Page images
PDF
EPUB

and the obvious inconsistency of these conclusions with the doctrine concerning the characteristics of mathematical or de-, monstrative evidence, which it was the chief object of this section to establish*.

* This doctrine is concisely and clearly stated by a writer whose acute and original, though very eccentric genius, seldom fails to redeem his wildest paradoxes by the new lights which he strikes out in defending them." Demonstratio est syllogismus vel syllogismorum series à nominum definitionibus usque ad conclusionem ultimam deriva"ta." (Computatio sive Logica, cap. 6.)

[ocr errors]

It will not, I trust, be inferred, from my having adopted, in the words of Hobbes, this detached proposition, that I am disposed to sanction any one of those conclusions which have been commonly supposed to be connected with it, in the mind of the author :I say supposed, because I am by no means satisfied, (notwithstanding the loose and unguarded manner in which he has stated some of his logical opinions) that justice has been done to his views and motives in this part of his works. My own notions on the subject of evidence in general, will be sufficiently unfolded in the progress of my speculations. In the meantime, to prevent the possibility of any misapprehension of my meaning, I think it proper once more to remark, that the definition of Hobbes, quoted above, is to be understood (according to my interpretation of it) as applying solely to the word demonstration in pure mathematics. The extension of the same term by Dr Clarke and others, to reasonings which have for their object, not conditional or hypothetical, but absolute truth, appears to me to have been attended with many serious inconveniences, which these excellent authors did not foresee. Of the demonstrations with which Aristotle has attempted to fortify his syllogistic rules, I shall afterwards have occasion to examine the validity.

The charge of unlimited scepticism brought against Hobbes, has, in my opinion, been occasioned, partly by his neglecting to draw the line between absolute and hypothetical truth, and partly by his applying the word demonstration to our reasonings in other sciences as well as in mathematics. To these causes may perhaps be added, the offence which his logical writings must have given to the Realists of his time.

It is not, however, to Realists alone, that the charge has been confined. Leibnitz himself has given some countenance to it, in a dissertation prefixed to a work of Marius Nizolius; and Brucker, in referring to this dissertation, has aggravated not a little the censure of Hobbes, which it seems to contain. "Quin si illustrem Leibnitzium audi

SECTION IV.

Of our Reasonings concerning Probable or Contingent Truths.

I.

Narrow Field of Demonstrative Evidence. Of Demonstrative Evidence, when combined with that of SENSE, as in Practical Geometry; and with those of Sense and of INDUCTION, as in the Mechanical Philosophy.-Remarks on a Fundamental Law of Belief, involved in all our Reasonings concerning Contingent Truths.

If the account which has been given of the nature of demonstrative evidence be admitted, the province over which it extends must be limited almost entirely to the objects of pure mathematics. A science perfectly analogous to this, in point of evidence, may indeed be conceived (as I have already remarked) to consist of a series of propositions relating to moral, to political, or to physical subjects; but as it could answer no other purpose than to display the ingenuity of the inventor, hardly any thing of the kind has been hitherto attempted. The only exception which I can think of, occurs in the speculations formerly mentioned under the title of theoretical mechanics.

66

mus, Hobbesius quoque inter nominales referendus est, eam ob causam, quod ipso "Occamo nominalior, rerum veritatem dicat in nominibus consistere, ac, quod majus est, "pendere ab arbitrio humano." Histor. Philosoph. de Ideis, p. 209. Augusta Vindelicorum, 1723.

But, if the field of mathematical demonstration be limited entirely to hypothetical or conditional truths, whence (it may be asked) arises the extensive and the various utility of mathematical knowledge, in our physical researches, and in the arts of life? The answer, I apprehend, is to be found in certain peculiarities of those objects to which the suppositions of the mathematician are confined; in consequence of which peculiarities, real combinations of circumstances may fall under the examination of our senses, approximating far more nearly to what his definitions describe, than is to be expected in any other theoretical process of the human mind. Hence a corresponding coincidence between his abstract conclusions, and those facts in practical geometry and in physics which they help him to ascertain.

For the more complete illustration of this subject, it may be observed, in the first place, that although the peculiar force of that reasoning which is properly called mathematical, depends on the circumstance of its principles being hypothetical, yet if, in any instance, the supposition could be ascertained as actually existing, the conclusion might, with the very same certainty, be applied. If I were satisfied, for example, that in a particular circle drawn on paper, all the radi were exactly equal, every property which Euclid has demonstrated of that curve might be confidently affirmed to belong to this diagram. As the thing, however, here supposed, is rendered impossible by the imperfection of our senses, the truths of geometry can never, in their practical applications, possess demonstrative evidence; but only that kind of evidence which our organs of perception enable us to obtain.

But although, in the practical applications of mathematics, the evidence of our conclusions differs essentially from that which belongs to the truths investigated in the theory, it does not therefore follow, that these conclusions are the less important. In proportion to the accuracy of our data will be that of all our subsequent deductions; and it fortunately happens, that the same imperfections of sense which limit what is physically attainable in the former, limit also, to the very same extent, what is practically useful in the latter. The astonishing precision which the mechanical ingenuity of modern times has given to mathematical instruments, has, in fact, communicated a nicety to the results of practical geometry, beyond the ordinary demands of human life, and far beyond the most sanguine anticipations of our forefathers *.

* See a very interesting and able article, in the fifth volume of the Edinburgh Review, on Colonel Mudge's account of the operations carried on for accomplishing a trigonometrical survey of England and Wales. I cannot deny myself the pleasure of quoting a few sentences.

"In two distances that were deduced from sets of triangles, the one measured by "General Roy in 1787, the other by Major Mudge in 1794, one of 24.133 miles, and "the other of 38.688, the two measures agree within a foot as to the first distance, and "16 inches as to the second. Such an agreement, where the observers and the instru"ments were both different, where the lines measured were of such extent, and dedu"ced from such a variety of data, is probably without any other example. Coinciden"ces of this sort are frequent in the trigonometrical survey, and prove how much more "good instruments, used by skilful and attentive observers, are capable of performing, "than the most sanguine theorist could have ever ventured to foretel.—

"It is curious to compare the early essays of practical geometry with the perfection "to which its operations have now reached, and to consider that, while the artist had "made so little progress, the theorist had reached many of the sublimest heights of ma"thematical speculation; that the latter had found out the area of the circle, and cal

This remarkable, and indeed singular coincidence of propositions purely hypothetical, with facts which fall under the examination of our senses, is owing, as I already hinted, to the peculiar nature of the objects about which mathematics is conversant; and to the opportunity which we have (in consequence of that mensurability * which belongs to all of them) of adjusting, with a degree of accuracy approximating nearly to the truth, the data from which we are to reason in our practical operations, to those which are assumed in our theory. The only affections of matter which these objects comprehend are extension and figure; affections which matter possesses in common with space, and which may therefore be separated in fact, as well as abstracted in thought, from all its other sensible qualities. In examining, accordingly, the relations of quantity connected with these affections, we are not liable to be disturbed by those physical accidents, which, in the other applications of mathematical science, necessarily render the result, more or less, at variance with the theory. In measuring the height of a mountain, or in the survey of a country, if we are at due pains in ascertaining our data, and if we reason from them with mathematical strictness, the result may be depended on as accurate within very narrow limits; and as there is nothing but the incorrectness of our data by which the result can be

"culated its circumference to more than a hundred places of decimals, when the for"mer could hardly divide an arch into minutes of a degree; and that many excellent "treatises had been written on the properties of curve lines, before a straight line of " considerable length had ever been carefully drawn, or exactly measured on the surface of the earth."

• See note (G.)

« PreviousContinue »