Technical Discussion and Philosophical Progress

In The Structure of Scientific Revolutions (p. 19-21), Thomas Kuhn remarks on the tendency of sciences to acquire a technical vocabulary and manner of discussion:

We shall be examining the nature of this highly directed or paradigm-based research in the next section, but must first note briefly how the emergence of a paradigm affects the structure of the group that practices the field. When, in the development of a natural science, an individual or group first produces a synthesis able to attract most of the next generation’s practitioners, the older schools gradually disappear. In part their disappearance is caused by their members’ conversion to the new paradigm. But there are always some men who cling to one or another of the older views, and they are simply read out of the profession, which thereafter ignores their work. The new paradigm implies a new and more rigid definition of the field. Those unwilling or unable to accommodate their work to it must proceed in isolation or attach themselves to some other group. Historically, they have often simply stayed in the departments of philosophy from which so many of the special sciences have been spawned. As these indications hint, it is sometimes just its reception of a paradigm that transforms a group previously interested merely in the study of nature into a profession or, at least, a discipline. In the sciences (though not in fields like medicine, technology, and law, of which the principal raison d’être is an external social need), the formation of specialized journals, the foundation of specialists’ societies, and the claim for a special place in the curriculum have usually been associated with a group’s first reception of a single paradigm. At least this was the case between the time, a century and a half ago, when the institutional pattern of scientific specialization first developed and the very recent time when the paraphernalia of specialization acquired a prestige of their own.

The more rigid definition of the scientific group has other consequences. When the individual scientist can take a paradigm for granted, he need no longer, in his major works, attempt to build his field anew, starting from first principles and justifying the use of each concept introduced. That can be left to the writer of textbooks. Given a textbook, however, the creative scientist can begin his research where it leaves off and thus concentrate exclusively upon the subtlest and most esoteric aspects of the natural phenomena that concern his group. And as he does this, his research communiqués will begin to change in ways whose evolution has been too little studied but whose modern end products are obvious to all and oppressive to many. No longer will his researches usually be embodied in books addressed, like Franklin’s Experiments . . . on Electricity or Darwin’s Origin of Species, to anyone who might be interested in the subject matter of the field. Instead they will usually appear as brief articles addressed only to professional colleagues, the men whose knowledge of a shared paradigm can be assumed and who prove to be the only ones able to read the papers addressed to them.

Today in the sciences, books are usually either texts or retrospective reflections upon one aspect or another of the scientific life. The scientist who writes one is more likely to find his professional reputation impaired than enhanced. Only in the earlier, pre-paradigm, stages of the development of the various sciences did the book ordinarily possess the same relation to professional achievement that it still retains in other creative fields. And only in those fields that still retain the book, with or without the article, as a vehicle for research communication are the lines of professionalization still so loosely drawn that the layman may hope to follow progress by reading the practitioners’ original reports. Both in mathematics and astronomy, research reports had ceased already in antiquity to be intelligible to a generally educated audience. In dynamics, research became similarly esoteric in the later Middle Ages, and it recaptured general intelligibility only briefly during the early seventeenth century when a new paradigm replaced the one that had guided medieval research. Electrical research began to require translation for the layman before the end of the eighteenth century, and most other fields of physical science ceased to be generally accessible in the nineteenth. During the same two centuries similar transitions can be isolated in the various parts of the biological sciences. In parts of the social sciences they may well be occurring today. Although it has become customary, and is surely proper, to deplore the widening gulf that separates the professional scientist from his colleagues in other fields, too little attention is paid to the essential relationship between that gulf and the mechanisms intrinsic to scientific advance.

As Kuhn says, this tendency has very well known results. Consider the papers constantly being published at arxiv.org, for example. If you are not familiar with the science in question, you will likely not be able to understand even the title, let alone the summary or the content. Many or most of the words will be meaningless to you, and even if they are not, their combinations will be.

It is also not difficult to see why this happens, and why it must happen. Everything we understand, we understand through form, which is a network of relationships. Thus if particular investigators wish to go into something in greater detail, these relationships will become more and more remote from the ordinary knowledge accessible to everyone. “Just say it in simple words” will become literally impossible, in the sense that explaining the “simple” statement will involve explaining a huge number of relationships that by default a person would have no knowledge of. That is the purpose, as Kuhn notes, of textbooks, namely to form connections between everyday knowledge and the more complex relationships studied in particular fields.

In Chapter XIII, Kuhn relates this sort of development with the word “science” and progress:

The preceding pages have carried my schematic description of scientific development as far as it can go in this essay. Nevertheless, they cannot quite provide a conclusion. If this description has at all caught the essential structure of a science’s continuing evolution, it will simultaneously have posed a special problem: Why should the enterprise sketched above move steadily ahead in ways that, say, art, political theory, or philosophy does not? Why is progress a perquisite reserved almost exclusively for the activities we call science? The most usual answers to that question have been denied in the body of this essay. We must conclude it by asking whether substitutes can be found.

Notice immediately that part of the question is entirely semantic. To a very great extent the term ‘science’ is reserved for fields that do progress in obvious ways. Nowhere does this show more clearly than in the recurrent debates about whether one or another of the contemporary social sciences is really a science. These debates have parallels in the pre-paradigm periods of fields that are today unhesitatingly labeled science. Their ostensible issue throughout is a definition of that vexing term. Men argue that psychology, for example, is a science because it possesses such and such characteristics. Others counter that those characteristics are either unnecessary or not sufficient to make a field a science. Often great energy is invested, great passion aroused, and the outsider is at a loss to know why. Can very much depend upon a definition of ‘science’? Can a definition tell a man whether he is a scientist or not? If so, why do not natural scientists or artists worry about the definition of the term? Inevitably one suspects that the issue is more fundamental. Probably questions like the following are really being asked: Why does my field fail to move ahead in the way that, say, physics does? What changes in technique or method or ideology would enable it to do so? These are not, however, questions that could respond to an agreement on definition. Furthermore, if precedent from the natural sciences serves, they will cease to be a source of concern not when a definition is found, but when the groups that now doubt their own status achieve consensus about their past and present accomplishments. It may, for example, be significant that economists argue less about whether their field is a science than do practitioners of some other fields of social science. Is that because economists know what science is? Or is it rather economics about which they agree?

The last point is telling. There is significantly more consensus among economists than among other sorts of social science, and consequently less worry about whether their field is scientific or not. The difference, then, is a difference of how much agreement is found. There is not necessarily any difference with respect to the kind of increasingly detailed thought that results in increasingly technical discussion. Kuhn remarks:

The theologian who articulates dogma or the philosopher who refines the Kantian imperatives contributes to progress, if only to that of the group that shares his premises. No creative school recognizes a category of work that is, on the one hand, a creative success, but is not, on the other, an addition to the collective achievement of the group. If we doubt, as many do, that nonscientific fields make progress, that cannot be because individual schools make none. Rather, it must be because there are always competing schools, each of which constantly questions the very foundations of the others. The man who argues that philosophy, for example, has made no progress emphasizes that there are still Aristotelians, not that Aristotelianism has failed to progress.

In this sense, if a particular school believes they possess the general truth about some matter (here theology or philosophy), they will quite naturally begin to discuss it in greater detail and in ways which are mainly intelligible to students of that school, just as happens in other technical fields. The field is only failing to progress in the sense that there are other large communities making contrasting claims, while we begin to use the term “science” and to speak of progress when one school completely dominates the field, and to a first approximation even people who know nothing about it assume that the particular school has things basically right.

