Statistical Laws of Choice

I noted in an earlier post the necessity of statistical laws of nature. This will necessarily apply to human actions as a particular case, as I implied there in mentioning the amount of food humans eat in a year.

Someone might object. It was said in the earlier post that this will happen unless there is a deliberate attempt to evade this result. But since we are speaking of human beings, there might well be such an attempt. So for example if we ask someone to choose to raise their right hand or their left hand, this might converge to an average, such as 50% each, or perhaps the right hand 60% of the time, or something of this kind. But presumably someone who starts out with the deliberate intention of avoiding such an average will be able to do so.

Unfortunately, such an attempt may succeed in the short run, but will necessarily fail in the long run, because although it is possible in principle, it would require an infinite knowing power, which humans do not have. As I pointed out in the earlier discussion, attempting to prevent convergence requires longer and longer strings on one side or the other. But if you need to raise your right hand a few trillion times before switching again to your left, you will surely lose track of your situation. Nor can you remedy this by writing things down, or by other technical aids: you may succeed in doing things trillions of times with this method, but if you do it forever, the numbers will also become too large to write down. Naturally, at this point we are only making a theoretical point, but it is nonetheless an important one, as we shall see later.

In any case, in practice people do not tend even to make such attempts, and consequently it is far easier to predict their actions in a roughly statistical manner. Thus for example it would not be hard to discover the frequency with which an individual chooses chocolate ice cream over vanilla.

Mixing Water and Wine

St. Thomas discusses what happens if you mix consecrated wine with another liquid:

I answer that, The truth of this question is evident from what has been said already. For it was said above (3; 5, ad 2) that the species remaining in this sacrament, as they acquire the manner of being of substance in virtue of the consecration, so likewise do they obtain the mode of acting and of being acted upon, so that they can do or receive whatever their substance could do or receive, were it there present. But it is evident that if the substance of wine were there present, then some other liquid could be mingled with it.

Nevertheless there would be a different effect of such mixing both according to the form and according to the quantity of the liquid. For if sufficient liquid were mixed so as to spread itself all through the wine, then the whole would be a mixed substance. Now what is made up of things mixed is neither of them, but each passes into a third resulting from both: hence it would result that the former wine would remain no longer. But if the liquid added were of another species, for instance, if water were mixed, the species of the wine would be dissolved, and there would be a liquid of another species. But if liquid of the same species were added, of instance, wine with wine, the same species would remain, but the wine would not be the same numerically, as the diversity of the accidents shows: for instance, if one wine were white and the other red.

But if the liquid added were of such minute quantity that it could not permeate the whole, the entire wine would not be mixed, but only part of it, which would not remain the same numerically owing to the blending of extraneous matter: still it would remain the same specifically, not only if a little liquid of the same species were mixed with it, but even if it were of another species, since a drop of water blended with much wine passes into the species of wine (De Gener. i).

Now it is evident that the body and blood of Christ abide in this sacrament so long as the species remain numerically the same, as stated above (4; 76, 6, ad 3); because it is this bread and this wine which is consecrated. Hence, if the liquid of any kind whatsoever added be so much in quantity as to permeate the whole of the consecrated wine, and be mixed with it throughout, the result would be something numerically distinct, and the blood of Christ will remain there no longer. But if the quantity of the liquid added be so slight as not to permeate throughout, but to reach only a part of the species, Christ’s blood will cease to be under that part of the consecrated wine, yet will remain under the rest.

Given the doctrine of transubstantiation, at least as St. Thomas understands it, so that it implies the existence of accidents without a subject, it is very difficult to understand how such a mixing would be possible at all. But his general position here is that a process analogous to substantial change necessarily happens if you mix anything into the consecrated wine, either according to a part of the wine, or according to the whole. He explains this kind of change in article five of the same question:

I answer that, Since “the corruption of one thing is the generation of another” (De Gener. i), something must be generated necessarily from the sacramental species if they be corrupted, as stated above (Article 4); for they are not corrupted in such a way that they disappear altogether, as if reduced to nothing; on the contrary, something sensible manifestly succeeds to them.

Nevertheless, it is difficult to see how anything can be generated from them. For it is quite evident that nothing is generated out of the body and blood of Christ which are truly there, because these are incorruptible. But if the substance, or even the matter, of the bread and wine were to remain in this sacrament, then, as some have maintained, it would be easy to account for this sensible object which succeeds to them. But that supposition is false, as was stated above (75, 2,4,8).

