Does using calculating devices weaken the ability to calculate without them?
Yes2
- 1762Rousseau, Emile
Too much apparatus, designed to guide us in our experiments and to supplement the exactness of our senses, makes us neglect to use those senses. The theodolite makes it unnecessary to estimate the size of angles; the eye which used to judge distances with much precision, trusts to the chain for its measurements; the steel yard dispenses with the need of judging weight by the hand as I used to do. The more ingenious our apparatus, the coarser and more unskilful are our senses. We surround ourselves with tools and fail to use those with which nature has provided every one of us. But when we devote to the making of these instruments the skill which did instead of them, when for their construction we use the intelligence which enabled us to dispense with them, this is gain not loss, we add art to nature, we gain ingenuity without loss of skill.
SummaryThe more ingenious our apparatus, the coarser our senses; instruments like the theodolite let us neglect judging angles and distances ourselves.
Summary disputed by reviewer: Rousseau adds that making the instruments oneself, with the skill they replace, is gain not loss.
Book III
- 1841Emerson, Self-Reliance
For every thing that is given something is taken. Society acquires new arts and loses old instincts. What a contrast between the well-clad, reading, writing, thinking American, with a watch, a pencil and a bill of exchange in his pocket, and the naked New Zealander, whose property is a club, a spear, a mat and an undivided twentieth of a shed to sleep under! But compare the health of the two men and you shall see that the white man has lost his aboriginal strength. If the traveller tell us truly, strike the savage with a broad axe and in a day or two the flesh shall unite and heal as if you struck the blow into soft pitch, and the same blow shall send the white to his grave. The civilized man has built a coach, but has lost the use of his feet. He is supported on crutches, but lacks so much support of muscle. He has a fine Geneva watch, but he fails of the skill to tell the hour by the sun. A Greenwich nautical almanac he has, and so being sure of the information when he wants it, the man in the street does not know a star in the sky. The solstice he does not observe; the equinox he knows as little; and the whole bright calendar of the year is without a dial in his mind.
SummaryOwning a watch and a nautical almanac, civilized man cannot tell the hour by the sun or name a star; each tool costs a native skill.
Essays, First Series
No3
- 1614Napier, Mirifici Logarithmorum Canonis Descriptio
Seeing there is nothing (right well beloved Students in the Mathematickes) that is so troublesome to Mathematicall practise, nor that doth more molest and hinder Calculators, then the Multiplications, Divisions, square and cubical Extractions of great numbers, which besides the tedious expence of time, are for the most part subiect to many slippery errors, I began therefore to consider in my minde, by what certaine and ready Art I might remove those hindrances. And having thought upon many things to this purpose, I found at length some excellent briefe rules to be treated of (perhaps) hereafter. But amongst all, none more profitable then this, which together with the hard and tedious Multiplications,
OCR corrected
SummaryLogarithms spare calculators the tedium of long multiplications and divisions and the slippery errors they cause.
Preface
Tu sçais comme, en operant par le jetton, le Calculateur (surtout lorsqu’il manque d’habitude) est souvent obligé, de peur de tomber en erreur, de faire une longue suite et extension de jettons, et comme la necessité le contraint apres d’abbreger et de relever ceux qui se trouvent inutilement estendus ; en quoy tu vois deux peines inutiles, avec la perte de deux temps. Cette machine facilite et retranche en ses operations tout ce superflu ; le plus ignorant y trouve autant d’avantage que le plus experimenté ; l’instrument supplée au defaut de l’ignorance ou du peu d’habitude, et, par des mouvemens necessaires, il fait luy seul, sans mesme l’intention de celuy qui s’en sert, tous les abbregez possibles à la nature, et à toutes les fois que les nombres s’y trouvent disposez. Tu sçais de mesme comme, en operant par la plume, on est à tous momens obligé de retenir ou d’emprunter les nombres necessaires, et combien d’erreurs se glis sent dans ces retentions et emprunts à moins d’une tres-longue habitude et en outre d’une attention profonde et qui fatigue l’esprit en peu de temps. Cette machine delivre celuy qui opere par elle de cette vexation ; il suffit qu’il ait le jugement, elle le releve du defaut de la memoire ; et, sans rien retenir ny emprunter, elle fait d’elle-mesme ce qu’il desire, sans mesme qu’il y pense.
