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conclusions in the other sciences. Even in physics, mathematicians have been led to acquiesce in conclusions which appear ludicrous to men of different habits. Thus, in the Mechanics of Euler, that illustrious man, after arriving at a result, which startled his own common sense from its apparent extravagance, professses, nevertheless, in the following memorable words, his implicit faith in the infallibility of the algebraical art : “ Sed “potius calculo Algebraico quam nostro judicio est fidendum.' The intrepidity with which the earliest writers on the arithmetic of infinites followed out their principles to their most paradoxical and revolting conclusions, affords a still more palpable illustration of the same remark.
The following instances of a misapplication of mathematical principles are mentioned by the first mathematician of the present age.
“ I rank also in the number of illusions, the application 66 which Leibnitz and Daniel Bernoulli have made of the calculus of probabilities to the summation of series. If we reduce " the fraction, whose numerator is l, and whose denominator " is 1 + x to a series, whose terms are arranged according to " the powers of x; it is easy to see, that, supposing x = 1, " the fraction becomes à ; and the series becomes +1-1+ 61-1, &c. &c. By adding the two first terms, the two 66 next, and so of the rest, we transform the series into another, " having each term = Zero. Hence, Grandi, an Italian Ju“ suit, had inferred the possibility of the Creation ; because the “ series being always equal to 1, he saw that fraction created 66 out of an infinity of Zeros ; that is, out of nothing." It was 66 thus that Leibnitz saw an image of the Creation in his binary " arithmetic, where he employed only two characters, Zero 6 and Unity. He imagined that Unity might represent God, " and Zero, nothing; and that the Supreme Being might have " brought all things out of nothing, as Unity with Zero ex
all numbers in this system of arithmetic. This idea “ pleased Leibnitz so much, that he communicated the remark "to the Jesuit Grimaldi, president of the Mathematical Board 66 in China, in the hope that this emblem of the creation would " convert to Christianity the reigning emperor, who was par6 ticularly attached to the sciences. I record this anecdote
* See Robin's Remarks on Euler's Treatise of Motion, Sections 27, 28, 29, 30. 59. † To readers unaccustomed to the algebraical notation, it may
proper to mention, that Grandi's inference amounted to this, that an infinite series of nothings is equal to one-half.
"only to show how far the prejudices of infancy may mislead " the greatest men”.
The misapplications of mathematical principles here pointed out by Laplace, are certainly extremely curious, and may furnish a subject for very important reflections to the philosophical logician; but while they serve to illustrate the influence exercised over the most powerful minds by the prejudices of infuncy, they may be considered also as examples of the absurdities into which mathematicians are apt to run, when they apply their predominant habits of thinking and reasoning to the investigation of metaphysical or moral truths. Some other examples of the same thing might, if I do not greatly deceive myself, be produced even from the Philosophical Essay on Probabilities.
In a very ingenious and learned article of the Supplement to the Encyclopædia Britannica, (commonly, and I believe justly, ascribed to one of my best friends,) the following passage occurs:t “ The formation of circulating decimals affords "a fine illustration of that secret concatenation which binds l the succession of physical events, and determines the various "though lengthened cycles of the returning seasons ;f a prin
ciple which the ancient Stoics, and some other philosophers, "have boldly extended to the moral world." This remark, I cannot help considering as a still finer illustration of the influence of mathematical habits of thinking on an understanding remarkable for its vigour and originality.
These inconvenient effects of mathematical studies are to be cured only by an examination of the circumstances which discriminate mathematics from the other sciences ; and which enable ús, in that branch of knowledge, to arrive at demonstrative certainty, while, in the others, nothing is to be looked for beyond probability: Had these circumstances been duly weighed by Pitcairn and Cheyne, they would never have conceived the extravagant project of compensating, by the rigour of a few mathematical steps, for the uncertainty which must necessarily attend all our data, when we reason on medical subjects. « Non dubito” (says the former of these writers) "me solvisse nobile problema, quod est, dato morbo, invenire “ remedium. Jamque opus exegi.” Other attempts, still more absurd, have been made to apply mathematical reasoning to morals.
* Essai Philosophique sur les Probabilités, par M. le Comte Laplace, pp. 194, 195. | See Article ARITHMETIC. Are we then to consider circulating decimals as physical events ?
The bias towards dogmatism, which I have been now imputing to mathematicians, is, I am sensible, inconsistent with the common opinion, that their favourite pursuits have a tendency to encourage a sceptical disposition, unfriendly to the belief of moral truths, and to a manly and steady conduct in the affairs of life. As no evidence is admitted by the mathematician in his own inquiries, but that of strict demonstration, it is imagined that there is a danger of his insisting on the same evidence with respect to some truths which do not admit of it. The late Dr. John Gregory himself, the early part of whose life was devoted to mathematical pursuits, and who possessed a considerable share of the mathematical genius which has been so long hereditary in his family, while he avows his own partiality for a science, which he with great truth calls “the most “ bewitching of all studies,” has given some countenance to this idea ; * and, in general, its justness seems to be admitted by the warmest admirers of mathematics.
That it has very little foundation, however, either in theory or in fact, the slightest consideration of the subject is sufficient to evince.
