Wednesday, December 18, 2013

Math is the language of the true culture of modern times

...and math skills are unavoidable side effects of skills that turned some humans into heroes of natural selection...
Copenhagen: I haven't been to theaters in Boston too many times ;-) but it was at least once when we went to see Copenhagen. You may watch Bohr and Heisenberg in a 90-minute BBC-4 movie now. Via Joseph S.
Three days ago, a German blog published a text about the role of mathematics in the society:
Mathematics, the language of nature. What are you sinking about?
Edward Measure's brief response proved that mathematics plays a much smaller role in the society of pigs.

Reading and writing is considered a normal skill of a person who is not a complete loser. Nevertheless, most TRF "readers" will be unable to write down the words from the pictures that start with a given letter – even though most 4-year-old kids around me can do it. ;-)

To summarize her text, the author is trying to answer the question whether mathematics was vital for our ancestors; whether it is needed for average people and for us today (these are two different questions); and whether mathematics should be included despite the stupid people's negative attitude towards it. I was sort of happy to see that the writer's answers are the correct No, No, Yes, Yes.

But let's start at the beginning and ask: Does the existence of mathematics skills contradict Darwin's theory of evolution?

Edward Measure's answer is Yes, Darwin has been debunked:
I think [the German writer] is a bit delusional on this point. Math, unlike language, is an unnatural activity in the sense that our remote ancestors almost never needed it.
Interesting. So why did these skills evolve?

Pig's comment would be nice except that it is at least morally wrong (wrong in the effect if not in his intent). Arun responded:
Equally unnatural are the finger and thumb motions used in keyboard entry or in video games.
Arun's and my point here is that we can do many things that our ancestors have never done before simply because the required skills and "hardware" needed to realize these skills are pretty much equivalent to what our ancestors did need (or at least found helpful).

And the brain able to do some reasoning that is closely analogous to mathematical reasoning was very helpful, indeed. Our species is one of those that have flooded the globe. There are other classes of species that have achieved a similar thing – various germs, insects, and rats (their number is close to the number of people) - but species that are too physically different from us can't be compared to us too well. Among similar enough species, humans are the winners.

Why is it so? The humans are rather mediocre when it comes to various physical parameters. We are much slower than cheetahs, much weaker than tigers, lions, and elephants, and so on, and so on. I think it is obvious that things related to our brain are the main reasons why we have reproduced much more than most other mammal species – and most other species, too.

Some reasoning and planning that is working "pretty much mathematically" was what has helped the individual people and their tribes or nations to spread, to become more resilient. The required amount of CPU power etc. was smaller than what you need to win the new \$3 million Milner-Zuckerberg Prize in Mathematics but there existed a clear regime in which upward fluctuations were often beneficial. So the brain size and power just happened to grow.

Pig would say
I wasn't talking higher math here, but rather lower math - counting, adding, multiplying, dividing... some primitive peoples only have words for one, two and many.
But the tribes whose counting system is "one, two, many" are not really exceptional winners of evolution (the number of members of these tribes doesn't even beat the llamas and tons of other species) so the basic claim about the importance of maths for the human survivor is actually confirmed. It's fashionable and politically correct to count "all humans" as the superior race but that's not how Nature actually works. If we interpret the term "humans" as "the tribes that count one, two, many", then humans are actually a rather mediocre species when it comes to all parameters. (These tribes are still significantly smarter than all other species on the globe, I believe, but this edge, however significant, is just not enough to overshadow the other signs of mediocrity.)

Genetically speaking, these tribes are still very close relatives of Albert Einstein and others. But the distance between them has an extremely strong impact on the survival skills; the tribes Pig mentioned – although we may love them – are simply not that exceptional among other species.

Moreover, the counting "one, two, many" may only be a proof of their inferior language skills. This simplified terminology for integers does not prove that these individuals are actually unable to think about greater integers or do operations with these integers.

OK, something similar to the math reasoning was a great gift for our ancestors and these skills just grew to a point that was no longer "vital" for the survival but these "enhanced" mathematics skills could rather easily develop from the material that was available. At least some limited amount of skills needed to do mathematics is an unavoidable consequence of skills that our ancestors found crucial for their success at the Darwinian job market.

