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Demonstrative vs Plausible Reasoning Patterns of Plausible Inference Volume II By G. Polya Princeton Univ. Press 1954. 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. N=1 then 8n+3=11 N=2 then 8n+3=19 N=3 then 8n+3=27

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

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**Demonstrative vs Plausible Reasoning**Patterns of Plausible Inference Volume II By G. Polya Princeton Univ. Press 1954**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**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**11=1+25**19=9+2 5 27=1+2 13 35=1+ 217=9+ 213=25+ 25 43=9+ 217 51=?**Does this prove Euler’s hypothesis?**No, Yet each verification renders the conjecture more credible**Let A denote some clearly formulated conjecture**For example: A is the conjecture 8N+3=x2+2p**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**The standard hypothetical syllogism**Demonstrative Reasoning (Aristotle) A implies B B false A false**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**Plausible Inference**Plausible Reasoning A implies B B true A more credible The verification of a consequence renders a conjecture more credible**Second example**The area of the lateral surface of the frustum is: Theorem: (R+r) sq rt [(R-r)2 + h2] Can you check this result by applying to some case you already know?**When R=r you get cylinder**Consequence B1: Area is (2 R) h**When r=0, and h=0**You get a circle Consequence B2: Area of circle is R2**Plausible Inference**Plausible Reasoning A implies Bn+1 Bn+1 is very different from the formerly verified consequences B1, B2, …, Bn of A Bn+1 is true A much more credible**Plausible Inference**Plausible Reasoning A implies Bn+1 Bn+1 is very similar to the formerly verified consequences B1, B2, …, Bn of A Bn+1 is true A just a little more credible**Plausible Inference**Plausible Reasoning A implies B B very improbable in itself B is true A very much more credible**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**Inference from Analogy**Perimeters of Figures Principal Frequencies of of Equal Area Membranes of Equal Area**Plausible Inference**Plausible Reasoning A analogous to B Bis true A more credible A conjecture becomes more credible when an analogous conjecture turns out to be true

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