Selective Rigor

Recently, I’ve been informed that very little is known about Pythagoras and in particular, it wasn’t known whether he even proved the theorem of his name and most of what’s known about him is myth. That got me to thinking how often I heard stories, namely about the death of the man who proved the irrationality of the square root of 2, and how I never really checked sources on whether this is true. I also recall (but can’t find) a portrait purported to be of a well-known mathematician (for some reason, I thought it was Lambert), turning out to be a portrait of someone else entirely.

Back in my undergraduate years, when I was first learning about mathematical rigor, I became a little zealous with the idea of rigorous thought. The unambiguity was so refreshing and beautiful that I felt it ought to be carried throughout life. To be precise and careful with everything one says seems like a great aspiration. Of course what actually ends up happening is I rarely talked and when I did talk, I would add conditional statements to everything and depend on exact wording that most people don’t hear. It’s been surprising frequent that I would make a conditional statement and people would hear only the conclusion. It also led to a skepticism and doubt regarding my own opinions. But despite the detriment, I feel it’s a net positive, since it taught me to try to hold my own opinions at a distance and that I can’t ultimately separate myself from my bias, but that if I try, I can go a long way to gaining just a little bit of objectivity.

Before I got to graduate school, I felt that this was an aspiration of all mathematicians. So I was quite surprised to realize that this wasn’t the case. Most people happily deal with imprecision and ambiguity. In some cases, I felt their mathematics also suffered from sloppiness. But in other cases, it seemed as though, despite an ability to be mathematically precise, there wasn’t any desire to extend this to other aspects of life and discussion. People seemed to hold strongly opinions, for example about politics, that seemed hardly justified. If they were doing math, I would think to myself, would they be so bold to make such unjustified claims?

I feel quite happy with my strange aspirations of precision and rigor, despite giving up on being able to speak in perfectly precise language. The more encounters I’ve had with people who don’t have this strange habit, including other mathematicians, the more I think of this as just another one of my own biases that I need to remember to take into account when trying to understand the world at large. And I need to remember that as a vocation, mathematicians can be very precise, but that need not extend to their personal lives.


One Response to Selective Rigor

  1. John Baez says:

    Here’s a great quote from Kitty Ferguson’s book The Music of Pythagoras: How an Ancient Brotherhood Cracked the Code of the Universe and Lit the Path from Antiquity to Outer Space:

    The tiny “core of truth” left after discounting all folk wisdom, semi-historic tradition, legend or what might be only legend, and blatant forgeries and inventions can be stated in one paragraph:

    Pythagoras of Samos left his native Aegean island in about 530 ac. and settled in the Greek
    colonial city of Croton, on the southern coast of Italy.Though the date of his birth is not certain, he was probably by that time about forty years old and a widely experienced, charismatic individual. In Croton, he had a significant impact as a teacher and religious leader; he taught a doctrine of reincarnation, became an important figure in political life, made dangerous enemies, and eventually, in about 500 B.C., had to flee to another coastal city, Metapontum, where he died. During his thirty years in Croton, some of the men and women who gathered to sit at his feet began, with him, to ponder and investigate the world. While experimenting with lyres and considering why some combinations of string lengths produced beautiful sounds and others did not, Pythagoras, or others who were encouraged and inspired by him, discovered that the connections between lyre string lengths and human ears are not arbitrary or accidental. The ratios that underlie musical harmony make sense in a remarkably simple way. In a flash of extraordinary clarity, the Pythagoreans found that there is pattern and order hidden behind the apparent variety and confusion of nature , and that it is possible to understand it through numbers. Tradition has it that, literally and figuratively, they fell to their knees upon discovering that the universe is rational. “Figuratively,” at least, is surely accurate, for the Pythagcreans embraced this discovery to the extent of allowing numbers to lead them, perhaps during Pythagoras’ lifetime and certainly shortly after his death, to some extremely farsighted and also some off-the-wall, premature notions about the world and the cosmos.

    One might assume that the above paragraph is a summary merely touching the highlights of
    what is known about events in sixth-century B.C. Croton, but it is, in fact, all that is known. Thoug you and I might wish to ask many more questions, the answers are irretrievably lost.

    Nobody knows if Pythagoras proved the Pythagorean theorem, and the story about the guy getting thrown out of a boat after proving the irrationality of the square root of two seems to have been made up in the 20th century.

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