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How real are real numbers? (2004)
31–40 of 108 posts
Re: How real are real numbers? (2004)
#32Earlier quoted context omitted.
"suppose that the minimum distance is one inch. well, that's one of something. so now imagine half of that! there you go: one half. a physically unrealizable number."
Except that you’re assuming that 1 must necessarily correspond to that minimum distance. Keeping that situation, we can just say that 1 corresponds to two inches and then realize 1/2 as that number. Classical geometry (a la Euclid) allows for constructing a lot of numbers. We get natural numbers pretty cheaply, negative integers through adding in a concept of directionality (zero is a bit of an imaginative leap which…
One could make the argument that the only numbers that actually “exist” are the natural numbers, but the question ultimately is can you model any real number in the physical universe. Modeling ½ is simply a question of picking a unit to be 1 and finding its midpoint (or for that matter, declaring two apples to be “1” and thus a single apple would be “½”, although it’s a bit of a challenge to use apples to model (2-√3)/5
Re: How real are real numbers? (2004)
#33I think people overindex on the continuity problem with the reals. I'm personally a bit of real-number denier myself as a constructivist / intuitionalist. But when we say things like "the rationals are discrete" or "the computable numbers are discrete" these are very specific claims in the domain of measure theory, a theory which yields almost nothing of value except endless paradoxes and naval-gazing nonsense. Simil…
> What do "real" numbers buy you?
They're a powerful abstraction - the base concept of a smooth continuous complete domain which encodes non-trivial relationships, and is a prototype for other analytic abstractions.
The reals are the philosophical base class for some very useful mathematical objects. Computability and physicality are both side issues.
Re: How real are real numbers? (2004)
#34It is too bad we don't have a pithy familiar term for the computable / definable / "nameable" / constructible reals. The most general class of undisputed numbers, consistent with the forms we actually use to represent quantities and perform numerical operations.
I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers.
Re: How real are real numbers? (2004)
#35I think people overindex on the continuity problem with the reals. I'm personally a bit of real-number denier myself as a constructivist / intuitionalist. But when we say things like "the rationals are discrete" or "the computable numbers are discrete" these are very specific claims in the domain of measure theory, a theory which yields almost nothing of value except endless paradoxes and naval-gazing nonsense. Simil…
> these are very specific claims in the domain of measure theory, a theory which yields almost nothing of value except endless paradoxes and naval-gazing nonsense
Measure theory is used for lots of practical things, for example probability theory.
Re: How real are real numbers? (2004)
#36Earlier quoted context omitted.
I don't think your position is silly, but this is not a great argument for it. > But when we say things like "the rationals are discrete" In the usual topology they are not? > In the practical sense the rational numbers are quite continuous -- between any two rational numbers there are an infinite (unbounded) number of rational numbers -- there's no notion of a "leap" the way there is with the integers. This characte…
No, the rationals are not discrete in the usual topology. They end up being discrete when we consider continuous mappings from R->Q though. That is the "technical" sense that I refer to. The rationals, as you say, are dense in R but they are also dense in the computables. The big Cantorian leap that we make is when we use the diagonal argument to argue that the rationals are countable. All the real construction techn…
The word you are looking for, probably, is "totally disconnected". Discrete always refers to the "discrete topology".
Re: How real are real numbers? (2004)
#37Re: How real are real numbers? (2004)
#38It is too bad we don't have a pithy familiar term for the computable / definable / "nameable" / constructible reals. The most general class of undisputed numbers, consistent with the forms we actually use to represent quantities and perform numerical operations. I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers.
> I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers.
They're worse. Just having another dimension is significantly more relevant to reality.
And the reals also ruin the word "normal".
Re: How real are real numbers? (2004)
#39Earlier quoted context omitted.
Except that you’re assuming that 1 must necessarily correspond to that minimum distance. Keeping that situation, we can just say that 1 corresponds to two inches and then realize 1/2 as that number. Classical geometry (a la Euclid) allows for constructing a lot of numbers. We get natural numbers pretty cheaply, negative integers through adding in a concept of directionality (zero is a bit of an imaginative leap which…
One could make the argument that the only numbers that actually “exist” are the natural numbers, but the question ultimately is can you model any real number in the physical universe. Modeling ½ is simply a question of picking a unit to be 1 and finding its midpoint (or for that matter, declaring two apples to be “1” and thus a single apple would be “½”, although it’s a bit of a challenge to use apples to model (2-√3…
Make a line of 10 apples and declare it to be 2 units long, then make a square that's 15 applies diagonal, finally measure how much longer the 10 apples are than the side of the square.
It's a pretty linear increase in complexity between the math and the apples.
Re: How real are real numbers? (2004)
#40This doesn't seem like a very good point to make, sure reals are uncountable and any set of them with labels is countable and of measure zero. That doesn't say anything about physics at all. In QM things are only discrete in certain ways, like energy levels, not positions. A wave function over space can take on any real valued value in its range. Probabilities are real numbers(norms of the wave functions), and there is no reason to believe they would be discrete. Take cosine squared, given any angle it takes on all values between zero and one at some point.