mayd

Many would disagree with your characterisation of pure mathematics, computer science, or both. And I am certain that computer science yields nothing to pure mathematics in her pursuit of abstraction.

As the late Martin Gardiner opined:

"Every few decades an unknown author brings out a book of such depth, clarity, range, wit, beauty and originality that it is recognised at once as a major literary event. This is such a work."

Nevertheless, Infinity and the Mind: The Science and Philosophy of the Infinite by Rudy Rucker is, in my opinion, a better choice for the mathematically inclined layman interested in Goedel's discoveries and plenty of related mathematics. Rucker's book even includes an account of his (somewhat over-enfusive) meeting with the great man.

Case 1: Suppose there are n doors and Monty does not know what is behind any of the doors. You choose one door at random. Then Monty opens (n-2) of the remaining doors AT RANDOM until there is only one other door left. By chance none of the doors he opened had the car behind it. Then he asks you if you want to switch. Should you switch or not?

Case 2: Suppose there are n doors and Monty knows what is behind every one of the doors. You choose one door at random. Monty deliberately opens (n-2) of the remaining doors from the left to right, skipping the door with the car. Then he asks you if you want to switch. Should you switch or not?

Whether n=3 or n=100, it seems to me that it does not matter whether Monty Hall has complete knowledge or zero knowledge of the location of the car. You are required to make a choice under the condition where there is only one other unopened door and all the other doors did not reveal a car. The player's original choice was correct with probability 1/n and the probability of the complementary event must be (n-1)/n. The player's strategy of switching will result in a win with probability of (n-1)/n.

For those who want to see quirky Australian children's television programming for older children may I suggest this? https://en.wikipedia.org/wiki/Round_the_Twist

It continues to surprise me that Principia Mathematica (PM) still gets mentioned so regularly in discussions related to computer science topics. As far as I can tell, PM was one of the least influential works in any branch of mathematics or logic or philosophy or computer science. It is must be one of the least-read books (3 volumes, 1994 pages) ever written. PM's sole claim to fame is that it was mentioned in Kurt Goedel's famous paper "Ueber formal unentscheidbare Saetze der Principia Mathematica und verwandter Systeme I". The emphasis strongly is on "verwandter Systeme" - related systems, of which PM is just one example.

The scientific method is the only method of enquiry that is valid if you are doing science. In science, skepticism of every theory, new or old, is expected, encouraged and valued, because, according to the scientific method, every theory is potentially falsifiable.

Skepticism and consensus are conflicting concepts. Consensus has no place in science as method of determining truth. You cannot form a scientific conclusion based on socially or politically manufactured consent.

Never has this been more clearly demonstrated than during and after the recent Coronavirus panic. Facts were ignored, or even suppressed, "settled science" and "scientific consensus" were used as weapons to silence dissenters, by trashing their reputations, or threatening their jobs.

We are living in an age of Cargo Cult Science. Richard Feynman must be rolling in his grave.

> I give Wolfram a ton of credit

The ability to give credit is something of which Steven Wolfram is in very short supply!

Never mind Wolfram's obvious debt to John Conway, John von Neumann and Alan Turing, among many.

From an interview with Steven Wolfram (FT.com, 11 April 2003. "Lunch with the FT: Stephen Wolfram" by Simon London):

   Does he have any heroes, I ask, anyone in science whose work he really
   admires?
   The question is met with a long pause. 
   "I think Alan Turing would have discovered a lot of the things I have
   discovered if he had lived another 20 years. He was pointed very much in
   the same direction."
   A hero?
   "The more you learn about these people the more difficult it is to idolise
   them. Turing was much more of a hacker than I am. He liked putzing around 
   with things in a way that is somewhat different from my interests."
   How about (John) von Neumann, then?
   "I'm sure I would have greatly enjoyed meeting him. He was a smart man. But
   I would say he fundamentally missed deeper points in a whole series of areas
   in his haste to figure things out faster than other people. It was almost
   like science as a competitive sport. He skated too quickly over many areas."
   Isaac Newton is summarised as "an able man who did interesting work", while
   Charles Darwin "had the good feature of living basically a happy, quiet 
   life"  although "the core of what he did is better than some of the
   bandwagons he let himself climb on to".
   Any living scientist he especially admires? I suggest Stephen Hawking, the
   Cambridge astrophysicist, or Benoit Mandelbrot, whose work on fractals seems
   adjacent to Wolfram's own interests in complexity.
   Again, the reply is careful and considered.
   "Hawking I view as a fascinating personal study but as a scientist...well,
   I suppose he has done some decent-quality stuff. Benoit Mandelbrot I know 
   quite well and some of his work is quite important. In fact, I am one of
   his bigger promoters. I'm just not sure he has completely internalised 
   quite how important it is."