What does this imply about progress in philosophy?

1. There is progress in the knowledge of topics that were once considered “philosophy,” but when we get to this point, we usually begin to use the name of a particular science, and with good reason, since technical specialization arises in the manner discussed above. Tyler Cowen discusses this sort of thing here.

2. Areas in which there doesn’t seem to be such progress, are probably most often areas where human knowledge remains at an early stage of development; it is precisely at such early stages that discussion does not have a technical character and when it can generally be understood by ordinary people without a specialized education. I pointed out that Aristotle was mistaken to assume that the sciences in general were fully developed. We would be equally mistaken to make such an assumption at the present times. As Kuhn notes, astronomy and mathematics achieved a “scientific” stage centuries before geology and biology did the same, and these long before economics and the like. The conclusion that one should draw is that metaphysics is hard, not that it is impossible or meaningless.

3. Even now, particular philosophical schools or individuals can make progress even without such consensus. This is evidently true if their overall position is correct or more correct than that of others, but it remains true even if their overall position is more wrong than that of other schools. Naturally, in the latter situation, they will not advance beyond the better position of other schools, but they will advance.

4. One who wishes to progress philosophically cannot avoid the tendency to technical specialization, even as an individual. This can be rather problematic for bloggers and people engaging in similar projects. John Nerst describes this problem:

The more I think about this issue the more unsolvable it seems to become. Loyal readers of a publication won’t be satisfied by having the same points reiterated again and again. News media get around this by focusing on, well, news. News are events, you can describe them and react to them for a while until they’re no longer news. Publications that aim to be more analytical and focus on discussing ideas, frameworks, slow processes and large-scale narratives instead of events have a more difficult task because their subject matter doesn’t change quickly enough for it to be possible to churn out new material every day without repeating yourself[2].

Unless you start building upwards. Instead of laying out stone after stone on the ground you put one on top of another, and then one on top of two others laying next to each other, and then one on top of all that, making a single three-level structure. In practice this means writing new material that builds on what came before, taking ideas further and further towards greater complexity, nuance and sophistication. This is what academia does when working correctly.

Mass media (including the more analytical outlets) do it very little and it’s obvious why: it’s too demanding[3]. If an article references six other things you need to have read to fully understand it you’re going to have a lot of difficulty attracting new readers.

Some of his conclusions:

I think that’s the real reason I don’t try to pitch more writing to various online publications. In my summary of 2018 I said it was because I thought my writing was to “too idiosyncratic, abstract and personal to fit in anywhere but my own blog”. Now I think the main reason is that I don’t so much want to take part in public debate or make myself a career. I want to explore ideas that lie at the edge of my own thinking. To do that I must assume that a reader knows broadly the same things I know and I’m just not that interested in writing about things where I can’t do that[9]. I want to follow my thoughts to for me new and unknown places — and import whatever packages I need to do it. This style isn’t compatible with the expectation that a piece will be able to stand on its own and deliver a single recognizable (and defensible) point[10].

The downside is of course obscurity. To achieve both relevance in the wider world and to build on other ideas enough to reach for the sky you need extraordinary success — so extraordinary that you’re essentially pulling the rest of the world along with you.

Obscurity is certainly one result. Another (relevant at least from the VP’s point of view) is disrespect. Scientists are generally respected despite the general incomprehensibility of their writing, on account of the absence of opposing schools. This lack leads people to assume that their arguments must be mostly right, even though they cannot understand them themselves. This can actually lead to an “Emperor has No Clothes” situation, where a scientist publishes something basically crazy, but others, even in his field, are reluctant to say so because they might appear to be the ones who are ignorant. As an example, consider Joy Christian’s “Disproof of Bell’s Theorem.” After reading this text, Scott Aaronson comments:

In response to my post criticizing his “disproof” of Bell’s Theorem, Joy Christian taunted me that “all I knew was words.”  By this, he meant that my criticisms were entirely based on circumstantial evidence, for example that (1) Joy clearly didn’t understand what the word “theorem” even meant, (2) every other sentence he uttered contained howling misconceptions, (3) his papers were written in an obscure, “crackpot” way, and (4) several people had written very clear papers pointing out mathematical errors in his work, to which Joy had responded only with bluster.  But I hadn’t actually studied Joy’s “work” at a technical level.  Well, yesterday I finally did, and I confess that I was astonished by what I found.  Before, I’d actually given Joy some tiny benefit of the doubt—possibly misled by the length and semi-respectful tone of the papers refuting his claims.  I had assumed that Joy’s errors, though ultimately trivial (how could they not be, when he’s claiming to contradict such a well-understood fact provable with a few lines of arithmetic?), would nevertheless be artfully concealed, and would require some expertise in geometric algebra to spot.  I’d also assumed that of course Joy would have some well-defined hidden-variable model that reproduced the quantum-mechanical predictions for the Bell/CHSH experiment (how could he not?), and that the “only” problem would be that, due to cleverly-hidden mistakes, his model would be subtly nonlocal.

What I actually found was a thousand times worse: closer to the stuff freshmen scrawl on an exam when they have no clue what they’re talking about but are hoping for a few pity points.  It’s so bad that I don’t understand how even Joy’s fellow crackpots haven’t laughed this off the stage.  Look, Joy has a hidden variable λ, which is either 1 or -1 uniformly at random.  He also has a measurement choice a of Alice, and a measurement choice b of Bob.  He then defines Alice and Bob’s measurement outcomes A and B via the following functions:

A(a,λ) = something complicated = (as Joy correctly observes) λ

B(b,λ) = something complicated = (as Joy correctly observes) -λ

I shit you not.  A(a,λ) = λ, and B(b,λ) = -λ.  Neither A nor B has any dependence on the choices of measurement a and b, and the complicated definitions that he gives for them turn out to be completely superfluous.  No matter what measurements are made, A and B are always perfectly anticorrelated with each other.

You might wonder: what could lead anyone—no matter how deluded—even to think such a thing could violate the Bell/CHSH inequalities?

“Give opposite answers in all cases” is in fact entirely irrelevant to Bell’s inequality. Thus the rest of Joy’s paper has no bearing whatsoever on the issue: it is essentially meaningless nonsense. Aaronson says he was possibly “misled by the length and semi-respectful tone of the papers refuting his claims.” But it is not difficult to see why people would be cautious in this way: the fear that they would turn out to be the ones missing something important.

The individual blogger in philosophy, however, is in a different position. If they wish to develop their thought it must become more technical, and there is no similar community backing that would cause others to assume that the writing basically makes sense. Thus, one’s writing is not only likely to become more and more obscure, but others will become more and more likely to assume that it is more or less meaningless word salad. This will happen even more to the degree that there is cultural opposition to one’s vocabulary, concepts, and topics.

Discount Rates

Eliezer Yudkowsky some years ago made this argument against temporal discounting:

I’ve never been a fan of the notion that we should (normatively) have a discount rate in our pure preferences – as opposed to a pseudo-discount rate arising from monetary inflation, or from opportunity costs of other investments, or from various probabilistic catastrophes that destroy resources or consumers.  The idea that it is literally, fundamentally 5% more important that a poverty-stricken family have clean water in 2008, than that a similar family have clean water in 2009, seems like pure discrimination to me – just as much as if you were to discriminate between blacks and whites.

