Diego Rivera
Michael Polanyi College
Semester Three Allopoïesis – Math Essay
January 20th, 2014
Michael Polanyi College
Semester Three Allopoïesis – Math Essay
January 20th, 2014
What is Measurement?
The Non-Conventional Way to Think About It
Have you ever measured how high you are? If you have, like all of us, what’s the first thing that comes to your mind? Probably you’re thinking in meters, something like 1.75 meters, or maybe 5 feet and 9 inches if you’re from the U.S. It’s our way of thinking of measurement. We’ve grown up all the way with established measurement systems that have changed little or nothing during our existence. So it’s normal to immediately think of a particular unit of measure when we’re trying to define the measurement of something. You would rarely hear someone say he measures eleven tenths of what another person measures. We don’t think like that, at least not directly. We forget or haven’t learned the nature of measuring something. Usually, we don’t think of measurement as a comparison of two or more things and neither do we know how we came up to have the measurement units we use today.
Before we move on and explain what measurement is, let’s explore what measurement is not. First, measurement is not mainly and exclusively about units. The use of units in measurement came after the discovery or exploration of this subject and people started using them in order to standardize their different measurements. So, when people refer to a measurement as a specific unit they are not referring to that unit, but to the relation between that unit with another. This leads to a second point. It’s easy to think there’s somehow a universal/objective/absolute measure for things. The reason for this is that we were able to make a good standardization of our units of measure, but nothing was given at first. When units emerged they were different in some places and had the same name. Through time we were able to eliminate these differences and have the same measures, but the principle is the same. Measurement is always relative. The last point of thinking what measurement is not is about the validity of measurement. As we had said before, measurement is relative and has evolved into units we all agree on using, but how exact can we measure something and what does this means? Again, we tend to think we are very accurate in measuring something, but history has proved us that we can improve on our measurements by creating new and better instruments and tools. The answer to the question of whether we would ever achieve an exact measurement or until what point we would be able to improve on this is somehow uncertain and circular. Uncertain because although we can see the improvement of our measurements through time, we cannot claim that this would keep on going, and circular because we would be trying to give a more exact unit of measure and this would be based in a comparison, which in turn has to be equally accurate, so the problem is in both cases. These are only some of the main problems we find in our conventional thinking about measurement.
Now, let’s explore what measurement is. More than a specific number, quantitative or qualitative relation, measurement is a way of thinking. Measurement is about seeing the relation between two things and know why it is the way it is. By this I mean that it’s not enough to know the what of things but you also have to know the how. The what refers to the specific relation or comparison between two things, and the how is the reasoning or process needed to achieve the what. They are both equally important when trying to understand the nature of measurement. When we understand both parts we approach to a more structured and mathematical way of thinking about problems. And problems are needed to answer the questions about nature and how we see the world and its patterns. With this, I’m not claiming that nature has patterns and we are discovering them, nor that there are not patterns. To be honest, I don’t know, but I like to think that nature just is and that our mind in its logical structure is what makes us look for patterns in order to understand something. In this search, we don’t stop comparing, making relations among things, and this simplification of reality is what turns out to evolve into the measurements we know today. In Measurement by Paul Lockhart, he talks about a mathematical thinking and its connection to the real world. In the mathematical and imaginary thinking, we use our creativity and divergent thinking to solve problems, i.e. to find patterns and relations. What’s most interesting is that mathematical thinking, this world that we create in our mind, is used to understand the different problems we encounter every day. It’s not only about measuring your height, but knowing and understanding practically every problem from knowing the area of your bedroom to knowing why planets orbit around the sun.
When we know what measurement is and how it works, we can have a better understanding of why great thinkers like Euclid and Newton used a geometrical method to prove their theories. By using a method based on ratios, i.e. relationships between things, you avoid the problem of accuracy in measuring something. You go directly into its relation without the difficulties that giving a specific number would give you. Of course, Newton uses data to give an approximation of his conclusions, but the basis of his work is on the ratios he was able to imagine. Euclid’s work is based entirely on relationships and you can see how this gives a theory more power and strength than basing it on specific quantitative numbers. This non-conventional way of thinking about measurement, about knowing the what and how of it, is what makes it fascinating and interesting. It opens a whole new world in your mind where you can explore and discover or create the patterns you need to understand and help others understand the world around you. To measure things is understanding their nature and knowing how to apply that reasoning to the problems you face. So, if you are always in the search for ratios, you’ll most certain to find the answers to your questions.