Thank you very much!

> I learned to trust nothing that man says.

I admit I did get a little caught up in Musk's web of hype in the early days but fortunately my natural scepticism was strong enough to resist him. Eventually, I found a few detailed analyses of some of his technological fantasies which convinced me that some of what he proposes will not be happening any time soon, if ever.

I am by no means an Elon Musk hater but he does speak a lot of nonsense, from time to time, and he seems to be able to persuade many to abandon their scientific scepticism.

I was persuaded by what Common Sense Sceptic says about some of Elon Musk's plans: <https://www.youtube.com/@commonsenseskeptic>, even though he is clearly biased against Elon Musk.

I am convinced that

1. A city-to-city Starship shuttle passenger service will never happen.

2. A self-sustaining colony on Mars will not happen in the foreseeable future.

I have strong doubts about other SpaceX and Tesla claims by Elon Musk, but these two events I would actually bet on.

I love corporate/financial/technology/mathematics thriller film genres but they seems to be a very small niche. Good ones are especially rare. My favourite is "Michael Clayton" (2007), and "Margin Call" (2011) is my second favourite. "Sneakers" (1992) is old but pretty good. "Disclosure" (1994) is also old but quite watchable. Dare I mention "Pi" (1998)?

Someone here mentioned the TV series "Billions" which is my guilty pleasure. The stories are so absurd but the great cast manages to carry it off successfully. The only criticism I would make is that, although the show is ostensibly about a hedge fund manager, it seems to contain almost zero actual finance or game theory stuff. The only occasion, that I can recall, when finance enters the plot in a semi-serious way is the episode "Short Squeeze", which was one of my favourite episodes.

I found this very watchable, and nostalgic, despite the sometimes glaring technical inaccuracies. Lee Pace elevates any show in which he appears.

You are incorrect. Natwest was forced to release internal documents about the debanking of Nigel Farage under an FOI request and it is clear from the evidence that he was debanked because of his politics.

From the Guardian 30 July 2023: <https://www.theguardian.com/business/2023/jul/30/uk-banks-cl...>:

"Farage used a subject access request to discover that, despite initial denials by NatWest subsidiary Coutts, his political views had played a part in the closure of his account."

the collective noun for cows is herd - not flock.

<https://en.wikipedia.org/wiki/List_of_animal_names>

I would definitely choose Strang to begin learning Linear Algebra. He develops the geometric intuition you need for applying it in other subjects such as Calculus or Statistics or Physics. Linear Algebra is not just Abstract Algebra, at least for most of us, including most mathematicians. After Strang, or even alongside him, if you want more rigour then you would benefit from a text like that of Anton, or Friedberg, or Curtis or, God forbid, Axler.

FYI I bought Axler's book (3rd edition).

> Isn‘t there anybody close to the Feynman of Linear Algebra?

That would probably be Gilbert Strang.

While, as a maths person I would prefer a bit more rigour, his choice of topics and his teaching skill make his the most outstanding introductory course I have seen.

I would run a mile from any course that disrespects determinants. And that includes Axler's!

Also I wish more Linear Algebra courses would cover Generalized Inverses.

It gives me little shock every time I realise I have lived long enough to see the world turned upside down. What was once regarded as an anodyne statement of fact has become increasingly contested and is even rejected by a sizeable section of society. That is, that free speech is a universal right, and a good thing. For a very long time, going back to the 19th century, at least, every British (and American) politician, or other public figure, would have agreed with Voltaire's position on freedom of speech: "I disapprove of what you say, but I will defend to the death your right to say it". How times have changed.

And Brazil.

Can't fix. Won't fix.

