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Quantum theory based on real numbers can be experimentally falsified

nature.com

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Re: Quantum theory based on real numbers can be experimentally falsified

#11

Post by midjji »

btilly wrote: post #7
PaulDavisThe1st wrote: post #4

Earlier quoted context omitted.

If you think you can describe the physical world without any irrational numbers, when one of the most basic (the ratio of the area of a circle to its diameter) is irrational, I think you're probably mistaken.

You can describe it to within measurement error without any irrational numbers. And with fewer decimal places than you'd probably imagine. See https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimal... for more.

That often works, but not always, some systems generate sequences of operations which can be symbolically simplified, or equivalently could be exactly computed using reals, but which if computed using any finite precision will fail. A simple example would be solving for the position of a planet in orbit under simple Newtonian gravity, a sufficient number of revolutions latter. For any finite precision the number of orbits can always be increased until failure occurs.

Its a cool how few bits of pi are required to compute the circumference of the earth to within an atoms width accuracy. But equally cool how bad it gets if you try the same to compute the position of earth 2^64 years later.

Re: Quantum theory based on real numbers can be experimentally falsified

#12

Post by katmannthree »

analog31 wrote: post #3

Can we go further and ditch the reals, relying instead on rational numbers or even IEEE floats? After all, the computers that we use for predicting empirical results all run on integers.

Can you? Most likely. Should you? You’ll need to reprove more than a handful of theorems, and for what? What advantage does using rationals instead of reals get you?

You might enjoy taking courses in real & complex analysis, the general purpose of which is to impart upon the receiver an understanding of why we’ve constructed those particular number systems and how despite the names they both describe things which are perfectly real in the philosophical sense.

Edit: recall that the rationals are simply defined as the set of numbers which can be represented in the form a/b where a is an integer and b is a nonzero integer. This isn’t some deep philosophical tie to an underlying reality, it’s just our first attempt at defining more useful numbers that lie between other useful numbers we already invented (the integers et al). The real and complex numbers are literally just the extension of that process, filling in holes between useful numbers with more numbers until the set is closed (i.e. there are no more holes, every operation between members of the set results in another member of the set). Closure, the real reason we care so much about the reals, is just a surprise tool that helps us later. For anyone who made it this far: if you find any of this interesting you should find a book or lecture on analysis. It’s not particularly difficult and presents deeper insights into the math you likely already learned.

Re: Quantum theory based on real numbers can be experimentally falsified

#13

Post by kadoban »

roywiggins wrote: post #10
PaulDavisThe1st wrote: post #4

Earlier quoted context omitted.

If you think you can describe the physical world without any irrational numbers, when one of the most basic (the ratio of the area of a circle to its diameter) is irrational, I think you're probably mistaken.

Pi is the ratio of a mathematical circle to its diameter, but there are no physical circles which have that ratio as exactly Pi, and even if one existed, you'd never be able to distinguish it from one that was merely equal to Pi to the accuracy you are capable of measuring, because you'd need to measure with infinite precision, which you can't.

Orbits over long stretches of time seem likely to need arbitrarily high amounts of accuracy in pi. Sure you can just pick a rational number close enough for the accuracy you need, but why should the definition of pi need to change based on what you're measuring?

Re: Quantum theory based on real numbers can be experimentally falsified

#14

Post by katmannthree »

roywiggins wrote: post #10
PaulDavisThe1st wrote: post #4

Earlier quoted context omitted.

If you think you can describe the physical world without any irrational numbers, when one of the most basic (the ratio of the area of a circle to its diameter) is irrational, I think you're probably mistaken.

Pi is the ratio of a mathematical circle to its diameter, but there are no physical circles which have that ratio as exactly Pi, and even if one existed, you'd never be able to distinguish it from one that was merely equal to Pi to the accuracy you are capable of measuring, because you'd need to measure with infinite precision, which you can't.

That’s not isomorphic to the discussion at hand though, which refers to modeling systems. Being able to physically construct an object within epsilon does not imply that you won’t run into precision issues when modeling said object at the same degree of precision.

