"The Order of Nature" (Champaign cover)

“The Order of Nature,” published June 1878, is the fifth installment of C.S. Peirce’s essay series “Illustrations of the Logic of Science.” It further describes induction, using cosmological speculation about what kind of universe is required for induction to work. It also describes Peirce’s take on a hot issue of his day: the relationship of science to religion.

Note: I have an annotated table of contents for my entire “cover band” reinterpretations of Peirce essays.

TL;DR

• The usefulness of logic depends on the nature of the universe.

• That nature is a mixture of chance and regularity, well suited to the mathematics of induction described in the last essay.

• Induction is not a process of collecting a sample, then wondering what properties the items share. Instead, it should require first selecting a property of interest, collecting the sample, and then asking what proportion of the sample shares that property.

• This is not the popular view of induction, which – following Mill – teaches that it requires nature to be uniform. This view is flawed because nature isn’t uniform.

• Our notions of time, space, and force are justified by induction. Indeed, they provide evidence that we have evolved so as to be good at creating hypotheses that match reality.

• But it is entirely possible we will never find regularities that apply to the whole of the universe, much less that those regularities will be benevolent, just, or wise – the kind that would evidence a Creator. It seems the details of various religions – especially as regarding the existence and nature of a Creator – are mere mysticism. But that does not destroy the essence of religion, which is a yearning toward perfection. It would be foolish to become so atheistical as not to find joy in religious celebration or feel ill at ease in churches.

I. The plan

Mr. Darwin’s theory is much discussed these days. Its account of the origin of species from chance workings, rather than God’s plan, provokes incredulity in many. That raises the question: what sort of universe do we inhabit? We students of logical inference care because the validity of induction (as outlined in the last essay) depends on us living in a universe that supports it. In this essay, I’ll discuss the nature of the universe (as it pertains to logic) and then discuss when and how induction is valid. I’ll finish by giving a different view of the implications of science for religion.

II. The chance-world

Archbishop Tillotson gives us a quote representative of those who reject Darwin:

“How often might a man, after he had jumbled a set of letters in a bag, fling them out upon the ground before they would fall into an exact poem, yea, or so much as make a good discourse in prose! And may not a little book be as easily made by chance as this great volume of the world?”

Is this an adequate description of the workings of chance? An adequate analogy for what a world of pure chance would look like? I think not.

In a world entirely governed by chance, there would be no regularities of the sort logic describes, no true statements of the form “Every A is B-like” or “No A is B-like” (or, equivalently, “Every A is non-B-like”).

In such statements, A represents an object of the world, and B represents some property that might be attached to such an object.

Examples of statements we could not logically make in a chance-world would be:

• Every ray of light is a non-curved line.
• Every dropped body accelerates toward the earth’s center.
• Every swan is white.
• And so on.

Consider a universe containing five properties; call them A, B, C, D, and E. We’ll further notate the lack of a property with a lower-case letter, viz., a, b, c, d, and e. (The lack of a property is itself a property.)

Any object might have any combination of properties, ranging from ABCDE through ABcDe to abcde, 32 possibilities in all.

Is it possible for two different objects to have the same properties? The philosophers have given us a doctrine, “the identity of indiscernables,” which declares this impossible. That seems puzzling until you realize that an object’s position in space is one of its properties, so two identical objects would have to be in the same location, fully overlapping with nothing to distinguish them. So how are they “separate objects”?

That given, this universe can have a total of 32 different objects in it, one for each combination of properties. For any object selected at random, we can predict nothing useful about which five properties we will find when we examine it. It might be hard, sweet, fragrant, and green, but not bright. Or it might be hard, sweet, and fragrant, but neither green nor bright.

While this world is entirely governed by chance, it is exceedingly ordered. We can list and even sort the objects:

• Let’s put the object with properties ABCDE first.
• ABCDe will be second.
• The third, ABCdE.
• And so on.

While it is, in a sense, orderly, let us say that a chance-world is in no way lawlike. In contrast, the Archbishop’s thought experiment contains elements that are predictable or even lawlike:

• When the letters are cast on the ground, they each obey Mr. Newton’s law and accelerate toward the center of the earth.
• There are, let us say, 26 possible letters and five possible punctuation marks. For there to be a possibility of them forming a poem, there must be duplicates – objects that share all important properties except their location. There is a regularity if we can recognize diverse tiles as representing the same letter.

  ⊕

We can say such a Scrabble-world is partly lawlike and partly a chance-world.

Consider, then, the Archbishop’s second argument from incredulity:

“How long might 20,000 blind men, which should be sent out from the several remote parts of England, wander up and down before they would all meet upon Salisbury Plains, and fall into rank and file in the exact order of an army? And yet this is much more easy to be imagined than how the innumerable blind parts of matter should rendezvous themselves into a world.”