My experience is similar. I got drawn into the I. T. security field in the early 1990s Internet era due to my technical background and experience. It was fun and rewarding at first. But by the late 2000s the huge burden of responsibility, general business hostility towards security, expensive credentialisation, a lack of corresponding compensation, and limited career options, led me stop doing it. Programming is still fun and way less stressful.

Great stuff! Mysticism and occult symbolism is what attracted me to mathematics in the first place. It's all great fun.

> We didn't want to believe that all these smart people really were into stupidity like turning lead into gold.

Alchemy was not stupid in the 17th century. You have the benefit of three centuries of subsequent scientific advances, to which geniuses like Isaac Newton, and those other smart people, contributed significantly.

Besides alchemy, Newton was deeply immersed in various occult studies. He was also a heretic, being a Unitarian, keeping his religious beliefs secret. Scientific research occupied only a part of his time. The seventeenth century was a time of religious and political turmoil, millenarianism and apocalyptic prophecy abounded. Newton was a man of his time.

Not in my zone of the international DRM prison system.

I strongly disagree. While rigorous mathematical proof (logic) is a big part of modern mathematical research output, it is an impenetrable barrier to learning for most students, including maths students. I doubt any mathematician ever learned their subject by mainly reading proofs. And I say that as someone who secretly enjoys reading and working through proofs of theorems. Understanding mathematics demands developing strong intuition of ideas before attempting their logical justification. That rigour and intuition in mathematics education are largely contradictory goals is widely acknowledged by maths teachers; 3BlueBrown, for example, has often mentioned this duality in his videos. It was only in latter part of the twentieth century that the idea of abandoning intuition in teaching mathematics was seriously attempted, by the Bourbakists I believe, and was embraced for a while, before fading away.

> This is the problem I find with the 3blue1brown videos. They're pretty but I never get any understand from them.

So relieved to learn that I am not the only one!

The difficulty of the Monty Hall Problem is not due to vague language (although some statements of it do suffer from vagueness). It is a genuinely astonishing result that many professional mathematicians, including statisticians, have tripped over. You should read about the story of Marilyn vos Savant and her columns discussing the Monty Hall Problem in Parade magazine, and the associated controversy. E.g. <https://chance.amstat.org/2022/11/monty-hall/>.

"vos Savant herself was flooded by a surge of disbelief. She received more than 10,000 letters from readers of her column, the vast majority of whom were absolutely convinced that she was wrong. Among them were many PhDs and a strikingly large number of mathematicians. One understanding mathematician kindly offered vos Savant some comforting words: “You made a mistake, but look at the positive side. If all those PhDs were wrong, the country would be in some very serious trouble.”"

A good introductory book is this one: "The Statistical Sleuth. A Course in Methods of Data Analysis" (3rd Edition) by Fred Ramsey and Daniel Schafer <http://www.statisticalsleuth.com/>.

Another good book at a more advanced level is this one: "Regression and Other Stories" by Andrew Gelman and Jennifer Hill <http://www.stat.columbia.edu/~gelman/arm/>.

Another good book at an even more advanced level is this one: "Data Analysis Using Regression and Multilevel/Hierarchical Models" by Andrew Gelman and Jennifer Hill <http://www.stat.columbia.edu/~gelman/arm/>.

These books are all very polished productions. What make makes them special is that they emphasis teaching the statistical way of thinking.

This discussion leaves unclear the the role of Pythagoras of Samos in the discovery of the Pythagorean Theorem. The statement of the theorem appears to have been known before the time of Pythagoras. He is credited with giving the first proof of the theorem, within the framework of Greek mathematics. On that claim his fame rests, although no record of his proof has survived.

Most would call it surreal.

Someone who is a Greek expat and pays attention to current events in Greece is more likely to be better informed than you (a non-Greek) simply because they know the history Greece better than you, and most importantly, they know the Greek language, which gives them access to more information from Greek language news sources, Greek language social media, and their Greek relatives back home. I find the English language media, provide very little coverage of countries where the language is not English. As a news junkie, I would love to be able to follow what is happening in places like the Balkans, Europe, etc; but I can't because I can't read them in their native languages.

Then I suppose it's a good thing that Einstein did not live long enough to have his own Physics Youtube channel.