« AnteriorContinuar »
a chemist announces the existence and properties of a newly-discovered substance, if we confide in his accuracy, we feel assured that the conclusions he has arrived at will hold universally, though the induction be founded but on a single instance. We do not withhold our assent, waiting for a repetition of the experiment ; or if we do, it is from a doubt whether the one experiment was properly made, not whether if properly made it would be conclusive. Here, then, is a general law of nature, inferred without hesitation from a single instance ; a universal proposition from a singular one. Now mark another case, and contrast it with this. Not all the instances which have been observed since the beginning of the world, in support of the general proposition that all crows are black, would be deemed a sufficient presumption of the truth of the proposition, to outweigh the testimony of one unexceptionable witness who should affirm that in some region of the earth not fully explored, he had caught and examined a crow, and had found it to be gray.
Why is a single instance, in some cases, sufficient for a complete induction, while in others, myriads of concurring instances, without a single exception known or presumed, go such a very little way toward establishing a universal proposition ? Whoever can answer this question knows more of the philosophy of logic than the wisest of the ancients, and has solved the problem of induction.
(pp. 431-432). What renders arithmetic the type of a deductive science, is the fortunate applicability to it of a law so comprehensive as “The sums of equals are equals”': or (to express the same principle in less familiar but more characteristic language), whatever is made up of parts, is made up of the parts of those parts. This truth, obvious to the senses in all cases which can be fairly referred to their decision, and so general as to be co-extensive with nature itself, being true of all sorts of phenomena (for all admit of being numbered), must be considered an inductive truth, or law of nature, of the highest order. And every arithmetical operation is an application of this law, or of other laws capable of being deduced from it. This is our warrant for all calculations. We believe that five and two are equal to seven, on the evidence of this inductive law, combined with the definitions of those numbers. We arrive at that conclusion (as all know who remember how they first learned it) by adding a single unit at a tiine : 5+1=6, therefore 5+1+1=6+1=7; and again 2=1 +1, therefore 5+2=5+1+1=7.
QUOTATIONS ON INTERPRETATION.
234. 1. From Davies and Peck, Dictionary of Mathematics.
Interpretation. [L. interpretatio, explanation). The process of explaining results arrived at by the application of mathematical rules. When, for example, an algebraic definition is laid down, there is frequently some restriction implied in making the definition, so that the result to which it leads presents more cases than can be explained by it, or even than was contemplated by it. Thus the abbreviation of a a, a a a, into a>, a', and the rules which spring from it, lead to results of the form.
a-, a', a", a - *, etc. These results, until interpreted, are without any intelligent algebraic meaning.
When such results arise, the province of interpretation begins ; their meaning and force are investigated and explained, and the definitions heretofore too narrow, are extended so as to cover these and other results.
The rule to be adopted in interpreting new expressions obtained by applying known processes, is to attribute to them such a meaning as to make
the whole of the process true by which they were obtained. For example : the formula
an x a" = amtn is perfectly intelligible so long as m and n are whole numbers. Suppose it were required to interpret the symbol a°, that is, to give to it such a meaning, that the above formula shall be true in that case. Making m=0, the formula becomes
ao x a" = a° + n = a"; hence, ao = 1. Again, suppose it were required to interpret the symbol ak. Make m = 1 and n = }, and the formula becomes
als xam = a+ to hence, ak = Va, for va x Va = a, by definition. =
, Besides the application of the principles of interpretation to the explanation of new symbols, another very important application consists in making suppositions upon certain arbitrary quantities which enter formulas, and then comparing the results with known facts, thus deducing new truths. As an example of this method of interpretation let us take the equation of the ellipse
a'ya + b 2 = a' ", and suppose x > a, finding the value of y in terms of X, we bave
6 y ==
Va’ – 2*,
from which we see that for all values of x greater than a, y is imaginary. Now an imaginary result indicates an impossibility in the assumption. Hence, we interpret the result as indicating that no point of the ellipse can lie at a greater distance from its conjugate axis than the extremity of the transverse axis.
In integrating the differential of a transcendental function by an algebraic rule, a result oo is reached, which is manifestly absurd, since no function can be c. We interpret this as indi- . cating that the rule fails in the case considered.
2. From Smith's Synonyms Discriminated.
Expound (Lat. expono) denotes sustained explanation ; while a mere word or phrase may be explained, a whole work or parts of it may be expounded. Exposition is continuous critical explanation. Interpret (Lat. interpres, an interpreter), beyond the mere sense of verbal translation from one language to another, conveys the idea of private or personal explanation of what is capable of more than one view. Hence interpretation is more arbitrary than exposition, and more theoretical than explanation. Expound relates only to words in series, interpretation is applicable also to anything of a symbolical character, as to interpret a dream or a prophecy. It is also, in common with explain, an application to anything which may be viewed in different lights, as the actions of men. In this way, to explain conduct would rather be to account for it ; to interpret it would be to assign motives or significance to it. Explanation deals with facts, interpretation with causes also.