Hence it is that others have said that the things generated have not sprung from the sacramental species, but from the surrounding atmosphere. But this can be shown in many ways to be impossible. In the first place, because when a thing is generated from another, the latter at first appears changed and corrupted; whereas no alteration or corruption appeared previously in the adjacent atmosphere; hence the worms or ashes are not generated therefrom. Secondly, because the nature of the atmosphere is not such as to permit of such things being generated by such alterations. Thirdly, because it is possible for many consecrated hosts to be burned or putrefied; nor would it be possible for an earthen body, large enough to be generated from the atmosphere, unless a great and, in fact, exceedingly sensible condensation of the atmosphere took place. Fourthly, because the same thing can happen to the solid bodies surrounding them, such as iron or stone, which remain entire after the generation of the aforesaid things. Hence this opinion cannot stand, because it is opposed to what is manifest to our senses.

And therefore others have said that the substance of the bread and wine returns during the corruption of the species, and so from the returning substance of the bread and wine, ashes or worms or something of the kind are generated. But this explanation seems an impossible one. First of all, because if the substance of the bread and wine be converted into the body and blood of Christ, as was shown above (75, 2,4), the substance of the bread and wine cannot return, except the body and blood of Christ be again changed back into the substance of bread and wine, which is impossible: thus if air be turned into fire, the air cannot return without the fire being again changed into air. But if the substance of bread or wine be annihilated, it cannot return again, because what lapses into nothing does not return numerically the same. Unless perchance it be said that the said substance returns, because God creates anew another new substance to replace the first. Secondly, this seems to be impossible, because no time can be assigned when the substance of the bread returns. For, from what was said above (4; 76, 6, ad 3), it is evident that while the species of the bread and wine remain, there remain also the body and blood of Christ, which are not present together with the substance of the bread and wine in this sacrament, according to what was stated above (Question 75, Article 2). Hence the substance of the bread and wine cannot return while the sacramental species remain; nor, again, when these species pass away; because then the substance of the bread and wine would be without their proper accidents, which is impossible. Unless perchance it be said that in the last instant of the corruption of the species there returns (not, indeed, the substance of bread and wine, because it is in that very instant that they have the being of the substance generated from the species, but) the matter of the bread and wine; which, matter, properly speaking, would be more correctly described as created anew, than as returning. And in this sense the aforesaid position might be held.

However, since it does not seem reasonable to say that anything takes place miraculously in this sacrament, except in virtue of the consecration itself, which does not imply either creation or return of matter, it seems better to say that in the actual consecration it is miraculously bestowed on the dimensive quantity of the bread and wine to be the subject of subsequent forms. Now this is proper to matter; and therefore as a consequence everything which goes with matter is bestowed on dimensive quantity; and therefore everything which could be generated from the matter of bread or wine, if it were present, can be generated from the aforesaid dimensive quantity of the bread or wine, not, indeed, by a new miracle, but by virtue of the miracle which has already taken place.

This is rather strange, because he seems to be saying that the subsequent substantial forms inhere in quantity as in a subject, and that there is no matter there. But if this is possible in any way, and in particular if things remain in this state permanently, as he seems to suggest, then there seems to be little reason not to adopt Descartes’s view of material substance in general, and say that quantity is always the subject of substantial forms, rather than saying that some parts of the world have matter as a subject, and other parts quantity. The account might be more reasonable if he were to accept that when a new substance is generated, matter again comes to be, not by being “created anew,” but because the being of matter in general is from substantial form.

As we can see, this discussion is especially complex on account of the doctrine of transubstantiation and St. Thomas’s account of that doctrine. But if we simply consider the mixing of two liquids in general, various difficulties will remain. Suppose we have a glass of water and a glass of wine, and mix the two together. What exactly will happen?

It is manifest to the senses that when we do this, there is a period of time when parts of the resulting liquid are water, just as it was, and parts are wine, just as it was, without any mixture. But what about the surface where the two are in contact? What is happening there?

According to St. Thomas, there will be a quantitative part which shares in the qualities of each. And this is pretty reasonable. Just as we can see that part is wine and part is water, at a certain point we can see that part is watery wine. But how exactly did that watery part get that way? If it is a certain size, was there a sudden transition of a part which was water into the watery wine? Or the like with the wine becoming watery? Or was there a continuous process with an expanding mixed region? The last possibility seems most consistent with what we see, but it might be difficult to analyze this in terms of substantial change, as St. Thomas does, because such a continuous process would have no first moment when the mixed substance came to be. For if it did, it would come to be with a definite size, and thus the process would not be continuous, but would imply that some part suddenly went from not being watery wine to being watery wine.

