I'm so sorry, I feel like everything gets mixed up in my head. — Twinkle221
Especially when you talk about intended interpretation, so this is why there is no contradiction between: (a) all polyhedral are Eulerian, (b) the picture-frame is not Eulerian — Twinkle221
So then, by changing language i.e., switching from L1 to L2, we're falling into this meta-language. "If we keep to the tacit semantical rules of our original language our counterexamples are not counterexamples — Twinkle221
And I guess we could add that concept formation cannot be separated from "definition formation" which relates to a more heuristic study of knowledge. — Twinkle221
Heuristic is concerned with language-dynamics, while logic is concerned with language-statics. — Twinkle221
So I assume, following your answer, that this is why by switching languages we can introduce "real" counterexamples that were not counterexamples before.I believe that's so, the kinds of polyhedron intuited in "all polyhedron are Eulerian" are conceptually distinct from the picture frame polyhedron (and the other monsters). — fdrake
What "allows the monsters to work as refutations" is a mismatch between how concept of polyhedron is articulated verbatim — fdrake
unformalised/unarticulated content of the intended interpretation of the terms. — fdrake
logical counterexamples — Twinkle221
That is, we may have two statements that are consistent in L1, but we switch to L2 in which they are inconsistent. Or, we may have two statements that are inconsistent in L1, but we switch to L2 in which they are consistent. As knowledge grows, languages change. — Twinkle221
Usually, when a counterexample is presented, you have a choice: either you refuse to bother with it, since it is not a counter-example at all in your given language L1, or you agree to change your language by concept-stretching and accept the counterexample in your new language L2… — Twinkle221
And yes, the word "metalanguage" felt odd so I'm glad to see we agree upon its use. — Twinkle221
But that works only when we switch between languages and get rid of the "ordinary language" right? — Twinkle221
So I assume, following your answer, that this is why by switching languages we can introduce "real" counterexamples that were not counterexamples before. — Twinkle221
However, if one takes the refutation as an opportunity to include the monster in the intension of the original term, it becomes a refutation, but the intension is also altered tacitly by taking this opportunity. — fdrake
That matching process has changed "the taxonomic, conceptual..." frameworks of (some of) the involved terms, and those frameworks are expressed in the statement of an L2 definition/theorem-statement in which the monster refutes the L1 statement since the monster now unambiguously counts as an example of the term it targets. EG "Polyhedra are Eulerian" with the intended interpretation of convexity and simplicity vs concave polyhedral monsters. — fdrake
In mathematics, at an opposite pole are extremely formal proofs by computer algorithms — jgill
Hello! I am a graduate student in philosophy of science — Twinkle221
I really want to thank you for your help, I can really see the bigger picture here now! — Twinkle221
In mathematics, at an opposite pole are extremely formal proofs by computer algorithms — jgill
Yes indeed, that's also why Lakatos' book is really interesting! He argues against formal mathematics byt his method of "proofs and reputations" may very well be applied to formal mathematics as well.
I am really interested in the use of heuristics as a research methodology :blush: — Twinkle221
Formal mathematics is simply that employed by research math people these days. You know all this of course. I've now read a bit about the topic here and I suppose what I have used might be called naive heuristics, in light of all the various types of heuristics described in Wiki. — jgill
Yes, an expectation of the reader filling in the gaps is common. It would be dreadful if that practice were abandoned! :cool: — jgill
What that does do, however, is make mathematical demonstrations heuristic in Lakatos' sense; more about displaying the concepts to a sufficient degree of obviousness than mandating that all proofs have every step of reasoning spelled out symbolically. — fdrake
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