Don’t even need to go that far. Just take e from logarithms and compound interests and you’re already in NOT-INTUITIVE-TO-HUMANS-land.
I.e. How does a “perfect sphere” even remotely make sense in the real world? What the hell does e correspond to in the universe? A ton of trig, logarithm and limit stuff can prove problematic to simpler philosophical analyses of mathematics. And it’s not like you can just throw out e and π either, since they yield accurate predictions so obviously there’s something “true” or at least valid about them.
There is no PRACTICAL difference between a theory that treats e and pi as “rational but not yet perfectly determined” and “irrational.” And by practical I mean as used by practitioners, people who build stuff, people who survey, even people who need to know the parallax to the most distant object in the universe between two relatively closely spaced telescopes on earth.
“infinite” broken down to its roots means “not finite.” The practical value of “not finite” is probably “so large that we need to be sure we always get the same answer when we assume it is a million times larger,” that is, we verify that we have a PRACTICAL solution that has converged as we make x larger and larger, and whether x is > 1 km (when designing a microcircuit) or x > 1 quadrillion light-years, we never need to know, PRACTICALLY, what happens when x finally reaches infinity.
Engineering principles which are WRONG when quantum and relativistic considerations are taken in to account stand firmly and valuably behind quadrillions of dollars worth of human infrastructure.
Philosophical theories which are limited in validity only to these principles are possibly not useless.
Sure, but I can’t name an accessible but deep reference about e or pi of the top of my head.
What the hell does e correspond to in the universe?
It’s a number than happens to have interesting mathematical properties—but it is no harder to explain physically than any other irrational number. Even if one thinks numbers are made of apples, one ought to be able to conceive of numbers of apples that aren’t integers or rationals.
In short, I don’t think the interesting constants are cleanest examples of the problems with mathematical pure physicalism.
Hah. The real hidden question was actually “How does one arrive at e specifically by looking at the universe, and why does it work like that?”, I think.
I agree that they’re not the most clear stuff, but I’ve listed them as the most accessible wonder-inducing mathematics-related points of interest.
That’s an interesting question, and I have no idea about the answer.. If aliens asked me to define e, I’d start talking about exponential functions that were their own derivative. But I have no idea if that’s the historical motivation for noticing e.
Pi is obviously much easier, since it is part of the ratios linking circle diameter to circle perimeter and circle area.
Pi is obviously much easier, since it is part of the ratios linking circle diameter to circle perimeter and circle area.
If I had to explain Pi to real aliens that somehow understood English but not our mathematics, I would start with straight lines of a fixed length (radius) that share one (fixed) endpoint and where the other (movable) endpoints get gradually closer and closer.
Some multiple of pi is the ratio you apparently get as you compare those lengths and extrapolate for infinitely-closer-and-closer lines.
Sounds simple enough, as far as explaining abstract concepts to real aliens goes.
In my imagination, I have a chalkboard, but no other ability to communicate. So, lots of drawing circles (with emphasis on diameters and circumferences).
Don’t even need to go that far. Just take e from logarithms and compound interests and you’re already in NOT-INTUITIVE-TO-HUMANS-land.
I.e. How does a “perfect sphere” even remotely make sense in the real world? What the hell does e correspond to in the universe? A ton of trig, logarithm and limit stuff can prove problematic to simpler philosophical analyses of mathematics. And it’s not like you can just throw out e and π either, since they yield accurate predictions so obviously there’s something “true” or at least valid about them.
There is no PRACTICAL difference between a theory that treats e and pi as “rational but not yet perfectly determined” and “irrational.” And by practical I mean as used by practitioners, people who build stuff, people who survey, even people who need to know the parallax to the most distant object in the universe between two relatively closely spaced telescopes on earth.
“infinite” broken down to its roots means “not finite.” The practical value of “not finite” is probably “so large that we need to be sure we always get the same answer when we assume it is a million times larger,” that is, we verify that we have a PRACTICAL solution that has converged as we make x larger and larger, and whether x is > 1 km (when designing a microcircuit) or x > 1 quadrillion light-years, we never need to know, PRACTICALLY, what happens when x finally reaches infinity.
Engineering principles which are WRONG when quantum and relativistic considerations are taken in to account stand firmly and valuably behind quadrillions of dollars worth of human infrastructure.
Philosophical theories which are limited in validity only to these principles are possibly not useless.
Sure, but I can’t name an accessible but deep reference about e or pi of the top of my head.
It’s a number than happens to have interesting mathematical properties—but it is no harder to explain physically than any other irrational number. Even if one thinks numbers are made of apples, one ought to be able to conceive of numbers of apples that aren’t integers or rationals.
In short, I don’t think the interesting constants are cleanest examples of the problems with mathematical pure physicalism.
Hah. The real hidden question was actually “How does one arrive at e specifically by looking at the universe, and why does it work like that?”, I think.
I agree that they’re not the most clear stuff, but I’ve listed them as the most accessible wonder-inducing mathematics-related points of interest.
That’s an interesting question, and I have no idea about the answer.. If aliens asked me to define e, I’d start talking about exponential functions that were their own derivative. But I have no idea if that’s the historical motivation for noticing e.
Pi is obviously much easier, since it is part of the ratios linking circle diameter to circle perimeter and circle area.
If I had to explain Pi to real aliens that somehow understood English but not our mathematics, I would start with straight lines of a fixed length (radius) that share one (fixed) endpoint and where the other (movable) endpoints get gradually closer and closer.
Some multiple of pi is the ratio you apparently get as you compare those lengths and extrapolate for infinitely-closer-and-closer lines.
Sounds simple enough, as far as explaining abstract concepts to real aliens goes.
In my imagination, I have a chalkboard, but no other ability to communicate. So, lots of drawing circles (with emphasis on diameters and circumferences).