Of course, it is one thing to say there are difficulties. It is quite another to say that they mean that the thing cannot happen. So none of this proves that the mixing of liquids is not a substantial change. Nonetheless, many of the ancient naturalists were moved by such considerations to adopt some form of atomic theory. If water and wine are each composed of atoms, the mixing process is easily understood — it is simply the movement in place of these atoms. Each part of the water remains as it was even qualitatively, and likewise each part of the wine, but the resulting mixture has different sensible qualities because one cannot distinguish the diverse qualities of each, just as mixing two very fine sands of different color may appear to result in a third color, even though the grains of sand are not changing qualitatively.

Modern atomic theory, of course, has far stronger arguments for it, but they are in principle, or at least were in the 18th and 19th centuries, of a very similar kind: atomic theory simply does a good job of explaining many of the things that we see happen in the world.

This is closely related to the discussion in the last post. When we construct a bicycle out of parts, it is manifest to the senses that the parts look just like they did before they were parts. And this is necessary, if it is true that those parts are governed by the same natural laws after they become parts that they were before they became parts. For however the parts “look,” they look this way because of how they act on the senses. So if their action does not change, the “way they look” will not change. Similarly, when we mix liquids, if the water parts and the wine parts do not change how they behave, the account one gives of the mixture must be an atomic theory or something very like it. That is, there must remain very small parts that act like water, and very small parts that act like wine. Or, given that wine and water are not in fact elements, at least the basic elemental parts must continue to act like those elemental parts.

Composing Elements

Suppose we have two elements, as for example water and earth (not that these are really elements.) How do we make something out of the elements? We can consider two different possible ways that this could happen.

Suppose that when we combine one part water and one part earth, we get mud, and when we combine one part water and two parts earth, we get clay. Thus clay and mud are two different composite bodies that can be made from our elements.

How do we expect clay and mud to behave? We saw earlier that the nature of the physical world more or less requires the existence of mathematical laws of nature. Now we could say, “Clay and mud are made of earth and water, and we know the laws governing earth and water. So we can figure out the behavior of clay and mud using the laws governing earth and water.”

But we could also say, “Although clay and mud are made of earth and water, they are also something new. Consequently we can work out the laws governing them by experience, but we cannot expect to work them out just from the laws governing earth and water.”

These two claims are basically opposed to one another, and we should not expect that both would be true, at least in any particular instance. It might be that one is true in some cases and the other is true in some cases, or it might be that one side is always true. But in any case one will be true and not the other, in each particular situation.

Someone might argue that the first claim must be always true in principle. If water behaves one way by itself, and another way when it is combined with earth, then you haven’t sufficiently specified the behavior of water without including how it behaves when it is beside earth, or mixed with earth, or combined in whatever way. So once you have completely specified the behavior of water, you have specified how it behaves when combined with other things.

But this way of thinking is artificial. If water follows an inverse square law of gravity by itself, but an entirely different mathematical law when it is combined with earth, rather than saying that the entirely different law is a special case governing water, we should just admit that the different law is a law governing clay and mud, but not water. On the other hand, it is not unreasonable to include various potential interactions in your laws governing water, rather than only considering how water behaves in perfect isolation. Thus for example one would want to say that water suffers gravitational effects from all other bodies, rather than simply saying that water attracts itself. Nonetheless, even if the distinction is somewhat rough, there is a meaningful distinction between situations where the laws governing the elements also govern the composites, and situations where we need new laws for the composites.

In one way, the second claim is always true. It is always the case that something is true of the composite which is not true of the elements in themselves, since the composite is a whole composed of elements, while the elements in themselves are not. This is true even in artificial compositions; the parts of a bicycle are not a bicycle, but the whole is. And I can ride a bicycle, but  I cannot ride the individual pieces of metal that form it. Likewise, it is evidently true of living things, which are alive, and in some cases have conscious experience, even though the individual elements do not.

In a second way, the second claim is almost always true. If we consider our laws as practical methods for predicting the behavior of a physical system, in practice we will almost always need special laws to predict the behavior of a complex composite, if only because it would be too complex and time consuming to predict the behavior of the composite using laws governing only the parts. Thus people who wish to predict the weather use generalizations based on the experience of weather, rather than trying to predict the weather simply by considering more general laws of physics, despite believing that the weather is in fact a consequence of such general laws.

In a third way, the first claim is true at least frequently, and possibly always. If we consider the behavior of a bicycle or a computer, not with respect to general questions such as “can I ride it?” or “can it calculate the square root of two?”, but with respect to the physical movement of the parts, there are good reasons to think that the behavior of the whole can be determined from the behavior of the parts of which it is composed. For these are human inventions, and although experience is involved in such inventions, people make guesses about new behavior largely from their understanding of how the parts behave which they plan to put together. So if the whole behaved in ways which are significantly unpredictable from the behavior of the parts, we would not expect such inventions to work. Likewise, as said above, there is little reason to doubt that the weather results from general principles of physics that apply to earth, air, water, and so on.

