onsdag 14 september 2016

Mathematics as Magics 1

Why does the subject of mathematics have such a prominent position in basic school forcing children through lengthy math hours during 9 years, while the fact is that most adults have forgotten most of their own school mathematics and get by very well with a bare minimum of simple arithmetics?

As an expression of this prominent position the total math hours on basic school in Sweden (grades 1-9) has recently been expanded from 900, in two steps with 120 hours in 2013 and additional 105 hours in 2016 to a total of 1.125 math hours, out of a total of about 6.685 in all subjects, thus roughly 1/6 math or almost one full day a week of math for 9 years.

The logic of the expansion is presented to be (i) math is important for both individual and society and (ii) the result of all the math hours invested is close to zero for many students (not even simple arithmetics mastered), from which the conclusion is drawn that (iii) more hours are required.

Of course the logic is a bit weak: if 900 hours gives no result, why would 1125 give better result?
But maybe this is irrelevant, since anyway most children when adults will not miss whatever math they missed to learn in school. But if 900 gives no result you could as well argue that cutting down to a half would give the same result and that would save hours to something more meaningful.

But this is not the way the argument goes. It is instead: mathematics is very important for both individual and society and thus no effort should be spared for the purpose of mathematical enlightenment of the minds of all young people of a nation (like Sweden or China). Of course we can expect another expansion in 2019 and so on until the school day is filled with math!

But why is mathematics viewed to be so important, when most people have little use of more than a bare minimum of arithmetics? Who is selling this idea? How come that it is so uncritically embraced by just about everybody? Take any subject and claim that it should be expanded by 225 hours and see if you can succeed! Math is the unique subject for which this is possible.

Let us see what answer we can find in the book Is God a Mathematician? by Mario Livio:
  • A few years ago, I was giving a talk at Cornell University. One of my PowerPoint slides read: “Is God a mathematician?” As soon as that slide appeared, I heard a student in the front row gasp: “Oh God, I hope not!”
  • My rhetorical question was neither a attempt to define God for my audience nor a shrewd scheme to intimidate the math phobics. Rather, I was simply presenting a mystery with which some of the most original minds have struggled for centuries—the apparent omnipresence and omnipotent powers of mathematics.
  • What is it that gives mathematics such incredible powers? Or, as Einstein once wondered: “How is it possible that mathematics, a product of human thought that is independent of experience [the emphasis is mine], fits so excellently the objects of physical reality?”
  • This sense of utter bewilderment is not new. Some of the philosophers in ancient Greece, Pythagoras and Plato in particular, were already in awe of the apparent ability of mathematics to shape and guide the universe, while existing, as it seemed, above the powers of humans to alter, direct, or influence it. 
  • The English political philosopher Thomas Hobbes (1588–1679) could not hide his admiration either. In Leviathan, Hobbes’s impressive exposition of what he regarded as the foundation of society and government, he singled out geometry as the paradigm of rational argument: 
  • Seeing then that truth consisteth in the right ordering of names in our affirmations, a man that seeketh precise truth had need to remember what every name he uses stands for, and to place it accordingly; or else he will find himself entangled in words, as a bird in lime twigs; the more he struggles, the more belimed. And therefore in geometry (which is the only science that it hath pleased God hitherto to bestow on mankind), men begin at settling the significations of their words; which settling of significations, they call definitions, and place them in the beginning of their reckoning
  • Millennia of impressive mathematical research and erudite philosophical speculation have done relatively little to shed light on the enigma of the power of mathematics. If anything, the mystery has in some sense even deepened.
  • Physics Nobel laureate Eugene Wigner (1902–95) was equally dumbfounded: (a success that Wigner dubbed “the unreasonable effectiveness of mathematics”): The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. 
  • The person who presents what may be the most extreme and most speculative version of the “mathematics as a part of the physical world” scenario is an astrophysicist colleague, Max Tegmark of MIT. Tegmark argues that “our universe is not just described by mathematics—it is mathematics”. 
We here find the idea of a mystery of an unreasonable effectiveness of mathematics as a positive answer to the question Is God a mathematician? We see this idea expressed by physicists (Wigner and Tegmark), while many mathematicians would take the (opposite) position of the British mathematician Godfrey Harold Hardy (1877–1947):
  • Hardy was so proud of the fact that his work consisted of nothing but pure mathematics that he emphatically declared: “No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.”  
Hardy's view of the uselessness of mathematics is supported by the mathematician and computer scientist Hamming:
  • “There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology.” 
We thus find a range of views concerning the usefulness or effectiveness of mathematics from unbounded admiration to controlled skepticism. Most adults would say that they believe that math is very important, as something they learned in school, while they have themselves found very little use math beyond (if they can) helping their children through the frustrations of school math.

We see that the idea of math as something of god-like quality gets stronger as we move away from pure mathematicians (Hardy), over physicists (Wigner, Tegmark) to ordinary people. A similar pattern may be found in religion with a priest as covert non-believer and a community of strong believers.

The less you know of mathematics, the more powerful you tend to believe it is. To convey this idea is in fact an explicitly stated goal of Swedish school mathematics, and in this respect the education is very successful: When finishing school all students, independent of success in math, are fully convinced that math is very powerful and important and in addition very beautiful!

Mathematics is thus viewed as truly magical by many, which makes rational reasoning about school mathematics very difficult, or simply impossible.  How to be rational about magics?

I will continue with some examples of the magical character of math....and eventually I will land on a standpoint of reasonable effectiveness,  between Wigner's unlimited optimism of unreasonable effectiveness and Hamming's deep pessimism of unreasonable ineffectiveness.

2 kommentarer:

  1. Mycket bra. Knyter (som du vet) an till vad jag försökte förstå i min avhandling och vad jag sedan jobbat vidare med. Kanske skulle vi försöka hjälpa varandra på något sätt, mer systematiskt, att förstå detta? Det finns så många frågor: dels historiskt, hur matematiken fick denna plats och "blev" detta oerhörda. Men också sociologiskt/antropologiskt: hur dess "väsen" som allsmäktig allestädes närvarande upprätthålls praktiskt, dvs hur människor ständigt indoktrineras att "veta" att matematiken är detta, märkliga.

  2. Ja, låt oss se om vi tillsammans kan förstå och förklara detta mysterium.