Similar observations hold about the contemporary mankind, too. Most people don't need any advanced mathematics for their survival – and even for their being wealthy. (And a long essay should be written about the maths that a theoretical physicist needs. The required maths is clearly vastly more extensive than the normal man's mathematics but there are still portions that remain irrelevant for a theoretical physicist, even one who wants to make one breakthrough after another.) But the fluctuations in the math skills are unavoidable. So some people are inevitably much better at maths than others. And this fact is ultimately useful for the whole society. To a large extent, math skills are extremely helpful for the survival of a stupid human, too – it's just the math skills of another human.

Finally, the German thinker rightfully argues that people are doing tons of other things that are not needed for the survival but they are viewed as the part of the "culture". People get frequently exposed to them and learn them through this exposure even though they are not needed. She says that it is appropriate to expose people to maths, too.

I completely agree with that. One of the most famous undergraduate textbooks in the mankind's history states that "physicist's way of looking at [this wonderful world...] is a major part of the true culture of modern times". Mathematics is the language underlying this way of looking at the Universe and this is the reason why people should constantly expose others – and be exposed – to the language of mathematics even though it is not really needed for their bare survival (and usually not even for their much-better-than-bare survival).

An anti-mathematical rule that some popular writers and publishers have internalized says that every equation included in a book/text reduces the number of its readers to one-half. You may calculate that if I have avoided equations, the number of TRF readers wouldn't be 10 million but 10 million times 2 to the power of 10,000 or so – vastly more than the number of electrons in the visible Universe. ;-)

I think that this rule is really counterproductive. Most of the things in physics – even the most fundamental and/or most elementary things – are impossible to be really conveyed without mathematics. It's questionable whether the popular presentations of physics without any mathematics are useful at all – or, in other words, whether they are more useful than some generic superstitions and paranormal stories that may look equally spiritually rich and justified (and equally emotionally satisfying for the reader) as physics if one decides not to see any maths.

Even if one doesn't understand all the things, it should be politically correct to include mathematics, its jargon, and equations into texts. Mathematics isn't some optional thing that one needs a permission for. Many people are clearly punching below their weight. They understand much less mathematics than they would be able to – and the reason is that they are not being exposed to mathematics. That should change.

The Czech alphabet (Q,W and mostly X are absent in truly native Czech words) along with a sentence including most of the diacritical sign-decorated letters: Let the sinful devils' saxophones sound through the halls by terrifying tones of waltz, tango, and quickstep (and I would add mathematical equations!).

Mathematics is sometimes avoided in popular books because "everyone should understand what is written in the book". But a dirty secret is that it is bullshit. A reader may know individual letters or may have seen individual words before but if he isn't thinking mathematically about the propositions, he just can't understand the whole sentences! He can't understand why they're right – or at least why they're more right than their negations. Sadly, too much focus is on the "syntactic understanding" but the "semantic understanding" – the understanding of the content and why the arguments are right or wrong – is something that almost no one seems to care about, either.

But the semantic understanding is even more important than the syntactic one; syntax is just a servant of semantics; the letters and words are just tools to convey something grander – the meaning, the content. Without mathematics, one doesn't really get any of the content so whether or not the individual letters or words of the text are syntactically comprehensible to the reader is an irrelevant detail.

The people's exposition to mathematics should be increased in order to improve at least some of these people's understanding of the meaning and the content – even though it makes the texts incomprehensible to some other people even at the syntactic level. One could argue that there should be a balance between the syntactic and semantic understanding but I have gone further: the syntactic understanding is really useless and worthless without the semantic one so if someone who doesn't understand the meaning is also led to misunderstand the text syntactically, the situation doesn't worsen at all.$J_\alpha(x) = \sum\limits_{m=0}^\infty \frac{(-1)^m}{m! \, \Gamma(m + \alpha + 1)}{\left({\frac{x}{2}}\right)}^{2 m + \alpha}$ The equation above was needed purely to eliminate most of the math-ignorant readers from the list of future commenters. ;-)

Completely off-topic: I completely agree with Larry that boycotts organized by Academics against other Academics are always wrong and that a specific targeted boycott of Academics in one country – Israel – is even worse than a general boycott. Some U.S. scholar institution named ASA has joined similar shameful trends in Europe. If the ASA head admits that there are others who are worse than Israel but they still choose to boycott Israel only (with a bonus hysterical applause resembling the applause after Adolf Hitler's speeches), do I need anything else to complete the proof that they're anti-Semites? If the territorial disputes are the reason, note that are hundreds of territorial disputes in the world. Someone's singling out Israel proves that he or she has something extremely special and unjustifiable against the Jews. It is pathological and ASA should be treated on par with Ku-Klux-Klan.

