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You can't count reals or complex numbers. Real numbers by definition describe objects with infinite precision. In the real world infinite precision cannot exist

You can't count reals or complex numbers. Real numbers by definition describe objects with infinite precision. In the real world infinite precision cannot exist therefore real numbers are the limit of an arbitrary precision measuring process not real themselves. Meanwhile natural numbers like "two" most certainly do exist as a quantitative attribute of a set. Show me how you can count with reals.

HelloNurse 5y ago
In practical terms, "counting and measuring" means well-behaved arithmetic operations like addition, subtraction, multiplication, division, roots etc. (and often specific algebraic structures like rings, fields etc.) Rational and real numbers represent the most intuitive concept of quantity with different cardinality; natural numbers are more basic in theory but a restricted special case in most application (they can only count "countable" objects and they aren't dense); complex numbers compensate unnatural weirdness with important applications; then there are other very number-like niches (e.g. p-adic numbers, quaternions, algebraic extension fields...) and number-based arbitrary constructions (multidimensional vector spaces, matrices and tensors...). > Show me how you can count with reals. Like you count with natural numbers, plus many other possibilities because there are more numbers to count with. The same applies to rational numbers (which exist more than real numbers), complex numbers, etc.
- TuringTest 5y ago
  > The same applies to rational numbers (which exist more than real numbers) Wait, did you say there are more rationals than reals? Isn't that the other way around? I don't know if that's a slip of the tongue or I'm missing something in my recall of basic math lessons
  - HelloNurse 5y ago
    Real numbers do not exist: they are convenient, but there is a high risk of crossing over from reality-relevant math to nonsense because of infinite calculations. Real numbers are of course more numerous than rational numbers, just like unicorn horns are more numerous than horse horns, but it's an entirely different question.
    - TuringTest 5y ago
      No numbers exist, they are abstraction on human-created procedures. Even naturals, which are an abstraction over pointing to things and counting.
  - gowld 5y ago
    They are "more existent" ("less fictional"), not "more in cardinality".
    - TuringTest 5y ago
      How are they more existent? They are both patterns of recognized concepts in human brains. Is it because more people understand rationals than reals? Ok, that would make rationals more existent, as they exist in more brains.
croes 5y ago
I don't need to, I can measure and label with real numbers. That's enough for being a number.
- gowld 5y ago
  You'll only ever use 0% of the real numbers for that. Unless you say that you count with "the reals", You don't need and won't use "the reals" for measurement and labelling, 100% of which are indescribable.
  - morpheos137 5y ago
    technically you will use an infintesimally small real number approaching 0%...
TuringTest 5y ago
At the end of the day, a number is a _reification_ (or thing-a-fication, watching a process as a separate entity) of the processes of mapping and ordering. You may begin with a very simple use of those processes, which get you the natural numbers. But you may use them in more creative ways, which will get you other different classes of numbers. Mathematician love to explore all the implications of using basic processes and combining already defined numbers to create new kinds, never seen before. For example, if you start with number '1' and apply operator _successor_ (or "adding one more"), you get the *natural numbers*, which are a mapping from the size of sets to strings of operators {'1', 'successor(1)', 'successor(successor(1))', ... }. You can use this process to define an order: number A is smaller than B if you can repeatedly apply _successor_ to A and generate B in a finite time. (Not the best definition, I know, but bear with me for a second). If you reverse the _successor_ operator, you get the _predecessor_, which can be used to dismount large numbers and make them smaller. If you apply _predecessor_ to '1', you get 'predecessor(1)' which doesn't match any of the *natural* numbers as defined in the paragraph above. However, this new number is useful because you can map it to a collection without elements, allowing you to count a set of size 'zero'. This set doesn't exist, but it has a well defined size thanks to the process explained above. (You may as well apply _predecessor_ to 'zero' and count *negative numbers*, which allow you to create a mapping with debt, and thus count _I owe you_ amounts that don't exist in the physical world either). Now as for real and complex numbers, they can't count physical objects as you said because at some point you can't measure smaller and smaller magnitudes (though you can create orders with them, even if it doesn't directly involve the _successor_ operator). But mathematicians use them to count the size of _infinite sets_, which being immaterial never lose precision; you just know you can repeat their defining processes once and again, showing that there exist a mapping between any step in the process and the instance of the real or complex number that correspond to its size.