I say “possibly always” above, because there is no case where the second claim is known to be true in this sense, and many instances, as noted, where the first is known or reasonably believed to be the case. Additionally, one can give reasons in principle for expecting the first claim to be true in this way, although this is a matter for later consideration.

An important objection to this possibility is that the fact that the second claim is always true in the first way mentioned above, seems to imply that the first claim cannot be true even in the third way, at least in some cases. In particular, the conscious behavior of living things, and especially human free will, might seem inconsistent with the idea that the physical behavior of living things is in principle predictable from laws governing their elements.

Hume’s Error on Miracles

After his comparison with the idea of the Real Presence, Hume continues by saying that we should proportion our beliefs to the evidence for them:

Though experience be our only guide in reasoning concerning matters of fact; it must be acknowledged, that this guide is not altogether infallible, but in some cases is apt to lead us into errors. One, who in our climate, should expect better weather in any week of June than in one of December, would reason justly, and conformably to experience; but it is certain, that he may happen, in the event, to find himself mistaken. However, we may observe, that, in such a case, he would have no cause to complain of experience; because it commonly informs us beforehand of the uncertainty, by that contrariety of events, which we may learn from a diligent observation. All effects follow not with like certainty from their supposed causes. Some events are found, in all countries and all ages, to have been constantly conjoined together: Others are found to have been more variable, and sometimes to disappoint our expectations; so that, in our reasonings concerning matter of fact, there are all imaginable degrees of assurance, from the highest certainty to the lowest species of moral evidence.
A wise man, therefore, proportions his belief to the evidence. In such conclusions as are founded on an infallible experience, he expects the event with the last degree of assurance, and regards his past experience as a full proof of the future existence of that event. In other cases, he proceeds with more caution: he weighs the opposite experiments: he considers which side is supported by the greater number of experiments: to that side he inclines, with doubt and hesitation; and when at last he fixes his judgement, the evidence exceeds not what we properly call probability. All probability, then, supposes an opposition of experiments and observations, where the one side is found to overbalance the other, and to produce a degree of evidence, proportioned to the superiority. A hundred instances or experiments on one side, and fifty on another, afford a doubtful expectation of any event; though a hundred uniform experiments, with only one that is contradictory, reasonably beget a pretty strong degree of assurance. In all cases, we must balance the opposite experiments, where they are opposite, and deduct the smaller number from the greater, in order to know the exact force of the superior evidence.

While Hume is right to say that convictions should in some way be proportionate to the evidence for them, we can already see here the cause of a serious error. As I have pointed out elsewhere, Hume does not have a developed mathematical theory of probability. Hence his talk of how one should “deduct the smaller number from the greater, in order to know the exact force of the superior evidence.” If one takes this literally, this would suggest that something with no experiments supporting it has a force of 0; something with 35 experiments supporting it and nothing against has a force of 35; something with 40 experiments supporting it and 5 against has the same force; and so on. All of this, of course, is evidently absurd.

He then brings up the example of the testimony of witnesses:

