MATHEMATICS AND THEOLOGY.
PROFESSOR DE MORGAN, in an amusing paper recently published in the Athenaum, has casually drawn attention to the blunders which rash theologians make in attempting to apply mathematical considerations to theology, and the subject has more bearings than it was possible for that profound mathematician and acute thinker to throw out in a mere paragraph or two upon paradoxes. The errors which arise from attempting to import quantitative ideas into ethics and theology are in fact much more than errors, they are due to a kind of moral blindness. We might well say of those who attempt it " this people's hearts are waxed gross and their ears are dull of hearing." Yet it may not be entirely unprofitable to answer them in their own language besides pointing out the true root of their mistake. "I have in my life," says Professor De Morgan, " heard from the pulpit, or 'read at least a dozen times, that all sin is infinitely great, proved -as follows : —The greater the Being the greater the sin of any -offence against him ; therefore the offence committed against an in- finite Being is infinitely great." As, however, our knowledge of in-
• dinity can never be immediately derived from immediate experience, and all we mean by a mathematical proposition being true of infinity is that it is more and more nearly true of larger quantities than of smaller ones, and this without any limit to the closeness of our ap- proxitnation,—the theologian has no right to reason to the infinitude . of sin against God on the ground of the infinitude of God, unless he can first prove that sin against finite beings is exactly proportional to the mental and moral magnitude of those finite beings, and this without any limit ;—so that, as Professor De Morgan says, " a sin against a being 4 7-10ths times as great as another is exactly, no more and no less, 4 7-10ths times as great as an offence against the -other," and so on without limit, however great the multiplier used. Is a given sin against a family, for instance, exactly pro- portioned to the number of the family, of course supposing its members to be of equal moral weight ? Is a sin against a corpora- tion the precise multiple of the same offence against a single member of it that the number of the corporation is of 1 ? Once put in this form,—and it is the only one giving meaning of any ;kind to a proposition that cannot be asserted at hazard or
. carelessly without exposing the preacher to the charge of real levity,—and the proposition is not only without justification, -but almost contrary to the truth. The sins which men sin against those lower than themselves in power, and conscience, and strength of resistance, are by the voice of conscience and of Christ alike asserted to be—at least up to a certain point—graver and less par- donable than those committed against beings above them. " Whoso shall cause one of these little ones to offend, it were better for him that a millstone were hanged about his neck, and that he were drowned in the depth of the sea :"—and this was said by Him who expressly declared that blasphemy against the Son of Man should be forgiven. Of course the mathematical theologian may answer that the offences against the weak and lowly are not intrinsically great, but great only because God has forbidden them, and declared them to be offences against Himself. Well, but why does He declare the offence against " little ones" to be a greater 'offence against Himself than the offence against equals, unless it be because intrinsically it is so? And, as we have shown, if sins which God has forbidden are supposed to be infinite only because they have thus become sins against Himself, then the word "infinite" loses all mathematical meaning; for unless we can prove in the case of finite beings that sin enlarges when it is committed against a greater nature, and that, too, in exact proportion to the magnitude of the nature offended, we know nothing about what may be true when -the object of the sin is infinite. Having no experience of quanti- tative infinitude, it expresses to us only the limit to which we
- approach always nearer and nearer, as we make our finite magni- tude larger and larger. If we have no power to ascertain any fixed
;proportion between sin and the object against whom the sin is
'.committed when dealing with finite natures, we cannot use the • word "infinite" in its mathematical sense without culpable and profane confusion.
Precisely the same kind of blunder, intellectual and moral, is made when it is asserted, as we have constantly heard it asserted, that as all sin is infinite, the suffering which it entails must be infinite also, and that therefore the only possible scheme of vicarious redemption is for an infinite being to suffer (though during a finite time) the equivalent for what finite beings would otherwise have to suffer for an infinite time. Now this statement is never made except in proof of the rigid
- and quantitative exactitude of God's justice, and for such a pur- pose we'venture to say that no statement was ever more entirely and absolutely delusive, since it leaves the question exactly where it found it, and where the slightest wisdom would always leave it, —one of pure faith. The demonstration, if it is worth anything at -all, rests upon the assumption that one infinitude is equal to any other infinitude,—that an infinitude arising from finite additions made throughout an infinite time must be the same as an infini- tude arising from infinite additions made throughout a finite time. Now so little is this certain, that the mathematician is familiar with as many relative values amongst infinitudes (so far as he un- derstands them) as amongst finite quantities. He habitually deals with infinitudes which are infinitely less than other infinitudes,— which are, in short, nothing in the comparison,—with infinitudes
which are infinitely larger than other infinitudes, and so forth. To take a very simple case, as five times five, or ten times ten, or a hundred times a hundred, must always be respectively five times,
ten times, or a hundred times as much as five, ten, and a hundred, — if you make the number infinite we must say that " infinity times infinity" is infinitely greater than the infinity so multiplied. In short we judge of the relative values of different infinitudes 'fiy the only test by which we can ever judge of infinitude at all, the relative values of the finite quantities out of which they were generated by increasing without limit. There are, then, an infinite number of relative values for infinityi—but the argument supposed can only have any weight at all in proving the justice of God, if the infinity arising from the prolongation of finite sufferings through an infinite time be precisely of the same order of infinitude as the infinity arising from the suffering of an infinite being through a finite time. If this be left to our faith after all,—as of course it must be,—since no one can gauge the " order" of God's infinitude, how far more consonant to the humility of truth would it have been to begin with faith in the absolute justice -of God instead of ending with it, after the production-of an argument which by its reckless abuse of quantitative metaphors leads us deep into the darkness instead of towards the light !
