## Deductive Reasoning

• What is the process of deductive reasoning used to come to the conclusion?

Usually deductive reasoning is logical, in that pieces of information are put together to come to a conclusion. One example would be syllogisms, where two axioms are known to be true, and based on those two axioms, a third thing must be true. It starts out more general and becomes more specific, e.g. all men are mortal, I am a man, therefore I am mortal. Broad axioms can lead to logical conclusions.

• What issues are there?

One example of such logical and deductive reasoning that I particularly enjoy is Plato’s definition of man as a ‘featherless biped’. The axioms this conclusion is based off are very true; man has no feathers, and man walks on two feet. However, as Diogenes points out, based on that logic, a plucked chicken is also a man. However, it is technically true, that man is a featherless biped (for the most part). Therefore, neither of the axioms is incorrect, and the conclusion is not incorrect either, but the conclusion can be misleading and therefore lead to a sort of half-truth.

In addition to logic not always being perfect when the axioms are correct and the system of thinking is correct, logic can be based off of flawed axioms. If axioms are broad generalizations, or just plain wrong, this can lead to a serious problem, as the conclusion will not be true. This can be seen in the St. Louis court case. The people condemning the judge are basing their judgement off of generalized axioms, which aren’t necessarily false in this scenario, and the Judge himself may have based his judgement off of his prior belief that “an urban heroin dealer not in possession of a firearm would be an anomaly”. Naturally in this case, urban can be construed to mean black, and therefore that axiom could be considered racism and therefore a bias rather than an actual fact.

## Intuition

Intuition is a very quick way of knowing. It helps us sum up all the data we currently have and make assumptions based off of it. However, these days the data we receive is rather skewed. On first thought we should rely on intuition when we’re in a dangerous situation. However, intuition isn’t the best for all situations that we may deem dangerous. For example, some people may view a low-income neighborhood as dangerous, and they wouldn’t necessarily be wrong, but that doesn’t mean they’d be right either. They could misjudge the people around them and maybe pepper spray a random innocent person who was out for a late jog. We should still rely on intuition when we have to make quick decisions and we’re in dangerous situations. There’s a reason we have it, it’s a useful way of knowing.

When acquiring knowledge we should not only be cautious in choosing our sources, but also careful about how much we trust them. Many new outlets which are considered reputable and reliable may not always be the best sources. The writers have bias, and the types of articles there are will give you bias. In order to have good intuition you must be careful of where you get your data and how accurate your data is. Otherwise, your intuition may be useless. That’s why we must view everything with a grain of salt, and take in information from a wide variety of sources.

## Bad Decision

One bad decision I’ve made several times before, is leaving homework until the literal morning before it’s due. I don’t just procrastinate until the night before, I go to bed and decide to wake up a little earlier than usual in order to do my homework. Similar to Dan Gilbert’s discussion of not considering all the different factors properly, I weigh how stressed and tired I feel the night before, as more important than how stressed and tired I’ll feel in the morning. Because I value my immediate comfort more than my future comfort. However this always makes me have a bad day afterwards. In addition, I don’t have as much time to do my homework in mornings so it could get cut short before I finish it. Part of the reason I don’t take this into account is that I use inductive logic to judge whether I’ll be able to finish my homework in the morning. In the past it has always worked, so why won’t it work this time? One of the ways of knowing is knowledge, and I think I consider it far more than I do the others such as reason and mathematics when I decide to do homework the morning right before school. In this situation I also do not properly weigh the validity of my premises. I think, I am tired, and tired people should go to bed, therefore I should go to bed. However, this syllogism is not valid, because while my two premises are true, they are not the only premises and I’m not taking all the data into account when making my decision.

## Real World Applications of Math

There are some types of math that are applicable to the real world, and some that aren’t. That’s why a person can spend their entire mathematical career on theoretical maths, and a different person can spend all that time on applied maths. Therefore it’s hard to figure out whether ‘math is applicable in the real world’.

On one hand math is intrinsic in what we do, even before we knew what math was we started counting, math existed before we created it. Math arises from the world naturally, and as such there are naturally a lot of applications of it. When we go to more abstract concepts it becomes less clear as to how they can be applied in the real world but maybe we’ll find a way they can be applied in the future. We get so caught up in notation we forget that math is very intrinsic to human nature and our world. In natural sciences we can use math to model everything. Because math arises from nature and nature has those mathematical patterns we can make use of those and apply them to different aspects of nature that may not seem mathematical. Part of the reason it may seem that some math doesn’t have practical applications is that the advancement of human mathematical development is very far ahead of technology and sciences. For example computer scientists use a lot of the factorization theorums first developed by Euclid who is long dead. The fact is that when Euclid formulated these theorums he didn’t know they would have such far reaching consequences, he had no idea what a computer is. That shows how many application math can have not just now, but far into the future. Maybe our ideas of research in modular theory, like solving for Fermat’s last theorum could potentially have far reaching consequences in modeling our world and certain things in the future of this world. This shows how math has many applications not only in the present, but also in things we haven’t even thought of. Even things that don’t seem to have applications in the real world may have applications in the future. In addition, it doesn’t necessarily need to have practical applications in that sense as math can help you develop your mental faculties, which can be applicable in may different things.

However there’s always the argument that math is a social construct and isn’t inherent on the world. In that sense, math isn’t real and can’t truly be applied to the real world. Math is perfect and the real world is not. Sometimes math can have models for things but they tend not to be perfect. That’s why uncertainties are so important and common in math. A lot of science is math, and in things like theoretical physics, it may be all math. In that sense, math is used to solve questions in physics, but it can’t be tested in the real world as of yet, so there’s no way of knowing whether certain maths are applicable. That’s why we can’t prove string theory. In a lot of math, data is required, but it’s never enough to have a full truth.

