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If there are infinitely many points in any interval of the real number line, how come it does not take infinitely many computations to compute a function over that interval?

Infinitely many computations is impossible assuming finite resources.

oolveea
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  • In terms of computer computation, you may not even be computing a function at any single point. In terms of mathematics, you do not prove theorems by computing functions, but by explioting formal properties of the functions as objects, or of the set of values they take. – Sassatelli Giulio May 02 '22 at 16:04
  • My question is motivated by signal processing, specifically, if there are infinitely many instants in any interval of time, how come it does not require the universe an infinite amount of resources to compute an analog (continuous time) signal. And if so, how come the signal arrives at all? – oolveea May 02 '22 at 16:08
  • Do you have evidence that a process mechanically analogous to human computation happens for "the universe", and specifically in the field of interest? – Sassatelli Giulio May 02 '22 at 16:09
  • You are right, there is no reason to believe that the universe is "human computational". – oolveea May 02 '22 at 16:46
  • What does it mean, really? We can make several simple examples: 1) $f(x)=1$ and 2) $f(x)=x$ and we may say that in both cases we "know" the respective values "over an interval" whatever. – Mauro ALLEGRANZA May 04 '22 at 07:15
  • If we mean that we have a formula defining the function and this formula is "computable", we may - for every number in the interval - use that number as input and compute the corresponding output. Obviously, in the real line we have uncountable many values in an interval, and thus we cannot compute explicitly all outputs, in the first place because we cannot "list" all inputs. – Mauro ALLEGRANZA May 04 '22 at 07:19

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