Inference To The Best Explanation

What is it?

A method for choosing between competing hypotheses.

Why is it important to what I’m working on?

Because Colyvan’s indispensability argument for a realist view of mathematical entities appears to be based on it.

Where is it found?

VOGEL, JONATHAN (1998). Inference to the best explanation. In E. Craig (Ed.), Routledge Encyclopedia of Philosophy. London: Routledge. Retrieved February 24, 2008, from http://www.rep.routledge.com.virtual.anu.edu.au/article/P025

Peter Lipton, Inference to the Best Explanation, Routledge, 2nd edition

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The Argument (or statement)

When choosing between competing hypotheses, one should choose the hypothesis that better explains the available data.

Explanation?

Factors involved in determining explanatory power in this sense include depth, comprehensiveness, simplicity and unifying power.

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The Main Counter Argument

Our main goal when choosing hypotheses should be to get closer to the truth. However, the usual explanatory criteria are unrelated to truthiness.

Responses

  1. Truth is not the goal.
  • Has problems re: use of IBE in everyday situations (e.g. jury).
  1. Truth is not sole goal. Explanatory quality is also legitimate basis for hypoth. selection.
  • Problem re: what to do when these two goals conflict with regard to theory choice.
  • Def.) An hypoth. is tested by comparing it with observed data.
  • P1) Successful testing increases the likelihood of truth.
  • P2) Better explanations are more testable.
    1. Theories of high explanatory quality are more likely to be true. — Problem re: whether testability is defined in terms of desire to confirm or disconfirm hypoth. as this leads to differ hypoth. choice in many cases.
  • P1) IBE is ampliative.
  • P2) Enumerative induction is ampliative.
  • P3) No pattern of ampliative inference can be shown antecendently to lead to true conclusions.
  • P4) We accept induction as a legitimate form of reasoning.
    1. We should also accept IBE as a legitimate form of reasoning. — Problem re: need additional argument to link IBE and induction as don’t want to argue that all forms of ampliative inference are acceptable. — Identity argument doesn’t seem to work as there exist cases in which inductive inference is not explanatory (e.g. people at shops on Tues.)

Another Issue Problem when using IBE with regard to theories involving inobservables: can formulate infinite theories involving inobservables - how can one ever verify that the IBE-based theory choice is the correct one? Seems then that one must restrict IBE usage to non-theoretical contexts.

Response - P1) In the past science has demonstrated convergence in theories. - P2) IBE has played a large role in science in the past. - C) IBE leads to the truth - Counter-args: — Above argument uses IBE and hence is circular. — {[red Some circular arguments are OK, especially when IBE is involved. Or so it’s claimed. Lipton’s book has stuff on this (an example involving snow-shoes, if I remember correctly). Jason ]} — IBE might not have been the property of sci. practice in the past that led to convergence. Indeed, it may have a deleterious effect but has just been cancelled out by some other aspect of the practice in the past. — {[red Excellent point. Jason ]}

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What do I think?

  • Most of the reasoning above seems to relate to scientific rather than mathematical theories. — Of course, one could argue that there is no distinction between the two but no such argument is made in the reading I have done. — For Jason: has such an argument been made?

    • {[blue I can’t think of anyone who’s ever said that maths was exactly the same as the sciences, but several people have said that it’s very similar, most famously Mill and Quine. (Mill in more detail than Quine.) It’s an unpopular position, but personally I think it’s right. Jason ]}
    • {[green Yeah, now that I’ve though about it a bit more, I think some interpretations of Wittgenstein would support this view as well. ]}
    • {[green I do have one difficulty with it however: while I think I would support the view that the processes from which scientific and mathematical theories arise are the same I do not think that these processes have lead to scientific and mathematical theories being structurally identical. In particular, I think that the they differ with regard to how they are associated with logical necessity (be that necessity socially constructed or otherwise). ]}
    • {[pink That’s tantalising! ]}
    • {[green Okay, I just sat in the NLA foyer and had a bit more of a think about this. If one views both mathematics and science as activities, then I guess the way one must distinguish them is by looking at the conditions under which it would be accepted that one’s activities qualify as “doing” science or mathematics. ]}
    • {[green In the mathematical case, I think there are probably three main conditions: 1.) the subject(s) of the activity is/are mathematical objects (i.e. ones that have been previously accepted as such, really statements any new math. objects must also involved references to pre-existing ones); 2.) the procedures used are valid in terms of deductive logic; 3.) the use of math. objects and procedures is accepted as valid by the math. community (this condition may not actually be necessary). ]}
    • {[green It seems then that the conditions for scientific activity would differ only with regrd to the what was considered an acceptable scientific procedure. I would argue that the conditions for this are not as stringent. In particular, there is not the requirement for conclusions to be necessary in a deductive logical sense. ]}
    • {[green So I guess my position that while I believe the way in which these conditions where arrived at (some form of social construction) is the same for both science and maths, the content of those conditions differs markedly. ]}
    • [[crimson Oh, I see. Yes, that makes complete sense. And I think your point 3 above is needed. Otherwise trivial logical manipulations of mathematical symbols would count as maths. Jason ]]

  • One could question whether or not the explanatory criteria detailing above are applicable in a mathematical context. — Although most of them do seem to be criteria that mathematicians would use to determine the “beauty” of a particular theory.

    • {[blue I agree. Although in my (limited) experience, it’s even harder to say what makes a mathematical theory beautiful than it is for a scientific theory … and even that is too hard! Jason ]}

  • Can the Riemann Hypothesis be used as an example to demonstrate how ampliative explanations are not acceptable in a mathematical context? — That is, the hypothesis is a very well empirically supported theory regarding the distribution of the primes that is still held to be an open problem despite this strong empirical support.

    • {[green I think your idea about replacing this with the twin prime hypothesis is probably a good idea. Although Riemann is probably “cooler” I think the twin prime will make the same point and allow my thesis to be more accessible. ]}
    • {[pink No problem with mentioning both. ]}

  • Testability stuff in response 3 (specifically P1) seems to be questionable due to the link between testability and the continuously developing state of observation technology.

  • For example, biological explanations involving the postulation of the existence of microscopic entities (cells, gametes, etc.) would not have been testable prior to the invention of the microscope and yet these are theories that we now believe to be true.

  • String theory (or M theory) seems like a current example of a situation in which a theory has great explanatory quality but is not yet testable.

— Note: this point is probably a bit rough and unclear - I will come back to it over the next couple of days and clean it up.

  • Is P4 of response 4 actually the case?

  • {[blue Yes. A few people disagree with it, most notably very strict Popperians … but even Popper himself eventually came to accept induction in some circumstances (see Schilpp ed. 1974, pp. 1192’1193). It’s not something you’d have to argue for unless it was the main topic of the thesis. Jason ]}

  • The another problem section certainly seems to apply in the mathematical context. One can easily modify mathematical theories (by adding a constant when differentiating for example) in ways that do not have observational consequences.

    • {[red I like this point a lot. You could write a lot of detail about it in the thesis, if you find it interesting. On the other hand, I expect somebody’s written some sort of counter-argument to it (not necessarily a persuasive one, but something). Should do a literature search later if this point turns out to be important. Jason ]}

{[red Up to you, but might want to turn this thread mode stuff into document mode soon. See http://www.c2.com/cgi/wiki?ThreadMode and http://www.c2.com/cgi/wiki?DocumentMode. Jason ]}

Chris Wilcox