To apply these principles to a particular instance; we may observe, that there is no species of reasoning more common, more useful, and even necessary to human life, than that which is derived from the testimony of men, and the reports of eye-witnesses and spectators. This species of reasoning, perhaps, one may deny to be founded on the relation of cause and effect. I shall not dispute about a word. It will be sufficient to observe that our assurance in any argument of this kind is derived from no other principle than our observation of the veracity of human testimony, and of the usual conformity of facts to the reports of witnesses. It being a general maxim, that no objects have any discoverable connexion together, and that all the inferences, which we can draw from one to another, are founded merely on our experience of their constant and regular conjunction; it is evident, that we ought not to make an exception to this maxim in favour of human testimony, whose connexion with any event seems, in itself, as little necessary as any other. Were not the memory tenacious to a certain degree; had not men commonly an inclination to truth and a principle of probity; were they not sensible to shame, when detected in a falsehood: were not these, I say, discovered by experience to be qualities, inherent in human nature, we should never repose the least confidence in human testimony. A man delirious, or noted for falsehood and villainy, has no manner of authority with us.
And as the evidence, derived from witnesses and human testimony, is founded on past experience, so it varies with the experience, and is regarded either as a proof or a probability, according as the conjunction between any particular kind of report and any kind of object has been found to be constant or variable. There are a number of circumstances to be taken into consideration in all judgements of this kind; and the ultimate standard, by which we determine all disputes, that may arise concerning them, is always derived from experience and observation. Where this experience is not entirely uniform on any side, it is attended with an unavoidable contrariety in our judgements, and with the same opposition and mutual destruction of argument as in every other kind of evidence. We frequently hesitate concerning the reports of others. We balance the opposite circumstances, which cause any doubt or uncertainty; and when we discover a superiority on any side, we incline to it; but still with a diminution of assurance, in proportion to the force of its antagonist.
This contrariety of evidence, in the present case, may be derived from several different causes; from the opposition of contrary testimony; from the character or number of the witnesses; from the manner of their delivering their testimony; or from the union of all these circumstances. We entertain a suspicion concerning any matter of fact, when the witnesses contradict each other; when they are but few, or of a doubtful character; when they have an interest in what they affirm; when they deliver their testimony with hesitation, or on the contrary, with too violent asseverations. There are many other particulars of the same kind, which may diminish or destroy the force of any argument, derived from human testimony.
Suppose, for instance, that the fact, which the testimony endeavours to establish, partakes of the extraordinary and the marvellous; in that case, the evidence, resulting from the testimony, admits of a diminution, greater or less, in proportion as the fact is more or less unusual. The reason why we place any credit in witnesses and historians, is not derived from any connexion, which we perceive à priori, between testimony and reality, but because we are accustomed to find a conformity between them. But when the fact attested is such a one as has seldom fallen under our observation, here is a contest of two opposite experiences; of which the one destroys the other, as far as its force goes, and the superior can only operate on the mind by the force, which remains. The very same principle of experience, which gives us a certain degree of assurance in the testimony of witnesses, gives us also, in this case, another degree of assurance against the fact, which they endeavour to establish; from which contradiction there necessarily arises a counterpoize, and mutual destruction of belief and authority.
I should not believe such a story were it told me by Cato, was a proverbial saying in Rome, even during the lifetime of that philosophical patriot. The incredibility of a fact, it was allowed, might invalidate so great an authority.
The Indian prince, who refused to believe the first relations concerning the effects of frost, reasoned justly; and it naturally required very strong testimony to engage his assent to facts, that arose from a state of nature, with which he was unacquainted, and which bore so little analogy to those events, of which he had had constant and uniform experience. Though they were not contrary to his experience, they were not conformable to it.

While we might disagree that someone would be reasonable in refusing to accept testimony concerning the effects of frost, Hume’s general points here are fairly reasonable.

But when he attempts to apply to this miracles, he basically attempts to reason from the invalid mathematical points in the previous text:

But in order to increase the probability against the testimony of witnesses, let us suppose, that the fact, which they affirm, instead of being only marvellous, is really miraculous; and suppose also, that the testimony considered apart and in itself, amounts to an entire proof; in that case, there is proof against proof, of which the strongest must prevail, but still with a diminution of its force, in proportion to that of its antagonist.
A miracle is a violation of the laws of nature; and as a firm and unalterable experience has established these laws, the proof against a miracle, from the very nature of the fact, is as entire as any argument from experience can possibly be imagined. Why is it more than probable, that all men must die; that lead cannot, of itself, remain suspended in the air; that fire consumes wood, and is extinguished by water; unless it be, that these events are found agreeable to the laws of nature, and there is required a violation of these laws, or in other words, a miracle to prevent them? Nothing is esteemed a miracle, if it ever happen in the common course of nature. It is no miracle that a man, seemingly in good health, should die on a sudden: because such a kind of death, though more unusual than any other, has yet been frequently observed to happen. But it is a miracle, that a dead man should come to life; because that has never been observed in any age or country. There must, therefore, be a uniform experience against every miraculous event, otherwise the event would not merit that appellation. And as a uniform experience amounts to a proof, there is here a direct and full proof, from the nature of the fact, against the existence of any miracle; nor can such a proof be destroyed, or the miracle rendered credible, but by an opposite proof, which is superior.
The plain consequence is (and it is a general maxim worthy of our attention), ‘that no testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavors to establish; and even in that case there is a mutual destruction of arguments, and the superior only gives us an assurance suitable to that degree of force, which remains, after deducting the inferior.’ When anyone tells me, that he saw a dead man restored to life, I immediately consider with myself, whether it be more probable, that this person should either deceive or be deceived, or that the fact, which he relates, should really have happened. I weigh the one miracle against the other; and according to the superiority, which I discover, I pronounce my decision, and always reject the greater miracle. If the falsehood of his testimony would be more miraculous, than the event which he relates; then, and not till then, can he pretend to command my belief or opinion.

