Importance of Mathematics

As an engineering student you could imagine that math is a large part of my education. As with most math students, my first impression was that it was all just a waste of time. It seemed like it was nothing more than number games. The teachers brought us through these exercises just to make us solve imaginary problems. The problems started to become increasingly frustrating and I started to question the purpose of these problems. Then the teacher starting assigning us word problems. These problems were relate to real life situations, and I started to realize that these problems may have a purpose behind them. I was measuring distances for bridges and finding the area of triangles among other things. As I started to do these problems I began to realize how influential math is in all of our modern technologies.

As with most industries, the genius of a very few people have allowed others to build upon those ideas and develop more advanced ideas. The history of mathematics is no exception, with one of the most notable examples being the work of Euclid. He developed a system for geometry that is still in use today. In fact, his book Elements, was the standard for geometry for over two thousand years after his death. His developments contributed greatly to modern civil engineering by allowing the development of Pythagoras theorem and the laws of sines and cosines. These rules allow surveyors to measure the distance between two points. By treating any problem as a triangle, the surveyor just needs to know the distance and angle between two other points to find the distance between two other points in the triangle. This allows the surveyor to have the necessary measurements required for constructing large buildings and structures.

Like the example of Euclid, there are many more great mathematicians that worked throughout the years, all contributing to their field in some way, all leading to further advancements. If you look to the history of technology you can see how new developments in mathematics lead to new products being produced. Such as Archimedes’s levers and pulleys, the arches of the Roman aqueduct, the French Trebuchet, Pascal’s mechanical calculator, the bridges and dams of the twentieth century, and our modern electronics.

This example of the progressive nature of mathematics helps a student because of how you learn. Just as ancient humans had to learn the basics of mathematics thousands of years ago, so does a student now. In order to understand the advanced concepts that will allow you engineer bridges, electronic circuits, engines, or airplanes, you have to learn the basics of math first. Those frustrating problems that students have to deal with that seem to have no purpose are quite the opposite. They give you the basic skills and knowledge so that you can solve the more advanced problems in the future.

Coconut-Almond (Paleo) Pancakes

This is a recipe I developed for Paleo diet friendly pancakes. It's primary ingredients are coconut milk and almond meal. I have not tried this with any other nut flour but I suppose it would be just as good.

For this recipe you need:

1 can coconut milk (about 13 fl. oz.)
3 eggs
1 tsp. vanilla extract
1 1/2 tbsp. baking powder
3 cups almond meal.
2 cups fresh berries or chopped fruit (optional)

These ingredients may seem odd when compared to a traditional recipe, but that is because you are using almond meal. When combined with liquid, almond meal tends not to bond well and can make very thin pancakes. Because of this you need lots of extra baking powder, and plenty of meal. This will help stop the pancakes from spreading out all over your pan. The amount of ingredients is based on the can of coconut milk because thats how you get it from the store.

Combine these ingredients one at a time in a large mixing bowl in the order listed above. Mix with a whisk until throughly combined. This will create a very thick mixture. Again, it needs to be very thick to stand up in the pan.

At this point you can add the berries or fruit. This step is completely optional, because the pancakes are quite tasty without the fruit. If you do, I recommend you use fresh berries or fruit. Make sure they are chopped up quite small, but not into a mush (the size of blueberries are fine). If you cannot use fresh, and need to use frozen that is ok, but make sure to defrost them the day before.

Now pour the mixture onto a hot non-stick griddle or pan. Make sure you pour small pancakes. (3-4 inches) No need to oil the pan unless you don't have a non-stick surface. Let these cook on the first side for awhile until the edges start to firm up around the sides. (about 1/4 of the way in) Flipping these pancakes is kind of tricky because they don't want to stay together. You have to move your spatula quickly under the entire pancake before you flip them or it will separate. You will end up breaking a few pancakes until you get the knack of it, so dont feel bad if you end up with a bunch of 1/2 pancakes. Cook them on the other side until they are golden brown and delicious.

Top them with some warm honey, coconut nectar or real maple syrup. This will make about 12-16 four-inch pancakes.

Consequences of invalid concept formation

I was just reading OPAR (Objectivism: The Philosophy of Ayn Rand) the other day and I came across a part about how we (human beings) form concepts by integrating and disintegrating properties of an object to create the concept. This made me think about how some people can become racist by forming invalid concepts about the races.

Basically, we notice the differences and similarities between objects to form unique concepts in our minds. For example, as a child you take all these different shapes and colors and start to notice similarities. Such as the fact that a ball looks different than a table. That's when you first start to notice these objects as separate entities apart from each other. Then you start to notice the similarities between two or more different objects. Such as how all tables have legs that support them, and a large flat surface on the top. This allows you to form the concept of "table". Because of this, we don't have to remember each instance of a table we've ever seen and can automatically know that an object is a table by observing the properties of a table in an object.

Now there are multiple levels of concepts. If we continue with our "table" concept you can see what I am talking about. There is a wider level concept that a table fits into, "furniture". Then below just "table" we have narrower concepts such as "coffee table", "picnic table", etc. I'll assume you all know the definition of these terms. In order for a table to be a table it still has to meet the definition of the wider level concept of "furniture".

Now this process is not flawless. We have to observe all the facts correctly and not ignore any wider level concepts that your object may fit into. If you ignore facts, or ignore the wider level concepts, you can end up with heinous contradictions.

So how this relates to racism? This method of concept integration allows us to form concepts for people of different races, including the derogatory terms we call them. So by observing the behavior of multiple people of a certain race you can forms concepts about these people. Say you notice a lot of people that are criminals, or are lazy, or dress sloppily all belong to the same race. Some people will use these behaviors in their definition of that race.

The big problem here is that anyone that is racist is ignoring the higher level concept of "man" and the properties that define man. One of the essential characteristics of man is that he has a volitional consciousness (free will). Each individual man has his own mind and can choose his own ideas and his own behavior. This implies that man isn't pre-determined to coincide with his racial stereotypes. He can choose not to commit crimes, not to be lazy, or to dress neatly.

The problem is not in the way that concept integration works, but in the fact that some men choose to ignore the wider levels concepts. In order to form any concept correctly you have to observe all the facts about the object, including its conceptual hierarchy.