What do you mean by can’t?

You can’t subtract 4 from 3 (and stay inside the natural numbers, but you can inside the integers).

You can’t divide 3 by 4 (inside the ring of integers, but you can inside the rational numbers).

You can’t take the square root of a negative number (in the real numbers, but in the complex numbers you can, once you pick a branch of the square root function).

You can’t divide by zero (in the field of real numbers, but you may be able to do something that could informally be referred to as dividing by zero, depending on the context, by reformulating your statement, often in terms of limits).

When people say a thing cannot be done, they may mean it cannot be done in some assumed context. They may mean that the thing is difficult, and assume that the listener is sophisticated enough to interpret their answer as hyperbole. Maybe they mean that they don’t know how to do it and presume it can’t be done.

When you hear that something can’t be done, it’s worth pressing to find out in what sense it can’t be done.

Related post: How to differentiate a non-differentiable function

4 thoughts on “What do you mean by can’t?

  1. I agree, perspective is important in the statements we make. Whether or not something is possible might depend entirely on the constraints or limits attached to it.

  2. This reminds me of
    (pdf link) by Philip J. Davis

    “To be known, to be knowable; to be proved, to be provable; to be computed, to be computable; to be decided, to be decidable; to be true, to be false; to be verified, to be verifiable; to intuit; to have evidence for; to doubt; to know that one knows; to know that one has proved; to know that one can’t prove — these are some of the epistemological environments (or clouds) out of which the states of mathematical knowledge are formed.”

  3. Something is just literally impossible.
    For example, you can’t find an algorithm for an undecidable problem.

  4. Has to do with requirements analysis and assumptions.
    Long ago I had a great argument with a friend about “needs”.
    Some things are needed only if you assume that sanity is necessary.
    Most of the things we think we need, we actually only Want, particularly in the First world.
    Physical needs are only “needs” if you assume continued functioning.
    If you remove the assumption that you want to live more than 30 seconds, you don’t even Need air anymore.

    Realizing that everything you “need” is really a debate about consequences can turn things around about what really matters most.

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