I don't consider this proof to be particularly elegant, but it gets the job done, so I thought I'd share.
p is Prime
p and n are integers
p is a factor of n^2
Thus, p is a factor of n
my proof
n^2 is not prime........................(can be factored by n)
p < n^2......................................(p is prime, p is not greater than or equal to n^2)
-----assume p is not a factor of n......(reductio ad absurdum)
-----p is not a factor of n*n...............(there's probably a rule for this)
-----n*n = n^2
-----p is not a factor of n^2
-----p is and is not a factor of n^2....(conjunction creates a contradiction)
p is a factor of n
Wednesday, November 30, 2011
Things to do
I'll go back and check these off as we go, and then eventually make a new list. Hopefully this will help with productivity!
Chapter 2 summery
Chapter 2 mini essay- What answer to the question "why philosophize" do I find most compelling?
Chapter 3 summery
Possible supplementary posts:
Vocabulary list
Interesting excerpts from reading
making up hypothetical situations (we had a conversation about how I'm bad at this)
Chapter 2 summery
Chapter 2 mini essay- What answer to the question "why philosophize" do I find most compelling?
Chapter 3 summery
Possible supplementary posts:
Vocabulary list
Interesting excerpts from reading
making up hypothetical situations (we had a conversation about how I'm bad at this)
Wednesday, November 16, 2011
A preface to Philosophy, Chapter 1
Well crap, I meant to put this up much sooner. So it goes.
Here are my responses to the study questions presented at the end of chapter 1.
1. Why is it difficult to develop an adequate description of philosophical problems?
The term philosophy has, over time, developed many alternative contextual meanings. The first step to creating a useful definition is to eliminate the more contemporary uses of the word. An accurate definition is hard to pin down, because philosophy covers a broad range of subject matter, and intersects with many other disciplines. Additionally, philosophy has no unique defining feature, but is instead a collection of characteristics, all of which can be found in other disciplines.
2. What characteristics of philosophical problems are cited in this chapter?
Philosophical problems are defined as having three necessary characteristics- They deal with fundamental ideas, question the meaning, truth, and logical relation of these ideas, and can not be answered through empirical methods.
3. What is a "fundamental idea"?
Fundamental ideas are broad beliefs, assumptions, and concepts that effect the truth of many more specific ideas.The meaning, truth, and interrelatedness of fundamental ideas make them relevant to philosophical discussion.
4. What does it mean to say that philosophical problems involve logical relations between ideas?
Philosophical problems explore fundamental ideas, and then expand out to all ideas logically connected to these fundamentals. Sets of ideas where one is dependent on another, or connect to each other through if-then statements, are logically related. Logical relations between ideas are useful in determining the truth of an idea or soundness of an argument.
5. Give some examples of kinds of logical relations.
A logical incompatibility is a relationship where two beliefs can not exist at the same time. p => ~q.
A logical implication is a relationship where one belief logically leads to the second. The second belief could be seen as a consequence of the first belief. p => q
6. Give an example of an incompatibility of beliefs that contributes to a philosophical issue.
Computers don't really think or create; they just do what they are programed to do.
True artificial intelligence is an attainable achievement.
Humans have free will and control of their destiny.
People's choices are constantly effected by unalterable forces, such as DNA and luck.
7. Distinguish between a question of principle and a question of fact.
Questions of principal are examined through logical relation. These questions lead to conclusions where known evidence is used to determine if an unknown is true or false.
Questions of fact encourage scientific investigation. They deal with things that can be proved to be true or false, whether right now or some time in the future.
8. What does it mean that philosophical problems are nonempirical?
Philosophical problems can not be conclusively solved though observation or experimentation. Philosophy is like science, but by necessity you have to stop after making the hypothesis. Empirical data, such as what can be determined through observation, can contribute to an argument or theory, but a philosophical theory is ultimately unprovable.
9. What does it mean to say that a belief is empirically falsifiable?
10. Why is it false to assert that science is about an observable world and philosophy is about an unobservable world?
11. Roughly describe the relation between empirical facts and philosophical arguments. Give Examples.
12. What is the difference between giving reasons and giving causes for a belief?
Here are my responses to the study questions presented at the end of chapter 1.
1. Why is it difficult to develop an adequate description of philosophical problems?
The term philosophy has, over time, developed many alternative contextual meanings. The first step to creating a useful definition is to eliminate the more contemporary uses of the word. An accurate definition is hard to pin down, because philosophy covers a broad range of subject matter, and intersects with many other disciplines. Additionally, philosophy has no unique defining feature, but is instead a collection of characteristics, all of which can be found in other disciplines.
2. What characteristics of philosophical problems are cited in this chapter?
Philosophical problems are defined as having three necessary characteristics- They deal with fundamental ideas, question the meaning, truth, and logical relation of these ideas, and can not be answered through empirical methods.
3. What is a "fundamental idea"?
Fundamental ideas are broad beliefs, assumptions, and concepts that effect the truth of many more specific ideas.The meaning, truth, and interrelatedness of fundamental ideas make them relevant to philosophical discussion.
4. What does it mean to say that philosophical problems involve logical relations between ideas?
Philosophical problems explore fundamental ideas, and then expand out to all ideas logically connected to these fundamentals. Sets of ideas where one is dependent on another, or connect to each other through if-then statements, are logically related. Logical relations between ideas are useful in determining the truth of an idea or soundness of an argument.
5. Give some examples of kinds of logical relations.
A logical incompatibility is a relationship where two beliefs can not exist at the same time. p => ~q.
A logical implication is a relationship where one belief logically leads to the second. The second belief could be seen as a consequence of the first belief. p => q
6. Give an example of an incompatibility of beliefs that contributes to a philosophical issue.
Computers don't really think or create; they just do what they are programed to do.
True artificial intelligence is an attainable achievement.
Humans have free will and control of their destiny.
People's choices are constantly effected by unalterable forces, such as DNA and luck.
7. Distinguish between a question of principle and a question of fact.
Questions of principal are examined through logical relation. These questions lead to conclusions where known evidence is used to determine if an unknown is true or false.
Questions of fact encourage scientific investigation. They deal with things that can be proved to be true or false, whether right now or some time in the future.
8. What does it mean that philosophical problems are nonempirical?
Philosophical problems can not be conclusively solved though observation or experimentation. Philosophy is like science, but by necessity you have to stop after making the hypothesis. Empirical data, such as what can be determined through observation, can contribute to an argument or theory, but a philosophical theory is ultimately unprovable.
9. What does it mean to say that a belief is empirically falsifiable?
10. Why is it false to assert that science is about an observable world and philosophy is about an unobservable world?
11. Roughly describe the relation between empirical facts and philosophical arguments. Give Examples.
12. What is the difference between giving reasons and giving causes for a belief?
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