Daniel Henry Gottlieb --- September 11, 2004
About 10 years ago, I suddenly thought of a possible definition of Mathematics. It occurred to me that mathematics was the study of Well-defined concepts. That is the study of absolutely unambiguous concepts. I was happy with this definition for awhile, however trouble soon arose. For there are two kinds of definitions. The usual type is a description, for example "mathematics is the study of numbers and algebra and calculus". And then there are mathematical definitions, from which the property and usage are determined by the definition. It seemed my definition was a mathematical definition. Thus I had to admit, according to my definition, the game of chess was mathematics. Its rules were completely well defined.
Now this proposition was outside of my experience. I knew of no mathematics department which had a course on chess, and I didn't think any of my colleagues would agree with me that chess was a branch of mathematics. But later it occurred to me that we mathematicians would be very interested if someone could prove that white always could win with best play. And I recalled a mathematician, Raymond Smullyan, who wrote a book illustrating mathematical ideas with with chess problems.
So I began to gain confidence in my definition. In order to continue, I used a common mathematical strategy. I let X stand for the study of well defined things. Chess was certainly part of the subject matter of X. Who could dispute that it was not a well-defined concept? I studied the history of mathematics. I learned that Plato was very excited that the diagonal of the square was not commensurate with the side. That is in more modern language, that the square root of 2 was not a rational number. He said that no man could be educated and not know this fact. I wondered why he thought that. Then it occurred to me: There is no way to know that fact by experience. No amount of measuring squares could ever show that the diagonal and the side do not have a common unit of measure.
So the subject X is like a sixth sense. We can learn things from it we cannot learn by experience. I began to look for possible ambiguities in the concept of well-definedness and I could not think of any. Finally I decided if I found an ambiguity it would also be a blow to mathematics itself. So I decided that X was really a discipline and of course I decided it contained arithmetic, Analysis, Algebra, Topology and any other subfield of mathematics, all of which deal with well-defined concepts. So it appeared that Mathematics is a subdiscipline of X.
I could not think of any criteria which would eliminate any well-defined concept from mathematics. So I thought I was prepared to declare that X was mathematics. But I paused. Mathematics was an ancient word, its history extends the length of the history of civilization. At times mathematics included subjects like astrology, which only employed mathematics, but were not well-defined. Physics was considered mathematics for most of history. Many authors see an underlying unity in Mathematics, whereas clearly X cannot have underlying unity. The meaning of mathematics changes according to the experience of the person using the word. To insist on X as the same as mathematics would put too much restriction on the usage of the word, cutting off its informal meaning used by people with various levels of experience of mathematics seemed to me to be too big a step.
So I decided not to propose that X equalled mathematics. But I am convinced that X is a discipline. So I want to give it a name. The "study of well-defined concepts" is too awkward a name. I thought I would hijack "Pure Mathematics" as the proper name for X. "Pure Mathematics is used often as an opposite to "Applied Mathematics" and many mathematicians don't think that that is a correct point of view. So there is no great loss in changing whatever is meant by pure mathematics to the study of well-defined concepts.
What does all this have to do with the gay marriage decision by the Massachusetts Supreme Court? A few years ago some people wanted to form partnerships to which the same laws and privileges accorded to partnerships of married people held. Rather than fight for the entitlements via the legislative process, they declared that their relationships, call them Y, were actually marriages and that the state, in not agreeing to bestow the name of marriage to Y were acting in a unfair manner discriminating against those people who wanted to form Y partnerships with all the entitlements of marriage. The Massachusetts supreme Court agreed that Y's were in fact marriages, and the state had to accord Y's the same status as marriages. In effect, the Court changed the legal meaning of the word marriage.
Now the Court did something analogous to my Hijacking the name Pure Mathematics for X. No one who disagrees with my position is required to use my definition, and he can continue to use the words Pure Mathematics for whatever he thinks it means. But the Supreme Court has rendered it impossible for the people's legislature to use the word marriage in its standard meaning without involving Y's. For example, suppose a brother and sister want to form a Y. In most societies they are prevented from being married because of the horrible afflictions which will probably be suffered by their offspring.Yet isn't it just as unfair to discriminate against the brother and sister as it is to discriminate against any one else who wants to form a Y?
Now in my desire to legitimize X, I nevertheless stopped before hijacking the word Mathematics for X due to the long history of the word and utility of having a descriptive definition for Mathematics. But whatever pedigree the word mathematics has, it comes from Classical Greece, the word Marriage has an order of magnitude more. I have never heard of a human culture, which did not have some type of marriage; although it always involved the two sexes, it was not always a partnership. Some marriages had several women and only one man, and a few had only one woman and several men. Whatever the grouping it always involved the opposite sexes. The most primitive tribes have marriages only involving the opposite sexes. It is hard to imagine a word which has so many analogues in every language all with the same meaning over so long a period of time as the word marriage does. It even has similarities in the animal kingdom.
Now if the Massachussets Supreme Court can change the milleniums old meaning of the word marriage, it can change the meanings of other words such as Freedom of Speech, Cruel and Unusual Punishment, Man, Women, or any language written in the Constitution to reestablish the traditional meaning of marriage. Most of those words do not appear from the beginning of mankind in some form and are not found in most present day cultures. This is a very frightening development and requires a new check and balance. I propose a Constitutional amendment to the effect that a jury of scholars be appointed yearly, people who know the history of words, and science and statistics and mathematics, and when they see hijackings of common words such as marriage, they can nullify the courts decision subject to a popular referendum of the people. And if the decision of the Court is over ruled by the people, then the court must resign and a new one be appointed.
Comments to gottlieb@math.purdue.edu
Return to Mathematics and the Law
Return to Daniel Henry Gottlieb's Home Page.