Mixing
What is the timeline of the sets of real, rational, integer, and natural numbers?
0
$begingroup$
Historically, did mathematicians:
1) construct the set of real numbers using rationals, integers, and naturals?
Or
2) did they already know the set of reals existed and partitioned it into rationals, integers, and naturals?
I'm curious to know the timeline of events. If #1 above is true, then I am assuming mathematicians could only define functions from $mathbb{Q},mathbb{Z},mathbb{N}$ such as $f: mathbb{Q} tomathbb{N}$. Then later on, once $mathbb{R}$ is constructed, they could define functions like $f: mathbb{R} tomathbb{R}$.
Also, where does the Completness Axiom fit into the timeline?
calculus math-history
asked Jan 15 at 1:21
user1068636user1068636
652719
$endgroup$
|
show 1 more comment
0
$begingroup$
Historically, did mathematicians:
1) construct the set of real numbers using rationals, integers, and naturals?
Or
2) did they already know the set of reals existed and partitioned it into rationals, integers, and naturals?
I'm curious to know the timeline of events. If #1 above is true, then I am assuming mathematicians could only define functions from $mathbb{Q},mathbb{Z},mathbb{N}$ such as $f: mathbb{Q} tomathbb{N}$. Then later on, once $mathbb{R}$ is constructed, they could define functions like $f: mathbb{R} tomathbb{R}$.
Also, where does the Completness Axiom fit into the timeline?
calculus math-history
asked Jan 15 at 1:21
user1068636user1068636
652719
$endgroup$
1
$begingroup$
Maybe one for hsm.stackexchange.com ?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 1:24
$begingroup$
This is a comment (as I am not a mathematical historian, and can't really speak authoritatively), but I would suspect that the answer is "neither." The Greeks worked with rational numbers, but also recognized that there were irrational numbers (hinting at the necessity of introducing the real numbers). Descartes and his contemporaries contrasted real numbers with imaginary numbers, so the concept has been around at least that long. The set theoretic constructions didn't come along until much later (late 19th or early 20th Century).
$endgroup$
– Xander Henderson
Jan 15 at 1:27
$begingroup$
@XanderHenderson - I think you pretty much answered my question. If I understand you correctly, you're saying that Greeks knew there have to be irrationals, but did not have the Completeness Axiom to point to, and so the set of real numbers were not formally constructed. But once the completeness axiom was recognized by mathematicians, the set theoretic constructions came along and formalized the set of reals.
$endgroup$
– user1068636
Jan 15 at 1:32
$begingroup$
No, it's not that simple. This is an extremely modern way of thinking of it that also is not clearly inevitable. The Greeks knew about decimal expansions of integers, but not of fractional numbers. There was no immediate reason for them to expect there to be any numbers other than the constructible numbers, which you can get by taking successive square roots. They were able to identify some numbers they couldn't figure out how to express this way, but they couldn't prove that it was impossible. They did not use negative numbers or zero.
$endgroup$
– Matt Samuel
Jan 15 at 2:02
$begingroup$
The Greeks wouldn't know the completeness axiom from a hole in the ground.
$endgroup$
– Matt Samuel
Jan 15 at 2:02
|
show 1 more comment
0
0
0
$begingroup$
Historically, did mathematicians:
1) construct the set of real numbers using rationals, integers, and naturals?
Or
2) did they already know the set of reals existed and partitioned it into rationals, integers, and naturals?
I'm curious to know the timeline of events. If #1 above is true, then I am assuming mathematicians could only define functions from $mathbb{Q},mathbb{Z},mathbb{N}$ such as $f: mathbb{Q} tomathbb{N}$. Then later on, once $mathbb{R}$ is constructed, they could define functions like $f: mathbb{R} tomathbb{R}$.
Also, where does the Completness Axiom fit into the timeline?
calculus math-history
asked Jan 15 at 1:21
user1068636user1068636
652719
$endgroup$
Historically, did mathematicians:
1) construct the set of real numbers using rationals, integers, and naturals?
Or
2) did they already know the set of reals existed and partitioned it into rationals, integers, and naturals?
I'm curious to know the timeline of events. If #1 above is true, then I am assuming mathematicians could only define functions from $mathbb{Q},mathbb{Z},mathbb{N}$ such as $f: mathbb{Q} tomathbb{N}$. Then later on, once $mathbb{R}$ is constructed, they could define functions like $f: mathbb{R} tomathbb{R}$.
