HUMA 1160 Lecture : 1001Study of Descartes - Lecture Notes.pdf

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greentoad340

27 Oct 2012

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What he things mathematics is about and how it can help learning and philosophy. The mind is reasoning is never at fault. Faculties from which all this knowledge can stem. A cognitive faculty given to us by god which utilizes the light of reason. Purpose is the see the connection between two relata. Moving down from first principles to a conclusion. Things that equal to the same thing are equal to each other. Whether you"re doing simple relata using intuition, it follows that there is only one cognitive faculty for knowledge. All the tools, and knowledge of mathematics come from god. Concerned with method focusing exclusively in the perspective of a mathematician. Level of certainty required by a philosopher exceeds the level required by a mathematician. What persuades each person that they have the right answer. What convinces each of them is the psychological component in mathematics. Unreliable source of deciding certainty and establishing truth.

Related Questions

Write about blancing chemical equations using linear algebra

Topic: Balancing chemical equations

https://www.wikihow.com/Balance-Chemical-Equations-Using-Linear-Algebra

Paper Specifications:

The writing component is minimally a five-page single space paper in size 12 font. The paper is expected to be written clearly and be free of grammatical mistakes in either the APA or IEEE format. The paper must include:

An abstract that gives a clear, concise description of the paper in less than 100 words. â¨

At least two pages that serve as an introduction to topic being discussed. Included in this introduction should be precise definitions and examples to help clarify any new concepts. The audience for this paper will be your peers, so make sure you write at an appropriate level. â¨

A few problems/examples completed and clearly written up. Make sure that you work a number of your own examples and don't just follow examples provided in your sources. The solutions need not be shown in there entirely, but must include a few key steps as well as an explanation of what the computation means. â¨

At least one additional (and interesting) application or example of this topic that was not found within the textbook. Be sure to cite youâre outside source using APA or IEEE style in your paper. â¨

A work cited page in either APA or IEEE style that is absence of questionable/inappropriate sources. â¨

Criteria

Introduction

*well developed; thoroughly explains the problem/process

*process clearly presented with lots of mathematical examples & reasoning

*needed definitions/ theorems are given

*fairly well developed

*informs the reader of problem/process

*process clearly presented

with mathematical support

*little development

*minimal information about problem/process

*process not clearly

presented and has little mathematical support

*little written

*confusing or inaccurate

*not all needed definitions/theorems given

Mathematical

Details

*proper mathematical reasoning employed & explained

*key steps shown with an appropriate level of detail

*lots of mathematical terminology correctly used

*some specific mathematical details that adequately explain

*most key steps are included

with a few unnecessary computations

*mathematical terminology used correctly

*either wrong or irrelevant details given

*too many confusing computations & steps provided

*some mathematical

terminology, but not used correctly

*no details given

*no mathematical terminology

Writing Organization & Style

*clearly organized throughout

*clearly guides the reader through the process using words such as âfirstâ, ârecallâ