Before we move on and explain what measurement is, let’s explore what measurement is not. First, measurement is not mainly and exclusively about units. The use of units in measurement came after the discovery or exploration of this subject and people started using them in order to standardize their different measurements. So, when people refer to a measurement as a specific unit they are not referring to that unit, but to the relation between that unit with another. This leads to a second point. It’s easy to think there’s somehow a universal/objective/absolute measure for things. The reason for this is that we were able to make a good standardization of our units of measure, but nothing was given at first. When units emerged they were different in some places and had the same name. Through time we were able to eliminate these differences and have the same measures, but the principle is the same. Measurement is always relative. The last point of thinking what measurement is not is about the validity of measurement. As we had said before, measurement is relative and has evolved into units we all agree on using, but how exact can we measure something and what does this means? Again, we tend to think we are very accurate in measuring something, but history has proved us that we can improve on our measurements by creating new and better instruments and tools. The answer to the question of whether we would ever achieve an exact measurement or until what point we would be able to improve on this is somehow uncertain and circular. Uncertain because although we can see the improvement of our measurements through time, we cannot claim that this would keep on going, and circular because we would be trying to give a more exact unit of measure and this would be based in a comparison, which in turn has to be equally accurate, so the problem is in both cases. These are only some of the main problems we find in our conventional thinking about measurement.
Now, let’s explore what measurement is. More than a specific number, quantitative or qualitative relation, measurement is a way of thinking. Measurement is about seeing the relation between two things and know why it is the way it is. By this I mean that it’s not enough to know the what of things but you also have to know the how. The what refers to the specific relation or comparison between two things, and the how is the reasoning or process needed to achieve the what. They are both equally important when trying to understand the nature of measurement. When we understand both parts we approach to a more structured and mathematical way of thinking about problems. And problems are needed to answer the questions about nature and how we see the world and its patterns. With this, I’m not claiming that nature has patterns and we are discovering them, nor that there are not patterns. To be honest, I don’t know, but I like to think that nature just is and that our mind in its logical structure is what makes us look for patterns in order to understand something. In this search, we don’t stop comparing, making relations among things, and this simplification of reality is what turns out to evolve into the measurements we know today. In Measurement by Paul Lockhart, he talks about a mathematical thinking and its connection to the real world. In the mathematical and imaginary thinking, we use our creativity and divergent thinking to solve problems, i.e. to find patterns and relations. What’s most interesting is that mathematical thinking, this world that we create in our mind, is used to understand the different problems we encounter every day. It’s not only about measuring your height, but knowing and understanding practically every problem from knowing the area of your bedroom to knowing why planets orbit around the sun.
When we know what measurement is and how it works, we can have a better understanding of why great thinkers like Euclid and Newton used a geometrical method to prove their theories. By using a method based on ratios, i.e. relationships between things, you avoid the problem of accuracy in measuring something. You go directly into its relation without the difficulties that giving a specific number would give you. Of course, Newton uses data to give an approximation of his conclusions, but the basis of his work is on the ratios he was able to imagine. Euclid’s work is based entirely on relationships and you can see how this gives a theory more power and strength than basing it on specific quantitative numbers. This non-conventional way of thinking about measurement, about knowing the what and how of it, is what makes it fascinating and interesting. It opens a whole new world in your mind where you can explore and discover or create the patterns you need to understand and help others understand the world around you. To measure things is understanding their nature and knowing how to apply that reasoning to the problems you face. So, if you are always in the search for ratios, you’ll most certain to find the answers to your questions.
Questions from Measurement
1. Suppose you are given both the sum and the difference of two numbers. How can you determine the numbers themselves? [pg. 56]
- We can determine the numbers by using the difference of squares formulas and a system of equations.
2. If two circles are arranged so that each passes through the center of the other, what are the area and perimeter of the overlap? [pg. 69]
- The perimeter would be the arc of one circle formed by the intersection of the other circle plus the arc formed by the other circle.
- The area can be obtained by the area under the curve of the circle with the intersecting points. This area would be half of the total area of the overlapping circles.
3. How can we measure the surface area of a cone? [pg. 81]
- We can apply the same method of exhaustion by using cylinders (cross-section) and measure the surface area of these cylinders and then add them up.
