Beeing aware of some of the various attitudes towards this out there, I think it's worth noting that these things - the ideas of induction - can be considered at different levels or abstractions.
First we have the meaning of induction applied to human reasoning, as an attempt to understand and analyse human reasoning. The fuzziest form of this is to dismiss this into psychology, and here the induction is more of a qualitative nature. Ie. it does not described by mathematics.
The next level is to quantify this, and considers "degrees of plausability" as real numbers, which ultimately are argued to follow the axioms of probability and where these things is equipped with mathemtics, and this may help explain a few real world pehenomena such as game theory applications, economics etc, but the idea that all players acts somewhat rationally on given information. This alone will predict various types of group behavour, and various game-type equilibria.
So far it's no news.
But, the next level, is to consider that even physics, physical systems, atoms particles are like players in a game, that does act upon information at hand only (note that this has similarities to the principle of locality!). But this then, comes with a range of new complications. For example, particles don't have brains (unlike players in a game), so it means that one needs to explain the process of selection between possible actions in a different way. Maybe something like random disturbances, that then due to the initial constratins do diffuse as per a particular distribution. This would suggest that the actions of physical interactions should "look like" systems interacting, but where the action of each part is determined by the "information it has" about it's environment. This will naturally give rise to things like inertia, as in resistance against change, depending on the complexity of the parts.
This is what I tried to convey in the other thread. This is very controversial and very non standard, but it's IMHO the natural extension to the spirit advocated by the mentioned scientists. But there are different variations of this. ET Jaynes argued in favour of real numbers from start, I think differently. Ariels suggestions - to suggest that GR is a physical consequence of thinking that physical interactions are like responses based on incomplete information, is a deep insight IMO and not as stupid as it first sounds. And if he is right, some of the current approaches to quantum gravity may be due to a akward way of presenting the problem. I adhere to that view. But I'm in minority and it's hard to convey what isn't a thery, but rather a special way of analysing the problem.
/Fredrik