1. Demonstrative vs Plausible Reasoning Patterns of Plausible Inference Volume II By G. Polya Princeton Univ. Press 1954

2. Conjecture Any integer of the form 8N+3, where N=1,2,3,… is the sum of a square and the double of a prime

3. N=1 then 8n+3=11 N=2 then 8n+3=19 N=3 then 8n+3=27 N=4 then 8n+3=35 N=5 then 8n+3=43 N=6 then 8n+3=51 N=7 then 8n+3=59 N=8 then 8n+3=67

4. 11=1+2  5 19=9+2  5 27=1+2  13 35=1+ 2  17=9+ 2  13=25+ 2  5 43=9+ 2  17 51=?

5. Does this prove Euler’s hypothesis? No, Yet each verification renders the conjecture more credible

6. Let A denote some clearly formulated conjecture For example: A is the conjecture 8N+3=x 2 +2p

7. Let B some consequence of A, which is neither proved or disproved For example: B is that the number 51 is the sum of a square and the double of a prime

8. The standard hypothetical syllogism Demonstrative Reasoning (Aristotle) A implies B B false A false

9. Plausible Reasoning What happens if B turns out to be true? 51=25+ 2  13 There is no demonstrative conclusion: the verification of its consequence B does not prove the conjecture A

10. Plausible Inference Plausible Reasoning A implies B B true A more credible The verification of a consequence renders a conjecture more credible

11. Second example The area of the lateral surface of the frustum is: Theorem:  (R+r) sq rt [(R-r) 2 + h 2 ] Can you check this result by applying to some case you already know?

12. When R=r you get cylinder Consequence B 1 : Area is (2  R) h

13. When r=0, and h=0 You get a circle Consequence B 2 : Area of circle is  R 2

14. Plausible Inference Plausible Reasoning A implies B n+1 B n+1 is very different from the formerly verified consequences B 1, B 2, …, B n of A B n+1 is true A much more credible

15. Plausible Inference Plausible Reasoning A implies B n+1 B n+1 is very similar to the formerly verified consequences B 1, B 2, …, B n of A B n+1 is true A just a little more credible

16. Plausible Inference Plausible Reasoning A implies B B very improbable in itself B is true A very much more credible

17. Plausible Inference Plausible Reasoning A implies B B quite probable in itself B is true A just a little more credible The verification of a consequence counts more or less according as the consequence is more or less improbable in itself

18. Inference from Analogy Perimeters of Figures Principal Frequencies of of Equal Area Membranes of Equal Area Circle3.55Circle4.261 Square4.00Square4.443 Quadrant4.03Quadrant4.551 Rec 3:24.08Rec 3:24.616 Semicircle4.10Semicircle4.624 Sextant4.21Sextant4.774

19. Plausible Inference Plausible Reasoning A analogous to B B is true A more credible A conjecture becomes more credible when an analogous conjecture turns out to be true