David Stove wrote an essay What is Wrong with Our Thoughts? that includes a list of 40 propositions about the number three, which are a small museum of the pathology of thought. He adds, "after the first two specimens, you will not know…in what way they have gone wrong."
Actually, I do. They’re good case studies, so let’s go through them, organized by error.
Before diving in, let’s clarify what three is. We can write down formal languages, which are sets of sequences of symbols, called sentences. Any other structure is a model of a formal language if there is a mapping of the symbols onto elements of the structure such that all the sentences hold. This is part of a field of mathematics called model theory.
We can define a formal language of 0, , and . If we can find some set of objects in a structure, call them , , , and , such that , , , and is undefined or something else besides these four, then we are justified in calling zero, one, two, and three. In other words, three is any element of a model of the formal language given by the Peano axioms which satisfies .
I’d like to pause and quote Dijkstra’s EWD1123 (warning: PDF):
"We now take what is a standard step in mathematical theory building. The step is taken after the introduction of a notational novelty – such as a new abbreviation or a "mathematical macro" – for formulae that were interpreted in a familiar domain of discourse. The step consists of starting with a clean slate and axiomatizing afresh the manipulations of the new formulae. In so doing, one creates a new domain of discourse; the role of the old, familiar domain of discourse, that used to constitute the subject matter, is thereby reduced to that of providing a possible model for the newly postulated theory. It is essential that the axioms of the new theory – which can be interpreted as theorems in the old universe of discourse – are clearly postulated as such and that the new theory is derived from them without reference to the model of the old universe of discourse. This is the only way to assure tha the axioms of the new theory provide an interface that is independent of the old universe of discourse and that, hence, the new theory is safely applicable to alternative models."
This is the inverse of the description of model theory above. After axiomatizing, we find an object which we have labelled three. It is all to easy to confuse whether we are talking about three in the axiomatized formal language, three in this or that model, or the symbol "3" that we write on paper, and many of the propositions fall into exactly this trap. Now let us turn to the propositions.
First, there is the statement of a classically false proposition.
1. Between 1960 and 1970 there were three US presidents named Johnson.
2. Between 1960 and 1970 there were three US presidents named Johnson, and it is not the case that between 1960 and 1970 there were three US presidents named Johnson.
9. There is no number three.
10. Three is the only number.
14. The sum of three and two is a little greater than eight.
29. Three is a positive integer, and the probability of a positive integer being even is 1/2, so the probability of three being even is 1/2.
33. How is the addition of numbers possible? Nothing can make the number three into four, for example. (Unless he means the object three in a model be the object four in another convention, which is just fine. Let 1 for one model of the integers be zero in another way of making the structure a model of the formal language.)
38. The unconscious significance of the number three is invariably phallic, nasal, and patriarchal. (I hesitated for a moment about this one, but then I thought, "boobies!")
Example (1) would be fine if you were writing an alternate history story. (2) is false in any structure anyone would call a logic. (9) is a statement that there is no model corresponding to the formal language. (14) is again logically wrong given the definition of three.
Then there is material that is simply undefined.
3. God is three persons in one substance, and one of these persons is Jesus, which is the lamb that was slain even from the foundations of the world. (So Jesus is one personality of someone with split personality disorder and…)
26. The tie which unites the number three to its properties (such as primeness) is inexplicable. (If he means a predicate, it’s easily explicable; if he’s trying to be profound, what does he mean by a tie?)
30. In some previous state of our existence we knew the number three face-to-face, as it is in itself, and by some kind of union with it. (I don’t kiss and tell…or whatever it is one does to a number and tell.)
35. We get the concept of three only through the transcendental unity of our intuitions as being successive in time. (Is this the same kind of union as in 30? Is someone a little kinky?)
36. One is identity; two is difference; three is the identity of, and difference between, identity and difference. (You defined identity and difference as one and two; now define identity of and difference between. This could also just be 3=1 and 3=2 and 2-1=3, all of which are false.)
There is the confusion between the label "three" and the object in a model which is denoted by three. If we speak of the label "three," then this is true:
7. Three lies between two and four only by a convention which mathematicians have adopted.
If we speak of the object we point to, and we use the usual ordering on natural numbers, then the statement is logically false. A convention is a choice among several ways in which a structure is a model for a formal language. Left-handed or right-handed coordinate systems are a convention. Ordering in the integers is not.
