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Tag Archives: Induction

what induction is and how it is justified

From generalizations to universals

John P. McCaskey’s Key to Induction shows what scientific induction is all about:

“[Scientists] want to know not only what is generally true but what is universally so, what is true without any possible exception. Below are three cases in which scientists were able to begin with general statements and progress to exceptionless universal ones. In each of the cases, scientists’ definitions evolved from being merely descriptive to identifying causes. That transition was crucial.”

He goes on to detail what happened with the science of cholera, electrical resistance, and tides, in which generalizations led to inductions. More famous examples would be the passage from the generalizations of Ptolemy and Kepler to the inductions of Galileo and Newton, or from the ad hoc transformations of Lorentz to Einstein’s principles.

As McCaskey puts it in Induction Without the Uniformity Principle, “The whole project of mature abstract thought is to identify similarities and differences, uniformities and changes, and to classify accordingly. And that—to Aristotle and followers such as Bacon and Whewell—is what induction is.”

Science progresses from data collection, to generalizations, to universals, to deductive hypotheses, and then new data collection to repeat the cycle. Sometimes “induction” is considered generalization (or testing) and universals are guessed by “abduction” (cf., C.S. Peirce) but the process of developing universals is the key element of induction.

In the series of posts on space and time, I have tried to show how approximate generalizations in transportation are exact inductions in physics.

The real scientific method

The real scientific method is the inductive method invented by Socrates and elaborated by Aristotle, Bacon, and Whewell. It is different from the hypothetico-deductive method invented by JS Mill in the 19th century which is passed off as the method of modern science.

Consider Francis Bacon. He called immature concepts “notions”. Induction starts with notions from common experience and iteratively improves them using sense experience until the form or essence is identified. This form is the cause in the full sense of the word; the form is what something truly is — and so should be defined as such. Thus the induction is true by definition. Sound circular or trivial? It’s not because getting the concepts right is what inductive science is all about.

William Whewell described two complementary processes, the explication of conceptions and the colligation of facts: To explicate a conception is to clarify it by identifying what it contains, by unfolding it, for example by surveying and examining examples. The end result is a careful definition of the conception. Colligation is the complementary process of binding facts together by means of a precise conception. The result is an induction, which is the narrowing of a generalization until it is exact and universal.

Yes, induction leads to hypotheses and testing but this is for the purpose of finding the consilience of inductions, the confirmation of inductions in different and multiple ways. The key step takes place before hypotheses and testing: the discovery of a conception of the facts that binds them together.

This understanding of induction was lost in late antiquity until Francis Bacon restored it and laid a foundation for science that lasted two centuries. Then in the 19th century Richard Whately and JS Mill replaced it with a different method, one that came to be called the hypothetico-deductive method, which depends on uniformity and naturalism, and is conceptually confused and logically deficient.

John P. McCaskey and others have explained the history of Socratic induction in science. As examples he gives cholera, electrical resistance, and tides (see here).

 

Reason and Risk: An Account of Induction

We begin with a non-empty, finite set of propositions that have already been accepted and compiled and now reside in a database of accepted propositions (DAP). Assume there is some known risk (possible zero) associated with accepting these propositions. Let the level of risk of the nth proposition Pn be a non-negative number, R(Pn), which is recorded in the DAP.

The question is, Given this DAP, what other propositions might be accepted with no increase in risk? Certainly there are enumerative propositions about the DAP, such as All x’s are y’s in the DAP, or If x is a y, then x is a z in the DAP. Such enumeration is pre-inductive but sets the stage for the inductive step.

Some pre-inductive propositions may contain terms that function as fields in a database. Whether the DAP is actually structured this way is not of consequence here. Some of these fields may have references beyond the DAP. For example, if the DAP contains propositions about employees, then it is possible that there are employees that exist but are not included in the DAP.

The point is that enumerative propositions that are true about the DAP might be true if the DAP were enlarged to contain more propositions. It is acknowledged that there would be risk associated with accepting a proposition that refers beyond the DAP, but that risk can be accounted for and ameliorated to some degree.

