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Does that mean the rest of the digits of pi are not "real," at least according to a realist rather than a Platonic philosophical position on the meaning and nat

by api 11y ago

Does that mean the rest of the digits of pi are not "real," at least according to a realist rather than a Platonic philosophical position on the meaning and nature of mathematics? Seems like you could argue that digits beyond what are needed to render measurement to within one Planck length are meaningless and therefore a kind of fiction... at least if you take that philosophical position.

21echoes 11y ago
Pi is far more than just the ratio of the diameter to the circumference, even as a constructivist: https://affinemess.quora.com/What-is-math-pi-math-and-while-were-at-it-whats-math-e-math?__snids__=1596646755&__nsrc__=2 https://affinemess.quora.com/What-is-math-pi-math-and-while-...
venomsnake 11y ago
Pi has other uses. The digits are real, but they just don't matter in practical engineering. But lets assume that there really are 10 dimensions - in that case a volume of a 10 dimensional sphere will require (pi^5)*r^10. If you want to measure the volume to 1 plank 10 dimensional cube, you will need more digits.
Al-Khwarizmi 11y ago
Pi appears in many more places than in ratio of circumferences to their diameter. For example, if you flip a coin n times, the probability of getting exactly n heads and n tails asymptotically tends towards 1 / sqrt(π*n). The probability that two randomly chosen integers are coprime is 6/π^2. Etc. There is a list of formulae where π appears (some related to circles, some not) here: https://en.wikipedia.org/wiki/List_of_formulae_involving_π https://en.wikipedia.org/wiki/List_of_formulae_involving_π
- ikeboy 11y ago
  But how many coins can you flip before the heat death of the universe, and does using 43 digits make your expectation off by more than say 1/1000000? In other words, if you were trying to calculate pi by checking the actual probabilities, how many digits can you get?
mc808 11y ago
It's been estimated that if the universe were a computer, it could have performed no more than 10^120 operations on 10^90 bits of data so far (based on the size, age, and total energy of the known universe). http://arxiv.org/abs/quant-ph/0110141 http://arxiv.org/abs/quant-ph/0110141 I think the number of physically relevant bits of pi would be represented in there somewhere. But there's a long road ahead. If the universe keeps "computing" forever, the precision of numbers involved could also keep growing.
- ycmbntrthrwaway 11y ago
  If the universe were a computer, it would not have to compute π to "simulate" physical processes. Mechanical processes don't depend on the value of π directly. If universe had to know every constant somehow involved in the process, it would not be able to simulate computers, because it involves uncomputable numbers: https://en.wikipedia.org/wiki/Chaitin%27s_constant#Uncomputability https://en.wikipedia.org/wiki/Chaitin%27s_constant#Uncomputa...
- xzephyr 11y ago
  And wave functions collapse only upon measurement...clever lazy initialization
andrepd 11y ago
But pi is not an empirical constant. It is a mathematical constant, with well-defined formulas such as pi/4 = 1-1/3+1/5-1/7+...

by api 11y ago

21echoes 11y ago

Pi is far more than just the ratio of the diameter to the circumference, even as a constructivist: https://affinemess.quora.com/What-is-math-pi-math-and-while-were-at-it-whats-math-e-math?__snids__=1596646755&__nsrc__=2 https://affinemess.quora.com/What-is-math-pi-math-and-while-...

venomsnake 11y ago

Pi has other uses. The digits are real, but they just don't matter in practical engineering. But lets assume that there really are 10 dimensions - in that case a volume of a 10 dimensional sphere will require (pi^5)*r^10. If you want to measure the volume to 1 plank 10 dimensional cube, you will need more digits.

Al-Khwarizmi 11y ago

Pi appears in many more places than in ratio of circumferences to their diameter. For example, if you flip a coin n times, the probability of getting exactly n heads and n tails asymptotically tends towards 1 / sqrt(π*n). The probability that two randomly chosen integers are coprime is 6/π^2. Etc. There is a list of formulae where π appears (some related to circles, some not) here: https://en.wikipedia.org/wiki/List_of_formulae_involving_π https://en.wikipedia.org/wiki/List_of_formulae_involving_π

ikeboy 11y ago
But how many coins can you flip before the heat death of the universe, and does using 43 digits make your expectation off by more than say 1/1000000? In other words, if you were trying to calculate pi by checking the actual probabilities, how many digits can you get?

mc808 11y ago

It's been estimated that if the universe were a computer, it could have performed no more than 10^120 operations on 10^90 bits of data so far (based on the size, age, and total energy of the known universe). http://arxiv.org/abs/quant-ph/0110141 http://arxiv.org/abs/quant-ph/0110141 I think the number of physically relevant bits of pi would be represented in there somewhere. But there's a long road ahead. If the universe keeps "computing" forever, the precision of numbers involved could also keep growing.

ycmbntrthrwaway 11y ago
If the universe were a computer, it would not have to compute π to "simulate" physical processes. Mechanical processes don't depend on the value of π directly. If universe had to know every constant somehow involved in the process, it would not be able to simulate computers, because it involves uncomputable numbers: https://en.wikipedia.org/wiki/Chaitin%27s_constant#Uncomputability https://en.wikipedia.org/wiki/Chaitin%27s_constant#Uncomputa...
xzephyr 11y ago
And wave functions collapse only upon measurement...clever lazy initialization

andrepd 11y ago

But pi is not an empirical constant. It is a mathematical constant, with well-defined formulas such as pi/4 = 1-1/3+1/5-1/7+...