This is actually one of the complaints made bout the modern world. It’s simply not possible to keep abreast of the advanced edges of more than one (perhaps two) fields any more, in the way that an 18th or 19th cent. bloke could be a polymath.

Klaus Roth, who has died aged 90, was the first British winner of the Fields Medal, the mathematical equivalent of the Nobel Prize, whose discoveries in number theory led to him being considered one of the greatest mathematicians of the second half of the 20th century.

Roth was, above all, a consummate problem-solver. His best-known work was in the field of Diophantine approximation, a branch of pure mathematics that deals with the approximation of real numbers by rational numbers. He was only 30 when he made a significant contribution to the Thue-Siegel theorem by proving that any irrational algebraic number has an approximation exponent equal to two. In 1958 he was awarded the Fields Medal on the strength of this breakthrough.

Undoubtedly great work to gain a Fields Medal. But I would wager that you’d need at least one degree in mathematics (or at least be able to pass the finals for one, whether you’ve done so or not) to even understand what it was that he had done, let alone replicate it or do it for the first time.

I am quite famously maths blind (statistics I can usually manage but not maths) so this might just be projection but I don’t think so. It’s all there in words and everything and I’ve not a clue what it all means. And while I’ve obviously got deficiencies in my education I’m not entirely stupid. I’m really pretty sure that this will be true or prize winning work in any field: those outside that field, without the underlying education, can no longer really follow what the arguments are.

I actually have some fun (for a given value of fun) with this each year with the Econ Nobel. Is it possible to put the point, in however bastardised a form, so that it can be understood by he layman. Krugman, Tirole, Deaton, yes, it can be and has been done. Those blokes who did heteroskedacity (Spelling?) not really, it was just possible to say they’d done something clever with numbers. Not what, just something.

Perhaps it’s not something to complain about, but something to celebrate? That we’re delving so deep into the secrets of the universe that no one human mind can encompass it all?

Your general point is undoubtedly correct; but, amusingly enough, in Roth’s case he was working in an area of number theory in which it’s not all that hard to explain the problems to a general audience. Basically, how well can certain sorts of numbers be approximated by particularly simple algebraic expressions.

Proofs were fiendishly hard, though.

I can think of someone who considers himself an expert in at least half a dozen fields, and will candidly tell you so.

Quite a lot of mathematical theorems are proving things that seem pretty straightforward – Fermat’s last theorem is easy to understand; it’s just Pythagorean triples (3:4:5, 5:12:13, etc) extended to other powers. But Andrew Wiles’ proof is incomprehensible to most people with a master’s in mathematics, never mind anyone without a university-level education in the subject.

If you ever want to be overwhelmed with how much maths you don’t know, go to http://mathoverflow.net/ which is the forum that research mathematicians use to discuss their research with each other.

I think it does pose a problem for children and young people setting out in life.

“That we’re delving so deep into the secrets of the universe that no one human mind can encompass it all?”

The human mind is nature becoming conscious of itself.

There have been many candidates offered as ‘the last man who knew everything’ but such a label couldn’t truthfully have been applied to anyone since about the start of the iron age. You would need the combined knowledge of several dozen experts to understand how an iPhone works down to a fundamental level. Transport them individually back in time to classical Greece and most of them would make a better living humming remembered fragments of Beethoven than by fostering incredible technological advancements.

@ Yorkie Mathmo – ok, in layman’s terms what is an approximation exponent? (genuine question by the way)

Easy to explain: every non-whole number can be approximated to a/b where a and b are whole numbers. Hard to prove.

Heteroscedasticity is what you’re thinking of. Way too many syllables.

Same but different. My wife did honours in pure maths (first class). She doesn’t understand her thesis ten years later. I don’t get past the intro without needing a dictionary.

That we’re delving so deep into the secrets of the universe that no one human mind can encompass it all?Except Ritchie, of course.

Rob beat me to it.

Bloke in Italy: “ok, in layman’s terms what is an approximation exponent?”

It quantifies how closely you can approximate certain types of numbers. In particular it gives a bound on how many “good” rational approximations there are to any particular irrational number.

We now know about things far too small to see, and far too big to properly comprehend. There’s just too much stuff now, and it’s all written down somewhere.

Understanding the workings and composition of your garden would have been possible once, if you were a polymath. Not so anymore.

The fact is, you need private wealth, no job, no distractions and a lifetime just to understand both baseball and cricket.

Ever since I became aware of relativity, quantum mechanics and that dude with his dead/alive cat in the box and all the spin offs from it, I’ve accepted that when it comes to maths, I’m simply not equipped to understand it.

But can you understand BOGOF deals?