Robin  Hanson disagreed, responding with this post:

But doesn’t discounting at market rates of return suggest we should do almost nothing to help far future folk, and isn’t that crazy?  No, it suggests:

  1. Usually the best way to help far future folk is to invest now to give them resources they can spend as they wish.
  2. Almost no one now in fact cares much about far future folk, or they would have bid up the price (i.e., market return) to much higher levels.

Very distant future times are ridiculously easy to help via investment.  A 2% annual return adds up to a googol (10^100) return over 12,000 years, even if there is only a 1/1000 chance they will exist or receive it.

So if you are not incredibly eager to invest this way to help them, how can you claim to care the tiniest bit about them?  How can you think anyone on Earth so cares?  And if no one cares the tiniest bit, how can you say it is “moral” to care about them, not just somewhat, but almost equally to people now?  Surely if you are representing a group, instead of spending your own wealth, you shouldn’t assume they care much.

Yudkowsky’s argument is idealistic, while Hanson is attempting to be realistic. I will look at this from a different point of view. Hanson is right, and Yudkowsky is wrong, for a still more idealistic reason than Yudkowsky’s reasons. In particular, a temporal discount rate is logically and mathematically necessary in order to have consistent preferences.

Suppose you have the chance to save 10 lives a year from now, or 2 years from now, or 3 years from now etc., such that your mutually exclusive options include the possibility of saving 10 lives x years from now for all x.

At first, it would seem to be consistent for you to say that all of these possibilities have equal value by some measure of utility.

The problem does not arise from this initial assignment, but it arises when we consider what happens when you act in this situation. Your revealed preferences in that situation will indicate that you prefer things nearer in time to things more distant, for the following reason.

It is impossible to choose a random integer without a bias towards low numbers, for the same reasons we argued here that it is impossible to assign probabilities to hypotheses without, in general, assigning simpler hypotheses higher probabilities. In a similar way, if “you will choose 2 years from now”, “you will choose 10 years from now,” “you will choose 100 years from now,” are all assigned probabilities, they cannot all be assigned equal probabilities, but you must be more likely to choose the options less distant in time, in general and overall. There will be some number n such that there is a 99.99% chance that you will choose some number of years less than n, and and a probability of 0.01% that you will choose n or more years, indicating that you have a very strong preference for saving lives sooner rather than later.

Someone might respond that this does not necessarily affect the specific value assignments, in the same way that in some particular case, we can consistently think that some particular complex hypothesis is more probable than some particular simple hypothesis. The problem with this is the hypotheses do not change their complexity, but time passes, making things distant in time become things nearer in time. Thus, for example, if Yudkowsky responds, “Fine. We assign equal value to saving lives for each year from 1 to 10^100, and smaller values to the times after that,” this will necessarily lead to dynamic inconsistency. The only way to avoid this inconsistency is to apply a discount rate to all periods of time, including ones in the near, medium, and long term future.

 

All Things are Water

In book I of his Metaphysics, Aristotle comments on earlier opinions about the first causes:

Evidently we have to acquire knowledge of the original causes (for we say we know each thing only when we think we recognize its first cause), and causes are spoken of in four senses. In one of these we mean the substance, i.e. the essence (for the ‘why’ is reducible finally to the definition, and the ultimate ‘why’ is a cause and principle); in another the matter or substratum, in a third the source of the change, and in a fourth the cause opposed to this, the purpose and the good (for this is the end of all generation and change). We have studied these causes sufficiently in our work on nature, but yet let us call to our aid those who have attacked the investigation of being and philosophized about reality before us. For obviously they too speak of certain principles and causes; to go over their views, then, will be of profit to the present inquiry, for we shall either find another kind of cause, or be more convinced of the correctness of those which we now maintain.

Of the first philosophers, then, most thought the principles which were of the nature of matter were the only principles of all things. That of which all things that are consist, the first from which they come to be, the last into which they are resolved (the substance remaining, but changing in its modifications), this they say is the element and this the principle of things, and therefore they think nothing is either generated or destroyed, since this sort of entity is always conserved, as we say Socrates neither comes to be absolutely when he comes to be beautiful or musical, nor ceases to be when loses these characteristics, because the substratum, Socrates himself remains. just so they say nothing else comes to be or ceases to be; for there must be some entity-either one or more than one-from which all other things come to be, it being conserved.

Yet they do not all agree as to the number and the nature of these principles. Thales, the founder of this type of philosophy, says the principle is water (for which reason he declared that the earth rests on water), getting the notion perhaps from seeing that the nutriment of all things is moist, and that heat itself is generated from the moist and kept alive by it (and that from which they come to be is a principle of all things). He got his notion from this fact, and from the fact that the seeds of all things have a moist nature, and that water is the origin of the nature of moist things.

Some think that even the ancients who lived long before the present generation, and first framed accounts of the gods, had a similar view of nature; for they made Ocean and Tethys the parents of creation, and described the oath of the gods as being by water, to which they give the name of Styx; for what is oldest is most honourable, and the most honourable thing is that by which one swears. It may perhaps be uncertain whether this opinion about nature is primitive and ancient, but Thales at any rate is said to have declared himself thus about the first cause.

It is possibly for polemical motives that Aristotle portrays Thales as asserting that water alone is the principle of all things, to the exclusion of other kinds of cause besides the material cause. That is, most materialists, ancient and modern, do not believe that matter is the sole principle of reality. They may think it is the most important principle, but they recognize that other principles are involved, much as Lucretius recognizes that his atoms alone are insufficient to explain the world, but he must add something:

We wish thee also well aware of this:
The atoms, as their own weight bears them down
Plumb through the void, at scarce determined times,
In scarce determined places, from their course
Decline a little- call it, so to speak,
Mere changed trend. For were it not their wont
Thuswise to swerve, down would they fall, each one,
Like drops of rain, through the unbottomed void;
And then collisions ne’er could be nor blows
Among the primal elements; and thus
Nature would never have created aught.

We can understand however that even if Aristotle’s account may not be a completely accurate account of Thales’s opinions, Aristotle likely has a charitable motive for his presentation, namely the education of the reader, as by beginning by discussing the position that matter alone is the first cause, it becomes easier to see the necessity of other principles. And it is also possible that Thales did not mention any other principles simply because his interest was in the material principle, rather than from the wish to deny other principles.

There is some probability to the opinion, mentioned by Aristotle, that Thales’s idea about water had ancient predecessors. For example, there may be something like this in Genesis 1:

In the beginning when God created the heavens and the earth, the earth was a formless void and darkness covered the face of the deep, while a wind from God swept over the face of the waters. Then God said, “Let there be light”; and there was light. And God saw that the light was good; and God separated the light from the darkness. God called the light Day, and the darkness he called Night. And there was evening and there was morning, the first day.

And God said, “Let there be a dome in the midst of the waters, and let it separate the waters from the waters.” So God made the dome and separated the waters that were under the dome from the waters that were above the dome. And it was so. God called the dome Sky. And there was evening and there was morning, the second day.

And God said, “Let the waters under the sky be gathered together into one place, and let the dry land appear.” And it was so. God called the dry land Earth, and the waters that were gathered together he called Seas. And God saw that it was good.

The text speaks of a number of principles, including God, and formless matter. But the formless matter is virtually identified with water; even when the earth is a “formless void,” there is still a “deep”, and there is still a “face of the waters.” And God makes the world by divisions in the waters, creating first sky and world, and then creating dry land by limiting the lower waters.

We can ask whether Thales was right in several different ways:

First, was he right if he is understood as Aristotle understands him? In this way, his position would involve denying all principles other than material principles. And in this way Thales was wrong, since there are other principles.