1. I hope to see your article on mathematics needed by theoretical physicists.

2. Every American university has a speech code that penalizes hate speech. The ASA resolution is plainly hate speech under those codes, and faculty who are members of ASA should be disciplined.

3. It was logical thinking that was necessary for the ancestors. Math is just a subset of that. An inevitable side effect. But what a side effect, it has driven most of the technological revolution since the agrarian start of the modern age.

4. If mathematics is not "natural" to man then how about music? There are many theories about the origins of music, but it has certainly been known to, in one form or another, to almost every human society. In the more developed ones, including the West, it was closely connected to ritual and religion and mathematics. All of these things are quite “natural” to humans, although they may exist only in very primitive form among the most primitive peoples.

The ancient Greeks considered the study of music to be a branch of mathematics. A very similar view could be found in other cultures (China, Babylon and so on): interest in music lead to the study of ratios. There is a well established correlation between musical and mathematical talent.

CIPig’s “evolutionary” view point is primitive and superficial because it identifies naturalness with “need” understood in the way that applies only to the less intelligent species like pigs. From that point of view, ritual, magic, religion, music and mathematics are all “unneeded” and indeed, they are surely unknown to pigs. CIPig and his like may think that this makes pigs superior but pigs have never built a cathedral, composed a symphony, proved a theorem or discovered quantum mechanics and these facts are not unrelated.

By the way, anyone interested in the relationship between mathematics and music will surely enjoy this beautiful book by Dave Benson:

http://homepages.abdn.ac.uk/mth192/pages/html/maths-music.html

5. It's worth to say that Leonard Euler wrote "Tentamen novae theoriae musicae", a book which is said to be an attempt to incorporate music theory as a full-fledged part of mathematics.

I agree completely that study of Mathematics should be a mandatory part of education. But I have a slightly off- topic , but relevant, question. If Math is a creation of human mind, why should it have so much to do
with how universe is made? I think many great
physicists, such as Wigner, have wondered about it. But as far as I know, no one has a satisfactory answer. Any takers amongst the readers?

7. "mathematics plays a much smaller role in the society of pigs." Your hometown swine are superior to ours. The Los Angeles Unified School District, 700,000 congenitally inconsequential students, has high schools averaging less than random choice scores on math standardized tests.

Proximity is not causality, 1965 legislated Liberalism, President Johnson's Great Society: "with your courage and with your compassion and your desire, we will build a Great Society. It is a Society where no child will go unfed, and no youngster will go unschooled," and nobody productive will keep what they earn. It succeeded beyond all estimated size, cost, and persistence - and 2pi steradians counter to every desired result. The numbers increasingly show hope, as Napolean's march back to Europe was quantitatively more efficient, better managed, and more tightly focused.

http://practicalaction.org/images/events/publicgood-king-7.gif
Lower standards, then dare anybody to pass under them.

8. Many thanks for that link to the book---My oldest son plays at a near concert level, and I can at least mangle stuff on my 1911 Bechstein grand.

9. While realizing that this is meant in a Swiftian sense, I disagree--just having a boycott legitimizes it in many peoples' minds, and the general populace ("dummer pobel") do not know or care that the instigators are Xstudies
scum and assorted whiners with a hugely undeserved sense of entitlement. It is the same lunacy that put Libya as the chair of the UN Human Rights Commission several years ago. Israel is a bastion against Islamic tribal fundamentalism in the region--or, in Carl Sagan's words, "A candle in the dark"...

10. ... "And a long essay should be written about the maths that a theoretical physicist needs."

I would love such an essay explaining what math is needed for what cool physics issues, spiced up with good hints about were one can find nice introductions to these (advanced) mathematical topics !