HelloNurse 5y ago

In practical terms, "counting and measuring" means well-behaved arithmetic operations like addition, subtraction, multiplication, division, roots etc. (and often specific algebraic structures like rings, fields etc.) Rational and real numbers represent the most intuitive concept of quantity with different cardinality; natural numbers are more basic in theory but a restricted special case in most application (they can only count "countable" objects and they aren't dense); complex numbers compensate unnatural weirdness with important applications; then there are other very number-like niches (e.g. p-adic numbers, quaternions, algebraic extension fields...) and number-based arbitrary constructions (multidimensional vector spaces, matrices and tensors...). > Show me how you can count with reals. Like you count with natural numbers, plus many other possibilities because there are more numbers to count with. The same applies to rational numbers (which exist more than real numbers), complex numbers, etc.

TuringTest 5y ago
> The same applies to rational numbers (which exist more than real numbers) Wait, did you say there are more rationals than reals? Isn't that the other way around? I don't know if that's a slip of the tongue or I'm missing something in my recall of basic math lessons
- HelloNurse 5y ago
  Real numbers do not exist: they are convenient, but there is a high risk of crossing over from reality-relevant math to nonsense because of infinite calculations. Real numbers are of course more numerous than rational numbers, just like unicorn horns are more numerous than horse horns, but it's an entirely different question.
  - TuringTest 5y ago
    No numbers exist, they are abstraction on human-created procedures. Even naturals, which are an abstraction over pointing to things and counting.
- gowld 5y ago
  They are "more existent" ("less fictional"), not "more in cardinality".
  - TuringTest 5y ago
    How are they more existent? They are both patterns of recognized concepts in human brains. Is it because more people understand rationals than reals? Ok, that would make rationals more existent, as they exist in more brains.

croes 5y ago

I don't need to, I can measure and label with real numbers. That's enough for being a number.

gowld 5y ago
You'll only ever use 0% of the real numbers for that. Unless you say that you count with "the reals", You don't need and won't use "the reals" for measurement and labelling, 100% of which are indescribable.
- morpheos137 5y ago
  technically you will use an infintesimally small real number approaching 0%...

TuringTest 5y ago

At the end of the day, a number is a _reification_ (or thing-a-fication, watching a process as a separate entity) of the processes of mapping and ordering. You may begin with a very simple use of those processes, which get you the natural numbers. But you may use them in more creative ways, which will get you other different classes of numbers. Mathematician love to explore all the implications of using basic processes and combining already defined numbers to create new kinds, never seen before. For example, if you start with number '1' and apply operator _successor_ (or "adding one more"), you get the *natural numbers*, which are a mapping from the size of sets to strings of operators {'1', 'successor(1)', 'successor(successor(1))', ... }. You can use this process to define an order: number A is smaller than B if you can repeatedly apply _successor_ to A and generate B in a finite time. (Not the best definition, I know, but bear with me for a second). If you reverse the _successor_ operator, you get the _predecessor_, which can be used to dismount large numbers and make them smaller. If you apply _predecessor_ to '1', you get 'predecessor(1)' which doesn't match any of the *natural* numbers as defined in the paragraph above. However, this new number is useful because you can map it to a collection without elements, allowing you to count a set of size 'zero'. This set doesn't exist, but it has a well defined size thanks to the process explained above. (You may as well apply _predecessor_ to 'zero' and count *negative numbers*, which allow you to create a mapping with debt, and thus count _I owe you_ amounts that don't exist in the physical world either). Now as for real and complex numbers, they can't count physical objects as you said because at some point you can't measure smaller and smaller magnitudes (though you can create orders with them, even if it doesn't directly involve the _successor_ operator). But mathematicians use them to count the size of _infinite sets_, which being immaterial never lose precision; you just know you can repeat their defining processes once and again, showing that there exist a mapping between any step in the process and the instance of the real or complex number that correspond to its size.