Ben,
I agree that space and time in quantum mechanics are continuous. But there is one issue that Peirce understood very well: the discrete lines in the spectrum of any chemical element that is heated on earth or in any distant star.
In the so-called "black body" radiation, the filament of an incandescent light bulb emits a continuous spectrum with a peak at a frequency that is determined by the temperature. But a photon emitted when an electron of an atom falls from a higher state to a lower state has a sharply defined frequency.
Peirce learned that fact when he studied chemistry, and he used it in his photometric studies of distant stars. Although it may be wrong to consider a continuous line as a set of discrete points, there are phenomena in physics whose measurements are discrete.
John
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