In other words, cutting your beams to +/- 1/2” may work for each individual beam in a building but that does not imply that your building as a whole can tolerate an average beam length being +.499” above nominal.

Re: Quantum theory based on real numbers can be experimentally falsified

#16

Post by p2t2p »

PaulDavisThe1st wrote: post #4
analog31 wrote: post #3

Can we go further and ditch the reals, relying instead on rational numbers or even IEEE floats? After all, the computers that we use for predicting empirical results all run on integers.

If you think you can describe the physical world without any irrational numbers, when one of the most basic (the ratio of the area of a circle to its diameter) is irrational, I think you're probably mistaken.

That is replacing actual reality with math all over again. In the actual reality this ratio is concrete, limited value. I would say that our numbers and approaches are limited that they are unable to precisely define it. It just we are so used to it that no one even thinks about challenging it. And even if they do - good luck finding funding.

Now we do have tools that overcome those limitations somewhat - like limits and stuff but doesn’t remove the need for better tools.

So I somewhat disagree with both of you - we need new, better numbers that are further from math abstraction and closer to actual reality.

Otherwise it is like this story with Poynting vector from Veritasium video - only confuses instead of explaining.

Re: Quantum theory based on real numbers can be experimentally falsified

#18

Post by EGreg »

MauranKilom wrote: post #9
PaulDavisThe1st wrote: post #4

Earlier quoted context omitted.

If you think you can describe the physical world without any irrational numbers, when one of the most basic (the ratio of the area of a circle to its diameter) is irrational, I think you're probably mistaken.

...but Planck length!

Perhaps that’s where quantum uncertainty reaches a threshold where you can’t make time based distinctions anymore.

Re: Quantum theory based on real numbers can be experimentally falsified

#19

Post by thereisnospork »

katmannthree wrote: post #14
roywiggins wrote: post #10

Earlier quoted context omitted.

Pi is the ratio of a mathematical circle to its diameter, but there are no physical circles which have that ratio as exactly Pi, and even if one existed, you'd never be able to distinguish it from one that was merely equal to Pi to the accuracy you are capable of measuring, because you'd need to measure with infinite precision, which you can't.

That’s not isomorphic to the discussion at hand though, which refers to modeling systems. Being able to physically construct an object within epsilon does not imply that you won’t run into precision issues when modeling said object at the same degree of precision. In other words, cutting your beams to +/- 1/2” may work for each individual beam in a building but that does not imply that your building as a whole can to…

>In other words, cutting your beams to +/- 1/2” may work for each individual beam in a building but that does not imply that your building as a whole can tolerate an average beam length being +.499” above nominal.

The stronger version of the argument is that the length of a steel beam cannot be more precise(-ish) than the radius of an iron atom, so only 10-12 decimal places (in meters) are required to fully describe a steel beam's length. Likewise an actual circle's area isn't a function of Pi, but is rather a 'really large number' regular polyhedron. Which could then be approximated by a fairly pedestrian number of decimal points of pi to atomic precision.

That said e.g. orbits are rather smooth, and could probably be considered to be fairly exact w.r.t. an arbitrary reference.

Re: Quantum theory based on real numbers can be experimentally falsified

#20

Post by drdec »

p2t2p wrote: post #16
PaulDavisThe1st wrote: post #4

Earlier quoted context omitted.

If you think you can describe the physical world without any irrational numbers, when one of the most basic (the ratio of the area of a circle to its diameter) is irrational, I think you're probably mistaken.

That is replacing actual reality with math all over again. In the actual reality this ratio is concrete, limited value. I would say that our numbers and approaches are limited that they are unable to precisely define it. It just we are so used to it that no one even thinks about challenging it. And even if they do - good luck finding funding. Now we do have tools that overcome those limitations somewhat - like limits…

Exactly. The square root of 2 had no more basis in reality than the so-called imaginary number i. Don't confuse the "real" in real numbers with reality, at this point it is just a name.
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