This is true, but in the actual world the blind men are, as far as we can see, not drawn up in any particular order at all. While there is a certain amount of order in the world, it is not as orderly as it could be – for instance, it is not as orderly as a world of pure chance must be (as shown above).

Let us descend from such abstractions to the perceptions and powers of living beings. Since it’s hard to imagine a world with no uniformities, let’s suppose it has no uniformities that are interesting or important to us. How would we experience such a world?

• We would have nothing to puzzle about, for there are no uniformities to be explained. Either nothing surprises us or everything does; to our way of thinking, the two have the same meaning.
• No action of ours, nor any work of Nature, would have important consequences in such a world. We would have no responsibilities, nothing to do but enjoy or suffer whatever happened to come along.
• With nothing to stimulate the mind or our will, we would neither act nor think.
• We would have no memory, because memory requires a predictable and repeatable organization of events.

We would have the consciousness of a rock, or of some animalcule at the very vanishing point of intelligence. To a polyp such as we would be, the world is very nearly a medley of chance.

But we are not polyps, because we are interested in the organization of Nature. Such interest can, in fact, serve as a measure of our intelligence.

III. The nature of induction

Induction, we saw last time, is a process of sampling. A number of specimens of a class are selected at random and their properties are examined. A great number of properties will be shared by those specimens.

If we believe that a second sample will also largely share a particular property, we might develop an inference that all such samples will share it in similar proportions. This, however, is not induction because most such properties will be accidental or insignificant. Consider the ages at death of the first five poets in Wheeler’s Biographical Dictionary. They are:

Aagard, 48.
Abeille, 70.
Abulola, 84.
Abunowas, 48.
Accords, 45.

A random enough sample. They have the following properties in common:

1. Add the two digits of the age and divide that sum by three. All will leave a remainder of one.

2. Raise the first digit to the power of the second and divide the result by three. All will leave a remainder of one.

3. Take the sum of the prime factors of the age, including 1. For example, 70 gives us 7 + 5 + 2 + 1 = 15. In all cases, the result will be divisible by three.

The number of such accidental agreements is boringly large.

To perform induction, we must select a property of interest and then take samples. When two such random samples are taken, it is extremely likely that the percentage of specimens with the property will be approximately the same. Thus, a definition of induction:

Induction is the inference that a previously chosen property that occurs with frequency f in a random sample of a class will occur at nearly the same frequency in the whole of the class.

IV. Another theory of induction

In J.S. Mill’s Logic, bk. 3, ch. 3, sec. 1, he claims that induction works because of the uniformity of nature. That is, whatever happens once will happen again if the circumstances are sufficiently similar. You examine a set of objects of a particular class (starfish, for example) in a particular circumstance. You observe they all have some property in common. You expect that if the circumstances recur, starfish will be observed to have the same property.

This doctrine has some imperfections.

First, reach into a bag and pull a handful of beans. You observe that 3/4ths of them are black. From this, you infer that about 3/4ths of the beans in the bag itself are black.

The inference procedure would be the same if half the beans in your hand had been black, or 9/10ths.

The point of this thought experiment is that the result (an observation of bean color) is not uniform. Sometimes the result is black, sometimes white. This is no problem for the induction process described in the previous essay, but it is for Mill’s.

Mill’s problem can be finessed by considering the whole sample to be a single object that is 3/4th black. But this generalization from a single instance gives the induction formula of last episode nothing to work with. Are we ready to abandon it for something with no calculational power, only aesthetical?

Second, consider the case of inferring that all swans are white, justified because Europeans have known only white swans for thousands of years. To Mr. Mills, this was “not a good induction” because the property of color was not known to be one that was uniform among all species of a particular genus, such as Cygnus.

But consider the advance of knowledge. When it was assumed that color was a property of genus, the inference was valid, subject to later contradiction.

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Then further investigation into the characteristics of different animal genera allowed us to infer – via induction – that color is not constant among species of a genus. This discovery would go a good way to weakening the white-swan conclusion and prepare us for the discovery of a black swan. We’ve exercised a form of forward-looking deduction: applying a new fact to reduce the certainty of an older conclusion. It is perfectly valid to apply such a deduction to an inductive conclusion.

Third, putting the weight of validity on Nature, rather than on our procedures of inference, makes the validity of induction depend on what we’re inducing about. For instance, the first edition of Mr. Mill’s book used as examples a number of scientific claims that have since been proven invalid. They had to be replaced in later editions with other examples. It’s a shame Mr. Mill did not infer (inductively) from the failed examples that there was something wrong with his theory of uniformity.

V. Why are we so good at induction?

Suppose we have concluded, via induction, that half of all births are male. We can always find a subset of births that are associated with some property we can name: the configuration of the planets, political events in Australia, and so on. (The property may be obscure; for example, it may be some property that none of the births share.)