Second, was he right if he is understood in contrast with other materialist philosophers such as Anaximenes, who said that all things are made of air? We can see that there is a certain difficulty in such a supposition from the beginning. For if there is only one material principle of all things, why call it “water” or “air” in particular? If water contains nothing but the first material principle, and air contains nothing but the first material principle, why is one of them identified with the principle rather than the other?

Nonetheless, we can understand the claim to be something like, “Water is the most basic and natural form of the first material principle.” In this case, what it means to be the most basic and natural would be a matter of investigation, but there is nothing impossible about it in principle. But if we understand it in this way, Thales’s position (along with that of Anaximenes and others) is refuted by modern science, since water is known to be made of other more basic things, namely oxygen and hydrogen.

But we can ask whether Thales and Anaximenes were both right in a third way, a more generic way. If we break composite material things down into their parts, and their parts into their parts, and so on, will we always arrive at one basic material “stuff” which all other things are made out of?

This would necessarily be different from the prime matter of Aristotle, because this matter is understood to be completely formless. So while it may be part of a substance along with substantial form, it is not a part in the way that an arm is a part of a human being, or in the way that oxygen is part of water. A part in the latter sense already has some actuality; an arm has a certain shape even apart from being a part of a human being, and oxygen has qualities that it has even apart from water, while prime matter has no actuality whatsoever.

Since every order of causes comes to a first cause, the same will be true if we follow the order of material causality found by asking, “What parts is this made out of?” This cannot be refuted even if it turns out that material things are infinitely divisible, because it will not actually be true that anything is made out of an infinite number of parts. If we say, “this is made out of two halves, and the halves out of more halves, and so on,” this is not an explanation at all, as was pointed out in the post linked above, and so it cannot be a true account of why the thing is as it is. It may be that the thing is potentially divisible in those ways; but it is not actually made out of those parts. Instead, we are asking what we will arrive at if we look at the material composition of things, looking only for things with some actuality, and stopping when we find something which is not made up of other things in this way.

We made a strong argument that there only one first efficient cause. But we cannot duplicate that argument to show that there must be only one first material part of things, since the first efficient cause could be an adequate explanation of two or more first material causes. The theory of the four (or five) elements is an explanation like this. The first material parts are thought be four or five, according to this account, and in this way Thales would have been mistaken.

We can see some motivation for holding such a position. If there is fundamentally only one kind of “stuff,” why do we see various things in the world, such as plants and animals, rocks, chairs, and people? An account with a number elements can say that they form various things by being combined in various proportions and ways, while it is not evident how this question can be answered, if there is only one element.

Nonetheless, this does not really refute the position that there is fundamentally a single material element. If that element can exist in various ways, then it would be possible to explain how it could be used to form more complex substances. It is true that this would still leave something to be explained, namely the nature of those various ways that the element can exist, and how and why they come to be.

The position is neither refuted nor established by modern science. The Standard Model of particle physics contains many elementary particles, but in any case it is not thought to be a complete theory of physics.

Considerations of simplicity would favor the position of Thales. Other things being equal, it should be thought that a theory with a single element is more likely than one with ten elements; one with ten more likely than one with a hundred, and so on. Nonetheless, there does not seem to be any proof of Thales’s general position, nor any refutation of it. But one way or another, there will be one or more simplest material parts that everything else is made out of.

Richard Dawkins and the Simplicity of God

Richard Dawkins concludes chapter 3 of his book The God Delusion with the following claim:

There is a much more powerful argument, which does not depend upon subjective judgement, and it is the argument from improbability. It really does transport us dramatically away from 50 per cent agnosticism, far towards the extreme of theism in the view of many theists, far towards the extreme of atheism in my view. I have alluded to it several times already. The whole argument turns on the familiar question ‘Who made God?’, which most thinking people discover for themselves. A designer God cannot be used to explain organized complexity because any God capable of designing anything would have to be complex enough to demand the same kind of explanation in his own right. God presents an infinite regress from which he cannot help us to escape. This argument, as I shall show in the next chapter, demonstrates that God, though not technically disprovable, is very very improbable indeed.

Throughout chapter 4, which is entitled, “Why There Almost Certainly is No God,” he struggles with the view of the theologians that God is simple, as opposed to his own idea that God, if he exists, must be extremely complicated. He begins the chapter:

The argument from improbability is the big one. In the traditional guise of the argument from design, it is easily today’s most popular argument offered in favour of the existence of God and it is seen, by an amazingly large number of theists, as completely and utterly convincing. It is indeed a very strong and, I suspect, unanswerable argument— but in precisely the opposite direction from the theist’s intention. The argument from improbability, properly deployed, comes close to proving that God does not exist. My name for the statistical demonstration that God almost certainly does not exist is the Ultimate Boeing 747 gambit.

The name comes from Fred Hoyle’s amusing image of the Boeing 747 and the scrapyard. I am not sure whether Hoyle ever wrote it down himself, but it was attributed to him by his close colleague Chandra Wickramasinghe and is presumably authentic. Hoyle said that the probability of life originating on Earth is no greater than the chance that a hurricane, sweeping through a scrapyard, would have the luck to assemble a Boeing 747. Others have borrowed the metaphor to refer to the later evolution of complex living bodies, where it has a spurious plausibility. The odds against assembling a fully functioning horse, beetle or ostrich by randomly shuffling its parts are up there in 747 territory. This, in a nutshell, is the creationist’s favourite argument— an argument that could be made only by somebody who doesn’t understand the first thing about natural selection: somebody who thinks natural selection is a theory of chance whereas— in the relevant sense of chance— it is the opposite.

There follows a discussion of evolution, creation, and intelligent design. He concludes the section by stating,

A deep understanding of Darwinism teaches us to be wary of the easy assumption that design is the only alternative to chance, and teaches us to seek out graded ramps of slowly increasing complexity. Before Darwin, philosophers such as Hume understood that the improbability of life did not mean it had to be designed, but they couldn’t imagine the alternative. After Darwin, we all should feel, deep in our bones, suspicious of the very idea of design. The illusion of design is a trap that has caught us before, and Darwin should have immunized us by raising our consciousness. Would that he had succeeded with all of us.

The argument here is basically that evolutionary theory has been fairly successful in explaining living things as having resulted from a slow and detailed process in which they became increasingly complex through natural causes. Consequently Dawkins is optimistic that this manner of explanation can in principle be applied to everything else. In fact, according to him, no one has ever offered any other plausible explanation of things:

Turning Watchtower’s page, we find the wonderful plant known as Dutchman’s Pipe (Aristolochia trilobata), all of whose parts seem elegantly designed to trap insects, cover them with pollen and send them on their way to another Dutchman’s Pipe. The intricate elegance of the flower moves Watchtower to ask: ‘Did all of this happen by chance? Or did it happen by intelligent design?’ Once again, no of course it didn’t happen by chance. Once again, intelligent design is not the proper alternative to chance. Natural selection is not only a parsimonious, plausible and elegant solution; it is the only workable alternative to chance that has ever been suggested. Intelligent design suffers from exactly the same objection as chance. It is simply not a plausible solution to the riddle of statistical improbability. And the higher the improbability, the more implausible intelligent design becomes. Seen clearly, intelligent design will turn out to be a redoubling of the problem. Once again, this is because the designer himself (/ herself/ itself) immediately raises the bigger problem of his own origin. Any entity capable of intelligently designing something as improbable as a Dutchman’s Pipe (or a universe) would have to be even more improbable than a Dutchman’s Pipe. Far from terminating the vicious regress, God aggravates it with a vengeance.

