Timeline for Nontrivial circular arguments?
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 9, 2012 at 23:55 | vote | accept | David Feldman | ||
| Jan 28, 2011 at 11:03 | comment | added | Terry Tao | tricki.org/article/… | |
| Jan 28, 2011 at 6:12 | comment | added | Aaron Meyerowitz | Given that $\sqrt{2}$ is irrational we get that for every rational with $\frac{p}{q} \le \sqrt{2}$ we have $\frac{p}{q} <\frac{p}{q}+\frac{1}{3q^2} <\sqrt{2}$ with something similar from above. | |
| Jan 28, 2011 at 1:50 | history | edited | David Feldman | CC BY-SA 2.5 |
added 1028 characters in body; deleted 1 characters in body; added 42 characters in body
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| Jan 28, 2011 at 1:33 | comment | added | David Hansen | The estimate $\psi(x)=x+O_{\varepsilon}(x^{1/2+\varepsilon})$ implies RH, which implies $\psi(x)=x+O(x^{1/2}\log^2{x})$. I've always found this rather amusing. | |
| Jan 27, 2011 at 19:04 | answer | added | Joel David Hamkins | timeline score: 34 | |
| Jan 27, 2011 at 18:13 | comment | added | Mitch | Every circular proof is also a proof of equivalence. That is, the unsatisfying circular proof, using an assumption that ends up depending on the thing to be proved, is also a proof the assumption is equivalent to the conclusion (they both imply each other). The hard part is to extract from the disappointment the bidirection or circle of implication. | |
| Jan 27, 2011 at 17:08 | comment | added | David Feldman | Doubtless one could vandalize Erdos-Selberg type arguments to get, uniformly, Chebyshev-type estimates of arbitrary strength, so I would say it is impossible to "rule out" this possibility. But if one found a weak theory of arithmetic where the known Chebyshev-type estimates had a formulation but not the Erdos-Selberg proof, then the result Diamond describes would have more than psychological import. I believe working in the theory of bounded induction it remains open even to prove the infinitude of the primes. So people do think about which theory just captures which argument. | |
| Jan 27, 2011 at 15:53 | comment | added | Qiaochu Yuan | Perhaps psychologically we feel that the infinite sequence of Chebyshev-type estimates could, in principle, be attacked by other methods. I don't think anyone has ruled out this possibility, in any case. | |
| Jan 27, 2011 at 14:48 | history | asked | David Feldman | CC BY-SA 2.5 |