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At The Universe of Discourse, Mark Dominus notes (with a followup) a gorgeously simple geometrical proof that the square root of 2 is irrational.

Irrational? Well, the mathematical use of that word is probably owed to the fact that such a number is not expressible as a ratio or fraction of two whole numbers. OED cites an early use of “to be rational to” as meaning “to be in a ratio with”:

1614 T. BEDWELL Nat. Geom. Numbers i. 2 The Base and Height are said to be rational one to another, when as the rate or reason of both may be expressed by a number of the same measure given.

But OED‘s first citation of numbers being called “irrational” predates Bedwell’s explanation by half a century, so it is not entirely clear that this is a watertight derivation.

Could we also ask, for example, whether the survival of the term “irrational” in mathematics also carries an echo of an ancient mystical superstition – that the existence of a number which could not be expressed as a fraction of two integers somehow represented a catastrophic flaw in the logical purity of the universe? (The legend goes that the Pythagorean who discovered √2’s irrationality was drowned by his comrades for this reason – although, as Myles Burnyeat shows in this week’s LRB, what we thought we knew about the Pythagoreans is mostly wrong.)

OED shows that, whether from distinct motivations or not, “irrational” was from the beginning applied both to beasts, or to men who couldn’t properly exercise the faculty of reason, and also to the “irrational numbers”. Interestingly, “rational” itself was for a time contrasted with “empirical”: the “Rational Physicians” or “Rational Psychologists” or practitioners of “Rational Mechanics” were those who worked from first principles rather than observation.

If an old superstitious despair at creation’s imperfection might illuminate the origin of the mathematical sense of “irrational”, we could also note that the use of the word in contemporary political arguments is, too, almost always superstitious and nihilistic. As I have previously pointed out, to call Islamist groups, for instance, “irrational”, is to take pride in intellectual laziness, to refuse the effort of understanding, and to shut down conversation as a prelude to more righteous violence. (Although arguably to do so is “rational” in the older sense of according a subordinate place to facts.)

A propos of the apparently endlessly ramifying links between mathematical and political terminology, meanwhile, the square-root sign, √, is also known as the “radical”. Thus, to save time, which is of the essence in such matters, “radical cleric” can be written: √Ω.

  1. 1  abb1  February 22, 2007, 10:41 pm 

    OK, but let’s not be negative here.

  2. 2  sw  February 22, 2007, 11:01 pm 

    I would note that today it was announced that women at Wimbledon will receive the same prize monies as men for each stage of the competition. This had been resisted because women play Best-of-Three sets, whereas men endure longer Best-of-Five sets (except Federer, who usually only bothers to play three sets). It was claimed before that it was rational to divide the winners’ spoils up in proportion to the amount of work they do; it is now claimed that this is a great victory for equality, even though women will have to work less for their money (except Federer, who only rarely even bothers to break into a sweat). Rationality does not always lead to equality.

  3. 3  Andrew Kenneally  February 22, 2007, 11:09 pm 

    The mathematical stuff suggests that what is
    Since the universe being itself cannot contradict itself, it is perhaps not the logical purity of our universe that is being confounded by such mathematics but an intellect incapable of surrendering to the infinite- the infinite by its nature beyond being rendered finite within mathematical systems. Also if an awareness truly subordinated itself to rationality and nothing but, it must then end in absolute nihilism, ie total scepticism about thought itself.

  4. 4  Andrew Kenneally  February 22, 2007, 11:34 pm 

    Oops, a bit of a screw up there..ignore first line.

  5. 5  Vronsky  February 22, 2007, 11:50 pm 

    I believe that what the Greeks (who thought in pictures) noted was that the diagonal of a square is not commensurable with its sides. So is the problem with Islam in fact incommensurability? Because that sets something from Khayyam vibrating in my memory. No, that’s Eliot. No, hold on…

  6. 6  Richard  February 23, 2007, 1:49 am 

    the only things I knew about Pythagoras was that he wasn’t responsible for the theorem, he miraculously dissuaded a cow from eating beans and he rose bodily into the air at the end of his life.

    I can now relax in my ignorance.

  7. 7  Andrew Kenneally  February 23, 2007, 2:05 am 

    Dissuading cows from eating beans a shamefully underrated ability.

  8. 8  Jeff Strabone  February 23, 2007, 8:24 am 

    On the matter of equal pay for male and female tennis players, there is another possible measure besides payment-for-quantity-of-work-performed. Does one gender or the other attract higher ratings and, thus, higher revenues? If so, the more popular gender could argue that they are entitled to an equal percentage share of higher revenues. A few years ago, it seemed to me that women’s tennis was much more popular. I’m not advocating that standard, merely pointing out that there is no rationally superior measurement of the value of men’s tennis versus women’s.

  9. 9  Steven  February 23, 2007, 2:07 pm 

    Vronsky’s ludically allusive comment is very pert…

    If we must turn to the subject of tennis, well, the chairman of the All England Club said:

    “This year, taking into account both the overall progression and the fact that broader social factors are also relevant to the decision, they have decided that the time is right to bring this subject to a logical conclusion and eliminate the difference. We believe this positive step will be widely welcomed.”

    As sw and Jeff had pointed out, this is hardly the only “logical” (or rational) conclusion. You could also, for instance, eliminate the whole idea of “men’s” and “women’s” tennis and just have, er, tennis.

  10. 10  Steven  February 23, 2007, 2:20 pm 

    Getting back on-topic slightly, “Advanced Geometry of Islamic Art”:

    “It’s absolutely stunning,” Harvard’s Peter Lu said in an interview.
    “They made tilings that reflect mathematics that were so sophisticated that we didn’t figure it out until the last 20 or 30 years.”

  11. 11  Richard  February 23, 2007, 2:55 pm 

    a space facility in Turkmenistan.

    ?????? That has to be the most Jamesbondian aside I’ve seen in print in years. I thought all the space stuff was in Kazakhstan, but maybe that country is too accessible now.

  12. 12  John Fallhammer  February 23, 2007, 4:36 pm 

    My impression is that, regardless of the babblings of mathematicians, the ratio thing is a red herring. Rationalism is etymologically a cousin rather than a child of the ratio.

    As it happens (dodgy link alert), I have a child. I’ve observed her ‘irrational’ behaviour over the last few years and, when carefully considered, it’s almost always rational within a certain kind of logic. It makes complete sense given the information and interpretative tools available to a small child (QED?).

    Which brings us back to the mathematicians. If your mathematical worldview is entirely built on integers, irrational numbers really are irrational. They make no sense at all. Mathematicians have had to invent totally new ways of counting to even start on them. It certainly makes yer fink.

    Anyway, on the topic of pythgoras and beans, if he ate all the beans himself, then I think I can guess the mechanism by which he later rose bodily into the air.

  13. 13  Vronsky  February 24, 2007, 12:30 pm 

    “Advanced Geometry of Islamic Art”

    Mention of the tilings in the Alhambra Palace usually crops up in mathematics courses in Group Theory (e.g. OU). Last I heard the Alhambra did not contain (as once claimed) examples of all 17 (from memory) possible symmetries (they scored 15). Still, beats family snaps of saints.

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