*transitions given between paragraphs

*organized most of the way through

*usually guides the reader through the process

*some transitions given between paragraphs

*little organization

*often does not guide the reader

*no organization evident

*does not guide the reader

Citations

*complete bibliography correctly formatted with reputable sources

*references appropriately made throughout the paper

*bibliography included with appropriate sources but not correct formatting

*sources referenced

*bibliography includes sources that are not

appropriate

*sparse references without proper APA or IEEE format

*bibliography not included

*no sources are references in the text

Completeness, Citations & Form

*abstract is clear, concise, and less than 100 words

*name & reference figures,

tables and equations

*all paper specifications are thoroughly satisfied

*abstract is is clear but does not accurately represent the paper

*many references to tables,

figures, etc were understandable

*paper specifications met

*abstract is not clear & does not represent the paper

*internal references made in

the paper to tables, etc

*most paper specifications are met

*abstract is poorly written and does not accurately represent the paper

*tables, equations, figures, or paragraph formatting was confusing

*not all paper specifications

were met

Problem 3

Math 151 - Sections 34, 35, 36 - Workshop 11 This workshop deals with an addition formula in section 5.1. This formula is related to an ld story about the great mathematician Gauss. Here is one version of the story Wh en Gauss was ten years old, he attended a certain math class with other kids. One fine day, the math teacher discovered that he was running out of time to grade the latest batch of math homeworks. This is why he decided to finish the grading during his math lecture period. To keep the students busy while he graded, he gave them the following task at the start of the period: Use your slate (the small personal blackboard that every student had in class) to add up the numbers 1,2,3,... , 100 Seconds after the teacher gave this in-class assignment, little Gauss pointed to his slate and yelled "Here is the answer." Then Gauss did nothing for an hour, while the other students labored with the additions and the teacher graded. At the end of the period Gauss showed the correct answer 5050 on his slate. Only Gauss got the correct answer Then Gauss told everybody that the sum 1+2+3+ + 100 is ( 1 + 100) + (2+99)+ (2+98)+--+ (50+51) = 101 + 101 + 101 +. . . +10 101. 50-5050 Problem 1. Explain carefully what Gauss had done. Include the reason why it is impor tant that 100 happens to be Problem 2. Let N be an even positive integer. Use the method of Problem 1 to prov the formula 1 +2+3+.+N) be an even number Problem 3. The formula is still valid when N is an odd integer. Prove this, using a variation on the argument given by Gauss The story continues: Gauss then gave the class a lecture on how his method could be used to sum an arithmeric series (a+1.b) + (a+2 b) + (a+3.b)+. + (a + N b), where a and b are constants Problem 4. Find a formula for the sum (a+1.b)+(a+2.b)+ (a +3.b)+(a+ N b) Prove this formula the way Gauss would have done it. You have to do the case when N is even and the case when N is odd Now let us consider a fictional alternate history where Gauss had not thought of this ve clever trick. We can imagine the ten year old Gauss going home and doing the sequential computations 1+2 1+2+3 1+2+3+4 1+2+3+4+5 and so forth, just to see if he could find a pattern in the sequence of answers. He would have discovered somethin like this: once he had the sum 1+2+3++N, he could simply add N+1 to that sum, and that would give him the sum 1+2+3 Now assume that, by trial and error, Gauss discovered that the formula 1+2+3+ +N an even numbe +) Include the reason why it is important that +Â·Â·Â· + (N +1) very quickly is true for several values of N. How could he prove that the formula is really true for ALL N? The insight described in the previous paragraph might have led Gauss to think the following: All he had to do was prove N(N+1) Â± (N + 1) other words, he only needed to show: Formula for 1 + 2+3+.+ N plus N +1 equals formula for 1+2+3+(N+1). He would have discovered the principle of mathematical induction Problem 5. Prove of the formula for 1+2 +3+... + N (N +1)(+1X++1) Show how this leads to a proof N(N+1 N+)

Complete problem 3 using the information above.

Math 151 Sections 34, 35, 36 Workshop 11 ted to arn This workshop deals with an addition formula in section 5.1. This formula is rel oft told story about the great mathematician Gauss. Here is one version of the story When Gauss was ten years old, he attended a certain math class with other kids. One fine day, the math teacher discovered that he was running out of time to grade the latest batch of math homeworks. This is why he decided to finish the grading during his math period. To keep the students busy while he graded, he gave them the following task at th of the period: Use your slate (the small personal blackboard that every student had start in class) to add up the numbers 1,2,3,... , 100 Seconds after the teacher gave this in-class assignment, little Gauss pointed to his slate Here is the answer." Then Gauss did nothing for an hour, while the other students labored with the additions and the teacher graded. At the end of the perioc Gauss showed the correct answer 5050 on his slate. Only Gauss got the correct answer. Then Gauss told everybody that the sum 12 +3.+ 100 is (1+100)+(2+99) + (3498)+ Problem 1. Explain carefully what Gauss had done. Include the reason why it is impor tant that 100 happens to be an even number Problem 2. Let N be an even positive integer. Use the method of Problem 1 to prove the formula 1 +2 +3+.. . +N(N. Include the reason why it is important that N +(50+51) = 101 + 101 +101 + +101-101.50-5050 an even number. Problem 3. The formula is still valid when N is an odd integer. Prove this, using a variation on the argument given by Gauss The story continues: Gauss then gave the class a lecture on how his method could be used to sum an arithmeric series (a +1.b) (a +2.b) (a +3.b)+. (a +N b), where a and b are constants Problem 4. Find a formula for the sum (a+1.b)+ (a+2.b) + (a +3.b)+ +(a+ N.b) Prove this formula the way Gauss would have done it. You have to do the case when N is even and the case when N is odd Now let us consider a fictional alternate history where Gauss had not thought of this very clever trick. We can imagine the ten year old Gauss going home and doing the sequential computations 1+2 1+2+3 1+2+3+4 1+2+3+4+5 and so forth, just to see if he could find a pattern in the sequence of answers. He would have discovered somethin like this: once he had the sum 1+2+3+...+ N, he could simply add N +1 to that sum and that would give him the sum 1+2+3+ + (N +1) very quickly Now ass N(N+1 ume that, by trial and error, Gauss discovered that the formula 1+2+3+ is true for several values of N. How could he prove that the formula is r for ALL N? The insight described in the previous paragraph might have led Gauss to other words, he only needed to show: Formula for 1+2+3+ + N plus N +1 equals formula for 1+2+3++N+1). He would have discovered the principle of mathema -.. + N eally true 1)+1) think the following: All he had to do was prove mm) + (N + 1) NN+1 (N+1)(( In ered the principle of mathematical induction Problem 5. Prove N(N+1) (N + 1 of the formula for 1+2+3+.+N. (Nt1MNHHI) Show how this leads to a uCtion a proof

HUMA 1160 Lecture : 1001Study of Descartes - Lecture Notes.pdf

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