4. What is the perimeter of a region formed by a moving stick? [pg. 101]
- With the path described by the moving stick, we can apply the method of exhaustion and arrange it so we can make any desired path. If we create a rectangle and apply the Pappus theorem, then “the area of a region swept out by a moving stick is the product of the length of the stick and the distance traveled by the center of the stick.” So, the perimeter of a region formed by a moving stick is the product of twice the length of the stick and twice the distance traveled by the center of the stick.
5. Do two lines in projective space necessarily intersect? [pg. 169]
- Yes, they necessarily intersect. Parallelism is vanished in projective space and even though these lines may seem to have no point of intersection, it’s because that point is infinitely far away. Nevertheless, they would always meet at some point.
6. If you connect lines in this evenly spaced pattern, a parabola appears. Why? [pg. 187]
- A parabola appears because it lies on the borderline between ellipses and hyperbolas; you get it when you fix one focal point of an ellipse and send the other one off to infinity.
7. How does the distance between two points in a plane depend on their coordinates? [pg. 208]
- A coordinate plane is a reference system that consists of an origin, a unit, and an orientation; so if we want to know the distance between two points in this plane we would need the coordinates of both points, since these may change from a coordinate plane to another.
8. How do you subtract two vectors? [pg. 210]
- You subtract one vector from another by reversing the direction of the vector you want to subtract and then add them.
9. What is the space‐time representation of a constant speed motion along a line? [pg. 222]
- In a graph where the vertical axis represent space and the horizontal axis represent time, the representation of a constant speed motion would be a straight line and its slope would depend on where it’s moving.
10. If two bugs crawl along the edge of a table in such a way that their motion has this space‐time diagram, what must have happened? [pg. 224]
- In this two-dimensional diagram of space-time, we can expect to see a straight line with positive slope.
11. What are the rectangular coordinates of the point with circular coordinate 3π/4? [pg. 237]
- x = (cosine)
- y = 1/√2 (sine)
12. Why does y =x2 carve a parabola? Where are its focal points? [pg. 260]
- It carves a parabola because of its squaring relation (it comes from a conic section), and it’s focal point would be inside the parabola, in line with the vertex, at a distance where if we take the distance from the focus to any point on the graph and from that point to the directrix or focal line, these distances would be the same. The directrix would be outside the parabola.
13. What is the largest circle that can sit at the bottom of a parabola? [pg. 261]
- The largest circle would be the one with a radius equal to the distance of the vertex to the focal point in the parabola.
14. What if we rotate it counterclockwise? [pg. 287 Coordinates of a velocity vector.]
- (-y, x)
15. Can you solve the differential equation 2x dy = y 2dx? [pg. 341]
- dx/dy = 2x/y
- dx/dy = 2/y
- dy/dx = y/2x
- dy/dx = 1/2x
16. Find the largest cylinder that fits inside a given cone. [pg. 359]
- One that has a radius equal to one third of the cone’s height.
17. Can you derive equation x2‐y2=1? [pg. 366]
- 2x + c
18. Show that am‐n = am/an for all m, n. [pg. 379]
- am-n = am/an
- (am)(a.n)
- (am)(1/an)
- am/an
19. Can you derive this elegant logarithm formula? [pg. 390]
- logzx = log x/log a
- (log x)(1/log a)
- (1/x)(1)(1/log a)
- (1/xlog a) + c
¿Cómo lo conecto con el MPC y lo que he aprendido acá?
- Más que todo con Euclid y Newton y poder entender más el método en el que probaron sus teorías. Entendí mejor que al derivar por medio del método geométrico, hablando de relaciones y proporciones, uno puede obtener una teoría más fuerte que al hacerlo con numéricamente.
- Inventamos la matemática o la descubrimos. Nuestra necesidad de encontrar patrones.
- “Un defecto es un tesoro.”
- “La ignorancia también es un tesoro.” Nos da una brecha a mejorar. Una necesidad es un tesoro también.
- “Know your limitations and that would make you stronger.”
- Go through your own process. Avances en matemática y no importa lo que estoy aprendiendo tangiblemente sino el proceso y la forma de pensar en que estoy solucionando un problema.
- Irracional, sin ratio, Mises – relación entre medios y fines.
- Matemática y su relación con filosofía, espiritualidad.
- We can determine the numbers by using the difference of squares formulas and a system of equations.
2. If two circles are arranged so that each passes through the center of the other, what are the area and perimeter of the overlap? [pg. 69]
- The perimeter would be the arc of one circle formed by the intersection of the other circle plus the arc formed by the other circle.
- The area can be obtained by the area under the curve of the circle with the intersecting points. This area would be half of the total area of the overlapping circles.
