Wrestling with Rosser: a hair-splitting experience

This notation

∫^a_o

represents the integral symbol with the limits of integration running from 0 to a. This notation

x²][a,0]

represents the symbol used to indicate that one substitutes the limits of integration for x.

J. Barkley Rosser (1907-1989) was a first-rank logician and mathematician who, for example, proved that Alonzo Church's first attempt at the lambda calculus was inconsistent and who simplified Goedel's incompleteness theorem. In 1936, he proved Rosser's trick, a stronger version of Gödel's first incompleteness theorem, showing that the requirement for ω-consistency may be weakened to consistency. Rather than using the liar paradox sentence equivalent to "I am not provable," he used a sentence that stated "For every proof of me, there is a shorter proof of my negation."

His book Logic for Mathematicians (McGraw Hill 1953) "succeeded in condensing most of Principia Mathematica's three large volumes" into a single 522-page book. I am very glad of that! The pioneering text by Bertrand Russell and Alfred North Whitehead is freighted with an abundance of obscure symbolism that I have no desire to learn.

Rosser wrote in a preface,

The subject matter in Principia Mathematica was admirably chosen for the needs of mathematicians, and we have followed the text closely with regard to subject matter. We have omitted a few topics which seem to be little-used nowadays, and instead have included treatments of such new developments as Zorn's lemma. We have improved on the symbolic machinery of Principia Mathematica, which is out of date and extremely unwieldy. By using techniques invented since its writing, we have succeeded in condensing most of Principia Mathematica's three large volumes into the present text.

This then is a valuable contribution, though these days readers complain that Rosser's notation is old-fashioned. But, as it is quite easy to pick up, there is no need for alarm.

Though PM has been hailed as an epochal event, only a few logicians read it, and not many of them once Goedel proved PM could not be both consistent and complete. Russell's paradox, that Russell worked so hard to exorcise, transmogrified into Goedel's theorem. Mathematicians surely did not read PM, preferring developments in set theory axiomatics -- though Goedel wrecked the dreams of the set theory set also.

In any case, a grand master of mathematical logic, Rosser, is about to clash with the fussbudget wood-pusher, yours truly. OK, I am being awkwardly melodramatic. So anyway, I happened to be perusing Rosser's book where he picks up on PM's discussion of exactly what is meant by a variable (somewhere in my bundles of ancient notes are my own thinking on this point, but the notes are at present inaccessible to me).

So let's cut to the chase. On page 82, Rosser gives examples of the use of x. (Comments with pale blue background are Rosser's, comments with pale green background are mine.)

(VI.1.1)  

 x² - 4x + 3 = 0

in which x denotes an unknown quantity whose value is to be determined.

Well, actually x represents two real values of the set {-1, -2} so we could argue that x e X such that X is the set in curly braces, in which case x is varying over that set; and then there are the complex values, of course. But even if it represented only one number, n_o, one could still claim that x varies over the set X, where (x) are in X, or, that is, where X = {n_o}. (Rosser uses the older PM notation (x) for all x.)

(VI.1.2)  

sin² x + cos ² x = 1

in which x denotes a variable for which we may substitute any angle (or any real number, or even any complex number).

No disagreement, but are we not talking about the size and finitude of the set from which we are drawing substitutes? Of course, we grant that to get a good definition of set, the logical groundwork must be laid first. On the other hand, as Russell's herculean efforts show, there really is no getting around accepting primitive sets of some sort (if by another name) in order to speak reasonably about logic.

We may also write

(VI.1.3)

∫ x² dx = x³/3

(VI.1.4)

∫³_ox² dx = 9

(VI.1.5)

∫^x_oy² dy = x²/3

(VI.1.6)

∫^x_ox² dx = x²/3

In these, x probably denotes an unknown (or indeterminate) in (VI.1.5), but a variable in the others. However, there is no universal agreement on this point. Usually, in (VI.1.4) one speaks of x as a variable of integration, but some writers insist that x does not really denote a variable (or unknown either) in (VI.1.4), and that (VI.1.4) is merely a conventionalized abbreviation of a very complicated definition. In any case, in (VI.1.3) to (VI.1.6), we certainly have at least two distinct usages of the letter x and perhaps as many as four distinct usages.

Again, the distinction between the concepts of variable and unknown or indeterminate seem mainly based on the size of the set of which x is a member.

To make matters still more confusing, in (VI.1.6), two of the occurrences of x are being used in one way and the other two in quite another way. Thus we can replace two of the x's in (VI.1.6) by 3's, and obtain (VI.1.4), but we cannot replace the other two x's by 3's, since this would result in the nonsense

∫^x_o3² d3 = x³/3

nonetheless, we can replace them by y's to get (VI.1.5).

The last equation is not as nonsensical as Rosser allows. The answer is that for any real x, the area under the defined curve is 0. To wit, d3/dx = 0, d3 = 0dx, 3²0 = 0. While true that one can substitute x for x, as in (VI.1.6), we concede that one cannot substitute anything for a constant left of the right brace. That is, writing

0]^x_o = 0

is indeed nonsense. So perhaps Rosser has a point. Yet I assert that since we know the area must be 0, it is unimportant whether we can actually make such an algorithmic substitution.

(VI.1.3) is an indefinite integral. Hence the right side of the equation should be x³ + 3C, where C is an indeterminate constant. In effect C is another variable ranging over the reals, inclusive of the number 0. Though variable, we "hold C constant" while dealing with x.

So (VI.1.2) and (VI.1.6) do not yield the same answer because we may write the answer to (VI.1.6) as  x³ + 3C, but insist that C must be 0, whereas in (VI.1.2), C is not necessarily 0.

My quibbles halt here.

Nevertheless, ye olde wood-pusher finds Rosser's  Logic for Mathematicians highly enjoyable and instructive.

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Discussion of free versus bound variables is beyond the scope of this note.

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