Also, where does the Completness Axiom fit into the timeline?
calculus math-history
calculus math-history
asked Jan 15 at 1:21
user1068636user1068636
652719
asked Jan 15 at 1:21
user1068636user1068636
652719
asked Jan 15 at 1:21
user1068636user1068636
652719
asked Jan 15 at 1:21
user1068636user1068636
652719
asked Jan 15 at 1:21
user1068636user1068636
652719
652719
1
$begingroup$
Maybe one for hsm.stackexchange.com ?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 1:24
$begingroup$
This is a comment (as I am not a mathematical historian, and can't really speak authoritatively), but I would suspect that the answer is "neither." The Greeks worked with rational numbers, but also recognized that there were irrational numbers (hinting at the necessity of introducing the real numbers). Descartes and his contemporaries contrasted real numbers with imaginary numbers, so the concept has been around at least that long. The set theoretic constructions didn't come along until much later (late 19th or early 20th Century).
$endgroup$
– Xander Henderson
Jan 15 at 1:27
$begingroup$
@XanderHenderson - I think you pretty much answered my question. If I understand you correctly, you're saying that Greeks knew there have to be irrationals, but did not have the Completeness Axiom to point to, and so the set of real numbers were not formally constructed. But once the completeness axiom was recognized by mathematicians, the set theoretic constructions came along and formalized the set of reals.
$endgroup$
– user1068636
Jan 15 at 1:32
$begingroup$
No, it's not that simple. This is an extremely modern way of thinking of it that also is not clearly inevitable. The Greeks knew about decimal expansions of integers, but not of fractional numbers. There was no immediate reason for them to expect there to be any numbers other than the constructible numbers, which you can get by taking successive square roots. They were able to identify some numbers they couldn't figure out how to express this way, but they couldn't prove that it was impossible. They did not use negative numbers or zero.
$endgroup$
– Matt Samuel
Jan 15 at 2:02
$begingroup$
The Greeks wouldn't know the completeness axiom from a hole in the ground.
$endgroup$
– Matt Samuel
Jan 15 at 2:02
|
show 1 more comment
1
$begingroup$
Maybe one for hsm.stackexchange.com ?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 1:24
$begingroup$
This is a comment (as I am not a mathematical historian, and can't really speak authoritatively), but I would suspect that the answer is "neither." The Greeks worked with rational numbers, but also recognized that there were irrational numbers (hinting at the necessity of introducing the real numbers). Descartes and his contemporaries contrasted real numbers with imaginary numbers, so the concept has been around at least that long. The set theoretic constructions didn't come along until much later (late 19th or early 20th Century).
$endgroup$
– Xander Henderson
Jan 15 at 1:27
$begingroup$
@XanderHenderson - I think you pretty much answered my question. If I understand you correctly, you're saying that Greeks knew there have to be irrationals, but did not have the Completeness Axiom to point to, and so the set of real numbers were not formally constructed. But once the completeness axiom was recognized by mathematicians, the set theoretic constructions came along and formalized the set of reals.
$endgroup$
– user1068636
Jan 15 at 1:32
$begingroup$
No, it's not that simple. This is an extremely modern way of thinking of it that also is not clearly inevitable. The Greeks knew about decimal expansions of integers, but not of fractional numbers. There was no immediate reason for them to expect there to be any numbers other than the constructible numbers, which you can get by taking successive square roots. They were able to identify some numbers they couldn't figure out how to express this way, but they couldn't prove that it was impossible. They did not use negative numbers or zero.
$endgroup$
– Matt Samuel
Jan 15 at 2:02
$begingroup$
The Greeks wouldn't know the completeness axiom from a hole in the ground.
$endgroup$
– Matt Samuel
Jan 15 at 2:02
1
1
$begingroup$
Maybe one for hsm.stackexchange.com ?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 1:24
$begingroup$
Maybe one for hsm.stackexchange.com ?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 1:24
$begingroup$
This is a comment (as I am not a mathematical historian, and can't really speak authoritatively), but I would suspect that the answer is "neither." The Greeks worked with rational numbers, but also recognized that there were irrational numbers (hinting at the necessity of introducing the real numbers). Descartes and his contemporaries contrasted real numbers with imaginary numbers, so the concept has been around at least that long. The set theoretic constructions didn't come along until much later (late 19th or early 20th Century).