3. How can we measure the surface area of a cone? [pg. 81]
- We can apply the same method of exhaustion by using cylinders (cross-section) and measure the surface area of these cylinders and then add them up.
4. What is the perimeter of a region formed by a moving stick? [pg. 101]
- With the path described by the moving stick, we can apply the method of exhaustion and arrange it so we can make any desired path. If we create a rectangle and apply the Pappus theorem, then “the area of a region swept out by a moving stick is the product of the length of the stick and the distance traveled by the center of the stick.” So, the perimeter of a region formed by a moving stick is the product of twice the length of the stick and twice the distance traveled by the center of the stick.
5. Do two lines in projective space necessarily intersect? [pg. 169]
- Yes, they necessarily intersect. Parallelism is vanished in projective space and even though these lines may seem to have no point of intersection, it’s because that point is infinitely far away. Nevertheless, they would always meet at some point.
6. If you connect lines in this evenly spaced pattern, a parabola appears. Why? [pg. 187]
- A parabola appears because it lies on the borderline between ellipses and hyperbolas; you get it when you fix one focal point of an ellipse and send the other one off to infinity.
7. How does the distance between two points in a plane depend on their coordinates? [pg. 208]
- A coordinate plane is a reference system that consists of an origin, a unit, and an orientation; so if we want to know the distance between two points in this plane we would need the coordinates of both points, since these may change from a coordinate plane to another.
8. How do you subtract two vectors? [pg. 210]
- You subtract one vector from another by reversing the direction of the vector you want to subtract and then add them.
9. What is the space‐time representation of a constant speed motion along a line? [pg. 222]
- In a graph where the vertical axis represent space and the horizontal axis represent time, the representation of a constant speed motion would be a straight line and its slope would depend on where it’s moving.
10. If two bugs crawl along the edge of a table in such a way that their motion has this space‐time diagram, what must have happened? [pg. 224]
- In this two-dimensional diagram of space-time, we can expect to see a straight line with positive slope.
11. What are the rectangular coordinates of the point with circular coordinate 3π/4? [pg. 237]
- x = (cosine)
- y = 1/√2 (sine)
12. Why does y =x2 carve a parabola? Where are its focal points? [pg. 260]
- It carves a parabola because of its squaring relation (it comes from a conic section), and it’s focal point would be inside the parabola, in line with the vertex, at a distance where if we take the distance from the focus to any point on the graph and from that point to the directrix or focal line, these distances would be the same. The directrix would be outside the parabola.
13. What is the largest circle that can sit at the bottom of a parabola? [pg. 261]
- The largest circle would be the one with a radius equal to the distance of the vertex to the focal point in the parabola.
14. What if we rotate it counterclockwise? [pg. 287 Coordinates of a velocity vector.]
- (-y, x)
15. Can you solve the differential equation 2x dy = y 2dx? [pg. 341]
- dx/dy = 2x/y
- dx/dy = 2/y
- dy/dx = y/2x
- dy/dx = 1/2x
16. Find the largest cylinder that fits inside a given cone. [pg. 359]
- One that has a radius equal to one third of the cone’s height.
17. Can you derive equation x2‐y2=1? [pg. 366]
- 2x + c
18. Show that am‐n = am/an for all m, n. [pg. 379]
- am-n = am/an
- (am)(a.n)
- (am)(1/an)
- am/an
19. Can you derive this elegant logarithm formula? [pg. 390]
- logzx = log x/log a
- (log x)(1/log a)
- (1/x)(1)(1/log a)
- (1/xlog a) + c
¿Cómo lo conecto con el MPC y lo que he aprendido acá?
- Más que todo con Euclid y Newton y poder entender más el método en el que probaron sus teorías. Entendí mejor que al derivar por medio del método geométrico, hablando de relaciones y proporciones, uno puede obtener una teoría más fuerte que al hacerlo con numéricamente.
- Inventamos la matemática o la descubrimos. Nuestra necesidad de encontrar patrones.
- “Un defecto es un tesoro.”
- “La ignorancia también es un tesoro.” Nos da una brecha a mejorar. Una necesidad es un tesoro también.
- “Know your limitations and that would make you stronger.”
- Go through your own process. Avances en matemática y no importa lo que estoy aprendiendo tangiblemente sino el proceso y la forma de pensar en que estoy solucionando un problema.
- Irracional, sin ratio, Mises – relación entre medios y fines.
- Matemática y su relación con filosofía, espiritualidad.