The approach is ameliorating risk is the natural kind. A natural kind is a set of things which are homomorphic with a proper subset of itself. An example would be a natural kind such as copper in which the properties of one sample of copper are homomorphic with those of another sample.

There is a risk as to whether or not something in the DAP is actually a member of a natural kind or which kind it is a member of. There is also a risk whether or not a particular property of the natural kind is one of the properties that is natural.

These risks are ameliorated by two strategies: (1) increasing the measure of risk associated with a proposition that refers beyond the DAP so as to capture this risk lest anyone be misled, and (2) searching for limits to the natural kind, both as to what properties are held throughout the kind and as to what the limits of the kind itself are.

For example, we might find in the DAP that All employees are female in the DAP. We may consider females a natural kind so that other females have like properties and infer that All employees are female, whether they are mentioned in the DAP or not, but we would have to increase the risk of this statement and search for limits to the set of female employees.

Until the limits to a proposition are found, its risk is higher than a proposition only about the DAP. If, for example, an employee outside the DAP is found to be male, then a limit to the inferred proposition has been found. In that case, the inferred proposition should be restricted to remain within the limit, and the risk lowered.

So if a set of things is proposed as a natural kind, then there is some risk whether or not it is in actuality the proposed natural kind. There is also the risk whether or not a property of a sample of the kind is a property of the rest of the kind. Everything in the kind must have some properties in common but there will always be some properties that are unique to each sample, otherwise the sample could not even be distinguished.

Looking at this situation globally we may say that every entity is a member of some natural kind. After all, there can be only one truly unique entity in the universe, that is, the universal entity. All other entities have properties in common with other entities, and to that extent they are members of the same natural kind. The question then becomes, Which other properties do they have in common?

January 2015

Dialogue on induction

Greek Coffee

Philario was sitting in the coffee shop, typing into his computer when he saw his friend Hector and greeted him.

Philario:  Hi, Hector.  What’s up?

Hector:  Well said, Philario.  What is up.  Who is down.

Philario:  Are you trying to Costello me?

Hector:  I wasn’t Abbott to do that.

Philario:  Very funny.  I’m searching on induction.  Can you tell me what it is?

Hector:  It depends on what kind of induction you want.

Philario:  I want the kind of induction used in natural science.

Hector:  OK, say we’ve got this large urn. You put your arm in and as far as you can tell it’s full of pieces of pottery.  Then you pull out one piece, and it’s painted blue.  What do you conclude about how the other pieces are painted?

Philario:  I don’t know; they could be painted anything.  Perhaps they’re from a beautiful urn that broke in pieces.

Hector:  Now think like a natural scientist.  What do natural scientists say about nature?

Philario:  They say nature is uniform.

Hector:  So if nature is uniform, how are all the balls painted?

Philario:  They must be painted the same way.

Hector:  That’s right!  So the natural scientist says they’re all painted blue.

Philario:  But they could easily be wrong!

Hector:  Did you ever notice how often natural scientists change their opinions?  They don’t seem to worry about being wrong.

Philario:  Well, I would worry about being wrong.

Hector:  Then you’re not a natural scientist!  Now suppose you pull out another piece, and it’s also painted blue.  What do you conclude?

Philario:  There’s beginning to be a pattern.  So it’s possible they could all be painted blue.

Hector:  You need more confidence if you want to be a natural scientist.

Philario:  I didn’t say I wanted to be a natural scientist.  I just want to know how they think.

Hector:  So try thinking like one.  What do you say?

Philario:  I suppose I should say they’re all painted blue.

Hector:  Now do you have any evidence to back that up?

Philario:  I don’t have much evidence; only two pieces.

Hector:  But is there any contrary evidence?

Philario:  No, not yet.

Hector:  There’s no contrary evidence so no-one can say you’re wrong yet.

Philario:  That’s not much consolation.

Hector:  You need more confidence, my man!  You can prove your case by appealing to all the available evidence.

Philario:  But someone else might take out other pieces and find they are painted differently.

Hector:  Has that happened yet?