This is a futile craving for certainty and uniformity. We should assume there are always exceptions. Thus, inductions are matters of probability, not certainty, and our job is to measure the probability.

However, there are certain inductions we all make. If they aren’t absolutely certain (probability 1.0), they’re so close that we are justified in treating them as such. The most remarkable of these are space and time.

With regard to space, Bishop Berkeley demonstrated that we do not see space, but rather infer it. He noted that the retina of the eye is a surface upon which is projected a two-dimensional image of the world. Via comparisons of the images of both eyes, we conclude that there is space.

Now that we know more physiology, we can go beyond Berkeley. The retina is not an undifferentiated surface upon which the image of the world is painted. It is, rather, a collection of nerve-needles that have their light-sensitive tips pointed toward the lens of the eye. Moreover, the receptive area at the tip of a nerve-needle is small in comparison to the spacing between needles. The excitation of a diverse number of nerve-needles cannot directly produce a perception of a flat image; that, too, must be manufactured by some action of the mind. There are relationships between the activations of the nerve-needles, and those relationships are the premises upon which the hypothesis of space rests, and from which it is inferred.

Such a mediate cognition is an inference subject to the normative (guiding) logic we are constructing in this series.

But wait. As we can judge by their actions, even newly-hatched chickens solve the seemingly-difficult task of inferring space from the impulses of nerve-needles. Chickens are not notably smart, so it must be true that both they and people have an inborn tendency to quickly infer the conception of space.

The same is true of time. Time is not directly perceived – at no point is an interval of time directly present to the senses, and nothing can be perceived that isn’t so present. Had we no idea of time, we could never perceive the flow of perceptions that bombard us every moment. So it’s hard to deny that we (and chickens) must have some inborn aptitude for conceiving of time.

The concept of force is similar. In at least its rudiments, it too is arrived at nearly immediately. Since force influences the actions of even the most primitive of creatures, the ability to infer it from perceptions must be innate.

But this innate ability to infer is not the same in all species. We must speak of degrees of ability, which I’ll characterize as the tendency of an idea to present itself to the mind – to become active and influential on our behavior. Some ideas, like that of space, present themselves irresistibly at the very dawn of every intelligence, while others are unavailable to lesser intelligences and require effort and time for even humans to infer.

Take Newton’s law of gravitation, which states its force varies inversely as the square of the distance. We think of this as a simple law, but that is merely to say that our minds are particularly adapted to a universe in which it is a law. If the idea of multiplication did not “fit” our minds so readily, would we have ever learned the inverse square law?

How can we explain this adaptation of the mind to the universe? The concepts of time, space, and force are useful to even the most limited intelligence. Without having a knack for geometrical, kinetic, and mechanical conceptions, no animal could find anything to eat or do anything else necessary to survival. That suggests Mr. Darwin’s natural selection has been at work.

Contrarily, we could say that the lowest animals have no conceptions of any kind, only instincts. (Which we can consider different, incorrect conceptions that happen to produce the same actions as do the conceptions of time, space, and force.) But instincts break down in novel situations, situations which even the most primitive animals must encounter from time to time. So there would be a constant selection pressure toward conceptions that have a better “fit” with reality.

VI. Science and religion

Science is concerned with reality, that which is independent of what any individual thinks about it. Science succeeds when it discovers regularities in reality that allow us to make successful predictions.

It is true that science is increasingly reducing the number of things of which we can say, “It is that way because a Creator made it so.” And the nature of the regularities we’ve found do not show them as characterized by the benevolence, justice, economy, or beauty we would expect from a Creator’s work.

However, religion can exist without a clear conception of a Creator:

• The religion with the most followers – and by no means the least intelligent ones – holds that the Divinity is distant from the world, wrapped in the most profound and eternal sleep. Hinduism? Peirce doesn’t say. As the difference between inertia and non-existence has no practical consequences, we can use either description of the Divinity.

• M. Vacherot is an earnest man, a devotee of religion with an earnest desire for Belief, but he holds that any description of the Ideal in terms of reality involves insoluble contradictions, leading him to the religious belief that perfection can have no characteristic of existence.

Science is compatible with such religions. It is an enemy to religion only insofar as it is an enemy to mysticism, the belief in astrology, numerology, the power of dreams, perpetual motion machines, and other relationships that are claimed to be separate from any purely mechanical principles. Such are not parts of reality (that which is independent of what any particular people believe about it). It is unfortunate that priests of many forms of religion say their creed cannot exist without the acceptance of certain mystical formulas. They might better teach the principles of religions in general.

Such teachings may indeed render those who cannot believe in mysticism irreligious. But it needn’t. Why should any man need feel excluded from participation in the aspiration toward the perfect that is the essence of religion, from the common feelings religion inspires, or from the public expression of them?