He says something similar while discussing multiverse hypotheses:

It is tempting to think (and many have succumbed) that to postulate a plethora of universes is a profligate luxury which should not be allowed. If we are going to permit the extravagance of a multiverse, so the argument runs, we might as well be hung for a sheep as a lamb and allow a God. Aren’t they both equally unparsimonious ad hoc hypotheses, and equally unsatisfactory? People who think that have not had their consciousness raised by natural selection. The key difference between the genuinely extravagant God hypothesis and the apparently extravagant multiverse hypothesis is one of statistical improbability. The multiverse, for all that it is extravagant, is simple. God, or any intelligent, decision-taking, calculating agent, would have to be highly improbable in the very same statistical sense as the entities he is supposed to explain. The multiverse may seem extravagant in sheer number of universes. But if each one of those universes is simple in its fundamental laws, we are still not postulating anything highly improbable. The very opposite has to be said of any kind of intelligence.

Beginning to address the response of theologians, he says:

But what attempts have theists made to reply? How do they cope with the argument that any God capable of designing a universe, carefully and foresightfully tuned to lead to our evolution, must be a supremely complex and improbable entity who needs an even bigger explanation than the one he is supposed to provide? The theologian Richard Swinburne, as we have learned to expect, thinks he has an answer to this problem, and he expounds it in his book Is There a God?. He begins by showing that his heart is in the right place by convincingly demonstrating why we should always prefer the simplest hypothesis that fits the facts. Science explains complex things in terms of the interactions of simpler things, ultimately the interactions of fundamental particles. I (and I dare say you) think it a beautifully simple idea that all things are made of fundamental particles which, although exceedingly numerous, are drawn from a small, finite set of types of particle. If we are sceptical, it is likely to be because we think the idea too simple. But for Swinburne it is not simple at all, quite the reverse. Given that the number of particles of any one type, say electrons, is large, Swinburne thinks it too much of a coincidence that so many should have the same properties. One electron, he could stomach. But billions and billions of electrons, all with the same properties, that is what really excites his incredulity. For him it would be simpler, more natural, less demanding of explanation, if all electrons were different from each other. Worse, no one electron should naturally retain its properties for more than an instant at a time; each should change capriciously, haphazardly and fleetingly from moment to moment. That is Swinburne’s view of the simple, native state of affairs. Anything more uniform (what you or I would call more simple) requires a special explanation. ‘It is only because electrons and bits of copper and all other material objects have the same powers in the twentieth century as they did in the nineteenth century that things are as they are now.’ Enter God. God comes to the rescue by deliberately and continuously sustaining the properties of all those billions of electrons and bits of copper, and neutralizing their otherwise ingrained inclination to wild and erratic fluctuation. That is why when you’ve seen one electron you’ve seen them all; that is why bits of copper all behave like bits of copper, and that is why each electron and each bit of copper stays the same as itself from microsecond to microsecond and from century to century. It is because God constantly keeps a finger on each and every particle, curbing its reckless excesses and whipping it into line with its colleagues to keep them all the same. But how can Swinburne possibly maintain that this hypothesis of God simultaneously keeping a gazillion fingers on wayward electrons is a simple hypothesis? It is, of course, precisely the opposite of simple. Swinburne pulls off the trick to his own satisfaction by a breathtaking piece of intellectual chutzpah. He asserts, without justification, that God is only a single substance. What brilliant economy of explanatory causes, compared with all those gigazillions of independent electrons all just happening to be the same!

Note that Richard Swinburne is not the only one who thinks it too much of a coincidence that electrons are not all different and randomly changing their properties from moment to moment. David Hume, praised by Dawkins, believes the same thing. In any case, in terms of the argument here, Swinburne is exactly right. There is only one first cause, and it does indeed explain why all electrons behave in the same way. Some such thing would have to be the case in any event, but the only way the activity of electrons (or of anything else) can be understood is in relation to a final cause, the formal aspect of an efficient cause.

Dawkins however objects that such an explanation is not simple at all, but supremely complex:

Swinburne generously concedes that God cannot accomplish feats that are logically impossible, and one feels grateful for this forbearance. Having said that, there is no limit to the explanatory purposes to which God’s infinite power is put. Is science having a little difficulty explaining X? No problem. Don’t give X another glance. God’s infinite power is effortlessly wheeled in to explain X (along with everything else), and it is always a supremely simple explanation because, after all, there is only one God. What could be simpler than that?

Well, actually, almost everything. A God capable of continuously monitoring and controlling the individual status of every particle in the universe cannot be simple. His existence is going to need a mammoth explanation in its own right. Worse (from the point of view of simplicity), other corners of God’s giant consciousness are simultaneously preoccupied with the doings and emotions and prayers of every single human being— and whatever intelligent aliens there might be on other planets in this and 100 billion other galaxies. He even, according to Swinburne, has to decide continuously not to intervene miraculously to save us when we get cancer. That would never do, for, ‘If God answered most prayers for a relative to recover from cancer, then cancer would no longer be a problem for humans to solve.’ And then what would we find to do with our time?

Outraged by this idea of simplicity, Dawkins considers another example of this position:

Not all theologians go as far as Swinburne. Nevertheless, the remarkable suggestion that the God Hypothesis is simple can be found in other modern theological writings. Keith Ward, then Regius Professor of Divinity at Oxford, was very clear on the matter in his 1996 book God, Chance and Necessity: “As a matter of fact, the theist would claim that God is a very elegant, economical and fruitful explanation for the existence of the universe. It is economical because it attributes the existence and nature of absolutely everything in the universe to just one being, an ultimate cause which assigns a reason for the existence of everything, including itself. It is elegant because from one key idea— the idea of the most perfect possible being— the whole nature of God and the existence of the universe can be intelligibly explicated.”

Like Swinburne, Ward mistakes what it means to explain something, and he also seems not to understand what it means to say of something that it is simple. I am not clear whether Ward really thinks God is simple, or whether the above passage represented a temporary ‘for the sake of argument’ exercise. Sir John Polkinghorne, in Science and Christian Belief, quotes Ward’s earlier criticism of the thought of Thomas Aquinas: ‘Its basic error is in supposing that God is logically simple— simple not just in the sense that his being is indivisible, but in the much stronger sense that what is true of any part of God is true of the whole. It is quite coherent, however, to suppose that God, while indivisible, is internally complex.’ Ward gets it right here.

Important things here are the statement that “Ward mistakes what it means to explain something,” and that “he also seems not to understand what it means to say of something that it is simple.” And lastly there is Dawkins’s attempt at doing theology when he says that “Ward gets it right here.” I will return to this shortly. In any case, Dawkins continues by recounting his experiences at a conference at Cambridge:

At a recent Cambridge conference on science and religion, where I put forward the argument I am here calling the Ultimate 747 argument, I encountered what, to say the least, was a cordial failure to achieve a meeting of minds on the question of God’s simplicity. The experience was a revealing one, and I’d like to share it.

