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> Such numbers must exist because there are only a countably infinite number of numbers that can be described using a finite collection of symbols, but there ar

by orlp 4y ago

> Such numbers must exist because there are only a countably infinite number of numbers that can be described using a finite collection of symbols, but there are an uncountably infinite number of real numbers. So not only are there numbers that cannot be described using a finite number of symbols, there are vastly more of these than numbers that can be so described. Why is almost no one more critical of this? A 'number' you can't describe or do computation with? Why do we still call this a number? If I ask Google to define number I get > an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations. Almost all (in the mathematical sense) 'real' 'numbers' can not be expressed, nor can be used for counting or making calculations. Over the years I'm becoming more and more convinced we should reject the 'real' numbers as our default mathematical object for numbers, and switch to a basis that has a computable foundation. There is some work being done in this area, but not nearly enough in my opinion: https://en.wikipedia.org/wiki/Computable_number#Use_in_place_of_the_reals https://en.wikipedia.org/wiki/Computable_number#Use_in_place...

dia80 4y ago
You are going to love this: How real are real numbers? https://arxiv.org/abs/math/0411418 https://arxiv.org/abs/math/0411418
deleted 4y ago
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mik1998 4y ago
Real numbers are interesting and math is not about computation, counting or calculation.
- hgsgm 4y ago
  math is not only about computation, counting or calculation.
hgsgm 4y ago
Mathematicians hijacked numbers. Computer scientist don't accept noncomputable objects. Mathematicians do. To a scientist, existence means "you, or someone, can find it, at least potentially". To a non-constructive mathematician (the common kind) existence means "you can't prove it doesn't exist, and I choose to believe it does, at least in the current game I am playing." Mathematicians use the term "real numbers" to describe precisely the set of numbers you get when you take all the numbers that could plausibly be computed, and then declare that all the holes between numbers we can individually compute/describe are completely full of numbers. This is justified because you can't prove that any specific one of them is impossible to write. (But there are some so called "real" numbers that are impossible to calculate, even approximately, like Chaitin's constant https://en.m.wikipedia.org/wiki/Chaitin%27s_constant https://en.m.wikipedia.org/wiki/Chaitin%27s_constant but you could write it in infinite time if God told you the digits.)
- drdeca 4y ago
  > Computer scientist don't accept noncomputable objects. People who study computability certainly do, and I would think they count as computer scientists.
- bmitc 4y ago
  > Computer scientist don't accept noncomputable objects. Mathematicians do. Computer scientists don't have a choice. And not all mathematicians do accept non-computable objects. Constructive and intuitionistic mathematics are two approaches that bring some questions (doubt?) of the continuum.
- shkkmo 4y ago
  > But there are some so called "real" numbers that are impossible to calculate, even approximately, like Chaitin's constant https://en.m.wikipedia.org/wiki/ https://en.m.wikipedia.org/wiki/ Just because you can create a definition, doesn't mean that definition names a thing. We can define "the set of all sets that do not contain themselves", but such a definition is nonsensical in that trying to apply it creates a contradiction. To me, the incomputability of Chaitin's constant seems very similar, except it's contradiction is buried inside the halting problem.

by orlp 4y ago

dia80 4y ago

You are going to love this: How real are real numbers? https://arxiv.org/abs/math/0411418 https://arxiv.org/abs/math/0411418

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[deleted]

mik1998 4y ago

Real numbers are interesting and math is not about computation, counting or calculation.

hgsgm 4y ago
math is not only about computation, counting or calculation.

hgsgm 4y ago

Mathematicians hijacked numbers. Computer scientist don't accept noncomputable objects. Mathematicians do. To a scientist, existence means "you, or someone, can find it, at least potentially". To a non-constructive mathematician (the common kind) existence means "you can't prove it doesn't exist, and I choose to believe it does, at least in the current game I am playing." Mathematicians use the term "real numbers" to describe precisely the set of numbers you get when you take all the numbers that could plausibly be computed, and then declare that all the holes between numbers we can individually compute/describe are completely full of numbers. This is justified because you can't prove that any specific one of them is impossible to write. (But there are some so called "real" numbers that are impossible to calculate, even approximately, like Chaitin's constant https://en.m.wikipedia.org/wiki/Chaitin%27s_constant https://en.m.wikipedia.org/wiki/Chaitin%27s_constant but you could write it in infinite time if God told you the digits.)

drdeca 4y ago
> Computer scientist don't accept noncomputable objects. People who study computability certainly do, and I would think they count as computer scientists.
bmitc 4y ago
> Computer scientist don't accept noncomputable objects. Mathematicians do. Computer scientists don't have a choice. And not all mathematicians do accept non-computable objects. Constructive and intuitionistic mathematics are two approaches that bring some questions (doubt?) of the continuum.
shkkmo 4y ago
> But there are some so called "real" numbers that are impossible to calculate, even approximately, like Chaitin's constant https://en.m.wikipedia.org/wiki/ https://en.m.wikipedia.org/wiki/ Just because you can create a definition, doesn't mean that definition names a thing. We can define "the set of all sets that do not contain themselves", but such a definition is nonsensical in that trying to apply it creates a contradiction. To me, the incomputability of Chaitin's constant seems very similar, except it's contradiction is buried inside the halting problem.