We think in continuous terms all the time when we use big O notation. If you have an O(n) (or O(n^2)) algorithm and make the input a tiny bit bigger, the computation becomes a tiny bit more complex. Dijkstra is too rigorous to let us get away with that, though. In theory, big O notation says nothing about any particular finite value, only about limits, and in practice, this manifests in big O notation's failure to predict big jumps in resource costs when internal limits such as cache sizes are exceeded. To create an abstraction as nice and well-behaved as big O notation, we have to gloss over the real behavior of the machine.
Is it not possible that one day we will invent a computer language that supports writing very "robust" (in the sense of source code sensitivity) programs?
If you apply an edit distance metric to source code and a behavior metric to running programs, it would take a ridiculous amount of work to write programs in any system where small changes in source code resulted in small changes in running systems. Also, very often people want small changes in input data to result in small changes in behavior, but just as often, they want small changes in input data to result in large changes in behavior, so we have external requirements to make some parts of our program "robust" and other parts extremely sensitive. A language would need to support both requirements.
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We think in continuous terms all the time when we use big O notation. If you have an O(n) (or O(n^2)) algorithm and make the input a tiny bit bigger, the computation becomes a tiny bit more complex. Dijkstra is too rigorous to let us get away with that, though. In theory, big O notation says nothing about any particular finite value, only about limits, and in practice, this manifests in big O notation's failure to predict big jumps in resource costs when internal limits such as cache sizes are exceeded. To create an abstraction as nice and well-behaved as big O notation, we have to gloss over the real behavior of the machine.
Is it not possible that one day we will invent a computer language that supports writing very "robust" (in the sense of source code sensitivity) programs?
If you apply an edit distance metric to source code and a behavior metric to running programs, it would take a ridiculous amount of work to write programs in any system where small changes in source code resulted in small changes in running systems. Also, very often people want small changes in input data to result in small changes in behavior, but just as often, they want small changes in input data to result in large changes in behavior, so we have external requirements to make some parts of our program "robust" and other parts extremely sensitive. A language would need to support both requirements.