My understanding of the topics in this thread are limited to the difference between rational and irrational numbers.

The former is my paycheque, the latter my wife’s credit card bill.

He was only 30 when he made a significant contribution to the Thue-Siegel theorem by proving that any irrational algebraic number has an approximation exponent equal to two.You can see that was written by someone who did not know much about the field of mathematics. It is unusual for someone that old to make a big breakthrough. Most of them do their good work before the age of 30. Very few continue to do good work past 40.

Einstein’s great year was 1905. He was 26. He did predict gravitational waves in 1916 – when he was 36. But after that most of his work was extending his previous work, or was work with others, or was trying to disprove quantum mechanics.

Ordinary folks are locked out of most science because they don’t have access to advanced equipment etc. Maths is still something that anyone can do if they are clever enough and put the time in. You still only need a pencil and paper.

Ian B – “Ordinary folks are locked out of most science because they don’t have access to advanced equipment etc. Maths is still something that anyone can do if they are clever enough and put the time in. You still only need a pencil and paper.”

Well if you give mathematicians massive grants, they can do things with advanced equipment too. See the Four Colour Problem – which only a computer can check.

Other than that, someone, perhaps Lord Kelvin or Lord Rutherford, said all science was mathematics or stamp collecting. Geology and biology are obviously stamp collecting. You do not need expensive equipment. You just have to know what you are looking at. Naturally the full time professional scientists have been erecting barriers to keep the hoi polloi out – credentials, expensive equipment and so on. But there is no reason for it.

That is true even in the harder fields. If someone does any decent physics they could do it with billions of pounds worth of equipment or with a pencil. But a real break through won’t involve CERN or Hubble. It will be some young man probably without a proper job working with pencil and paper.

I can understand what Roth’s theorem is saying, but I wouldn’t have the first clue about how he approached the problem. One of the biggest problems in modern science and mathematics is how sharp the knee in the curve is. Most tolerably bright people can follow most things up to about halfway through an undergraduate degree, but after that the level of difficulty takes off like a skyrocket.

Arthur the Cat:

“Heteroscedasticity is what you’re thinking of. Way too many syllables.”Apparently it can be spelled “heteroskedasticity” or “heteroscedasticity, depending on how one wishes to transliterate from Greek roots to English. Econometricians seem to go with “k” more often than not, but not exclusively. The account at that link by Dr Cox is pleasantly droll.

BiCR

“One of the biggest problems in a R Murphy political economics degree is how sharp is the knee in the curve, the break in the ankle, the right angle of the foot in the mouth and the doubling back on themselves of the arguments. Most tolerably bright people can follow most things up to about halfway through a normal undergraduate degree, but in a Murphy degree, the non sequiturs, failures in logic, inconsistencies and hypocritical positions are likely to leave the average student, let alone the brighter one, utterly baffled.”

“the approximation of real numbers by rational numbers”: that bit is schoolboy stuff – surely you understand that bit, Tim?

P.S. There’s no point banging on about Einstein in a discussion of maths. The man was a mathematical physicist – a quite different beastie. He said himself that he was no mathematician, and that was not false modesty.

As for the proposition in Tim’s headline: it depends, as Yorkie Mathmo and others have said.

In some fields, it is even the case that the falseness of advanced work is penetrable by the layman – e.g. Global Warmmongering.

No, I don’t get that bit. Numbers are numbers to me, don’t know the difference between real (well, OK, so not square root of minus 1) and rational. Also don’t know the different between ordinal and cardinal. Was explained to me once but it never stuck.

dearieme – “In some fields, it is even the case that the falseness of advanced work is penetrable by the layman – e.g. Global Warmmongering.”

But that has nothing to do with the science. The average intelligent person can see solid science works by saying “So you disagree with me? Well, what does the evidence say?” It is not hard for them to figure out that whatever title is on the box, people who respond to the slightest criticism with cries of Heresy and calls for them to be fired are not doing science.

Although obviously the science is garbage. There is no way that models can predict anything.

Tim, that’s a bit worrying considering that the basis of modern theory of value is that values are ordinal rather than cardinal.

Don’t feel bad about not knowing the difference between real and rational numbers. Although I once knew off the top of my head I just realized that I can’t explain it anymore without a refresher. Learning something isn’t enough. That knowledge has to be used or it can very easily be lost.

I argue that there has never been an expert in more than a couple fields even in the stone age. While the flint knapper might have enough general information to build a rabbit trap, skin, cook the meat, and tan the hide he would not qualify as an expert in any field except the one or two he used the most.

The beauty of the human mind is that we are able to acquire a broad base of knowledge and use this to know if the “expert” in another field knows what they are talking about or is just full of it. Without the ability to compartmentalize information in different minds nothing in modern society is possible. Concepts such as economics, education, and industry simply could not happen if someone can be an expert in everything.

