Do the Real Numbers Exist?
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Do the Real Numbers Exist?
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Re: Do the Real Numbers Exist?
#2The hypotenuse of a triangle with unit length sides is real enough.
Re: Do the Real Numbers Exist?
#3There are the useful ones we know about, like pi or e, which can be described to arbitrary precision, but you cannot "write them out" bc that would require infinite paper and a lot of pencils. But they're useful, they have meaning to us, so we give them nicknames and use those.
But what about all those other ones? Is it possible that there are certain ones, of future usefulness, that we haven't yet recognized? Are there, perhaps, different classes of such special numbers?
Maybe it's just a trick of our notational system; writing infinite strings of digits does let us describe some numbers in a vast uncountable sea of possibilities, but just bc the notation makes it possible, doesn't mandate that for any arbitrary string there is ultimately some succinct description, that elaborates to that string. In effect, the names are the numbers.
Yeah, these are fun things to think about.
Re: Do the Real Numbers Exist?
#4The hypotenuse of a triangle with unit length sides is real enough.
I assume you mean in the physical sense?
What if you get down to the quantum level? You don't have a straight line in the Euclidean sense any longer.
Re: Do the Real Numbers Exist?
#5Numbers aren’t real.
Re: Do the Real Numbers Exist?
#6There are the useful ones we know about, like pi or e, which can be described to arbitrary precision, but you cannot "write them out" bc that would require infinite paper and a lot of pencils. But they're useful, they have meaning to us, so we give them nicknames and use those. But what about all those other ones? Is it possible that there are certain ones, of future usefulness, that we haven't yet recognized? Are th…
The subset of numbers for which you can compute successively better approximations, such as pi and e, is countable. It’s called the set of computable numbers and contains everything that we could even in theory ever describe or “use” somehow. The actually uncountable subset of reals is by definition forever out of our reach, which is what makes uncountable sets… rather ontologically suspicious.
Re: Do the Real Numbers Exist?
#7The hypotenuse of a triangle with unit length sides is real enough.
I assume you mean in the physical sense? What if you get down to the quantum level? You don't have a straight line in the Euclidean sense any longer.
Yeah instead you have probability theory baked right in, real numbers and all!
Re: Do the Real Numbers Exist?
#8The hypotenuse of a triangle with unit length sides is real enough.
Algebraic reals are the easy part, as are the few familiar classes of transcendentals that include numbers like pi and e. There’s only countably many of those. The set of all reals that can be described is countable. It’s the literally undescribable ones that are the problem, ontologically speaking
Re: Do the Real Numbers Exist?
#9The hypotenuse of a triangle with unit length sides is real enough.
Is it? Can you measure it? Do distances with arbitrary precision even exist?
Re: Do the Real Numbers Exist?
#10The hypotenuse of a triangle with unit length sides is real enough.
Algebraic reals are the easy part, as are the few familiar classes of transcendentals that include numbers like pi and e. There’s only countably many of those. The set of all reals that can be described is countable. It’s the literally undescribable ones that are the problem, ontologically speaking
Ontologically speaking describing members of any uncountable set is a problem.