In mathematics, we have things called "natural" and "unnatural" numbers, "rational" and "irrational," "real" and "imaginary."
Number theorists, however, recognize one universal truth about all numbers - they are made up.
Numbers are just a system humans made up and imposed on the natural world. They are just something we use to label, categorize, and hopefully understand the natural order of things.
A long conversation several days ago led me to realize how lucky we are that mathematics and number theory is no Tower of Babel. The example I used went something like this:
-
The year is roughly 500 BCE, and in some bizarrely unprecedented historical event, people from all over the world are coming together to share their knowledge and collectively better the human race. A Greek mathematician, a Chinese scholar, and a Native American wise man find themselves standing next to each other observing the same landscape. The Greek says there are "tría" trees. The Chinese man shakes his head and says there are "san." The Native American looks a bit confused, and says something in a somewhat gutteral tone that no one understands.
These three people, of course, cannot understand each other. The Greek, being used to unintelligible foreigners, decides to demonstrate how he reached that conclusion. He points to each tree in turn, and as he points, verbally labels them: "Éna. Dýo. Tría." The Chinese scholar catches on, and does the same thing, saying instead "Yi, er, san." The Native American begins to understand that these strange, pale, and somewhat hairy people have developed systems identical to his own culture's method of keeping track of quantities of things. He grabs a stick and draws a pictogram in the dirt. The Chinese person smiles at this overly-complicated procedure, and draws three stacked parallel horizontal lines. The Greek's eyes light up when he sees this, and he in turn draws three tick marks (vertical), excitedly gesturing that his system is similar to the Chinese one.
An Egyptian comes and joins the three, noticing the Native American's pictogram. He takes the stick and draws a hieroglyphic character indicating the quantity of trees before them. The Greek is a bit confused as to why people would use art to represent such mundane things. The Chinese man thinks it overly complicated, but recalls that his culture once used pictographic characters (and remembers their writing system is in fact based on them). Gradually, all four begin to understand that whatever they wrote or said means the same thing.
Then an Arabic guy comes over, rolls his eyes, and writes "3."
-
At any rate, this "verbal labelling" is of course what we call "counting." Each of these hypothetical people come from completely different cultures that never interacted as they were developing, leading to their very different forms of communication. Some Native American cultures, for instance, did not exactly have a writing system, whilst their contemporaries in some parts of China felt they had too many. But at some point, they all realized they needed a way to keep records and indicate quantities, or else trade and record-keeping would be hopeless.
-
The Greek thought to himself: after all, despite being from different parts of the world, we all have only ten fingers. What if that dark-skinned man with long flowing hair wants eleven of something? Does he indicate that in a manner similar to the fellow with the narrow eyes?
The Chinese scholar sees a man who looks similar to himself, and rushes over, thinking that perhaps he would finally be able to talk to someone. But as he approaches, he instantly knows the man is Japanese - clearly, he had spent a good deal of time on a fishing boat, not a terraced farm. But the scholar remembers something he learned about Japan once, and instead of talking to the Japanese man, he deftly scratches some characters in the dirt.
The Japanese man thinks for a moment, then smiles and writes a few characters as well, though he is restricted by his language's limited use of Chinese characters. He asks the Chinese scholar how many trees are in that spot, and the Chinese man says "san." The Japanese man looks confused, and writes a character identical to the one the Chinese man had written, saying "mi."*
It dawns on the Chinese man that this is a bit odd, having a different way of saying the exact same thing. We even write it the same way! Curious, he writes a sequence of characters, saying them as he goes along. "Yi, er, san, si, wu, liu, qi, ba, jiu, shi." The Japanese man looks puzzled. Why does he say it like that? He picks up a stick and points to each of the Chinese characters, saying "hita, huto, mi, yo, itu, mu, nana, ya, kokono, towo."*
The Arabic man was observing all of this smugly. Someday, these backwards Easterners will realize my people's system is far superior.
-
If a situation like this had actually arisen, there likely would have been a lot more frustration. The East Asians would all hang out with each other anyway, since with only a few exceptions, they can all recognize enough classical Chinese (or more accurately, Han) characters to communicate reasonably well. The Native Americans would be cautious of these strange pale people who seemed to claim "possession" of "their" strange systems of drawing series of lines and whatnot to represent quantities. The Greeks, however, would be fascinated by these different cultures and the ways in which they applied order to the world around them. The Romans would eavesdrop on the Greeks so they didn't have to come up with anything brand-new. The Egyptian wouldn't understand why these cultures were so proud of their "number" systems, since his culture used theirs to build monuments none of the others could dream of.
And if the Arabic guy was just sitting there laughing at the way these various people were writing numbers - come on, lines? pictures? - well, who could blame him?
-
The Martian watching these Earth-dwellers realized that despite not being able to understand each other, they at least recognized they were trying to express the same exact idea. He grabbed his holo-tablet and jotted some quick notes, writing a quick description of each Earth being (so vastly different depending on where they came from!), and then did his best to replicate each of the characters or pictograms they had written or drawn. What was it they were trying to say? Oh, right, the amount of trees! He counted them up and, at the top of the holo-page, wrote his version of what Earth beings hundreds, maybe thousands of years later would call "three." Well, that's more trees than we've got, at any rate.
--
* That's Old Japanese. Remember, I said this was around 500 BCE! The "ichi, ni, san..." system didn't come around til later. It's still different from Chinese, but they do use the exact same characters for numbers.