With this in mind Tim doesn’t need to understand the difference between ordinary and cardinal numbers. What is important is that his mind is liberal* enough to accept the fact that these exist so he can gain enough useful information from a mathematician(expert) to explain a concept he is an expert in such as rare earth processing.

*Tim I hope you aren’t insulted by me calling you liberal. For the definition of the word I trust an expert(dictionary writer) and not some yahoo! that decided to create an arbitrary meaning. The fact that you are willing to admit when you don’t know something is the first step in becoming a literal,not political, liberal. Since you also ask others instead of making something up completes the process so you are a (non-political) liberal to me.

Tim, you do know the difference between 1st, 2nd, 3rd and 1, 2, 3 and there’s no point pretending otherwise. The mathematicians’ version of ordinals and cardinals is a lot hairier but also of absolutely no importance to anyone not a mathematician (and even to most of those, the ones who are not set theoreticians).

Well of course I’m a liberal. A classical liberal. What you in the US would call a libertarian but without all that Ayn Rand idiocy.

Yes, but I didn’t know that’s what it meant.

A rational number is the ratio of two integers. Some irrational numbers, including the square root of two, are algebraic – they are the solutions of polynomial equations. The others, such as pi, are called transcendental numbers.

The existence of irrational numbers was first proved in about 400BC, perhaps by Hippasus, who is said to have been drowned for it. The existence of transcendental numbers was first proved in 1844 by Liouville, who was awarded a chair in Paris for it.

Roth proved an property of algebraic numbers – how accurately they can be approximated using rational numbers with increasingly large denominators – which is useful (in some sense) because it tells you that irrational numbers which can be approximated more accurately must be transcendental.

It’s worth knowing that pi is transcendental if you’re trying to ‘square the circle’. Because it tells you to give up.

I’ve neglected to report on my visit to the local Libertarian party meeting.

Long story short I was not exactly welcomed. One has to believe in Ayn Rand just as scientologists insist on believing L Ron Hubert is a god.

I seriously pissed off a rather pretty young London (expat yank) Randite. She was talking about how unlimited sex with whoever you want was just a necessary part of human rights. Or something, I wasn’t following the argument all that well. She got most pissed off when I made her an offer she could refuse.

Ayn Rand was a nut-job. And a hypocrite for claiming welfare.

Rand’s Objectivism isn’t the definition of libertarianism, it’s not even really libertarian. It’s like saying to be A Christian you have to be a Calvinist.

Randians for instance are decidedly lukewarm on the Austrian/classical economics of libertarianism and some reject it entirely; because it is based on the subjectivity of value while Rand tries to derive an objective theory of value which in a nutshell says that value is defined by how Randian something is.

Hence Rothbard’s short play “Mozart Was A Red” which is a total pisstake of the Rand Cult.

Rand called her movement “Objectivism” for a reason. It is not at all compatible with the Subjectivism that underpins (the economic theory, at least of) Libertarianism.

Mind you, it’s a Sysyphean task trying to get most Libertarians to recognise that subjective value isn’t limited to economic exchanges but in fact encompasses all values, including morals and ethics etc, of which economics are just a subset (the values of the shit we trade in a market).

Anyway, Libertarianism and Obectivism are not synonyms.

Ordinality of values is, by the way, why people can’t really answer those “rank this on a scale of 1 to 10” questions meaningfully. It’s asking for a cardinal answer to an ordinal question.

“Rank the beauty of person X from 0 to 10” is not what we’re good at. But “Is X more or less beautiful than Y?”. That we can do.

https://xkcd.com/435/

In a brief article in Econometrica [1985b], J. Huston McCulloch said that “[t]he most

pressing issue in econometric orthography today is whether heteros*edasticity should be

spelled with a k or with a c.”

Tim Worstall – “I seriously pissed off a rather pretty young London (expat yank) Randite. She was talking about how unlimited sex with whoever you want was just a necessary part of human rights. Or something, I wasn’t following the argument all that well. She got most pissed off when I made her an offer she could refuse.”

Well there’s your mistake. If I understand Ms Rand’s writings, personality and political beliefs correctly, the True Objectivist Super Man does not make a pretty young thing an offer she can refuse. He is such a supreme example of manhood that she is powerless to resist as he takes her right there on the public bar. Or anywhere else he fancies.

So she wasn’t entirely nuts then.

There is no way that models can predict anything.So it would be impossible for James Clerk Maxwell’s vortex model of electric and magnetic fields to predict the existence of electromagnetic waves and their velocity. How unilluminating.