Frederik, lists,
That is a general explanation attempt of why humans are capable of > abduction - that does not say anything about particular cases such as > Wegener's. Hmm. I'm not sure what you could be looking for here. In general, any semiotic being capable of abduction must have a natural instinct. In particular, any given abduction will be the result of that natural instinct meeting with the observation of given phenomena. On the other hand, if you mean it seems that with this idea we can't really get into a detailed analysis of just how this hypothesis was achieved and none of the others, I too share some such frustration, and would try to offer a more robust account than Peirce offered. You are right that discoveries are often seen or suspected prior to > demonstration - but it is too little to say demonstrations are only for > communication and persuasion purposes. Yes, you are right, it is too little to say. Certainly it helps to check one's work to be really convinced of the idea, and make sure it doesn't turn out somehow self-contradictory or incoherent. But I stand by the contention that the method of discovery of the idea is typically separate from its demonstration in the context of mathematical research. Certainly - and that is where P argues that theorematic deduction is called > for - Yes, I know, though of course I am saying instead that this is when abduction is called for. Theorematic reasoning should describe the whole process, both abductive and deductive, in my opinion. But I think that this is as far as we will get in discussion about it. I'll just have to agree to disagree with Charles on this one. -- Franklin On Sun, May 3, 2015 at 3:55 AM, Frederik Stjernfelt <stj...@hum.ku.dk> wrote: > Dear Franklin, lists - > > > It is classically described as such in the literature. The formal >> structure af abduction (the proposition A explains the occurrence B as a >> matter of necessity, therefore A can be chosen as a hypothesis to explain >> B) does not explain why A should be chosen over infinitely many other >> propositions with the same property. (see e.g. Michael Hoffmann's papers on >> abduction) > > > Though Peirce did address this issue in terms of Galileo's il lume > naturale, with the qualification that it has to do with a natural instinct. > I have my own ideas about why we can happen upon the right hypotheses, but > this is not the thread for such a discussion. > > > That is a general explanation attempt of why humans are capable of > abduction - that does not say anything about particular cases such as > Wegener's. > > > And this is where the trial-and-error phase of theorematic reasoning >> differs from ordinary abduction. The latter is standardly seen as a step in >> empirical research, from data to hypothesis. But all P's examples of >> theorematic reasoning are non-empirical, there is no data, for the whole >> problem considered is purely formal (like when selecting the right >> auxiliary lines in the triangle proof). That is a trial-and-error thing >> without procedural necessity - you may have to experiment with different >> lines until you find the right ones permitting you to conduct the proof. >> In that sense it is an "abductive" phase of theorematic reasoning. But it >> is not abductive in the sense that its starting point is data and its >> conclusion is a hypothesis. The right auxiliary lines are not at all a >> hypothesis explaining anything. For that reason, I do not think the >> proposal of saying that theorematic reasoning is just trivial deduction >> interspersed with abduction is satisfactory. > > > I'm not sure about abduction being characterized as a move from data to > hypothesis. Peirce's early account of abduction is somewhat close to that > idea, but not so much his later account. Rather, it is typified by the move > from a surprising fact, something which does not fit available data, to a > hypothesis explaining the surprising fact. > > > Correct, and that fact is a part of data. > > > Suppose a case where the conclusion of the theorematic proof is > considered the first premiss of an abductive argument, and the second > premiss is the introduction of a hypothesis that would explain the > conclusion of the theorematic proof. Then the conclusion of such an > abduction would be the theorem introduced into the proof. So the "data" is > simply the desired conclusion itself. In later discussions of abduction, > Peirce does put it as something like this: There is a surprising fact. But > if A were true, then the surprising fact would be a matter of course. > Therefore A is true. Peirce admits though that not every case of abduction > involves a surprising fact, but simply something that calls for > explanation. I would suggest in this case that the desired conclusion is > what is in need of explanation. > > It should be noticed that the way mathematicians make new discoveries is > not typically through mathematical demonstrations; rather, the > demonstrations are produced after the fact to communicate and prove the > discovery to the satisfaction of other mathematicians. > > > You are right that discoveries are often seen or suspected prior to > demonstration - but it is too little to say demonstrations are only for > communication and persuasion purposes. > > Considered in the larger context of the difference between discovery and > demonstration in mathematics, it may very well be the case that every such > major theorem in theorematic reasoning started off as a hypothesis to > explain a desired conclusion, and the demonstration was produced after the > fact. Of course, it would be very difficult to prove this as a general > rule. But it is an alternative explanation which bears merit. It should > also be noticed that all of this doesn't change the necessity of the > conclusion in the theorematic reasoning, once proven. > > I suppose it could be replied that nevertheless, diagram experimentation > would be required to develop the hypothesis. Well, my suggestion would be > that, having certain propositions already, and a desired conclusion, but > not being able to reach that conclusion from the given propositions alone, > the diagram is put on hold while the mathematical mind starts thinking > about what would explain the conclusion. > > > Certainly - and that is where P argues that theorematic deduction is > called for - > > Best > F > >
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