Tag Archives: Michael Spivak


If you roll out a lump of clay into a snake, then tie a string loosely around it, can you contort the ends of the snake, without actually pressing them together, so that you cannot get the string off?

You can stretch the clay into a Medusa’s head of snakes, and tangle them as you like, again without letting them touch. If you are allowed to rest the string on the surface of the clay, then you can get it off: you just slide it around and over what was an end of the original snake.

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What Mathematics Is

Mathematics “has no generally accepted definition,” according to Wikipedia today. Two references are given for the assertion. I suggest that what has no generally accepted definition is the subject of mathematics: the object of study, what mathematics is about. Mathematics itself can be defined by its method. As Wikipedia currently says also,

it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.

I would put it more simply. Mathematics is the science whose findings are proved by deduction.

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This is about the ordinal numbers, which (except for the finite ones) are less well known than the real numbers, although theoretically simpler.

The numbers of either kind compose a linear order: they can be arranged in a line, from less to greater. The orders have similarities and differences:

  • Of real numbers,

    • there is no greatest,

    • there is no least,

    • there is a countable dense set (namely the rational numbers),

    • every nonempty set with an upper bound has a least upper bound.

  • Of ordinal numbers,

    • there is no greatest,

    • every nonempty set has a least element,

    • those less than a given one compose a set,

    • every set has a least upper bound.

One can conclude in particular that the ordinals as a whole do not compose a set; they are a proper class. This is the Burali-Forti Paradox.

Diagram of reals as a solid line without endpoints; the ordinals as a sequence of dots, periodically coming to a limit Continue reading

On the Idea of History

Our environment may influence our feelings, but what we make of those feelings is up to us. Thus we are free; we are not constrained by some fixed “human nature”—or if we are, who is to say what its limits are?

Rembrandt van Rijn (and Workshop?), Dutch, 1606-1669,
The Apostle Paul, c. 1657, oil on canvas,
Widener Collection, National Gallery of Art

Insofar as we humans have come to recognize our freedom, we have done so after thinking that what we did depended on our class—our kind, our sort, even our “race.” We might distinguish three stages of thought about ourselves.

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This is about G. H. Hardy and Sylvia Plath: Hardy quâ author of A Mathematician’s Apology (1940); Plath, The Bell Jar (1963).

Photo: the Hardy and Plath books

I first read Plath only recently, after encountering The Bell Jar by chance in the Istanbul bookshop called Pandora. Continue reading

Learning mathematics

This is mostly reminiscences about high school. I also give some opinions about how mathematics ought to be learned. This article originally formed one piece with my last article, “Limits.”

I learned calculus, and the epsilon-delta definition of limit, in Washington D.C., in the last two years of high school, in a course taught by a peculiar fellow named Donald J. Brown. The first of these two years was officially called Precalculus Honors, but some time in that year, we started in on calculus proper. Continue reading