They had not the habit of definition

The question of whether 0.9999…. is equal to 1 is, and probably always will be, a source of heated disagreement among people who know a certain amount of math, but not too much. High school math teacher Polymathematics delivered a magisterial series of posts on this question last year, which covers with admirable thoroughness every one of the many, many strange trails this argument likes to wander down. So I’ll leave that to him, and just use the question as an excuse to copy in one of my favorite quotes from G.H. Hardy, from his 1948 book Divergent Series:

“…it does not occur to a modern mathematician that a collection of mathematical symbols should have a ‘meaning’ until one has been assigned to it by definition. It was not a triviality even to the greatest mathematicians of the eighteenth century. They had not the habit of definition: it was not natural to them to say, in so many words, `by X we mean Y.’ … it is broadly true to say that mathematicians before Cauchy asked not ‘How shall we define

1 – 1 + 1 – 1 + …

but

‘What is 1 – 1 + 1 – 1 + …

and that this habit of mind led them into unnecessary perplexities and controversies which were often really verbal.”

The connection to 0.9999… is quite direct. To ask what 0.9999… is is to miss the point. Rather, you ought to ask “How can we choose a real number which deserves to be called ‘the value of 0.9999…?'” And once you have done this, you realize you are not at all sure what the definition of a real number is … and before long, you’ve learned the first few weeks’ material of a course in real analysis and all confusion has departed.

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4 thoughts on “They had not the habit of definition

  1. John Cowan says:

    The following fable was published anonymously in the September 1954 issue of the Mathematical Gazette:

    Once upon a time there was a teacher who set his class an examination to perform. And when the youths had finished he marked their scripts. But at the end of his labours he found that, by evil chance, he had worked with a total of 99. And, being an industrious man, he converted all the marks into
    percentages.

    So it was that a pupil with 58 marks gained 58.585858… per cent., and a pupil with 73 marks gained 73.737373. .. per cent., and others likewise.

    And when the time was come that he should return the scripts to his class, being an honest man as well as an industrious, he confessed what he had done and delivered to them their marks in the form of percentages.

    Until he came to one named Smith whose work was perfect, to whom perforce he had awarded the percentage 99.999999.. per cent.

    “So, Smith minor,” saith he, “Though I find no fault in you, yet your percentage falls short of the full total of 100. What say you? ”

    “Sir,” saith Smith minor, moved to anger, ” I call that the limit!”

  2. JmSR says:

    .9999… = 1 is one of the two things that make my job of explaining things hard to some of my semi-numerate friends.

    The other is that a curve that stretches to infinity could have an area something less than infinity.

  3. […] They had not the habit of definition Meaning “High school math teacher Polymathematics delivered a magisterial series of posts on […]

  4. I think that I have pretty clear explanation of why 0.99999..=1 here

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