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This is a follow on question from my question here

Consider the hypotenuse of a right angle triangle where the opposite and adjacent sides both have length 1. The hypotenuse has length sqrt(2)... until you measure it, at which point it takes on a discrete value, based on the precision of your ruler.

I am wondering if this is related to the question/conundrum as in quantum mechanics, where multiple states exist at the same time until measured. In both cases doing a measurement defines the infinite discretely. Does it matter that my triangle is an abstract geometric shape, removed from the physical world, whereas a sub-atomic particle is something real, but unmeasured by the universe?

Are uncertainty and infinity / certainty and discreteness philosophically related? Is the whole thing a question of the precision of the ruler?

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    No, finite measurement precision has nothing to do with quantum uncertainty. The latter only precludes you from measuring two non-commuting observables with infinite precision simultaneously. Both Euclidean geometry and quantum mechanics are idealized models, so that makes no difference either. Infinity is related to precision, not certainty, and those are two different things. One can be certain about an imprecise measurement and uncertain about a precise one.
    – Conifold
    Commented Aug 23, 2019 at 18:17
  • When Conifold says that one can be uncertain about an imprecise measurement, he is expressing the concept of accuracy. Note both accuracy and precision express certainties of different sorts. The former expresses the certainty of the correspondence between the measurement and the length measured. Precision expresses the degree of certainty of the lengths compatibility with other lengths. A 4.1 m length (rounded from 4.09? 4.14?) may exceed a specification that a 4.1000 length will not depending on a required tolerance.
    – J D
    Commented Aug 23, 2019 at 18:48
  • It's not clear why you suppose that measuring a triangle's side changes its length. A right triangle with legs of length 1 has a hypotenuse that has length sqrt(2), full stop. You can measure that with a simple yardstick or with the most advanced atomic ruler in the world, but it won't change the length of the thing you're measuring - the only thing that changes is what your measurement device reads.
    – Nuclear Hoagie
    Commented Aug 23, 2019 at 18:49
  • He's struggling to articulate the difference between a physical length and the representation of the length. And, technically, at the atomic level, all lengths are in flux because of vibrations and thermal expansions and contractions.
    – J D
    Commented Aug 23, 2019 at 18:51

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Uncertainty and infinity are related but different concepts.

On the one hand, uncertainty is a concept that expresses a state of modality, or whether or not knowledge or information is complete or certain. This is related to such concepts as probability and determinism. Infinity is a concept related to cardinality and ordinality, that is to say counting and ordering.

Precision is a concept that bridges the two concepts, by expressing the modality of a numerical quantity. The square root of two serves as a perfect example to explain how modality, precision, and quantity are related.

In the abstract, the square root of two is a number that is defined as an operation performed on a natural number. In essence, mathematical philosophy states that the square root of two is irrational, that is to say cannot be expressed as a ratio of two integers, and any attempt to find it therefore must be a non-terminating, non-repeating decimal. (Note non-terminating, repeating decimals can be expressed as fractions.)

However, in practical application, say one is building a truss for a roof whereby the angle at the ridgeboard is 45 degrees, one needs to actually cut rafters and the bottom chord, and so having a length of lumber and cutting the square root of two needs to, and can be done. Does it make a difference of discussing the root of 2 as a mathematical object and a practical measurement with a ruler with limited precision? Absolutely. It is frequently said that an engineer is a mathematician that rounds off numbers, and in computer science the misconstruction of circuitry to conduct floating-point operations can lead to errors in computation.

This is where the concept of precision comes in. The precision of a measurement is a quantity that expresses a degree of certainty about the measurement. If the chord needs to be cut to a relative length of square root of two, the square root of two can be expressed to some degree or number of decimal places. For instance, if the rafters are 1.000 meter, then the chord can be cut to a precision of 1.4 meters or 1.41 meters or even 1.414 meters. The more precise the measurement, the more certainty that the cut will function in the system.

To answer the second part of your question, yes, certainty and discreteness are related, because the more precise a measurement, the more information you possess. Small differences in measurement can lead to big differences in the outcome of physical systems that are constructed. That's why NASA uses multiple teams all of whom do calculations with as much precision as possible and sometimes averages their results.

As all measurements have a degree of precision, they by definition also have a degree of imprecision. The question of how much precision is needed or possible is a subject that is covered under the debate over the nature of infinity, with some people rejecting it (no such thing as infinity), some people admitting it occurs in a practical, physical context and is limited (potential infinity), and yet others believing infinity is actual characteristic of the universe itself (actual infinity).

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the idea of an irrational number like sqrt(2) has nothing to do either mathematically or physically with quantum uncertainty (i.e., the noncommutativity of the position and momentum operators); the concept of irrationality can be demonstrated in the absence of quantum mechanics and vice versa.

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It does , it does not... Clearly it depends on the type of question so let's say if I asked a question to u whether a solution is positive or not.. u get solution as infinity ofc it is positive. On the case if I tell u find the value of sinx at infinity it is uncertain

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