From the link: “To count means to determine the cardinality of some finite set. Technically, since the natural numbers are usually defined as sets, that means to determine the natural number such that There is a bijection with the given finite set.” Perhaps someone could translate that into English?
I am sorry about that,Mike
All I want to emphasize is two different aspects of mathematics if you look right back in history those were two different things that were happening counting your animals or your wives or your children and measuring your clothing your Fields etc
I will try to find a simpler article or something myself but it is explains why there was a big problem with the Greeks when they’re discovered that a right angle triangle could have a hypotenuse that was not a rational number for example if two sides of a right angle thrown of lengths one unit the hypotenuse will be the square root of 2 and unfortunately they were able to prove that the square root of 2 cannot be a rational number.
So they stopped using numbers and just went to measurement because you can have an irrational number as a measurement because measurements are much more complicated than counting aren’t they?
But rational dirsnt necessarily mean what we think it means
It meant numbers were either whole numbers or ratios of two whole numbers and the Greeks are risk for us all that all numbers were of that type unfortunately their infinitely many numbers which of neither type
Pi is another number which can’t be expressed as a ratio.
Thank you Katherine. You make some good points. Although I have used the term for years, I had never stopped to think that the word ‘irrational’ is a direct reference to ‘ratio’.
It is curious to observe how the meanings of words change over time, such that their original derivation is lost. I struggle now with words like ‘cardinality’ and ‘canonical’, which raise images of religious hierarchy rather than anything to do with mathematics.
Yes it’s very interesting about the words and we take rational 90 in sensible but what is originally was that it was understandable you could have a number like one and a half which was three divided by 2 but a number which exists because you can construct the triangle with two sides of equal length 1 at right angles to each other and they hypotenuse will be the square root of 2 and there is no rational number that can be found that will give you two when squared
Of course the language of any speciality is off putting whether it’s mathematics or theology. But it came to me one day that many people could understand the difference between counting and measuring even if they don’t want to go any further that there are two different ways in the history of mankind in which numbers came into use and probably the measurements side was very very important and it’s also more complicated
From whole numbers it’s not difficult to accept fractions but once you go beyond that it does get more difficult. But it’s also very interesting and that’s what I would like people to know if you couldn get there it gets really really interesting.
You can read about like a concise account of the history of mathematics by
Struck. I think he might have been a Marxist but he’s certainly relax everything to economics rather than just to rational tools or anything else and I think that economics obviously was a major influence and of course the other major influence was war and weapons…
How my heart sings 
From the link: “To count means to determine the cardinality of some finite set. Technically, since the natural numbers are usually defined as sets, that means to determine the natural number such that There is a bijection with the given finite set.” Perhaps someone could translate that into English?
I am sorry about that,Mike
All I want to emphasize is two different aspects of mathematics if you look right back in history those were two different things that were happening counting your animals or your wives or your children and measuring your clothing your Fields etc
I will try to find a simpler article or something myself but it is explains why there was a big problem with the Greeks when they’re discovered that a right angle triangle could have a hypotenuse that was not a rational number for example if two sides of a right angle thrown of lengths one unit the hypotenuse will be the square root of 2 and unfortunately they were able to prove that the square root of 2 cannot be a rational number.
So they stopped using numbers and just went to measurement because you can have an irrational number as a measurement because measurements are much more complicated than counting aren’t they?
But rational dirsnt necessarily mean what we think it means
It meant numbers were either whole numbers or ratios of two whole numbers and the Greeks are risk for us all that all numbers were of that type unfortunately their infinitely many numbers which of neither type
Pi is another number which can’t be expressed as a ratio.
Thank you Katherine. You make some good points. Although I have used the term for years, I had never stopped to think that the word ‘irrational’ is a direct reference to ‘ratio’.
It is curious to observe how the meanings of words change over time, such that their original derivation is lost. I struggle now with words like ‘cardinality’ and ‘canonical’, which raise images of religious hierarchy rather than anything to do with mathematics.
Yes it’s very interesting about the words and we take rational 90 in sensible but what is originally was that it was understandable you could have a number like one and a half which was three divided by 2 but a number which exists because you can construct the triangle with two sides of equal length 1 at right angles to each other and they hypotenuse will be the square root of 2 and there is no rational number that can be found that will give you two when squared
Of course the language of any speciality is off putting whether it’s mathematics or theology. But it came to me one day that many people could understand the difference between counting and measuring even if they don’t want to go any further that there are two different ways in the history of mankind in which numbers came into use and probably the measurements side was very very important and it’s also more complicated
From whole numbers it’s not difficult to accept fractions but once you go beyond that it does get more difficult. But it’s also very interesting and that’s what I would like people to know if you couldn get there it gets really really interesting.
You can read about like a concise account of the history of mathematics by
Struck. I think he might have been a Marxist but he’s certainly relax everything to economics rather than just to rational tools or anything else and I think that economics obviously was a major influence and of course the other major influence was war and weapons…
Struck