Number theorists, however, recognize one universal truth about all numbers - they are made up.
Numbers are just a system humans made up and imposed on the natural world. They are just something we use to label, categorize, and hopefully understand the natural order of things.
A long conversation several days ago led me to realize how lucky we are that mathematics and number theory is no Tower of Babel. The example I used went something like this:
-
The year is roughly 500 BCE, and in some bizarrely unprecedented historical event, people from all over the world are coming together to share their knowledge and collectively better the human race. A Greek mathematician, a Chinese scholar, and a Native American wise man find themselves standing next to each other observing the same landscape. The Greek says there are "tría" trees. The Chinese man shakes his head and says there are "san." The Native American looks a bit confused, and says something in a somewhat gutteral tone that no one understands.
These three people, of course, cannot understand each other. The Greek, being used to unintelligible foreigners, decides to demonstrate how he reached that conclusion. He points to each tree in turn, and as he points, verbally labels them: "Éna. Dýo. Tría." The Chinese scholar catches on, and does the same thing, saying instead "Yi, er, san." The Native American begins to understand that these strange, pale, and somewhat hairy people have developed systems identical to his own culture's method of keeping track of quantities of things. He grabs a stick and draws a pictogram in the dirt. The Chinese person smiles at this overly-complicated procedure, and draws three stacked parallel horizontal lines. The Greek's eyes light up when he sees this, and he in turn draws three tick marks (vertical), excitedly gesturing that his system is similar to the Chinese one.
An Egyptian comes and joins the three, noticing the Native American's pictogram. He takes the stick and draws a hieroglyphic character indicating the quantity of trees before them. The Greek is a bit confused as to why people would use art to represent such mundane things. The Chinese man thinks it overly complicated, but recalls that his culture once used pictographic characters (and remembers their writing system is in fact based on them). Gradually, all four begin to understand that whatever they wrote or said means the same thing.
Then an Arabic guy comes over, rolls his eyes, and writes "3."
-
At any rate, this "verbal labelling" is of course what we call "counting." Each of these hypothetical people come from completely different cultures that never interacted as they were developing, leading to their very different forms of communication. Some Native American cultures, for instance, did not exactly have a writing system, whilst their contemporaries in some parts of China felt they had too many. But at some point, they all realized they needed a way to keep records and indicate quantities, or else trade and record-keeping would be hopeless.
-
The Greek thought to himself: after all, despite being from different parts of the world, we all have only ten fingers. What if that dark-skinned man with long flowing hair wants eleven of something? Does he indicate that in a manner similar to the fellow with the narrow eyes?
The Chinese scholar sees a man who looks similar to himself, and rushes over, thinking that perhaps he would finally be able to talk to someone. But as he approaches, he instantly knows the man is Japanese - clearly, he had spent a good deal of time on a fishing boat, not a terraced farm. But the scholar remembers something he learned about Japan once, and instead of talking to the Japanese man, he deftly scratches some characters in the dirt.
The Japanese man thinks for a moment, then smiles and writes a few characters as well, though he is restricted by his language's limited use of Chinese characters. He asks the Chinese scholar how many trees are in that spot, and the Chinese man says "san." The Japanese man looks confused, and writes a character identical to the one the Chinese man had written, saying "mi."*
It dawns on the Chinese man that this is a bit odd, having a different way of saying the exact same thing. We even write it the same way! Curious, he writes a sequence of characters, saying them as he goes along. "Yi, er, san, si, wu, liu, qi, ba, jiu, shi." The Japanese man looks puzzled. Why does he say it like that? He picks up a stick and points to each of the Chinese characters, saying "hita, huto, mi, yo, itu, mu, nana, ya, kokono, towo."*
The Arabic man was observing all of this smugly. Someday, these backwards Easterners will realize my people's system is far superior.
-
If a situation like this had actually arisen, there likely would have been a lot more frustration. The East Asians would all hang out with each other anyway, since with only a few exceptions, they can all recognize enough classical Chinese (or more accurately, Han) characters to communicate reasonably well. The Native Americans would be cautious of these strange pale people who seemed to claim "possession" of "their" strange systems of drawing series of lines and whatnot to represent quantities. The Greeks, however, would be fascinated by these different cultures and the ways in which they applied order to the world around them. The Romans would eavesdrop on the Greeks so they didn't have to come up with anything brand-new. The Egyptian wouldn't understand why these cultures were so proud of their "number" systems, since his culture used theirs to build monuments none of the others could dream of.
And if the Arabic guy was just sitting there laughing at the way these various people were writing numbers - come on, lines? pictures? - well, who could blame him?
-
The Martian watching these Earth-dwellers realized that despite not being able to understand each other, they at least recognized they were trying to express the same exact idea. He grabbed his holo-tablet and jotted some quick notes, writing a quick description of each Earth being (so vastly different depending on where they came from!), and then did his best to replicate each of the characters or pictograms they had written or drawn. What was it they were trying to say? Oh, right, the amount of trees! He counted them up and, at the top of the holo-page, wrote his version of what Earth beings hundreds, maybe thousands of years later would call "three." Well, that's more trees than we've got, at any rate.
--
* That's Old Japanese. Remember, I said this was around 500 BCE! The "ichi, ni, san..." system didn't come around til later. It's still different from Chinese, but they do use the exact same characters for numbers.
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