There are various ways to read this, but each way leads to problems. Hume has told himself that he believes that he has found a conclusive proof that accounts of miracles should never be accepted; and this implies that he must be saying that his condition, that the testimony should “be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavors to establish,” can never be satisfied.

But there is no reasonable understanding where this condition can never be satisfied. Hume seems to be equating “more miraculous” with “less probable,” but there is no degree of probability that could not be established by witnesses in principle. Even if each witness has only a small chance of telling the truth, multiple independent witnesses could in principle establish any degree of probability whatsoever.

The basic problem here seems to be Hume’s mathematically incorrect understanding of probability. If something has never been seen to happen, he says, this is a full proof that it cannot happen. Thus he seems to imply that there is a 0% chance of it happening. But this is evidently unreasonable. In reality, of course, we often see particular things happen which never happened before. And similarly, it is simply not true that a miracle is only called a miracle because it “has never been observed in any age or country.” There have been many reports, in many ages and many countries, of dead people coming to life again. So the only way Hume could say that a dead person coming to life is a miracle in this sense, is by assuming that all of these reports are false. This is simply to assume what he is trying to prove, and in any case we think that resurrection is a miracle whether or not these reports are true. In other words, to say that resurrection is a miracle is not to say that these reports are false, but that if they are true, they are reports of miracles.

Taken in another way, Hume seems to be saying, “A miracle requires a suspension of natural laws. But false testimony does not. Therefore if we have the choice of believing that there was a miracle or of believing that there was false testimony, we should always choose to believe that there was false testimony.” The problem is that, again, if you evaluate this in terms of probabilities, the suspension of natural laws might well be more probable than a particular possibility which does not suspend natural laws. If someone predicts the result of a coin flip 100,000 times in a row, it does not violate natural laws to think that this happened by chance, with a fair coin. But it is much more probable that natural laws were violated, than that this happened by chance with a fair coin.

While it would only relate to Hume personally, we might also note that according to Hume, induction cannot even establish a probability, let alone a necessity.  So according to his position, the experience of dead people remaining dead does not make it improbable that one would rise, let alone excluding it as impossible.

Hume himself seems to sense that there is something wrong with his position, even if he cannot quite work out what it is, again probably on account of the lack of a mathematical theory of probability. Consequently he adds a number of arguments:

