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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