After some discussion of the background of the conference, Dawkins explains his experience with his argument against the existence of God:

For better or worse, I attended two days at the Cambridge conference, giving a talk of my own and taking part in the discussion of several other talks. I challenged the theologians to answer the point that a God capable of designing a universe, or anything else, would have to be complex and statistically improbable. The strongest response I heard was that I was brutally foisting a scientific epistemology upon an unwilling theology. Theologians had always defined God as simple. Who was I, a scientist, to dictate to theologians that their God had to be complex? Scientific arguments, such as those I was accustomed to deploying in my own field, were inappropriate since theologians had always maintained that God lay outside science. I did not gain the impression that the theologians who mounted this evasive defense were being willfully dishonest. I think they were sincere. Nevertheless, I was irresistibly reminded of Peter Medawar’s comment on Father Teilhard de Chardin’s The Phenomenon of Man, in the course of what is possibly the greatest negative book review of all time: ‘its author can be excused of dishonesty only on the grounds that before deceiving others he has taken great pains to deceive himself’. The theologians of my Cambridge encounter were defining themselves into an epistemological Safe Zone where rational argument could not reach them because they had declared by fiat that it could not. Who was I to say that rational argument was the only admissible kind of argument? There are other ways of knowing besides the scientific, and it is one of these other ways of knowing that must be deployed to know God.

There are basically three possibilities here. Either Dawkins did not understand the theologians, the theologians did not understand Dawkins, or the theologians did not understand their theology. The third possibility is very plausible given the criticism of St. Thomas by Keith Ward and Sir John Polkinghorne mentioned by Dawkins earlier. Most likely all three are the case.

Dawkins continues to what perhaps is the heart of the issue between himself and the theologians:

Time and again, my theologian friends returned to the point that there had to be a reason why there is something rather than nothing. There must have been a first cause of everything, and we might as well give it the name God. Yes, I said, but it must have been simple and therefore, whatever else we call it, God is not an appropriate name (unless we very explicitly divest it of all the baggage that the word ‘God’ carries in the minds of most religious believers). The first cause that we seek must have been the simple basis for a self-bootstrapping crane which eventually raised the world as we know it into its present complex existence. To suggest that the original prime mover was complicated enough to indulge in intelligent design, to say nothing of mindreading millions of humans simultaneously, is tantamount to dealing yourself a perfect hand at bridge. Look around at the world of life, at the Amazon rainforest with its rich interlacement of lianas, bromeliads, roots and flying buttresses; its army ants and its jaguars, its tapirs and peccaries, treefrogs and parrots. What you are looking at is the statistical equivalent of a perfect hand of cards (think of all the other ways you could permute the parts, none of which would work)— except that we know how it came about: by the gradualistic crane of natural selection. It is not just scientists who revolt at mute acceptance of such improbability arising spontaneously; common sense baulks too. To suggest that the first cause, the great unknown which is responsible for something existing rather than nothing, is a being capable of designing the universe and of talking to a million people simultaneously, is a total abdication of the responsibility to find an explanation. It is a dreadful exhibition of self-indulgent, thought-denying skyhookery.

I am not advocating some sort of narrowly scientistic way of thinking. But the very least that any honest quest for truth must have in setting out to explain such monstrosities of improbability as a rainforest, a coral reef, or a universe is a crane and not a skyhook. The crane doesn’t have to be natural selection. Admittedly, nobody has ever thought of a better one. But there could be others yet to be discovered. Maybe the ‘inflation’ that physicists postulate as occupying some fraction of the first yoctosecond of the universe’s existence will turn out, when it is better understood, to be a cosmological crane to stand alongside Darwin’s biological one. Or maybe the elusive crane that cosmologists seek will be a version of Darwin’s idea itself: either Smolin’s model or something similar. Or maybe it will be the multiverse plus anthropic principle espoused by Martin Rees and others. It may even be a superhuman designer— but, if so, it will most certainly not be a designer who just popped into existence, or who always existed. If (which I don’t believe for a moment) our universe was designed, and a fortiori if the designer reads our thoughts and hands out omniscient advice, forgiveness and redemption, the designer himself must be the end product of some kind of cumulative escalator or crane, perhaps a version of Darwinism in another universe.

We can see here what Dawkins means when he says that Ward mistakes what it means to explain something. “The very least that any honest quest for truth must have in setting out to explain such monstrosities of improbability as a rainforest, a coral reef, or a universe is a crane and not a skyhook.” Otherwise, according to Dawkins, you haven’t explained anything. And what does he mean by a crane rather than a skyhook? A skyhook, identified with what he considers a complex God, would be something that already has such complexity within itself. A crane is something simple, and simple in the sense intended by Dawkins. Explanation, therefore, according to Dawkins, requires an original simplicity, this being understood as he understands it.

In reality, attempting to explain things is to look for their causes. And correspondingly, there are different kinds of explanation and different kinds of causes. But Dawkins is identifying certain types of causality and explanation in particular, namely those that are found in Darwinian evolution. It is likely that he is doing this because he feels satisfied by such explanations, and therefore tends to think that other accounts are not real explanations, since they leave him dissatisfied. In reality, however, there are various types of explanation and thus various types of cause.

What did Dawkins mean when he said that Ward “seems not to understand what it means to say of something that it is simple”? And why does he say that “Ward gets it right here” when Ward opposes St. Thomas on the understanding of the simplicity of God?

St. Thomas asserts that God is simple in the sense that he is not composed of parts. Given his supposed activities, Dawkins considers this absurd, and thus he says that Ward gets it right when he admits that God is “internally complex.” In other words, despite believing that God does not exist, Dawkins is making the theological claim that God cannot be simple in the sense asserted by St. Thomas, but must be composed of parts.

Why does he say this? Why doesn’t he think that since he doesn’t believe in God, this is none of his concern and he should just leave it to the theologians as they apparently told him?

Dawkins is reasoning from the supposed activities of God to his nature. God is supposed to be “a being capable of designing the universe and of talking to a million people simultaneously.” Designing the universe seems to involve planning, which involves a plan, which has various parts. Talking to people seems to involve words and sentences, which are distinct from one another, and also thoughts, which seem to be distinct insofar as they are thoughts about diverse things. In other words, it is obvious that when we design and plan things, and when we speak with people, we are capable of doing so because we consist of parts. Consequently if God can do these things, he must have parts as well.

In fact, in terms of the argument for a first cause, Dawkins nearly admits that he cannot refute the argument:

Yes, I said, but it must have been simple and therefore, whatever else we call it, God is not an appropriate name (unless we very explicitly divest it of all the baggage that the word ‘God’ carries in the minds of most religious believers). The first cause that we seek must have been the simple basis for a self-bootstrapping crane which eventually raised the world as we know it into its present complex existence.

His problem is not the argument for a first cause, therefore, but the things that are typically said of that cause, and he objects to these things because they seem to him to imply that the first cause is not simple.

We already saw that Dawkins objects to the idea that God has no parts. But is this his real objection? Simply that he thinks that the first cause must be partless, and therefore that it cannot do things like designing, planning, and talking that seem to involve parts?

This is not his real objection, whether or not he understands this fact himself. For the correct response to this objection, from a theological point of view, is exactly that God is simple in the sense defined by St. Thomas. And he does not perform the activities mentioned by Dawkins in the way that he supposes. God does not pass from one thought to another. He does not think of one part of a plan, and then another. If he speaks, he does not go from word to word in his mind. To the extent that parts are implied by such things, they are to be denied of God, and the theologian only believes that they exist in God by analogy.

But Dawkins will still have a problem with this response, if it implies that God still performs those activities, even in an analogous way. If for example God ever directly produces a voice in my mind telling me to do something, Dawkins will have a problem with this, even if I say that God does not have parts. Only the voice has parts. Dawkins will still insist that this explanation is “not simple.”

