded to my new ideas and suggested its own. It reached a level of understanding of the issues involved that frankly would have taken a new student embarking on their PhD months to grasp. Using ChatGPT, I have now discovered very plausible evidence of the behaviour I conjectured. Because of this experiment, I was not altogether surprised by OpenAI’s recent announcement of its AI model’s successful progress on the Navier-Stokes equation — one of the seven unsolved maths problems known as the Millennium Prize Problems. "

It is not clear which version of ChatGPT he was using, possibly one that is paid for but is still not clear how the models are creating proofs by a process of working out what might come next.  

As a human mathematician my memory is quite limited but when I tackle a problem it si important to know what is already known, obviously a machine can input all the relevant data rapidly and then store it. This can be done much more efficiently and almost certainly more completely than humans.  Having gone through and hopefully understood this material you then try to see how to use it to solve the problem at hand.  Many attempts will fail but as  a human you know that and you know when you have used an argument incorrectly. This is where I get left behind, what does the AI do when it has collected all the data? I think it would be helpful to have some idea how this works, my limited knowledge come up with a few ideas but they are probably rubbish

One other concern is that as humans we come up with questions and they often feed to development of a subject.  If AI is capable of AGI then can we get it to come up with new ideas and questions. Here is a possible test: Create and AI with now knowledge after 1900, it would have to be isolated from the internet. It would have to have input data in a specified way. As the question I want to pose is about physics so perhaps it would only need to have a database of all published physics up to 1900. Would it be able to come up with relativity?

 

North Sea Oil & Gas

 There is a discussion taking place about exploitation in the North Sea and whether the UK government should give permission. The reasons agains are clear, any further development will add to global warming. The argument against says that the developments will create jobs, reduce the UK's reliance on foreign sources and bring down resources.

The last two arguments are rather questionable. As I understand the situation any one producing the gas and oil can sell at world market prices and is there any restriction on who they can sell the oil or gas to. So if someone makes a offer above global prices what happens.

If the country was really serious about the cost then the licences should only be granted on two condition. The first being that the UK has total control over where the oil and gas go. even more important must be the price. If they are only sold at world prices than the UK consumer has no benefit of chaeper prices because it is UK oil and gas.  So the second and most important condition would be a variation on a fixed price contract. That is the oil and gas would be sold at a price negotiated with he government, perhaps fixed at the level preceding the invasion of Ukraine or at world prices whichever is lower. 

This would obviously restrict the profit and presumably to taxes that the government would get but if properly implemented could lead to significantly  lower prices for the UK consumer and thus lower the cost of living.

Let be clear I am not an expert in the way prices paid at the petrol pump and by the energy industry are calculated and regulated so this is a broad outline of a plan . It would need to be carefully pl;anned as private industry is very adept at looking and using loopholes. It might even be that the companies would not with to do the extraction with these conditions but when they first started oil prices were much lower so there would still be a sizeable profit, just not gargantuan!

Nigel Farage and 5 million pounds

 For some time I have wanted to write about the different value money has to different people. My work with Norwich Credit Union made it very clear that people have no concept of how much others earn and why £5 can be a lot of money to some whist £5 million or almost trivial to some. Most people in Britain are somewhere in the middle. 

The fact that Nigel Farage was given five million pounds should seem like a huge amount of money to many people. If you earned £50 thousand  a ear it would take 100 years to earn 5 million. How many of Nigel Farage's constituents earn even £30 thousand a year. Why does Christopher Harborne want to give so much money to  Nigel Farage. Recall this was not a gift to Reform, the political party, but a personal gift to one man, a gift he does not have to pay tax on. So Farage has just been given the equivalent to a hundred years salary at a not unreasonable wage.

This leads to the question of why Mr Harborne would give away this apparently large some of money. According to Wikipedia he is worth over £18 billion. This means that Harborne has given away less that 0.00001% of his wealth. So presumably he wouldn't even notice.For some earning £50,000 that is less that £15, a couple of pints in a London pub. So what does Harborne get for his money?

Notice that he didn't give the money to the Reform party so it is not just about politics. So why would Harborne, who lives in Thailand but maintains a house in the UK so he can donate money to political parties, want for his money. Why make it a personal gift, is he keeping Farage as his own personal poodle. A well trained dog who rolls over and has his tummy tickled as a reward for performing some trick at his maters request. Or perhaps the treat is just another million somehow gifted to Farage. 

One conclusion from that story is that Farage cannot be trusted, he will jump whichever way his master commands.  The other is that most economics does not seem to understand that £5 million is an enormous sum to some and trivial to others. This should really have an enormous impact on the way politicians and economists see the way taxation and benefits affect different people.

A sad day for British politics

 Hi,

I am very sad to see Keir Starmer resign, he did itb with some dignity but it depresses me. Starmer is a good man who has tried his best and not has not been rewarded for it. He never really had 100% support from all his MP's, perhaps less than he deserved for bringing back the Labour Party from the disaster of the Corbyn era.  He made mistakes but rather than the PM's congratulating him for his ability to recognise and correct errors they enjoyed attacking him and not helping.