$endgroup$
– Xander Henderson
Jan 15 at 1:27
$begingroup$
This is a comment (as I am not a mathematical historian, and can't really speak authoritatively), but I would suspect that the answer is "neither." The Greeks worked with rational numbers, but also recognized that there were irrational numbers (hinting at the necessity of introducing the real numbers). Descartes and his contemporaries contrasted real numbers with imaginary numbers, so the concept has been around at least that long. The set theoretic constructions didn't come along until much later (late 19th or early 20th Century).
$endgroup$
– Xander Henderson
Jan 15 at 1:27
$begingroup$
@XanderHenderson - I think you pretty much answered my question. If I understand you correctly, you're saying that Greeks knew there have to be irrationals, but did not have the Completeness Axiom to point to, and so the set of real numbers were not formally constructed. But once the completeness axiom was recognized by mathematicians, the set theoretic constructions came along and formalized the set of reals.
$endgroup$
– user1068636
Jan 15 at 1:32
$begingroup$
@XanderHenderson - I think you pretty much answered my question. If I understand you correctly, you're saying that Greeks knew there have to be irrationals, but did not have the Completeness Axiom to point to, and so the set of real numbers were not formally constructed. But once the completeness axiom was recognized by mathematicians, the set theoretic constructions came along and formalized the set of reals.
$endgroup$
– user1068636
Jan 15 at 1:32
$begingroup$
No, it's not that simple. This is an extremely modern way of thinking of it that also is not clearly inevitable. The Greeks knew about decimal expansions of integers, but not of fractional numbers. There was no immediate reason for them to expect there to be any numbers other than the constructible numbers, which you can get by taking successive square roots. They were able to identify some numbers they couldn't figure out how to express this way, but they couldn't prove that it was impossible. They did not use negative numbers or zero.
$endgroup$
– Matt Samuel
Jan 15 at 2:02
$begingroup$
No, it's not that simple. This is an extremely modern way of thinking of it that also is not clearly inevitable. The Greeks knew about decimal expansions of integers, but not of fractional numbers. There was no immediate reason for them to expect there to be any numbers other than the constructible numbers, which you can get by taking successive square roots. They were able to identify some numbers they couldn't figure out how to express this way, but they couldn't prove that it was impossible. They did not use negative numbers or zero.
$endgroup$
– Matt Samuel
Jan 15 at 2:02
$begingroup$
The Greeks wouldn't know the completeness axiom from a hole in the ground.
$endgroup$
– Matt Samuel
Jan 15 at 2:02
$begingroup$
The Greeks wouldn't know the completeness axiom from a hole in the ground.
$endgroup$
– Matt Samuel
Jan 15 at 2:02
|
show 1 more comment
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1
$begingroup$
Maybe one for hsm.stackexchange.com ?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 1:24
$begingroup$
This is a comment (as I am not a mathematical historian, and can't really speak authoritatively), but I would suspect that the answer is "neither." The Greeks worked with rational numbers, but also recognized that there were irrational numbers (hinting at the necessity of introducing the real numbers). Descartes and his contemporaries contrasted real numbers with imaginary numbers, so the concept has been around at least that long. The set theoretic constructions didn't come along until much later (late 19th or early 20th Century).
$endgroup$
– Xander Henderson
Jan 15 at 1:27
$begingroup$
@XanderHenderson - I think you pretty much answered my question. If I understand you correctly, you're saying that Greeks knew there have to be irrationals, but did not have the Completeness Axiom to point to, and so the set of real numbers were not formally constructed. But once the completeness axiom was recognized by mathematicians, the set theoretic constructions came along and formalized the set of reals.
$endgroup$
– user1068636
Jan 15 at 1:32
$begingroup$
No, it's not that simple. This is an extremely modern way of thinking of it that also is not clearly inevitable. The Greeks knew about decimal expansions of integers, but not of fractional numbers. There was no immediate reason for them to expect there to be any numbers other than the constructible numbers, which you can get by taking successive square roots. They were able to identify some numbers they couldn't figure out how to express this way, but they couldn't prove that it was impossible. They did not use negative numbers or zero.
$endgroup$
– Matt Samuel
Jan 15 at 2:02
$begingroup$
The Greeks wouldn't know the completeness axiom from a hole in the ground.
$endgroup$
– Matt Samuel
Jan 15 at 2:02