Philario:  No.

Hector:  So you’ve made your case for now.  No-one can prove you wrong.

Philario:  Now suppose you put your hand in and pull out another piece, and it’s painted red.  What do you say?

Hector:  I would say I was wrong about all of them being blue because some of them are red.

Philario:  That’s weak, much too weak.

Hector:  I could say based on the evidence two-thirds are probably blue and one-third are probably red.

Philario:  That’s what statisticians say!  You’re trying to think like a natural scientist.

Hector:  So what should I say?

Philario:  You should say there are two kinds of pieces in the urn.  One kind are all painted blue and the other kind are all painted red.  You might say that the blue kind are from a piece of blue pottery and the red kind are from a piece of red pottery.

Hector:  That sounds like a hypothesis.

Philario:  Yes, it is a hypothesis!

Hector:  So natural scientists make bold statements based on flimsy evidence and call them hypotheses.

Philario:  You might put it that way.  But remember they are careful not to contradict evidence, unless they want to say the evidence is erroneous.

Hector:  Why would they say evidence is erroneous?

Philario:  Because it gets in the way of a good hypothesis!

Hector:  So it’s all about making up hypotheses that sound good.

Philario:  You’re catching on!

Hector:  I think I’m too cautious to be much good at that.

Philario:  Have you considered becoming a statistician?

Hector:  No, do they like to be cautious?

Philario:  Boy, do they like to be cautious!  That’s probably all they do.

Hector:  They must eat sometimes.

Philario:  Probably.  But you can’t be 100% certain.

Hector:  I think I can be 100% certain about some things.

Philario:  Like what?

Hector:  I can be 100% certain that the sun will rise tomorrow.

Philario:  OK, let’s consider that.  What do you base that assertion on?

Hector:  I base it on the fact that it’s risen every time in the past.

Philario:  I didn’t know you were as old as time!

Hector:  Well, I haven’t personally witnessed the sun rising every day, but someone has.

Philario:  Who has?

Hector:  Other people.  There are records that go back to Babylon.

Philario:  What about before Babylon?

Hector:  Well, I suppose it must have risen before that, too.  We’ve got thousands of years’ worth of evidence that the sun rises every day.

Philario:  So there’s a high probably the sun will rise tomorrow.

Hector:  That’s what I said!

Philario:  No, you said you were 100% certain the sun will rise tomorrow.

Hector:  That’s virtually the same thing.  You’re not going to split hairs, are you?

Philario:  Of course I am!  We’re thinking like statisticians now.

Hector:  Oh no.  You mean statisticians are super cautious?

Philario:  Professionally, yes.  They’re paid to be hedge their bets.

Hector:  I don’t think I’m cut out to be a statistician either.

Philario:  You could always be a philosopher.

Hector:  Why is that?

Philario:  They can take any side of an argument!

Hector:  I think you’re better at that than I am.

Philario:  Study philosophy and you’ll get better at it.

Hector:  I’d rather have a latte.

 

All horses are the same color?

Recall that mathematical induction has two requirements: a base case and an inductive step. Show that a statement is true for x=1 and show that if it is true for x=n, then it is true for x=n+1. The induction follows as a falling series of dominoes.

Evolutionists try to do something similar. They show that evolution is true in some cases (microevolution) and they show that if it is true up to a certain level of adaptation or complexity, then they can show a plausible scenario in which it is true of the next higher level.

But consider a ‘proof’ that all horses are the same color as proposed by Joel E. Cohen: The base case is n=1 which is trivially true: a single horse has the same color as itself. Then if we assume that all groups of n horses have the same color, we can show that all groups of n+1 horses have the same color. First remove the last horse so only n horses are left; they must have the same color. Then remove the first horse so only n horses are left; they must have the same color too. Thus the (n+1)th horse has the same color as the others.

The flaw in this ‘proof’ is that the base case does not match the inductive step. If the base case were n=2, it would be valid but n=1 does not have a valid comparative.