Why should a non-mystic, a scientist, exclude himself from the common joy of the ceremonies of Easter and Christmas just because priests make claims – scientific, logical, or metaphysical – that are untenable? Doing so would be to believe those temporal errors of more consequence than the eternal truth – which few would admit. It is rare to find people who do not believe what are really the fundamental principles of Christianity, and all but these few ought to feel at home in the churches.

Liner notes for the 2026 reissue

It seems to me that Peirce couldn’t resist including his developing evolutionary cosmology and theories of religion into a piece that’s really about something else. I would have advised against it.

Religion

Perhaps Peirce thought he was reassuring readers of a popular magazine who were likely torn between their childhood religion and some of the recent scientific innovations that were reducing the role for God. (Darwin, for example, leaves room for God as the creator of the first species, but makes Him superfluous for all the rest.)

However, bits that I left out of my cover version above likely would not have accomplished his purpose:

• Some people like and admire their religious leaders. Referring to such leaders with phrases like “[the] unphilosophical narrowness of those who guard the mysteries of worship” strikes perhaps a poor tone, as does describing people like himself as “minds emancipated from the tyranny of tradition.” Nor would such a reader likely be comforted by “It would be extravagant to say that science can at present disprove religion; but it does seem to me that the spirit of science is hostile to any religion except such a one as that of M. Vacherot.“ A kind of thoroughgoing Deism combined with something that reminds me of Apophatic theology.

• In Peirce’s time, as in ours, an argument for religion is that, without it, people would become immoral monsters. Peirce’s answer does not reassure: “This, too, must be admitted; such a revolution of thought could no more be accomplished without waste and desolation than a plantation of trees could be transferred to new ground, however wholesome in itself, without all of them languishing for a time, and many of them dying.” Or that “one thing is certain: that the state of the facts, whatever it may be, will surely get found out, and no human prudence can long arrest the triumphal car of truth—no, not if the discovery were such as to drive every individual of our race to suicide!“ Sounds like the CEO of a frontier AI firm.

Cosmology

Peirce’s cosmology, though only partly developed at this point, is worth sketching.

1. In the beginning (or, for an infinite amount of past time if there was no beginning), the universe was entirely unpredictable. What I take that to mean is: remember the universe of five properties and 32 entities? Pick a sample of entities that all have property A. Knowing that, what else can you say about that subset of entities? The answer is: nothing.

2. However, chance is a real thing in the universe – more real, at this early point, than time and space are. Which means that entities will occasionally temporarily adopt new properties in a random way. One of those properties might happen to be the property of tending to retain or repeat properties. Or put differently: the property of forming habits. This adds a tiny bit of predictability to the universe, one which can only increase over time. I think Peirce is drawing from the mathematics of probability, such as that random variation tends to produce a normal distribution.

3. So habits will gradually become more and more reliable, making deviations both rarer and smaller. They will eventually “solidify” enough to be called natural laws.

4. Over time, law will become increasingly dominant, and the role of chance will diminish. At the limit, the universe will become entirely law-bound: “[the] complete triumph of law and absence of all spontaneity.”

As theories of evolution go, this is fairly close to Darwinism, with the exception that it leans toward Lamarckism (the inheritance of acquired characteristics). Note that Darwin allowed for Lamarckism as a factor in evolution in The Origin of the Species and even proposed a mechanism in 1868. It was still credible at the time Peirce wrote, though Galton’s 1869-71 experiments had put it in doubt. But it is missing one thing: a source of selection pressure.

Later in his life, Peirce identifies that pressure as agapism from the Greek “agape.” I’m not sure if he’s using the word in the classical Greek sense, the Christian sense, or some mixture. (I’d guess the latter.)

• The classical Greek sense is of affection or fondness, as in “to greet with affection.”

• The Christian sense is the love of God for humans, humanity’s reciprocal love for God, and people’s consequent love for each other. I myself prefer the translation of “agape” into “charity” or “sympathy” rather than “love.” The latter has too many unhelpful connotations.

It is agape – for Peirce, another reality out there in the world – that provides the universe’s selection pressure in the way that survival does for Darwin. Peirce:

“The agapastic development of thought is the adoption of certain mental tendencies, not altogether heedlessly […] nor quite blindly by the mere force of circumstances or of logic […] but by an immediate attraction for the idea itself, whose nature is divined before the mind possesses it, by the power of sympathy, that is, by virtue of the continuity of mind.”

There’s a lot hidden in “development of thought” (Peirce’s “objective idealism") and “continuity of mind” (something that feels panpsychist), but explaining that will have to wait until I get to the relevant essays.

Finally, to Peirce, God is something like agape itself and something like the limit point of cosmic evolution. Think of Roko’s Basilisk without assuming God is an enormous asshole, or de Chardin and Tipler’s later theory of God as an “omega point.” Given how people deploy such alternatives, maybe Peirce’s notion of love, however defined, as foundational isn’t so bad.