10th September 2006
From these two philosophers’ perspective, we can think that numbers have been created by human mind to answer some important questions of this life. As the example goes, a shepherd needed to make sure that no sheep missing in the flock. Then he uses little rocks for each sheep and when a sheep enters the farm, he takes one rock from the basket and put into another basket. At the end, if all the sheep return back, there would be no rock left in the first basket. This is actually a very basic example for one to one and onto relationship between two finite sets. Mathematicians call these kinds of sets bijective, which basically means equivalent (equal number of cardinal numbers) Let’s say this theory is true and let’s accept that numbers are born as a solution for general life problems. Then shouldn’t we argue that in a very different world, very different universe, the aliens might have different kind of number sense?
Firstly, this basic theorem does not explain the concept of infinity. In Math, we can construct bijective relationship between two infinite sets. How did we find out infinity of numbers? It can not be explained by daily life problems and the solutions which are barely answering them. So what makes us to think that numbers are universal?
Secondly, I believe that sense of numbers and Euclidian Geometry has a strong link. Before the theory of relativity people believed that the space is flat and the shortest distance between two points in the space is a line. Kant considers this truth as an a priori and synthetic knowledge. This means, our mind is created or formed by such a strange way that we can not think the other way. We know this as an idea innate and there is no way to reject it. However, Einstein showed that the shortest instance between two points in the space is not a line but a curve (the radius of curvature is determined by the gravity force around the space). Although, our Euclidian space assumption is not true, it works with 99.99% accuracy for basic Physics problems. Now, let’s imagine another planet in which gravity is much higher than 9.86. Then, we will have enough right to think that the aliens living there probably had discovered the concept of curved space before assuming that the space is flat. Then, can’t we say the geometry they develop will be very different?
Epistemologically there are two main rivals about the source of knowledge, those who think all knowledge comes from our mind and those who think all knowledge comes from experiments. John Locke defines human mind as a ‘tabula rasa’, an empty page. When I was in university, we had Ali Ulger as our Algebra teacher. He taught us Fundamental Principles of Mathematics (Math 161). I remember one of his lectures, he mentioned some tribes in the Amazon forests. He said that they did not need numbers after 2 because they did not need to count things which are more than 2. They had total three numbers: 1,2 and many. Anything more than two is called many. If we go to their tribes and mention them about 5 or 11, they will probably know their difference but they will ignore it. This brings one big question into my mind: If we consider Kuhn’s paradigm theory, incommensurability becomes an unavoidable problem. How can we communicate with the people of this tribe? Can we teach them addition and subtraction without letting them leaving their own paradigm? It seems impossible! To teach them modern algebra, we have to teach them numbers higher than 2 are not equal to each other. Another question is more complicated: If they learn Modern Algebra, can they teach it to their descendants since they think it is useful? Because it is useless to them, they will forget it soon.
Now, let’s go back to movie. Prime numbers have been sent because they are universal. Although we can not make everybody believe that numbers are useful and there are numbers more than 2, how can we suppose that prime numbers are universal? It sounds like an anthropocentric assumption. Because we discovered prime numbers and we made them pillars of our number theory, we want to think that those who live outside also think by the same way. The similar assumption can be found even in biology. Why do scientists look for Carbon Oxygen as a basis of life in other planets? A few years ago, I read an article about a type of bacteria that lives hundreds of meter down under the ice in Antarctica. There is no oxygen, no light and no food! Scientists discovered that this bacterium consumes iron as food. It does not need light, oxygen, heat or carbon! So do we have right to think that life is possible with the elements we can easily find on this earth?
I think it is time to use our imagination to create something non-anthropocentric. By this way, we might get more closed to other civilizations, if they exist!
I am adding some links for the review of the movie. It is an old movie but still remarkable…
http://www.ram.org/ramblings/movies/contact.html
http://www.coseti.org/klaescnt.htm
http://www.reelviews.net/movies/c/contact.html
http://www.maths.ex.ac.uk/~mwatkins/isoc/sagan.htm
From these two philosophers’ perspective, we can think that numbers have been created by human mind to answer some important questions of this life. As the example goes, a shepherd needed to make sure that no sheep missing in the flock. Then he uses little rocks for each sheep and when a sheep enters the farm, he takes one rock from the basket and put into another basket. At the end, if all the sheep return back, there would be no rock left in the first basket. This is actually a very basic example for one to one and onto relationship between two finite sets. Mathematicians call these kinds of sets bijective, which basically means equivalent (equal number of cardinal numbers) Let’s say this theory is true and let’s accept that numbers are born as a solution for general life problems. Then shouldn’t we argue that in a very different world, very different universe, the aliens might have different kind of number sense?
Firstly, this basic theorem does not explain the concept of infinity. In Math, we can construct bijective relationship between two infinite sets. How did we find out infinity of numbers? It can not be explained by daily life problems and the solutions which are barely answering them. So what makes us to think that numbers are universal?
