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I got into a discussion with someone stemming from the set of uncomputatble numbers and how they claimed that such numbers like $\pi$ (not uncomputable but you'll see in a second) don't exist.

I was pretty perplexed and asked for their explanation to which I got the following (abriged) explanation

The universe we live in is built on the idea of Constructive Mathematics in which everything that we use must be explicited computable and constructable in order to exist. Numbers like $\pi$ and $\sqrt{2}$ cannot every be explicitely computed. We can use finite approximations as those can be constructed, but since irrationals cannot be we cannot use them in this form of math. This is in line with the universe in which there is no such thing as an irrational number, as you cannot have a distance smaller than the Plank Length

I am neither a physicist nor an expert in this axiomatic field. However, my quick scan of material makes me 99% sure that this person is completely incorrect about reals in Constructive Math. Constructive Math is just ZF without Excluded Middle, and I have seen the reals to be defined to be used as the limits of Cauchy Sequences.

Am I correct that this person is completely misunderstanding what "constructive" means here and that the real numbers do exist in this set of axioms, or am I wrong in how I am interpreting the axioms?

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    $\begingroup$ What background does this person have? $\endgroup$
    – eyeballfrog
    Commented Mar 14, 2022 at 1:45
  • $\begingroup$ It sounds like more of a philosophy question. $\endgroup$
    – Lee Mosher
    Commented Mar 14, 2022 at 2:18
  • $\begingroup$ See Constructive Mathematics. $\endgroup$
    – Mauro ALLEGRANZA
    Commented Mar 14, 2022 at 12:50
  • $\begingroup$ And see this post for a quote about well-know constructivist Errett Bishop's point of view. $\endgroup$
    – Mauro ALLEGRANZA
    Commented Mar 14, 2022 at 12:55

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I would class that kind of viewpoint as a strong form of finitism. Most constructivists would consider $\pi$ and $\sqrt 2$ as fine since you can explicitly give finite rules (e.g. as algorithms on a computer) to find them to any desired degree of accuracy.

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