And why not? Because it is not the kind of explanation that is pleasing to him, where complexity comes from simplicity, not just in the sense that a partless being causes beings with parts, but in the sense that mathematical complexity is caused by mathematical simplicity. This is ultimately what he means when he talks about a crane rather than a skyhook. If we give a mathematical explanation of the voice in my head, it will be a mathematically complex one, and if the only cause is God, it may not be clearly possible to reduce that mathematical complexity to something mathematically simple. Evolutionary explanations, on the other hand, allow something mathematically complex to be explained in terms of laws which are mathematically simple. And this is the only kind of explanation that Dawkins considers reasonable, satisfying, or true.

We can divide all of this discussion into various questions:

  1. Is there a first cause at all? We have established that there is, and Dawkins does not deny it.
  2. Does the first cause have parts? We have established that it does not, and in principle Dawkins does not assert that it does. To some extent he could be taken to be conceding that it does not, since his objection is that if God exists, he has many parts and is extremely complicated, and therefore cannot be the first cause.
  3. Does the first cause produce mathematically complex things from mathematically simple ones? It is certain that it does in general. Our discussions of mathematical laws in nature and of the order of the world are both relevant, as well as the issue of simplicity and probability. Dawkins agrees with this, and in fact his position is that this is the only way that mathematical complexity is ever produced.
  4. Does the first cause ever produce mathematical complexity without doing this through mathematically simple things? Nothing in our discussions establishes that such a thing is impossible, nor that it is actual. Dawkins denies that this is possible or at least that it is reasonable, but he does not seem to have a particular argument for this other than the fact that such a claim leaves him feeling dissatisfied, feeling that something has been left unexplained which should be explained. But as we have seen, this is not a question about the nature of explanation in general, but the kind of explanations which are pleasing to him.
  5. Is a first cause which does not directly produce such mathematical complexity worthy of being called God? This is mainly a question about the meaning of words, although there also could be questions about what that being would be like. Dawkins denies that this is a reasonable way to use the word “God”, because, according to him, God is always understood to intervene directly in the world, causing things which are meaningful on a human level and consequently which are already mathematically complex.
  6. Do God’s activities imply that he has parts? Dawkins assumes that they do, and apparently the theologians at the conference that he attended were unable to explain otherwise.

It is problematic to discuss the question of “whether God exists” with someone like Richard Dawkins because these separate questions end up being mixed together. Dawkins gives a negative response to question 5, but if this is in fact a reasonable way to use the name “God,” then Dawkins should not deny that God exists, even if the rest of his position is correct. Likewise, Dawkins assumes an affirmative answer to question 6, and therefore concludes that if the answer to question 2 is negative, God cannot be the first cause, and therefore that if he exists he must be caused. Discussing these questions with him separately would possibly be much more productive.

The First Cause and The World

Bertrand Russell, in a passage quoted earlier, affirms that if there is a first cause, it might as well be the world:

If everything must have a cause, then God must have a cause. If there can be anything without a cause, it may just as well be the world as God, so that there cannot be any validity in that argument.

As we saw at the time, Russell misunderstands the argument, since he supposes that it depends on saying that “everything has a cause.” But in any case, by the argument regarding the first cause and distinction, there is only one first cause, and that cause is not the world. It is not the world because the world has things in it which are distinct from one another, and the first cause cannot have anything within it distinct from anything else within it, since otherwise at least one of the two distinct things would have a cause. Instead, the first cause is absolutely simple. St. Thomas makes this argument, saying, “Every composite has a cause, for things in themselves different cannot unite unless something causes them to unite. But God is uncaused, as shown above, since He is the first efficient cause.”

There are two things that should be noted about this argument relative to Catholic theology. First, as was already stated, the first cause at which this argument arrives would be the person of the Father; otherwise it would be wrong to say that there is nothing in the first cause distinct from anything else within it.

Second, this argument does not prevent one from saying that the first cause is both a part of the world, and the cause of the whole world. My discussion of whole and part does not prevent any two distinct things from being taken as parts of a whole, as long as we can think of something that would include them both. And in the case under consideration, we can think of such a thing: “reality”, which which is intended to include both causes and effects. Thus the first cause is a part of reality. Nonetheless, it is also the cause of reality as a whole. This is not hindered by the fact that nothing can be the cause of itself, since a part is not the whole, and the whole is not the part. Rather, if we think of it in this particular way, the first cause causes the whole of reality by causing other things distinct from itself, and by causing them to be also in some way united with itself, in other words, by causing them to be part of the whole of reality. In a similar way the Council of Constantinople stated that “the Father is the source of the whole Trinity.”

It is not customary in Catholic theology to say that God is a part of anything else. But in order to avoid saying this, one would deal with the issue of “reality as a whole” by distinguishing between real and conceptual wholes, and saying that “reality as a whole” is a conceptual whole rather than a real whole.

I have not made such a distinction mainly because it is not clear to me what such a distinction would mean. I pointed out that distinction always involves something conceptual, but we can distinguish between real distinctions and conceptual distinctions insofar as it is one thing to say, “this thing is not that thing,” and another to say, “the concept of this is not the concept of that.” The idea of distinction leads to the ideas of parts and wholes, and the distinction between real distinctions and conceptual distinctions would allows us to distinguish between “real wholes” and “conceptual wholes” if we intended to say that a conceptual whole is something composed of parts which are conceptually distinct but not really distinct. But this does not apply to the case of the first cause and its effects, since these are really distinct from one another. Thus it is not clear to me what one would be intending to say if one asserted that “reality as a whole” is only a conceptual whole.

In any case, nothing opposed to Catholic doctrine follows of necessity from the argument. If God is a part of reality as a whole, it does not follow that reality is better than God. It does not follow that God created of necessity, nor that anything other than God is necessary or uncaused, and so on.

Extraordinary Claims and Extraordinary Evidence

Marcello Truzzi states in an article On the Extraordinary: An Attempt At Clarification“An extraordinary claim requires extraordinary proof.” This was later restated by Carl Sagan as, “Extraordinary claims require extraordinary evidence.” This is frequently used to argue against things such as “gods, ghosts, the paranormal, and UFOs.”

However, this kind of argument, at least as it is usually made, neglects to take into account the fact that claims themselves are already evidence.

Here is one example: while writing this article, I used an online random number generator to pick a random integer between one and a billion inclusive. The number was 422,819,208.

Suppose we evaluate my claim with the standard that extraordinary claims require extraordinary evidence, and neglect to consider the evidence contained within the claim itself. In this case, given that I did in fact pick a number in the manner stated, the probability that the number would be 422,819,208 is one in a billion. So readers should respond, “Either he didn’t pick the number in the manner stated, or the number was not 422,819,208. The probability that both of those were true is one in a billion. I simply don’t believe him.”

There is obviously a problem here, since in fact I did pick the number in the way stated, and that was actually the number. And the problem is precisely leaving out of consideration the evidence contained within the claim itself. Given that I make a claim that I picked a random number between one and a billion, the probability that I would claim 422,819,208 in particular is approximately one in a billion. So when you see me claim that I picked that number, you are seeing evidence (namely the fact that I am making the claim) which is very unlikely in itself. The fact that I made that claim is much more likely, however, if I actually picked that number, rather than some other number. Thus the very fact that I made the claim is strong evidence that I did pick the number 422,819,208 rather than some other number.