Secondly, I believe that sense of numbers and Euclidian Geometry has a strong link. Before the theory of relativity people believed that the space is flat and the shortest distance between two points in the space is a line. Kant considers this truth as an a priori and synthetic knowledge. This means, our mind is created or formed by such a strange way that we can not think the other way. We know this as an idea innate and there is no way to reject it. However, Einstein showed that the shortest instance between two points in the space is not a line but a curve (the radius of curvature is determined by the gravity force around the space). Although, our Euclidian space assumption is not true, it works with 99.99% accuracy for basic Physics problems. Now, let’s imagine another planet in which gravity is much higher than 9.86. Then, we will have enough right to think that the aliens living there probably had discovered the concept of curved space before assuming that the space is flat. Then, can’t we say the geometry they develop will be very different?
Epistemologically there are two main rivals about the source of knowledge, those who think all knowledge comes from our mind and those who think all knowledge comes from experiments. John Locke defines human mind as a ‘tabula rasa’, an empty page. When I was in university, we had Ali Ulger as our Algebra teacher. He taught us Fundamental Principles of Mathematics (Math 161). I remember one of his lectures, he mentioned some tribes in the Amazon forests. He said that they did not need numbers after 2 because they did not need to count things which are more than 2. They had total three numbers: 1,2 and many. Anything more than two is called many. If we go to their tribes and mention them about 5 or 11, they will probably know their difference but they will ignore it. This brings one big question into my mind: If we consider Kuhn’s paradigm theory, incommensurability becomes an unavoidable problem. How can we communicate with the people of this tribe? Can we teach them addition and subtraction without letting them leaving their own paradigm? It seems impossible! To teach them modern algebra, we have to teach them numbers higher than 2 are not equal to each other. Another question is more complicated: If they learn Modern Algebra, can they teach it to their descendants since they think it is useful? Because it is useless to them, they will forget it soon.
Now, let’s go back to movie. Prime numbers have been sent because they are universal. Although we can not make everybody believe that numbers are useful and there are numbers more than 2, how can we suppose that prime numbers are universal? It sounds like an anthropocentric assumption. Because we discovered prime numbers and we made them pillars of our number theory, we want to think that those who live outside also think by the same way. The similar assumption can be found even in biology. Why do scientists look for Carbon Oxygen as a basis of life in other planets? A few years ago, I read an article about a type of bacteria that lives hundreds of meter down under the ice in Antarctica. There is no oxygen, no light and no food! Scientists discovered that this bacterium consumes iron as food. It does not need light, oxygen, heat or carbon! So do we have right to think that life is possible with the elements we can easily find on this earth?
I think it is time to use our imagination to create something non-anthropocentric. By this way, we might get more closed to other civilizations, if they exist!
I am adding some links for the review of the movie. It is an old movie but still remarkable…
http://www.ram.org/ramblings/movies/contact.html
http://www.coseti.org/klaescnt.htm
http://www.reelviews.net/movies/c/contact.html
http://www.maths.ex.ac.uk/~mwatkins/isoc/sagan.htm
Merhaba Ali,
YanıtlaSilI am still experimenting with your blog to try to learn how to work with it. I did finally see that on the left there is a list of older blog issues that I can go back to. Now I am still trying to figure out how to interline my comments with what you wrote. I thought I had it for a minute just now, but I was wrong.
This example from my own experience actually relates to your September 10 thoughts on numbers. I think that the idea that numbers developed strictly and exclusively for problem-solving is wrong because
it it too limited. In relation to my comments above on your blog: it is true that I worked to solve the problems I have in communicating with you when it is at the center of things. But I did more than work to solve a problem. I was also playing, exercising myself for the sheer joy of doing it. I was also creating -- creating a solution where nothing had existed, creating something new in an empty field. Other beings could use numbers for things other than practical purposes. They could use them for the joy of discovery, for their aesthetic beauty, and for the other reasons I mentioned.
I agree that we are "anthropocentric, all too anthropocentric" some
times, but numbers are the most abstract things we have, the ones with the most of every day human reality stripped from them. Our universe has been removed from them as much as possible, and at the same time that leaves them as the most universal of concepts that we can come up with. Prime numbers and similar things represent our best effort to
communicate in a non-anthropocentric way. Perhaps creatures from another world would find them to be the same. They too must be charmed
with the notion of developing an internally consistent system.
The ancient Egyptians had developed a practical kind of geometry to allocate land after the flooding by the Nile had wiped out boundaries. They used 45 degree right triangles. But it was Euclid and other Greeks who looked at what the Egyptians were doing and were drawn to seeing if they could develop a system of general principles that dealt with earth measuring (geo-metry). I think they did it for intellectual recreation, not to help with the fair allocation of land. Your poor Amazon tribesman who counts "one, two, many..." would never enjoy that recreation -- and neither would most of the Ancient Egyptians who were
laying out land, although some would. Even today, most people in the West probably think that a prime number has something to do with prime time on television. Mathematical sophistication is for the few. But doing algebra is pleasurable to some people. Some of the Amazonians who learned it might want to keep it alive and pass it on to others. I know a Turk who is doing that in Vietnam right now.
The discovery of the iron-eating bacteria under the ice in
Antarctica delighted me as it did you. It forces us to re-think out
systems and broaden out outlook. It forces us to engage in an
intellectual adventure -- and now that I think about it, that sense of adventure is probably a greater motivator for some people than iguring out how many sheep came home. Rather than put stones in one bowl or another to keep the count, the shepherd could have counted the number of sheep legs and divided by four.
Best wishes,
Allan