The same confusion taken to an extreme is seen here:
8. There is an integer between two and four, but it is not three, and its true name and nature are not to be revealed.
If we refer to three, the name of a symbol, then call it whatever you like, but until you and your cult get around to inducting me, I shall go on calling it three. If we refer to the element corresponding to the term in our formal language, then the statement that it is not three is logically false.
Similarly, the following statements are all false because three lies between two and four by construction. If an element in a particular model did not satisfy this, it would not be called three.
4. Three lies between two and four only by a particular act of the Divine Will.
5. Three lies between two and four by a moral and spiritual necessity inherent in the nature of numbers.
6. Three lies between two and four by a natural and physical necessity inherent in the nature of numbers.
And then there is the idea that three is a unique thing, rather than anything that we care to so label that satisfies our fragment of a formal language, though some verge into undefined.
16. Three is a real material object.
17. Three is a real spiritual object.
18. Three is an incomplete object, only now coming into existence.
19. Three is not an object at all, but an essence; not a thing, but a thought; not a particular, but a universal.
20. Three is a universal all right, but it exists only, and it exists fully, in each actual triple.
21. Actual triples possess threeness only contingently, approximately, and changeably, but three itself possesses threeness necessarily, exactly, and immutably.
22. The number three is only a mental construct after all, a convenience of thought.
15. Three is a real object all right: you are not thinking of nothing when you think of three.
24. The number three is that whole of which the parts are all and only the actual inscriptions of the numerals, ‘three’ or ’3′.
27. The number three is nothing more than the sum of its properties and relations.
32. Since the properties of three are intelligible, and intelligibles can exist only in the intellect, the properties of three exist only in the intellect.
34. What the number three is in itself, as distinct from the phenomena which it produces in our minds, we can, of course, never know.
37. The number three is not an ideal object of intellectual contemplation, but a concrete product of human praxis.
39. The three members of any triple, being distinct from and merely related to one another, would fall helplessly asunder, if there were not some deeper non-relational unity of which their being three is only an appearance.
40. It may be – though I don’t really believe in modalities – that in some other galaxies the sum of three and two is not five, or indeed is neither five nor not five. (Don’t laugh! They laughed at Christopher Columbus, you know, and at Copernicus; and even the logical law of excluded middle is being questioned nowadays by some of the sharper young physicists.)
Some of the statements are experimentally false.
13. Three is a lucky number. (i.e., in all random situations where you have a choice of numbers, your average payoff will be greater if you choose three.)
28. The number three is neither an idle Platonic universal, nor a blank Lockean substratum; it is a concrete and specific energy in things, and can be detected at work in such observable processes as combustion. (There is no form of energy which it is sensible to call three.)
Some of the statements can be true, depending on context. If you are working in a very restricted set where only the fragment of the formal language up to three has a model, then this is true:
11 Three is the highest number.
And if you’re talking about the number of times you have to die before someone will forgive you, then this also may very well be true:
12 Three is a large number. (Is this the number of times you have to die before someone will forgive you? Starts to look pretty big, then…)
31. How can I be absolutely sure that I am not the number three? (If you are the object corresponding to three in a model of the natural numbers, then you most certainly are the number three, as someone’s drill sergeant has doubtless informed him.)
I think the sense Stove meant is that they are everywhere true, that is, "for all contexts, three is a large number." Then these are logically false.
This one makes the same mistake, but in a particulary odd manner:
25. Five is of the same substance as three, co-eternal with three, very three of three: it is only in their attributes that three and five are different.
If you have a model of the natural numbers where "substance" is defined, then it may be that they are of the same substance, and it may not. Or you may just not have any way of defining substance in a useful manner.
The computer scientists will note the immediate problem with this one:
23. The proposition that 3 is the fifth root of 243 is a tautology, just like ‘An oculist is an eye-doctor.’
There are two relevant definitions of tautology. One is used in formal logic, but since we are comparing to the English language, that isn’t the one we are interested in. The other is "redundant use of words." Anyone who has to compute the fifth root of 243 will disagree that being told that it is three is redundant.
At least in this list, it’s very clear how all these thoughts go wrong. And the philosophers wonder why mathematicians and scientists largely ignore them?
June 29, 2010