For first, there is not to be found, in all history, any miracle attested by a sufficient number of men, of such unquestioned good sense, education, and learning, as to secure us against all delusion in themselves; of such undoubted integrity, as to place them beyond all suspicion of any design to deceive others; of such credit and reputation in the eyes of mankind, as to have a great deal to lose in case of their being detected in any falsehood; and at the same time, attesting facts performed in such a public manner and in so celebrated a part of the world, as to render the detection unavoidable: all which circumstances are requisite to give us a full assurance in the testimony of men.
Secondly. We may observe in human nature a principle which, if strictly examined, will be found to diminish extremely the assurance, which we might, from human testimony, have in any kind of prodigy. The maxim, by which we commonly conduct ourselves in our reasonings, is, that the objects, of which we have no experience, resembles those, of which we have; that what we have found to be most usual is always most probable; and that where there is an opposition of arguments, we ought to give the preference to such as are founded on the greatest number of past observations. But though, in proceeding by this rule, we readily reject any fact which is unusual and incredible in an ordinary degree; yet in advancing farther, the mind observes not always the same rule; but when anything is affirmed utterly absurd and miraculous, it rather the more readily admits of such a fact, upon account of that very circumstance, which ought to destroy all its authority. The passion of surprise and wonder, arising from miracles, being an agreeable emotion, gives a sensible tendency towards the belief of those events, from which it is derived. And this goes so far, that even those who cannot enjoy this pleasure immediately, nor can believe those miraculous events, of which they are informed, yet love to partake of the satisfaction at second-hand or by rebound, and place a pride and delight in exciting the admiration of others.
With what greediness are the miraculous accounts of travellers received, their descriptions of sea and land monsters, their relations of wonderful adventures, strange men, and uncouth manners? But if the spirit of religion join itself to the love of wonder, there is an end of common sense; and human testimony, in these circumstances, loses all pretensions to authority. A religionist may be an enthusiast, and imagine he sees what has no reality: he may know his narrative to be false, and yet persevere in it, with the best intentions in the world, for the sake of promoting so holy a cause: or even where this delusion has not place, vanity, excited by so strong a temptation, operates on him more powerfully than on the rest of mankind in any other circumstances; and self-interest with equal force. His auditors may not have, and commonly have not, sufficient judgement to canvass his evidence: what judgement they have, they renounce by principle, in these sublime and mysterious subjects: or if they were ever so willing to employ it, passion and a heated imagination disturb the regularity of its operations, their credulity increases his impudence: and his impudence overpowers their credulity.
Eloquence, when at its highest pitch, leaves little room for reason or reflection; but addressing itself entirely to the fancy or the affections, captivates the willing hearers, and subdues their understanding. Happily, this pitch is seldom attains. But what a Tully or a Demosthenes could scarcely effect over a Roman or Athenian audience, every Capuchin, every itinerant or stationary teacher can perform over the generality of mankind, and in a higher degree, by touching such gross and vulgar passions.
The many instances of forged miracles, and prophecies, and supernatural events, which, in all ages, have either been detected by contrary evidence, or which detect themselves by their absurdity, prove sufficiently the strong propensity of mankind to the extraordinary and the marvellous, and ought reasonably to beget a suspicion against all relations of this kind. This is our natural way of thinking, even with regard to the most common and most credible events. For instance: There is no kind of report which rises so easily, and spreads so quickly, especially in country places and provincial towns, as those concerning marriages; insomuch that two young persons of equal condition never see each other twice, but the whole neighbourhood immediately join them together. The pleasure of telling a piece of news so interesting, of propagating it, and of being the first reporters of it, spreads the intelligence. And this is so well known, that no man of sense gives attention to these reports, till he find them confirmed by some greater evidence. Do not the same passions, and others still stronger, incline the generality of mankind to believe and report, with the greatest vehemence and assurance, all religious miracles?
Thirdly. It forms a strong presumption against all supernatural and miraculous relations, that they are observed chiefly to abound among ignorant and barbarous nations; or if a civilized people has ever given admission to any of them, that people will be found to have received them from ignorant and barbarous ancestors, who transmitted them with that inviolable sanction and authority, which always attend received opinions. When we peruse the first histories of all nations, we are apt to imagine ourselves transported into some new world; where the whole frame of nature is disjointed, and every element performs its operations in a different manner, from what it does at present. Battles, revolutions, pestilence, famine and death, are never the effect of those natural causes, which we experience. Prodigies, omens, oracles, judgements, quite obscure the few natural events, that are intermingled with them. But as the former grow thinner every page, in proportion as we advance nearer the enlightened ages, we soon learn, that there is nothing mysterious or supernatural in the case, but that all proceeds from the usual propensity of mankind towards the marvellous, and that, though this inclination may at intervals receive a check from sense and learning, it can never be thoroughly extirpated from human nature.
It is strange, a judicious reader is apt to say, upon the perusal of these wonderful historians, that such prodigious events never happen in our days. But it is nothing strange, I hope, that men should lie in all ages. You must surely have seen instances enough of that frailty. You have yourself heard many such marvellous relations started, which, being treated with scorn by all the wise and judicious, have at last been abandoned even by the vulgar. Be assured, that those renowned lies, which have spread and flourished to such a monstrous height, arose from like beginnings; but being sown in a more proper soil, shot up at last into prodigies almost equal to those which they relate.

Hume is making some reasonable points here. But note that all of these things are contingent. They could have been otherwise in general, and they might well be otherwise in particular cases, even actual ones. Consequently they cannot possibly amount to a full proof that miraculous accounts should not be accepted. The fact that Hume feels the need to point to these contingent facts shows that at some level he is aware of the fact that his argument is not conclusive, although he wishes it to be.

In the end, Hume’s argument does not establish anything, but only expresses his own incredulity, as in this example:

There is also a memorable story related by Cardinal de Retz, which may well deserve our consideration. When that intriguing politician fled into Spain, to avoid the persecution of his enemies, he passed through Saragossa, the capital of Arragon, where he was shewn, in the cathedral, a man, who had served seven years as a doorkeeper, and was well known to every body in town, that had ever paid his devotions at that church. He had been seen, for so long a time, wanting a leg; but recovered that limb by the rubbing of holy oil upon the stump; and the cardinal assures us that he saw him with two legs. This miracle was vouched by all the canons of the church; and the whole company in town were appealed to for a confirmation of the fact; whom the cardinal found, by their zealous devotion, to be thorough believers of the miracle. Here the relater was also contemporary to the supposed prodigy, of an incredulous and libertine character, as well as of great genius; the miracle of so singular a nature as could scarcely admit of a counterfeit, and the witnesses very numerous, and all of them, in a manner, spectators of the fact, to which they gave their testimony. And what adds mightily to the force of the evidence, and may double our surprise on this occasion, is, that the cardinal himself, who relates the story, seems not to give any credit to it, and consequently cannot be suspected of any concurrence in the holy fraud. He considered justly, that it was not requisite, in order to reject a fact of this nature, to be able accurately to disprove the testimony, and to trace its falsehood, through all the circumstances of knavery and credulity which produced it. He knew, that, as this was commonly altogether impossible at any small distance of time and place; so was it extremely difficult, even where one was immediately present, by reason of the bigotry, ignorance, cunning, and roguery of a great part of mankind. He therefore concluded, like a just reasoner, that such an evidence carried falsehood upon the very face of it, and that a miracle, supported by any human testimony, was more properly a subject of derision than of argument.