The problem with evolutionary induction is similar: the base case does not match the inductive step. Variation within a kind is not variation between kinds. If it could be shown that variation between kinds exists, then they might have a case but they have only shown that variation within a kind exists. By focusing on ‘species’ they have used variation within and between ‘species’ to promote their case but that begs the question of whether species are proper kinds or not.

December 2014

Convergent induction

History of Chemistry, Simplified

The simplest universe has only one kind of substance, which was the first scientific theory, that of Thales (ca 585 BC) who stated that the origin of all matter is water. Then there was Anaximenes, who held that everything in the world is composed of air. Xenophanes said, It’s all Earth. No, it’s all fire, said Heraclitus.

Empedocles combined them all in his theory that all matter is made up of four elemental substances – water, air, earth, and fire – in fixed quantities. The Pythagoreans taught that all things are composed of contraries. Aristotle combined the four elements of Empedocles and the contraries of the Pythagoreans and said that every substance is a combination of two sets of opposite qualities – hot and cold, wet and dry – in variable but balanced proportions.

Leucippus and Democritus disagreed with this approach and took the opposite position that all matter is made up of imperishable, indivisible entities called atoms. The atomic approach languished until many years later it was revived during the Renaissance. It was further developed by Dalton and others in the 19th century. Basic combinations of atoms, called elements, came to be seen as the building blocks of all substances. The list of elements was expanded into the Periodic Table which is key to the very successful science of chemistry today.

The so-called Ockham’s Razor, which may be stated “entities must not be multiplied beyond necessity”, is usually understood as a preference for simplicity. But it ignores trade-offs between, for example, a plurality of substances and a plurality of entities. Which is simpler, Thales’ single substance in many forms or Democritus’ single form but many entities?

Convergent Induction

There are three related lessons to be taken from this brief historical review: (1) science starts with simple, extreme positions; (2) for every simple, extreme position there is an opposite simple, extreme position; and (3) science develops complex, intermediary positions between simple extremes.

(1) It is well-known that science follows a principle of simplicity (parsimony) which leads it to start with overly simplified ideas, find their empirical weaknesses, and then gradually add complexity. As A.N. Whitehead said, “Seek simplicity and distrust it”.

(2) Simplicity comes in pairs. This is demonstrated in the case of chemistry between the extremes of one or a few substances and the opposite extreme of many atoms. There is the simplicity of a few unique entities vs. the simplicity of many uniform entities. There is also the simplicity of simple stasis vs. simple dynamics. These contrary simplicities have loomed large in the history of science, and many other subjects.

(3) While science begins with simple, extreme positions, it does not stay there. It progresses toward complexity. In an analogue to the mathematical theorem that any bounded increasing (or decreasing) sequence is convergent, simple extremes provide the bounds that ensure progressive induction converges.

Thus science proceeds via convergent induction, which is bounded by simple extremes and seeks empirical adequacy by progressively converging toward a complex mean. The process is progressive in that each step introduces a complexity not present before. Convergence is ensured by bounding the progression with simple extremes.

It is most usual to begin with one extreme and work in the direction of the other extreme rather than to oscillate between opposites in a convergent way. In general, there are two strategies for inductive logic: (1) assume the most about what is unknown and (2) assume the least about what is unknown. Natural science takes approach (1) and statistical science takes approach (2).

There are two directions for each of these approaches. Statistical science may be approached from the direction of maximal or minimal knowledge. For example, if there is knowledge of the physical source of variability such as by examining a pair of dice, then a frequentist direction may be best. If little is known except some empirical data that are gradually available over time, then a Bayesian direction may be best.

Natural science also has two directions. The most that can be assumed about what is unknown is that it is like what is known. But that may be either because it is a different form of the same thing or because it is a different combination of the same constituents. The former direction is top-down, macrocosmic, whereas the latter direction is bottom-up, microcosmic or atomic. The atomic direction has proved to be the most fruitful for natural science.

Since the convergent is a kind of mean between the initial extremes, that leads to the question of whether it would be possible to follow means instead of extremes. One could start with a simple mean between the extremes and then adjust it to another mean as need be. Perhaps this would be more efficient.

October 2010