[US] T. Dantzig: “Numbers: The Language of Science”

7,279 characters2007.07.18

[美]T·Danzig: “Numbers: The Language of Science,” translated by Su Zhongxiang, Shanghai Education Press, December 2000, 17.1 yuan

This book is a classic, though unfortunately it was written a little early (1938). Of course, read today it does not feel outdated; it’s just that in some places it seems to leave one wanting a bit more. Among the various popular mathematics books I have read in translation, this one is neither especially good nor especially bad—though it is also possible that, having already read so many related books, I’ve long since grown numb to much of the content…

Page10, p12 Numerals are usually remarkably stable; even as everything else undergoes fundamental change over the passage of time, we find that the vocabulary of numerals is hardly affected at all. In fact, linguists make use of this stability to infer the kinship between language families that seem, on the surface, to be very far apart.////—I’ve learned something~

Page11, p14 Among the most primitive peoples of Australia and Africa, there still exists a way of counting that is neither based on five, nor on ten, nor on twenty, but on two: a binary system. These savages have not yet reached the stage of counting on their fingers; their independent numerals are only one and two, their compound numerals go only as far as six, and beyond six everything is simply called “many.” Page12: Kohl, mentioned earlier in connection with Australian tribes, claimed that most of these peoples count by twos. Their habit has become so deeply ingrained that if we remove two from a row of seven needles, they will hardly notice; but if we remove only one, they notice at once.///—Another thing learned~

Pages26~27, p26 There is a story about a German merchant in the fifteenth century which, though I cannot prove that it actually happened, presents the conditions of the time so vividly that I cannot help telling it. It goes like this: the merchant had a son, and he wanted to have the boy receive a good advanced commercial education. So he went to ask a famous university professor what school he should send his son to. The professor replied that if the young man’s mathematics curriculum was to be limited to addition and subtraction, then he could study those subjects at a university in his own country; but as for multiplication and division, he said, Italy was still the most advanced, and in his view only there could one obtain such higher education.///—I cannot help excerpting this as well~

Page30, p37 Those who are accustomed to explaining cultural history from a utilitarian point of view will conclude that arithmetic came before number theory; yet on the contrary, the theory of integers is one of the oldest branches of mathematics, while modern arithmetic has a history of less than four hundred years. P38 The origins of all science can be traced back to reflections on this mysterious influence. Astrology preceded astronomy; chemistry emerged from alchemy; the predecessor of number theory was a kind of magic numerology, and even today it still persists in other incomprehensible byways and omens.

Pages82~83, p97 I admit that I personally cannot endorse the extreme formalism of the Peano–Russell school, and I admit that their formal logical method has never interested me. I admit that my repeated efforts to grasp their complex symbolic system have always ended in dizziness and disappointment. Such a personal quirk of temperament will no doubt color my opinions—and that is precisely why I should not be voicing my bias here. Still, I am certain that these prejudices will not lead me to underestimate the role of mathematical symbolism. My own view is that the importance of such a symbolic system lies not in its hopeless effort to drive intuition out of the realm of human thought, but rather in its limitless power to assist intuition and create new forms of thinking.////—I agree.

Page193, p232~233 Religion is the mother of the various sciences; when the children grow up, they leave their mother. Philosophy stays at home to entertain the old mother in her twilight years. And in the long years of close companionship, the daughter’s story has become even longer than the mother’s. Even now, philosophy’s central problems still bear a theological flavor. To my mind, what philosophy most lacks is a principle of relativity. The principle of relativity is only a limiting rule; it defines the range within which a discipline may operate, and frankly admits that there is no way to determine whether a given set of facts is a manifestation of the observed object or an illusion of the observer.////—That’s quite interestingly put; poor old mother… In fact, in a certain sense, Kantian philosophy and all kinds of modern philosophy can both be said to have, in different ways, added this so-called principle of relativity.

Page196, p236 The domain of natural numbers is built on the assumption that the operation of adding one can be repeated an infinite number of times, but it explicitly stipulates that the last step of this process itself cannot count as a number. When we extend this to the real numbers, we not only extend the validity of infinite repetition to arbitrary rational operations; in fact, we discard this restriction altogether and regard the limit of such a process as a bona fide number. And this is precisely the irony of the term: so-called real numbers are obtained by sacrificing part of the reality we ascribe to natural numbers.///—Makes sense~

July 18, 2007

Latest comments

UNIC

2007-07-18 23:05:46 Anonymous 222.82.73.112 [Reply]

And this is precisely the irony of the term: so-called real numbers are obtained by sacrificing part of the reality we ascribe to natural numbers
I don’t quite understand….
So natural numbers are finite? Numbers that are repeatedly added without end don’t appear in nature, then?

Gu Xu

2007-07-19 09:47:03 Anonymous 123.112.105.153 [Reply]

This point is mainly aimed at irrational numbers. “The natural number domain is built on the assumption that the operation of adding one can be repeated infinitely many times”—natural numbers are constructed one after another starting from “0,” through the operation of “adding one.” However, within the natural numbers, the “last term” of this infinite construction, namely “infinity,” does not belong to the natural numbers. But if we move to the real numbers, then if we say that the real number domain is an extension of the natural numbers, this extension requires that every irrational number can be constructed from rational numbers—that is, from natural numbers. And now we find that any real number can be represented by a rational sequence that converges infinitely, and at the same time any rational sequence can be represented as an infinitely long decimal; in this way, the real number domain is closed under infinite operations. But what the author of this book says is that this step of extension is essentially different from the extension from integers to rational numbers, because it requires taking the result of an infinite sequence itself (the limit) as a number.
In brief, what is meant here is nothing other than the introduction of “actual infinity.” The infinity of natural numbers is potential infinity; it does not require affirming that this infinite process of adding one has already been completed. But real numbers require that the infinite operation be “carried through to completion.”
Note: As for what it means for a rational sequence converging infinitely to represent an irrational number, one may think of π/4=1-1/3+1/5-1/7+1/9-1/11+……; as for what it means for the real number domain to be “closed” under infinite operations, one can compare this with the fact that the natural number domain is closed under addition and multiplication but not under subtraction and division (that is to say, the sum of two natural numbers is still a natural number, but the difference need not be), whereas the integer domain is closed under subtraction, and the rational number domain, except for the special case of 0, is closed under division.

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2008-07-05 19:45:53 Anonymous 116.19.92.64 [reply]