You know how, when working with counters, the Calculator (above all when he lacks practice) is often obliged, for fear of falling into error, to lay out a long series and spread of counters, and how necessity then forces him to shorten and gather up those that turn out to be laid out uselessly; in which you see two useless labours, with the loss of two lots of time. This machine eases and cuts out of its operations all this superfluity; the most ignorant finds as much advantage in it as the most experienced; the instrument makes up for the defect of ignorance or of little practice, and, by necessary movements, it alone makes, without even the intention of the person using it, all the shortcuts possible in nature, every time the numbers are so disposed. You know likewise how, when working with the pen, one is at every moment obliged to carry or borrow the necessary numbers, and how many errors slip into these carryings and borrowings without very long practice and, besides, a deep attention that tires the mind in a short time. This machine delivers the person who works with it from this vexation; it is enough that he have judgment, it relieves him of the defect of memory; and, without carrying or borrowing anything, it does of itself what he wishes, without his even thinking of it.
machine translation
SummaryThe machine does every arithmetic operation without effort and relieves the mind of the fatigue of reckoning with counters or pen.
- 1671Leibniz, Description of his calculating machine
It is unworthy of excellent men to lose hours like slaves in calculation that could be given to anyone else if machines were used.
English in D. E. Smith, A Source Book in Mathematics (1929), pp. 180–181
Calculation need not require skill1
- 1832Babbage, On the Economy of Machinery and Manufactures
Third section. The members of this section, whose number varied from sixty to eighty, received certain numbers from the second section, and, using nothing more than simple addition and subtraction, they returned to that section the tables in a finished state. It is remarkable that nine-tenths of this class had no knowledge of arithmetic beyond the two first rules which they were thus called upon to exercise, and that these persons were usually found more correct in their calculations, than those who possessed a more extensive knowledge of the subject. 245. When it is stated that the tables thus computed occupy seventeen large folio volumes, some idea may perhaps be formed of the labour. From that part executed by the third class, which may almost be termed mechanical, requiring the least knowledge and by far the greatest exertions, the first class were entirely exempt. Such labour can always be purchased at an easy rate. … but when the completion of a calculating engine shall have produced a substitute for the whole of the third section of computers, the attention of analysts will naturally be directed to simplifying its application, by a new discussion of the methods of converting analytical formulae into numbers.
SummaryDe Prony's tables were computed by workers who knew only addition and subtraction and were more accurate than skilled mathematicians; this shows calculation can be done by machinery.
Chapter 20, §§ 241–248
It must not be set down as a fault if at first he counts with his fingers. Let him do so by all means if he likes. The faculty of abstracting numbers, and of learning to do without visible and tangible illustrations comes more slowly to some children than others. So long as they get the answer right, let them have what help they want till this power comes. It is sure to come ere long. At first, too, the little questions and problems which are given to children may fitly refer to marbles or apples, or to things which are familiar to them. But the mistake made by many teachers is to continue using these artifices too long; to go on shewing an abacus, or talking about nuts and oranges after the children have fully grasped the meaning of 6 + 5 in the abstract, and are well able to do without visible help. It is a sure test of a good teacher that he knows when and how far to employ such artifices, and when to dispense with them. The moment that concrete illustrations have served their purpose, they should be discarded.The ordinary purchase of household supplies requires a practical ability in the mental arithmetic of daily life, and this ability comes to the mind only through repeated exercise. As will be seen later, it is a fair inference from statistical investigations that a person may be rapid and accurate in written work but slow and uncertain in oral solutions. Therefore, it will not do, from the practical standpoint, to drill children only in written arithmetic if we expect them to be reasonably ready in purely mental work.In fact, the tradesman can calculate mentally just as well as, or perhaps even better than, his customer; only he will not. If he makes use of the soroban in the simplest possible kind of addition or subtraction, that is only an illustration of the irresistible force of habit, and we can only blame him for his being strictly consistent. Are we not apt to make slips and to be tripped up in matters too small to rouse our earnest attention? Machines, too simple to get out of order, are they not more trustworthy than the average human brain, crowded nowadays with the perplexities of modern civilization? The world is becoming more and more busy and supreme efforts should be made to economize our mental work as much as possible.