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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