There was no realisation that the situation the government found itself in was very difficult, much worse than they expected. This should have caused the MP's to gather round and support the cabinet 100% but did they. No of course not, they complained if their own particular concern was not addressed. The problem is that to correct all the failures of the previous fourteen years was not going to be easy. Especially those that go back as far as Thatcher's  regime. 

Things that need correcting that go back that far has been to sale of council housing which lead to a dramatic shortfall in affordable rented housing and a consequential transfer of public funds to private landlords. This government wants to build more council owned properties but has lost all the expertise that it used t have as well as the skilled builders needed to actually build the houses. Another issue which many feel needs sorting is the water industry which was nationalised by Thatcher which many want to take under state control. sadly millions have been wasted by transferring money out of the public purse to fund private equity and, to be fair, private pension funds. It is an interesting that the first thing that happened after privatisation was the directors doubly their income! But what is the best way to get the privatised companies back under public control. Their are various solutions proposed and it will need an act of courage to find the right way forward as well as convincing the public that it is the right way.

Another issue which I find concerning is what will now happen. Assuming Andy Burnham becomes the leader, I voted for him in earlier leadership elections though I don't know that I would this time, can he deliver, Many of those supporting him expect he will change the course of the government but how will he manage that. Being a mayor does not involve handling 400 so-called supporters with their own agendas let alone the issue of a vast agenda to control, many of which are not under your control. For example wars in Ukraine and a complete nut leading USA and the difficulties in the Middle East. The trouble is that if he does not deliver then the trust in politicians will drop even further. This tens to lead to a rise in totalitarian leaderships, a desire for the "strong man". In my life, which is probably passed its used by date, I have never seen a strong man whose rule has been good in the long run.  

To end this diatribe I would like to recommend Neil deGrasse Tyson a very interesting american astrophysicist who does some interesting popularising stuff. Tow comments which are relevant to the current situation. I am not sure the accuracy of the quotes but this is my memry of them which is probably what I want to convey.

Politicians lie because if they told the truth they would not get elected.

Simple slogans are easy to understand but take a lot of thinking about to counter.

Regards

Alan 

 

Houses of Parliament renew?

 We all know that this building is being neglected and will probably fall down soon without action. Some of you will know this is not a building I admire and if it were to fall down I would not feel particularly upset. However many people think it should be preserved and much discussion has taken place but no action has followed. The longer the the parliamentarians take to wake up the more expensive the process renovation will be and probably the longer it will take.

 

One of the causes of dispute is whether or not the Houses of Parliament shouls consider sitting whilst the renovation is being carried. It is obvious to any sensible person that closing the whole building down and renovating it whilst empty be the would be the best option. The question now is what would happen and how will parliament function if the Houses of Parliament are closed. It would be a great opportunity for a serious rethink of how they work.

 The first major step would be for the hoiuses to sit most of the time virtually. Anyone who watches parliament live will rea;lise that for most of the time most MP's and probably Lords do attend the chamber to listen to discussions. A bell rings and they all troop along to vote, I sometimes wonder if they even know what they are voting about, they certainly have not listened to the debate but have been told how to vote. Nost of this could be done online, some of this was done during the Covid emetgency. my poroposal would be that parliament should only meet in person one week per month, that shoud be sufficient to talk in perspn to other MP's. 

More controversially they should meet for differnt weeks in different places. There would be twelve weeks for paliament to meet, I suggest two each  in Belfast, Cardiff, Edinburgh and London, in alphabetical order and one each in Birmigham, Leeds, Manchester and Newcastle. I believe that in the past the sovereign would process around the country. Given a timescale of about 10 years this would give plenty of time for everone to get used to the idea. It would also move the focus away from London and England, note that parlimant would only meet in Enland half the time. 

This should be combined with some other reforms. One that is needed are the rules about doing work outside of parliamentary duties. It would be a mistake to ban this completely because some MP's do useful work and it keeps them in touch with their previous lives. However anything they earn would be put into the fund for the renovation of the Houses of Parliament.  It would then be clear how many are just doing it for the money rather than for helping them keep in touch. I am sure that being on TV programmes is good for their profile but that will happen whether the pocket the money or not.  

Another beef I have had  for a long time is the provision for office and secretarial staff for MP's in their constituencies. These should be provided by the civil service and not used as slush funds. If the office is provided centrally it will stay in the same place and at least one or two wokers should be there permanently. It has never seemed to be good practice for work an MP has done for a constituent of business should be tied to that individual. I realise that sometimes a new MP may not feel that they wish to support somehtinga previous MP has been involved with but they should have the integrity to say so.

 

 

 

 

Who defends the antisemites?

https://www.theguardian.com/commentisfree/2026/may/01/antisemitism-racism-jews-britain, this is one of the most depressing articles I have read for many years.

Like many jews, not religious, but clearly of jewish heritage, it had seemed to me that many of those who supported Palestinians and opposed their treatments by the Israeli state as being genuinely concerned about the Palestinian people. In discusions which I did manage to have with the supporters it seemed they could never answer the question of what about the Jewish population of Israel. There seemed a blank in their brains or their thinking when faced with this question. 