It was already said, that the speculative propositions of mathematics do not relate to facts; and that all we are convinced of by any demonstration in the science, is of a necessary connexion subsisting between certain suppositions and certain conclusions. When we find these suppositions actually take place in a particular instance, the demonstration forces us to apply the conclusion. Thus, if I could form a triangle, the three sides of which were accurately mathematical straight lines, I might affirm of this individual figure, that its three angles are equal to two right angles : but, as the imperfection of my senses puts it out of my power to be, in any case, cer. tain of the exact correspondence of the diagrams which I delineate, with the definitions given in the elements of Geometry, I never can apply with confidence to a particular figure, a mathematical theorem. On the other hand, it appears from the daily testimony of our senses, that the speculative truths of geometry may be applied to material objects with a degree of accuracy sufficient for the purposes of life; and from such applications of them, advantages of the most important kind have been gained to society. It is only in cases of this description that a mathematical proposition is expressive of a fact; and it must be remarked, that wherever this happens, the assertion partakes more or less of that uncertainty which has
* See his Lectures on the Duties and Qualifications of a Physician. Lecture III.
been so often complained of in the other sciences. It partakes of that uncertainty which arises from the imperfection of our organs of perception; and it is exposed to all the sceptical cavils which have been suggested with respect to the deceptions of the senses. In some of the practical applications of mathematical truths, the uncertainty is wonderfully increased. In proof of the greater part of astronomical facts, on which we every day reason, we have only the evidence of testimony; and this evidence relates to facts which cannot be ascertained without an uncommon degree of skill and attention. I never heard of any mathematician who was a sceptic in astronomy or physics; and yet, there are few branches of knowledge which lie more open to metaphysical quibbles. On the contrary, do we not daily see men, on the faith of some calculation, founded perhaps on observations made by others, predict, with the most perfect confidence, phenomena which are to happen many years afterwards? In this case, there is a vast accumulation of uncertainties, arising from the possibility of mistake in the original observer; from the fallibility of testimony; from our want of evidence with respect to the uniformity of the laws of nature ; and from several other sources. Yet a mathematician would treat any man with ridicule, who should so much as suggest a doubt concerning the probability of a solar or lunar eclipse taking place at the precise instant of time which had been predicted for that event by a skilful astronomer.
It appears, therefore, that in every case in which the mathematician can be said to believe fucts, in matters connected with his own science, he acknowledges the authenticity of those sources of evidence which are admitted by the philosophers who have turned their attention to other inquiries. A still stronger argument in proof of the same conclusion might be derived from those calculations concerning probabilities, on which some of our most eminent mathematicians have exercised their genius. In all these calculations it is manifestly assumed as a principle, that the conduct of a prudent man ought to be guided by a demonstrated probability, not less than by a demonstrated certainty; and that, to act in opposition to the former species of evidence, would be as irrational and absurd, as to deny the conviction which is necessarily produced by the latter. The only effect which can reasonably be expected from such studies on the mind of the mathematician, is a cautious, and, on the whole, a salutary suspense of judgnent on problematical questions, till the evidence on both sides is fully weighed; nor do I see any danger to be apprehended from this quarter, but a disposition in some weak understandings to compute, with arithmetical precision, those probabilities which are to be estimated only by that practical sagacity which is formed in the school of the world. But I must content myself with suggesting these topics as hints for examination.
If the foregoing observations be duly considered, it will not be found easy to conceive in what manner mathematical studies should have any tendency to encourage a sceptical bias concerning the sources of evidence in other sciences.
To myself so very different does the truth seem to be, that, in some particular cases of scepticism, I should be disposed to recommend these studies as the most effectual remedy for that weakness of mind in which it originates. When a person reads the history of Natural Philosophy prior to the time of Lord Bacon, and observes the constant succession of chimeras, which, till then, amused men of science, he is apt to imagine that they had been applying to a study which is placed above the reach of human genius. Similar conclusions are likely to be formed, and with still greater verisimilitude, by those who have confined their attention to the unintelligible controversies of scholastic metaphysicians, or to the vague hypotheses of medical theorists. In mathematics, on the other hand, and in natural philosophy since mathematics was applied to it, we see the noblest instances of the force of the human mind, and of the sublime heights to which it may rise by cultivation. An acquaintance with such sciences naturally leads us to think well of our faculties, and to indulge sanguine expectations concern. ing the improvement of other parts of knowledge. To this I may add, that, as mathematical and physical truths are perfectly uninteresting in their consequences, the understanding readily yields its assent to the evidence which is presented to it; and in this way, may be expected to acquire a habit of trusting to its own couclusions, which will contribute to fortify it agai the weaknesses of scepticism, in the more interesting inquiries after moral truth in which it may afterwards engage.
These observations are confirmed by all the opportunities I have had of studying the varieties of intellectual character. In the course of my own experience, I have never met with a mere mathematician who was not credulous to a fault ;-credulous not only with respect to human testimony, but credulous also in matters of opinion; and prone, on all subjects which he had not carefully studied, to repose too much faith in illustrious and consecrated names. Nor is this wonderful. That propensity to repose unlimited faith in the veracity of other men, which is plainly one of the instinctive principles of