In this sense, extraordinary claims are already extraordinary evidence, and thus do not require some special justification.

However, we can consider another case, a hypothetical one. Suppose that in the above paragraphs, instead of the number 422,819,208, I had used the number 500,000,000, claiming that this was in fact the number that I got from the random number generator.

In that case you might have found the argument much less credible. Why?

Assuming that I did in fact pick the number randomly, the probability of picking 422,819,208 is one in a billion. And again, assuming that I did in fact pick the number randomly, the probability of picking 500,000,000 is one in a billion. So no difference here.

But both of those assume that I did pick the number randomly. And if I did not, the probabilities would not be the same. Instead, the fact that simpler things are more probable would come into play. At least with the language and notation that we are actually using, the number 500,000,000 is much simpler than the number 422,819,208. Consequently, assuming that I picked a number non-randomly and then told you about it,  is significantly more probable than one in a billion that I would pick the number 500,000,000, and thus less probable than one in a billion that I would pick 422,819,208 (this is why I said above that the probability of the claim was only approximately one in a billion; because in fact it is even less than that.)

For that reason, if I had actually claimed to have picked 500,000,000, you might well have concluded that the most reasonable explanation of the facts was that I did not actually use the random number generator, or that it had malfunctioned, rather than that the number was actually picked randomly.

This is relevant to the kinds of things where the postulate that “extraordinary claims require extraordinary evidence” is normally used. Consider the claim, “I was down in the graveyard at midnight last night and saw a ghost there.”

How often have you personally seen a ghost? Probably never, and even if you have, surely not many times. And if so, seeing a ghost is not exactly an everyday occurrence. Considered in itself, therefore, this is an improbable occurrence, and if we evaluated the claim without considering the evidence included within the claim itself, we would simply assume the account is mistaken.

However, part of the reason that we know that seeing ghosts is not a common event is that people do not often make such claims. Apparently 18% of Americans say that they have seen a ghost at one time or another. But this still means that 82% of Americans have never seen one, and even most of the 18% presumably do not mean to say that it has happened often. So this would still leave seeing ghosts as a pretty rare event. Consider how it would be if 99.5% of people said they had seen ghosts, but you personally had never seen one. Instead of thinking that seeing ghosts is rare, you would likely think that you were just unlucky (or lucky, as the case may be.)

Instead of this situation, however, seeing ghosts is rare, and claiming to see ghosts is also rare. This implies that the claim to have seen a ghost is already extraordinary evidence that a person in fact saw a ghost, just as my claiming to have picked 422,819,208 was extraordinary evidence that I actually picked that number.

Nonetheless, there is a difference between the case of the ghost and the case of the number between one and a billion. We already know that there are exactly one billion numbers between one and a billion inclusive. So given that I pick a number within this range, the probability of each number must be on average one in a billion. If it is more probable that I would pick certain numbers, such as 500,000,000, it must be less probable that I would pick others, such as 422,819,208. We don’t have an equivalent situation with the case of the ghost, because we don’t know in advance how often people actually see ghosts. Even if we can find an exact measure of how often people claim to see ghosts, that will not tell us how often people lie or are mistaken about it. Thus although we can say that claiming to see a ghost is good evidence of someone actually having seen a ghost, we don’t know in advance whether or not the evidence is good enough. It is “extraordinary evidence,” but is it extraordinary enough? Or in other words, is claiming to have seen a ghost more like claiming to have picked 422,819,208, or is it more like claiming to have picked 500,000,000?

That remains undetermined, at least by the considerations which we have given here. But unless you have good reasons to suspect that seeing ghosts is significantly more rare than claiming to see a ghost, it is misguided to dismiss such claims as requiring some special evidence apart from the claim itself.

Simplicity and Probability

Given some reasonable postulates regarding the formulation of explanatory hypotheses, it can be mathematically demonstrated that a probability distribution over all possible explanations will be biased toward simpler explanations — in an overall way the simpler explanations will be more probable than the more complex ones, although there may be individual exceptions.

We make the following postulates:

1) The explanatory hypotheses are described by a language that has a finite number of different words, and each hypothesis is expressed by a finite number of these words. That this allows for natural languages such as English, but also for computer programming languages and so on. The proof in this post will be valid for all cases. This is a reasonable assumption since human beings do not use any infinite languages, nor do they use an infinite number of words to make a point.

2) A complexity measure is assigned to the hypotheses in such a way that there are or may be some hypotheses which are as simple as possible, and these are assigned the complexity measure of 1, while hypotheses considered to be more complex are assigned higher integer values such as 2, 3, 4, and so on. Note that apart from this, we can define the complexity measure in any way we like, for example as the number of words used by the hypothesis, or in another way, as the shortest program which can output the hypothesis in a given programming language. Many other definitions would be possible. The proof is valid for all definitions that follow the conditions laid out, even the ones which would be intuitively somewhat distant from the idea of something simple. This again is a reasonable assumption given what we mean by simplicity — we do not think it is possible to make a thing infinitely simpler, but there is always something simplest.

3) The complexity measure is also defined in such a way that there are a finite number of hypotheses given the measure of 1, a finite number given the measure of 2, a finite number given the measure of 3, and so on. Note that this condition is not difficult to satisfy; it would be satisfied by either of the definitions mentioned in condition 2, and in fact by any reasonable definition of simplicity and complexity. If there are an infinite number of hypotheses that are supposedly absolutely simple (with the measure of 1), and we describe these hypotheses in English, an infinite number of them will not be able to be described without using at least 10,000 words, or without using at least 100,000 words, and so on. This seems very remote from the idea of a simple explanation.

With these three conditions the proof follows of necessity. To explain any data, in general there will be infinitely many mutually exclusive hypotheses which could fit the data. Suppose we assign prior probabilities for all of these hypotheses. Given condition 3, it will be possible to find the average probability for hypotheses of complexity 1 (call it x1), the average probability for hypotheses of complexity 2 (call it x2), the average probability for hypotheses of complexity 3 (call it x3), and so on. Now consider the infinite sum “x1 + x2 + x3…” Since all of these values are positive (and non-zero, since we consider each hypothesis to be at least possible), either the sum converges to a positive value, or it diverges to positive infinity. In fact, it will converge to a value less than 1, since if we had multiplied each term of the series by the number of hypotheses with the corresponding complexity, it would have converged to exactly 1, since the sum of all the probabilities of all our mutually exclusive hypotheses should be exactly 1.

Now, x1 is a finite real number. So in order for this series to converge, there must be only a finite number of terms in the series equal to or greater than x1, and therefore some last term which is equal to or greater than x1. There will therefore be some complexity value, y1, such that all hypotheses with a complexity value greater than y1 have an average probability of less than x1 (the average being taken over the hypotheses with the same complexity value, as above). Likewise for x2: there will be some complexity value y2 such that all hypotheses with a complexity value greater than y2 have an average probability of less than x2. Leaving the derivation for the reader, it would also follow that there is some complexity value z1 such that all hypotheses with a complexity value greater than z1 have a lower probability than any hypothesis with a complexity value of 1, some other complexity value z2 such that all hypotheses with a complexity value greater than z2 have a lower probability than any hypothesis of complexity values 1 or 2, and so on.

From this it is clear that as the complexity tends to infinity, the probability of the hypothesis will tend toward zero in the limit. This will happen in such a way that for any particular probability (e.g. one in a billion), there will be some degree of complexity such that every hypothesis at least that complex will be less probable than the chosen probability (e.g. less than one in a billion.)