Another similar abuse of mathematical language is made when thinkers—like Mr. Mansel, for example--assert that direct know- ledge of God is impossible in consequence of His infinitude. The argument is then put in this shape,—that the finite cannot know the Infinite, because knowledge implies a subject knowing dis- tinguishable from the object known, and the Infinite ex vi termini cannot be discriminated from the finite which must be swallowed up and contained in it. All this is a gross abuse of quantitative language for purposes to which it has no sort of appli- cation. It is true undoubtedly that in the measurement of mere magnitudes, it is not worth while to take into account a very small addition to a very large number, say a penny to the fortune of a Rothschild, or a barleycorn to the distance between the earth and the sun ; that the greater the _number to which you add, the smaller is the value of a given finite increase, and all this is compressed into saying that a finite number makes no addition to an infinite number,—meaning, that is, none that admits of appreci - ation. But when this comes to be applied to mutual knowledge, —before we can have any right at all to say that knowledge of the infinite by the finite is impossible, we must satisfy ourselves that knowledge of character necessarily becomes less and less as the being to be known is greater and greater,—and that, too, whether the knowledge be supposed to be attained by the upward efforts of the smaller, or by the condescension of the larger, mind. Now this is obviously not true, so far as there is any real meaning attached to magnitude of character at all. No one would say that David, or Marcus Aurelius, or Socrates, or Shakespeare, or Mahomet, or any other man of the greatest weight of human character was harder to know or less competent to make his epiritual nature visible and felt by smaller• men, than men of comparatively small and insignificant character, such as Rehoboam, or Caligula, or Mr. Perceval, or Lord Amberley. So far as there is any differ- ence at all, the greater, the deeper, the stronger the character of a man the more power he has to show himself to weaker men as he is, and to mould them to his own nature. No one would main- tain that .St. Paul had not impressed himself on the world more distinctly than St. Peter, and St. Peter more distinctly than St. James. No one would deny that of characters called finite, the ' greater'' (if that word has any meaning) have shown also the greater power of self-revealing. If, then, the proposition that the greater the mind the less capable is its moral nature of revelation to us be not more and mote nearly true in the region of finites where we can test it,—it is an abuse of mathematical language to say it is true in the region of infinitude, of which we know abso- lutely nothing except what is derived from reasoning up towards it. In point of fact probably no mind ever revealed such depths of itself to the world as that of Christ, who, as even Mr. Mansel would assert, was Himself infinite, if infinity in relation to character have any precise meaning. Infinity is never used by the mathematician at all except as an expression for something larger than anything that can be named,---and nothing is asserted of infinity that is not more and more nearly true as the magnitude increases, and this with- out limit. If this wise limitation of the term were kept in theology, we should have a vast deal less ignorant and mischievous reasoning.
Another set of abuses of mathematical language by theologians depends on the manipulation of eternity. Professor De Morgan quotes from a tract which argues in this way :—" Let x be the present value of the future estate (eternal happiness) and a of all that this world can give, then x being infinitely larger than a, x+a may be considered as equal to a." If this means anything it means something the reverse of true. ' Present value' expresses ma- thematically the sum which put out to interest would yield either a given sum or a series of successive payments at the end of a certain time. Now the present value' of an endless series of payments is certainly not infinite. One hundred pounds will yield five pounds, or four pounds, or three pounds a year for ever if the minimum rate of interest may be taken at five, or four, or three per cent. So the mere endlessness of the series of happy moments has nothing to do with infinite value. If the analogy be worth anything,—which it is not,—every bad act would buyan infinite series of unhappy moments, and every good act an equally infinite series of happy moments in the future. What the author of the argument probably meant was that the rate of interest allowed for certain good acts (perhaps beliefs) was itself infinite, while the rate of interest allowed for certain bad acts (perhaps failures to believe) was also infinite. But the illus- tration, so far as it rests upon the mere idea of a perpetuity, is thoroughly bad. Its idea is that an in finite future value can be bought by a finite present pain,—but that idea receives no illustra- tion at all from mathematics. An endless series of gains no doubt can be bought by a finite payment, but the finite payment and the endless series of gains are absolutely equal in value. Even if a man could live on earth for ever and receive all the instalments of interest himself, he would not find it to his advantage to invest all his money in perpetual annuities,—or else it would be equally to his advantage to invest the yearly instalments of those annuities again in further annuities, and finally to spend nothing whatever on immediate enjoyment. He must choose somewhere between spending 1001. and having five pounds a year for ever, and the first choice may be quite as wise as the last. The advantage (if it be an advantage) of deliberately buying Heaven by a small invest- ment in earthly pain, receives no sort of illustration from the theory of compound interest.
It is only astonishing that this mathematical language has got into the treatment of theological questions at all. It is always and uniformly misleading, and has no vestige of foundation in the Bible. We believe that even Eternity' and ' eternal' are almost always purely qualitative words in the Gospels,—and that if they are over used in the simple sense of endless duration, the endlessness (for example) of happiness or anguish is derivative, and supposed to be a consequence of their divine origin, not the adequate mea- sure or expression of it.
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