Overall, I think that if certain areas of math don’t seem to have applications in the real world, they may in the future, and that’s why I can’t say that math does or doesn’t have real-world applications with absolute certainty.

## The Mathematical Method

THIS PART IS KIND OF A RANT, NO NEED TO READ, JUST STUFF I DIDN’T SAY IN CLASS                     (feel free to stumble through my rant if you want to though)

We’re supposed to describe the ‘Mathematical Method’, but I’m not even sure what math is. There’s a part of me that thinks that it’s simply a social construct. One could counter that thought rather easily, because if you thought about math in an extremely one dimensional fashion, math exists, numbers work, and they repeat. Therefore math can’t be a social construct because it exists and developed separately in different cultures. However, we only experience the world in three dimensions, four if you count time, or more because I’m not exactly smart. The thing is, we don’t really understand our world, and it could all be coincidence that math works for us the way it does. That does mean it’s right or true. We could learn one more simple thing about the world, we could learn something about the way math works considering higher dimensions, and our entire system of math going down to one plus one could be wrong.

In class we discussed how in math, things can be proven mathematically and physically. However, we had an extremely flawed discussion about it that really annoyed me. We discussed the idea that triangles can have different sums of angles based on the shape of the plane they’re on. For example, we discussed triangles on spheres. I think the problem was that the people in the class didn’t understand that mathematically, spheres are two-dimensional objects, balls are the ones that are three-dimensional. Someone in class used the idea that if you cut out a triangular piece of pie crust from a curved pie, it’s a three dimensional object and therefore not a triangle, and rather a prism. Therefore, continually proving that the sum of angles on all triangles is 180 degrees? That got me extremely frustrated but I’ve been working on holding my tongue. Anyways, the entire basis of that argument is wrong because the pie crust is three-dimensional, not two dimensional. Also, the idea that a plane must be perfectly flat is kind of frustrating. However, this was a good introduction into the idea of whether there are mathematical ideas that can’t be proven physically, only mathematically.

THIS IS WHERE I ACTUALLY DO MY HOMEWORK

I guess the mathematical method is Euclidean logic, as in the seven books of Euclid that Lincoln took off five years to read and ponder. Based on that assumption (I’m using Euclidean logic here), the mathematical method is the use of known facts, to extrapolate new facts. For example, I know the sky is blue, because my mother told me so, therefore, anything that looks like the sky is also blue. It’s annoying that I never got to study logic in math class in school, we should really put it into the curriculum. However, I still use this in school, it’s how we learn every single subject. Take science, if mixing water and acid results in an exothermic reaction, and I know that exothermic reactions give off energy. We constantly learn things, and put them together to realize new things. In a way, that’s all learning is, and therefore, everything is the mathematical method.

In that way, one could say that the scientific method, is part of the mathematical method. In the way that people argue the natural sciences are branches of math, simply with dimensions, the scientific method is a branch of the mathematical method. The scientific method involved the use of prior knowledge to gain further knowledge. However, at its’ very base the mathematical method doesn’t have an empirical element. Whereas is science, you test theories, making science seem very different than math, and in my opinion, much less concrete. Because of the empirical nature of science, the use of the scientific method involves degrees of certainty, whereas math doesn’t. However, I’m not completely sure where the line between science and math is. Depending on your definition of math, it does have degrees of certainty. In fact the mathematical method could also be something within the scientific method. Often times you have prior knowledge on a subject (knowledge + observation), which leads to a question, which leads to an experiment, data collection, and conclusion. Math also has all of these elements, except the experiment part is usually the process of solving an equation, and the data and conclusion are the answers. That mostly assumes that math is equations and it’s definitely more than that. Still, this illustrates the fact that math and science are so intertwined it’d be impossible to find a proper difference for “strengths and weaknesses”.

The “strengths and weaknesses” of the mathematical and scientific methods boil down to one main thing, truth. They both attempt to find truth in slightly different ways. One could say that the mathematical method is flawed because it makes assumptions, such as “the sky is blue”. However, the scientific method also makes assumptions, if you’re testing the density of two different balls, you don’t question whether they’re balls, or that they’re what you’re actually measuring. In that sense, they’re equal. One could argue that a strength of the scientific method is trials, however it can never find a perfect truth, and it’s always based off of flawed fact, just like the mathematical method. On the other hand, a strength of the mathematical method is that it doesn’t have degrees of certainty and therefore can have perfect truths if you don’t take into account the degrees of certainty of the original truths.

## Scientific Controversy

The many-worlds-theory which some argue is more philosophical than scientific is a point of scientific controversy as it’s unprovable. It is the theory that (in essence) parallel universes exist. Some argue it’s true from the string theory point of view, and the quantum mechanics approach, both of which one could only prove with math and not empirical data, this is partially where the idea that it’s philosophical comes from. Regardless of the type of, say, multiverse theory people argue (these are all very similar), they are only provable within the clear world of mathematics. This means that it’s impossible to even conceive an experiment or question whether scientists have been cheating. Still, the mathematical route to proving these questions is kind of an experiment in its own way. The theoretical physicists start out with question and then they get a hypothesis and then they test out mathematically. I personally choose to believe that there may be other universes or versions of our world out there, or there is simply so much universe that it’s almost the exact same thing as have alternate realities.

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