The population of Israel and Palestine combined, "the land between  the river and the sea", is about 15 million split roughly equally between Jews and Arabs. These figures are all disputed by various sources. Thje Jewish population is also split roughly equally between those whose me recent heritage is Europe/USA and the middle east, coumtries like Iran, Iraq, Morocco, Egypt, Turkey amongst others.

How to resolve the conflict between the two main groups of of the population is not easy. One solution which seems to be popular with extremist is simply "genocide", eliminate the other side, this is the stand of Hamas, Hezbollah and the Iranian state on one side and of the present Israeli government and many of the settlers on the West Bank. Inevitably if someone chooses one side or the other they are forced down the difficult path of seeming to support genocide.Is this the reason that that those who join marches in support of the Palestinans have difficulty answering the quastion, what ahppens to the Jewish population. 

I had liked to think they were embarrassed  by this conclusion, perhaps they still are but more worryingly perhaps they support the genocide of jews. To put it differently may be they are just antisemitic. Jeremy Corbyn has expressed his horror on both Instagram and Twitter I am glad to say, just as he did over the attack on the synagogue in Manchester on Yom Kippur. Is there any action, perhaps a dem,and for a march in support of Jewish people in Britain. It is interesting to compare his reaction to the terrorist attack oin Finsbiry Park some yearts ago, His defence might be that he is the MP for the area but his comments on the recent antisemitic attacks are by his standards very mute and generalist, they can be found on Facebook and Instagram. The organisation "hope not hate" been vocal in supporting British jews, a statement on Instagram, again not exactly running to the barricades.

Before finishing this blog it seemed like a good idea to look at some other comments. One was from one f the organisers of "Stop the War Coalition". I had never looked at their website before and it horrified me. It doesn't seem to be against war, merely against Israel. The title seems innocent but the content is entirely aimed at Israel and is a platform for the more extreme plastinian causes. There seem,s to be no recognitioom that  Hamas, Hezbollah, Iran aand the Houthis have been waging war on Israel for the last twenty years or so. Israelis have been driven from their homes in the north of Israel by constant rocket attacks. Hamas have been firing rockets from Gaza into Israel for sometime and as we saw on October 7th would cheerfully murder Israeli civilians and burn them in their homes. What is also shows that if the Palestinian groups had the same fire power as the Israeli grovernment they would behave with the same diregard for human life. Not a very comforting thought. If "Stop the War Coalition". is not just a front for the war mongers on the Palestinian side where is the balance in their stand. It is difficult not to see them as AntiSemitic organisation. 

If peace is to be achieved in the the Middle East it will not be though one-sided marches or campaigns, it will have to come through dialogue and  a realisation that evryone who lives there jas a right to live in peace.The tragedy is that the actions of Hamas on October 7th has put thos ehope back by 50 years, that is an optomistic guess.

I do not see how anything that the groups supporting thge Palestinian cause will do anything in Britain except encourage antisemitism. Probably Netanyahu would like nothing better. Harass jews in Britain, attack the current government, and hope for more emigration to Israel. Perhaps the antisemites in these organisation would like to drive jews from Britain.

Starmer, McSweeney and Mandeson

 

My thinking, for what it is worth as I have no inside track, is that many people thought appointing Mandelson was very clever, that includes Kemi Bandenoch and Nigel Farage. Mandelson was just the slightly sleazy character with links to Trump who might be good at keeping in with Trump.

Almost certainly not true but it is not clear quite how nasty and vicious Trump is.

It is also true that Mandelson had been caught out in dodgy activities in the past but nothing suggested that he would sell government secrets to outsiders. Many politicians are influenced by friends and large donors but most do not sell secrets, they just make appropriate decisions.

Another important figure is Morgan McSweeney who was Starmer's chief of staff when he became PM but was also crucial to Starmer becoming leader of the Labour party. He his a protogee of Mandelson, definitely on the right of the Labour party . He help set up a faction within the Labour Party called Labour Together which campaigned against Jeremy Corbyn. It might be that McSweeney identified Starmer as the person to take on Corbyn and made himself an essential part of Starmer's team.

The problem is that Starmer is not a natural politician and does not come across as a man of the people. He is very much the lawyer who argues his cases very carefully. There is a feeling that his knighthood makes him more of a toff. He got it for being head of the DPP, and goes with the job. It might be his difficulty with the civil service goes back to having been head of a big government service and possibly being used to his orders just being obeyed and trusting those below him just to getting on with things. Perhaps he trusted McSweeney too much, others assumed that when he spoke it was always with the support of the PM but he might not have been  consulted, not clear how much McSweeney always cleared with Starmer.

The other issue appears to be the claim that the Foreign Office does not have to worry about vetting of their staff. There might be some truth especially if they were using double agents in some of their posting:) 

It could be messy but I think it unlikely that Starmer was aware of all that was going on, and jst assumed that things would happen. Starmer does not appaer to make friends abd be very clubable. Once this may not have mattered but with television and modern media polticians are much more exposed as people. It is not competence does not a valuable talent anymore but Starmer seems to have shown both a level