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Fantods, whole no. 9, Winter 1945

Page 4

page 4     EFTY-NINE

to consider populations like those found in practice, which are simply classes of quantities measured all in the same units (for instance, the lengths of peapods from a field,or the number of divorces in successive years) ; or classes of pairs of quantities (for instance, the heights and weights of Army inductees). Instead, he deals with "continuous populations," which he claims are populations with an infinite number of members (!) but which actually are not populations at all in the above sense. This might seem abstract enough to be called pure math., yet few results obtained in mathematical statistics have even theoretical interest if they are not seen to be applicable to practical results of work with finite populations.
Certainly, though, in some branches of math. the validity of a new method or theorem is not judged by consistency with experimentation. Postulational geometry comes first to  mind, and Euclid is the readiest example. Euclid's construction theorem on the bisection of an angle is not considered "correct" because it is found to work in practice; if this were the criterion, Rankine's method of squaring the circle would also be correct, since it is so accurate the error cannot be detected by the best draftsman. Further, a mathematician who calls the theorem as it stands correct would call it incorrect itself if, say, one of Euclid's Axioms were changed to read, "If A equals B, B cannot equal A" -- because now the theorem would not follow logically from the given axioms. A new criterion has been introduced, namely that of logical rigor.
So now we have to figure out exactly what the criterion of logical rigor is. On this subject I'll give a short digest from an article by R. L. Wilder int he American Mathematical Monthly, entitled The Nature of Mathematical Proof.

"I cannot refrain from expressing my protest against what I like to call, variously, the 'mathematical dogmatist' or the 'mathematical fascist.' He may have no religious philosophy or affiliation in the ordinary sense, yet if you venture to doubt the validity of some one of his favorite proof methods, you may find yourself in danger of physical violence.

"Dealing with a subject that must be kept scrupulously abstract, except in 'applied mathematics,' we must ever be on guard against dogmatism. It is natural for the layman to think of us as the possessors of absolute truth,since we have been singularly successful in avoiding contradiction in applied mathematics. But we must not allow this to tempt us to set up our own mathematical cults. I think that the worth of new ideas can safely be left,in the long run, to the judgment of the mathematical electorate. A mathematical idea that never 'takes' is probably not worth taking.

"To put the matter bluntly, it is a case of our knowing mathematics when we see it. And we don't set out to prove a theorem in the first place unless we think it is worth proving.

"And this brings me to a consideration of the source of the thing we set out to prove - the theorem. Where do we get it? I think most of us would say from our institution. With proof or without proof, the mathematical theorem is not necessarily a statement of fact or truth in the ordinary sense.

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page 4 EFTY-NINE to consider populations like those found in practice, which are simply classes of quantities measured all in the same units (for instance, the lengths of peapods from a field,or the number of divorces in successive years) ; or classes of pairs of quantities (for instance, the heights and weights of Army inductees). Instead, he deals with "continuous populations," which he claims are populations with an infinite number of members (!) but which actually are not populations at all in the above sense. This might seem abstract enough to be called pure math., yet few results obtained in mathematical statistics have even theoretical interest if they are not seen to be applicable to practical results of work with finite populations. Certainly, though, in some branches of math. the validity of a new method or theorem is not judged by consistency with experimentation. Postulational geometry comes first to mind, and Euclid is the readiest example. Euclid's construction theorem on the bisection of an angle is not considered "correct" because it is found to work in practice; if this were the criterion, Rankine's method of squaring the circle would also be correct, since it is so accurate the error cannot be detected by the best draftsman. Further, a mathematician who calls the theorem as it stands correct would call it incorrect itself if, say, one of Euclid's Axioms were changed to read, "If A equals B, B cannot equal A" -- because now the theorem would not follow logically from the given axioms. A new criterion has been introduced, namely that of logical rigor. So now we have to figure out exactly what the criterion of logical rigor is. On this subject I'll give a short digest from an article by R. L. Wilder int he American Mathematical Monthly, entitled The Nature of Mathematical Proof. "I cannot refrain from expressing my protest against what I like to call, variously, the 'mathematical dogmatist' or the 'mathematical fascist.' He may have no religious philosophy or affiliation in the ordinary sense, yet if you venture to doubt the validity of some one of his favorite proof methods, you may find yourself in danger of physical violence. "Dealing with a subject that must be kept scrupulously abstract, except in 'applied mathematics,' we must ever be on guard against dogmatism. It is natural for the layman to think of us as the possessors of absolute truth,since we have been singularly successful in avoiding contradiction in applied mathematics. But we must not allow this to tempt us to set up our own mathematical cults. I think that the worth of new ideas can safely be left,in the long run, to the judgment of the mathematical electorate. A mathematical idea that never 'takes' is probably not worth taking. "To put the matter bluntly, it is a case of our knowing mathematics when we see it. And we don't set out to prove a theorem in the first place unless we think it is worth proving. "And this brings me to a consideration of the source of the thing we set out to prove - the theorem. Where do we get it? I think most of us would say from our institution. With proof or without proof, the mathematical theorem is not necessarily a statement of fact or truth in the ordinary sense.