Yes, I find Lakatos' book very interesting! Turns your head upsidedown but I enjoyed reading it :smile: I sometimes have some troubles doing thought experiments especially as I am terrible with geometrical concepts...

Heyyyy

Wow I'm really glad this discussion is interesting you all, sharing is the best really! :blush:

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:

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

Yes I agree !

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

:up: :100: I think that's the conclusion I too can draw from all this

I really want to thank you for your help, I can really see the bigger picture here now!

And so heuristic counterexamples bring about a change in mathematical meaning!

Don't know whether I was clear on this point so I'm writing it again

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

I think that what appeared to be logical counterexamples were heuristic counterexamples and only when we switched language under some kind of heuristic pressure did we end up with logical counterexamples, used to revise a given theorem.

Thanks! It's all clearer now! and I think I can fully appreciate the last sentence:

Heuristic is concerned with language-dynamics, while logic is concerned with language-statics. — Twinkle221

And yes, the word "metalanguage" felt odd so I'm glad to see we agree upon its use.

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

So I assume, following your answer, that this is why by switching languages we can introduce "real" counterexamples that were not counterexamples before.

What "allows the monsters to work as refutations" is a mismatch between how concept of polyhedron is articulated verbatim — fdrake

But that works only when we switch between languages and get rid of the "ordinary language" right?

unformalised/unarticulated content of the intended interpretation of the terms. — fdrake

And so when this content gains structure by being incorporated in a language that's when counterexamples (that are all heuristics) get "real" by concept-stretching

logical counterexamples — Twinkle221

And I assume these are heuristic still

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

And this is just to say that concept-stretching does his job through refutations, at least I believe so!

And I guess we could add that concept formation cannot be separated from "definition formation" which relates to a more heuristic study of knowledge.

Hello, Thank you so much for taking the time to answer and, indeed, I believe it helps!
Especially when you talk about intended interpretation, so this is why there is no contradiction between: (a) all polyhedral are Eulerian, and (b) the picture-frame is not Eulerian.
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" means that switching to L2 or L3 (?) we use concept-stretching to add counter-examples into the proof and contribute to the growth of knowledge (heuristic power). Thus using L1/L2 and L3 is just a way of describing the extension of knowledge/increase in content ?
I'm so sorry, I feel like everything gets mixed up in my head.