Timeline for Model of an argument
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Sep 19, 2023 at 21:03 | comment | added | J D | Let us continue this discussion in chat. | |
| Sep 19, 2023 at 20:41 | comment | added | causative | Essentially, circularity is the formal fallacy "A implies B; B implies A; therefore, A and B." In informal reasoning, if you're dealing with credences you run into the problem of where to stop. "A increases the credence of B; B increases the credence of A." After enough cycles back and forth between A and B, wouldn't that bring your credence for both A and B arbitrarily close to 1? So it doesn't seem informally useful either - unless, possibly, strong safeguards are put in place to somehow prevent that "stopping problem." | |
| Sep 19, 2023 at 19:33 | comment | added | causative | To resolve that question at least one of the two propositions (e.g. "3SAT is not in P") must be proved on its own merits starting from the axioms, which has not happened yet. The circular justification, ungrounded in the axioms, is not enough to establish either proposition. | |
| Sep 19, 2023 at 19:33 | comment | added | causative | In mathematics everything is done according to a formal system, and formal systems do not have circularity. They begin from axioms and a proof proceeds with the structure of a DAG. It may be that theorem T1 can be used to prove theorem T2 and vice versa, but this alone does not support T1 or T2. For example, in computer science we may say, "If 3SAT is not in P then subset-sum is not in P. If subset-sum is not in P then 3SAT is not in P." But neither of these statements shows 3SAT to be outside P, nor does it show subset-sum to be outside P; that remains the open question, "P=NP?" | |
| Sep 19, 2023 at 15:49 | comment | added | J D | corresponds to the truths our PA-based language show to be true also. Making math rigorous is about showing the circular justification between multiple domains. That's why calculus was intuitive up until analysis was fully in place. Vicious circularity is fallacious NOT because it is false (circular arguments are tautological), but because IT abstracts too much from the argument and relies the triviality of the Principle of Identity. In more complex forms of justification, it is desirable because it is demonstrative of a lack of contradiction. Quine's web of belief and Popper's falsifibility. | |
| Sep 19, 2023 at 15:47 | comment | added | J D | but that the arithmetic arguments can be seen as supporting geometric conclusions, and neither domain is foundational to the other. The Greek construction to prove the Pythagorean theorem can be used as a foundation to demonstrate the arithmetic principle of the measures of legs and the hypotenuse, or you can introduce proof in the other directional. It is the coherence (the justificational circularity) between the domains that fully characterizes and endorses the legitimacy of the Pythagorean theorem. Why? Because it shows that what we see with our eyes as sensible experience... | |
| Sep 19, 2023 at 15:44 | comment | added | J D | I just read the conversation with Conifold, and I see you seem reluctant to accept that informal necessarily maintains some loose degree of circularity to maintain coherence. Perhaps the motivations might make it more palatable. Knowledge is often constructed in dependent domains independently. Think about how analytical geometry relies on both arithmetic and geometric reasoning which started off as separate domains. Eventually, as the two disciplines were integrated by the introduction of the Cartesian plane, you have a situation where geometric arguments support arithmetic ones... | |
| Sep 19, 2023 at 15:27 | history | answered | J D | CC BY-SA 4.0 |