And many other people, who got their questions closed on Physics SE when asking for the mathematical prerequisits to understand different advanced theoretical physics topics, would certainly be thankful too ...!
Dimension10 (and I in a few days too in principle) could the put the link to the TRF essay below the corresponding questions ...

Please Lumo, this would be a very nice Christmas present ... ;-)

Going to read the rest of the artilce now, but I just had to say this immediately ...

11. Being at near concert level can be a big problem ;-)

Here is a curious story which I once heard from one of the protagonists.

Some years ago two young Australians came to one of the most famous British universities to do a PhD in mathematics, with two of the most prestigious scholarships. One of them did his PhD and eventually became a very well known mathematician. The other (who actually had the somewhat more prestigious of the two scholarships) decided however that he would much rather be a concert pianist than a mathematician. So he have up mathematics and spent all his time practicing. Various famous musicians came to hear him and were impressed. But eventually when the time came to try to give a concert in public the young Australian discovered that it was an impossible task for him. He just could not perform in public. So that was the end of his career as a pianist. He returned to mathematics, having been away for several years, and to make things worse, ended up as a student of the other Australian. He did some good work, but for many years found it difficult to get a full-time academic position (what’s even worse, his wife, who was also a mathematicians had one and had to support him) (eventually he did succeed getting a position at one of the best British universities).

I heard the story from the other Australian (not the pianist). “Of course he hates me” -, were his concluding words.

12. "You may calculate that if I have avoided equations, the number of TRF
readers wouldn't be 10 million but 10 million times 2 to the power of
10,000 or so – vastly more than the number of electrons in the visible
Universe. ;-)"

Maybe if all equations were removed (Nope, TRF articles containing equations does NOT mean they are about homework ;-)...), all the Boltzmann brains would read TRF too ... ;-D?

Ok enough jokes for today, I really like and agree with this TRF article :-)

13. I know you are not a big fan of Roger Penrose, Lubos, but in this post you match in 100% with the criteria of Roger Penrose for writing "The Road to Reality" as stated in his preface.

"...I have thought seriously about this question, and have come to the conclusion that what I have to say cannot reasonably be conveyed without a certain amount of mathematical notation and the exploration of genuine mathematical concepts..."

And so he dedicates as a third of the book to give us a crash course and unorthodox of Mathematical Methods of Physics. I get the impression that anyone without a good foundation in mathematics passed the line marked by the (brilliant) prologue. The conclusion is immediate, to understand physics, mathematics are mandatory... as well as to understand a musical score you will need to know music theory. There is no middle ground.

14. Hooray, the Bessel Function! I just learned about this beauty several weeks ago, though I am still shaky on exactly what the gamma function is/does (at this point all I really understand is that it is a generalization of the factorial function to the entire complex plane minus the negative integers).

Just chiming in here, but I would also be much in debt if you were to write an extensive article outlining the necessary mathematics to understand, think about, and of course extend, physics as it is currently practiced. One of my medium-term personal goals (~2-3 year time frame) is to learn this set of mathematics. I've been basing my pursuits off of Gerard 't Hooft's page http://www.staff.science.uu.nl/~hooft101/theorist.html up until now, but I am certain that your perspective would help me in structuring my learning plan.

15. I read Penrose's book "The Emperor's New Mind. Concerning Computers, Minds, and the Laws of Physics" when I was like 14, and while I later found out that it contained way too much speculation weakly founded in reality for my taste, the book taught me some maths school didn't teach me until years later.

I am still thankful for that, because I think the sooner one comes into contact with somewhat "advanced" mathematical concepts the better. I firmly believe all the mathematics I learned in school could have easily been taught to me in half the time, and what I could have done with those riches!

Too bad I was too lazy to teach mathematics to myself, but whenever the opportunity presents itself, I try to pass some of my knowledge on to adolescents, in the hopes that it falls on fertile ground, like it would have with me.

16. Yep, I like best the first mathematical part of the "Road to Reality", sometimes I even go back there when I have forgotten what a trivial fibre bundle is and want to know for some reason, etc ... :-D
The nLab is then something do dig deeper ...

http://ncatlab.org/nlab/show/HomePage

However there are things I disagree with Roger Penrose in the physics part too ...

17. This is a good Chrismas story with a nice happy end ... ;-)

The other comment is very funny to read too.

