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I can understand how the verification of hypotheseses based on inductive reasoning can be problematic, and I understand that a lot of prominent figures in philosophy are in opposition to this (Popper, for example).

But it is not entirely clear to me whether the problem is induction or inductivism? By 'induction', I mean, for example, that we observe that the non-trivial zeros of the Riemann Zeta function all lie in a certain interval, and thus we propose the hypothesis that all non-trivial zeros of this function lie in that interval... and then we can attempt to verify or falsify that hypothesis based on other scientific methods (that is, not by simply finding more zeros in the same interval which would be inductivism). That is, induction is only part of the chain, the part that leads to the formulation of a hypotheses (which can then be studied in more appropiate manners).

Is this latter kind of thinking also grouped along with the inductivism that is opposed by many? If yes, how would they then come up with their hypotheses to begin with (there are other ways, yes, but induction seems to be the most obvious one)?

  • It's not clear to me that this is an induction; it's certainly not mathematical induction; but rather a conjecture based on what is understood of the underlying mathematical architecture. – Mozibur Ullah Feb 8 '16 at 19:51
  • Induction as a thought that is truth productive, seems to be more associated with the physical sciences; at least Mach calls Physics an inductive science. – Mozibur Ullah Feb 8 '16 at 19:53
  • I am assuming that the Riemann hypothesis was derived from induction. Whether it really is or isn't doesn't matter. What matters is, is deriving hypotheses based on inductive reasoning considered logically unsound by the figures who are known to completely oppose induction? – Mikaoe Feb 8 '16 at 21:28
  • I agree with @MoziburUllah, I don't see an induction here; because no decision is made. Obviously, if you are going to test theories, there must be theories, and so there need to be techniques for generating theories. Observation of a pattern is one of them, and you can see that as 'inductive', but that is not related to the objection to making a decision primarily upon unstructured observations – user9166 Feb 8 '16 at 21:28
  • But decisions are made based upon the studies of hypotheses. If your hypotheses come, in part, from inductive reasoning, then decisions are being made as a result of inductive reasoning, right? – Mikaoe Feb 8 '16 at 21:32
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If you use induction to create conjectures, e.g. the Riemann Hypothesis, that's quite OK. Every method is allowed to create conjectures. Using induction to create conjectures is the method of generalisation.

The crucial point is how to confirm the conjecture.

Obviously in mathematics the only method is to prove the conjecture - or to disprove it by generating a counter example. But in science one cannot prove general results. A finite number of confirmed cases does not increase the probability that the general result is true. That's the problem of induction.

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But it is not entirely clear to me whether the problem is induction or inductivism? By 'induction', I mean, for example, that we observe that the non-trivial zeros of the Riemann Zeta function all lie in a certain interval, and thus we propose the hypothesis that all non-trivial zeros of this function lie in that interval... and then we can attempt to verify or falsify that hypothesis based on other scientific methods (that is, not by simply finding more zeros in the same interval which would be inductivism). That is, induction is only part of the chain, the part that leads to the formulation of a hypotheses (which can then be studied in more appropiate manners).

Your observation that the known Riemann Zeta function zeros all lie in one interval does not lead to your guess that all of the zeros lie in that interval. For example, you could have some explanation that contradicts the idea that the zeros all lie in that interval without knowing any values lying outside the interval. Rather, your idea that all the zeros lie in the relevant interval is a guess.

You can call your guessing induction if you want. But that induction has nothing at all to do with induction in the sense discussed by Popper and others. If you want to understand the objections to induction your best option is to read Popper. A relatively short essay by Popper explaining the situation is in "Objective Knowledge" Chapter 1. Also worth reading: "On the sources of knowledge and of ignorance", the introduction to "Conjectures and Refutations" by Popper, "Realism and the Aim of Science", Chapter I by Popper and Chapter 7 of "The Fabric of Reality" by David Deutsch, which is called "A conversation with a crypto-inductivist".

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