Philosophy assignment 10 questions

1 8 : T h e U G R u l e P H I L 1 1 0 ; S p r i n g 2 0 1 9 ; To m D o n a l d s o n

PHIL 110 and COVID 19

• There will be a final exam of some kind – I’m not sure yet how this will be done. I will do my very best to ensure that the assessment is fair to all of you.

• The TAs will deliver the graded midterm exams to me – I have them in my office. If you want to see your midterm, let me know.

• I will omit some of the more challenging material from this iteration of the course. This is to ensure that you all have adequate time to prepare for the final, despite the unusual obstacles that 2020 has produced.

PHIL 110; Spring 2020; Lecture 18 2

1 : T h e U G R u l e I n A r i t h m e t i c

Square numbers: Rectangle numbers:

1  1 = 1 1  2 = 2

2  2 = 4 2  3 = 6

3  3 = 9 3  4 = 12

4  4 = 16 4  5 = 20

5  5 = 25 5  6 = 30

Hypothesis:

• If you add together two consecutive rectangle numbers, the result is always twice a square.

• For any n, the sum of the nth rectangle number and the (n+1)th

rectangle number is always twice a square.

PHIL 110; Spring 2020; Lecture 18 4

Let n be any arbitrarily chosen natural number.

Then the nth rectangle number is: n(n+1)

Also, the (n+1)th rectangle number is: (n+1)(n+2)

So, the sum of the nth rectangle number and the (n+1)th rectangle number is:

n(n+1) + (n+1)(n+2)

= (n2 + n) + (n2 + n + 2n + 2)

= 2n2 + 4n + 2

= 2(n+1)2

This is indeed twice a square!

Therefore:

For any n, the sum of the nth rectangle number and the (n + 1)th rectangle number is twice a square.

PHIL 110; Spring 2020; Lecture 18 5

The UG Rule

• The statement we just proved is a universal generalization:

For any n, the sum of the nth rectangle number and the (n + 1)th rectangle number is twice a square.

• We proved it by proving that an “arbitrary instance” is true.

• This is an example of the UG rule at work.

PHIL 110; Spring 2020; Lecture 18 6

2 : T h e U G R u l e i n G e o m e t r y

Alternate Angles

Assuming that the red lines are parallel, the angles b and c are equal!

PHIL 110; Spring 2020; Lecture 18 8

a b

c d

Alternate Angles

Assuming that the red lines are parallel, the angles b and c are equal!

PHIL 110; Spring 2020; Lecture 18 9

ab

cd

PHIL 110; Spring 2020; Lecture 18 10

a

b

c

PHIL 110; Spring 2020; Lecture 18 11

a

b

c

PHIL 110; Spring 2020; Lecture 18 12

a

b

c

a

PHIL 110; Spring 2020; Lecture 18 13

a

b

c

a

PHIL 110; Spring 2020; Lecture 18 14

a

b

c

a c

PHIL 110; Spring 2020; Lecture 18 15

a

b

c

a c

PHIL 110; Spring 2020; Lecture 18 16

a

b

c

a c

We’ve shown that the angles inside any triangle add up to 180°.

We will acknowledge only those proofs in which one can appeal step by step to preceding propositions and definitions. If, for the grasp of a proof, the corresponding figure is indispensable, then the proof does not satisfy the requirements that we imposed on it. … in any complete proof the figure is dispensable. (Pasch, 1882)

PHIL 110; Spring 2020; Lecture 18 17

[B]e careful, since [the use of diagrams] can easily be misleading. A theorem is only proved when the proof is completely independent of the diagram. The proof must call step by step on the preceding axioms. The making of figures is [equivalent to] the experimentation of the physicist … (Hilbert, 1894)

PHIL 110; Spring 2020; Lecture 18 18

PHIL 110; Spring 2019; Lecture 15 19

3 : T h e U G R u l e Wi t h i n N a t u r a l D e d u c t i o n