18. No problem--the piano is now a hobby. He does AI, Machine Learning...ie math.

19. No, see, the speech codes are made to be used to silence people with the "wrong" views, and amplify the speech of the actual hatemongers. You have fallen for the newspeak, comrade.

20. Mathematics is the language underlying *reality*. It's actually provincial in the extreme to call it merely a language of the modern age.

21. Physics and Math are the universe trying to describe itself using its own built-in rules. Something like this http://www.ci.westminster.co.us/Portals/0/Repository/Images/ExploreWestminster/PublicArt/SelfMadeMan.jpg

22. Or, it is the language Reality most keenly listens to and obeys (when spoken clearly).
;-)

23. Of course I agree but would you rather be boycotted or praised by a bunch of lunatics?

"President-elect Lisa Duggan, of New York University is also the President-elect of the Council, the Chair of the Board of Trustees, the Chair of the Finance Committee, and the Councilor ex officio of the Women's Committee.Duggan writes for The Nation, is a Professor of gender and sexuality and of lesbian and gay studies at New York University.
Professor Duggan published "Sapphic Slashers: Sex, Violence, and American Modernity", "Sex Wars: Sexual Dissent and Political Culture" and "The Twilight of Equality? Neoliberalism, Cultural Politics and the Attack on Democracy".

Judging particularly by this last work, could professor Duggan be our Cynthia?

24. Tegmark has proposed something in this direction, a hypothesis which may be called "The Mathematical Universe", as you can see here http://arxiv.org/abs/0704.0646.

Roughly speaking, he assumes that there exists an external physical reality which is nothing more than a mathematical structure. That is, our physical world is a mathematical structure. So, it would be no surprise why mathematics is so good in describing the reality, since that, in such a view, reality is mathematical and physics is constructed by mathematical structures approaching the actual mathematical structure which constitutes the physical reality itself.

I have some skepticism about all this because it seems to me to be a strong form of Platonism, since its assumes that the elements of our theories (in particular, the mathematical structures) are really elements of the world (or at least approximations to the more fundamental ones). So, its smells like metaphysics... and I prefer to take the positivist point of view in this matter.

[ However, Tegmark claims that such hypothesis can have some testable predictions, as the reader may read in his paper. Maybe a detailed TRF topic about this would be very nice. ;) ]

You may want to read an opposite point of view (which I also see with skepticism) that all mathematics can be derived from some "conceptual metaphor". Take a look in: http://www.amazon.com/Where-Mathematics-Come-From-Embodied/dp/0465037712/ref=sr_1_1?ie=UTF8&qid=1387561375&sr=8-1&keywords=where+mathematics+comes+from

PS: One of the authors (G. Lakoff) is a hardcore leftist, and he used his ideas about the role of "conceptual metaphor" in reasoning to help the democrats to win an election!!!

25. Nah, give Cynthia some credit. I have yet to see her go on a LGBT or feminist rant--hers are economic and political, and on those, I mostly agree with her. She seems more like a progressive libertarian. People like Duggan are all for stricter government control of improper thinking, with "improper" being defined as anything that deviates from her thinking.
WTF is a "professor of gender and sexuality"?
Instead of making such creatures Presidents of the university, they should be getting restraining orders to keep them off the campus.

26. "I have yet to see her go on a LGBT or feminist rant—hers are economic and political, and on those, I mostly agree with her."

Is this supposed to be the decisive argument ;-)

"She seems more like a progressive libertarian."Yes, indeed, as so was Karl Marx, in exactly the same sense. I am not joking, of course Marx believed in the ultimate complete abolition of the State. The logic was cynthiaish: first you make the state all powerful, “euthanise” all independent centres of power and once you have done that, naturally, the State abolishes itself. You have to have the brains of a “progressive libertarian” to believe in that.

27. Making the state all powerful is the antithesis of "libertarian". I may be wrong about her because I seem to remember her calling herself a ____-anarchist (fill in the blank).
I am sure she can and will speak for herself.
I certainly am totally opposed to a "nanny" centralised government that spies on and controls its citizens.
The bull spread by the NSA about all the spying only being used to root out terrorists is negated by, for example, the McChrystal affair. Inconvenient people are simply rebranded.