This is a distorted account of the miracle of Calanda. Ironically, Hume’s position is actually supported to some extent by the errors contained in his own account of the miracle: we cannot “trace its falsehood,” in the sense that we cannot determine whether the account has been distorted by Hume himself, by the Cardinal, by the residents of Zaragoza, or by others, or some combination of these, but it is easy enough to determine the fact that it has been so distorted. Nevertheless, Hume is not proving anything here, but simply asserting that he would not believe in a miracle no matter how good the testimony brought in its favor.

This is not a reasonable attitude, but sheer stubbornness.

Mathematics and the Laws of Nature

In his essay The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Eugene Wigner says, “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” But in reality, it can be proved that a physical world — a world which has an order of place, with one part beside another, and an order of time, with one thing before another — must of necessity either follow mathematical natural laws, or it must be more or less intentionally designed in order to avoid this.

For example, suppose we attempt to determine how long it takes a ball to fall a certain distance. We do not need any particularly exact method to measure distances; for example, we could be measuring a fall of ten feet, taking foot in the presumably original sense of “the length of an adult human foot,” despite the noisiness of this measure. Nor do we need any particularly exact method to measure time; we could for example measure time in blinks. Something took 10 blinks if it took so long that I blinked 10 times before it was over. This would be even noisier than measuring in feet. But the point is that it does not matter how exact or inexact the measures are. If we have a world with place and time in it, we can find ways to make such measurements, even if they are inexact ones. Nor again do we need a way to get an extremely precise measure in blinks or in feet or in whatever of the physical quantity we are measuring; it is enough if we get a best estimate.

Now suppose we repeatedly measure, in some such way, how long it takes for a ball to fall a certain distance. After we have made many measurements, we can add them together and divide by the total number of measurements, getting an average amount of time for the fall. The question that arises is this: as we increase the number of measurements indefinitely, will that average converge to a finite value? or will it diverge to infinity or go back and forth infinitely many times?

Evidently it will not diverge to infinity. It is difficult to see any reason in principle why it could not go back and forth infinitely many times, for example the average fall time might tend toward 1/4 of a blink for a long time, then start tending toward 1/5 of a blink for a long time, and then go back to 1/4, and so on. But we should notice the kind of pattern that is necessary in order for this to happen. Suppose the average is 1/4 of a blink after 100 measurements. In order to get the average to 1/5, there must be a great many measurements 1/5 or below, or at least many measurements which are very much below 1/5. And the more measurements we have taken to get the average, the more such especially low measures are needed. So if we are at an average of 1/4 of a blink after 1,000,000 measurements, this average will be very stable, and it will require an extremely long series, more or less continuous, of especially low measurements in order to get the average down to 1/5 again. And the length of the “especially low” or “especially high” series which is needed to move the average will be increasing each time we want to move it again. In other words, in order to get the average to go back and forth infinitely many times, we need to have a rather pathological series of measurements, namely one that looks like it was designed intentionally to prevent the series from converging to an average value.

Thus the “natural” result, when things are not designed to prevent convergence to an average, is that such measures of distance and time and basically anything else we might think of measuring, like “how much food does an adult eat in a year”, will always converge to an average value as we increase the number of measurements indefinitely. Given this result it follows that it is possible to express the behavior of the physical world using mathematical laws.

Several things however do not necessarily follow from this:

It does not follow that such laws cannot have “exceptions”, since they are only statistical laws from the beginning, and thus are only expected to work approximately. So it is not possible to rule out miracles in the way supposed by David Hume.

It also does not follow that such laws have to be particularly simple. A simpler law will be more likely than a more complex one, for the reasons given in a previous post, but theoretically the laws governing a falling body could have 500 variables, which would be simpler than ones having 50,000 variables. In practice however this does not tend to be the case, or at least we can find extremely good approximate laws with very few variables. It may simply be the case that in order to have a world with animals in it, the world needs to be fairly predictable to them, and this may require that fairly simple laws work at least as a good approximation. But a mathematical demonstration of this would be extremely difficult, if it turns out to be possible at all.