Inference to the Best Explanation for Mathematical Claims

Following up my last post, I want to outline an argument for why a reasonable restriction on IBE will block the use of IBE to provide much justification for mathematical claims. The main "mathematical explanations" which are discussed by advocates of this sort of justification, like Baker and Colyvan, involve explanations of patterns observed in the physical world. These examples include the life-cycle of cicada and the shape of the cells of a honeycomb. One of the problems with these explanations is that they bring in complications associated with explanations via natural selection. Another problem is that they may involve mathematical terms in the description of what is to be explained.

To avoid these problems, I will focus on the bridges of Konigsberg case (see here for some background). The explanation could be reconstructed as

(1) The bridges of Konigsberg form a graph of type O.
(2) There is no Euler path through a graph of type O.
(3) Therefore, there is no Euler path through the bridges of Konigsberg.

An Euler path is a circuit through the graph that crosses each edge exactly once. For someone who worries that even this begs the question by using a mathematical term we can offer to extend the explanation to include "(4) Therefore, it is impossible to cross each of the bridges exactly once."

I claim that Sensitivity blocks the use of IBE to support (2). This is because an agent who was genuinely in doubt about the truth of (2) would also have as a relevant epistemic possibility that (2') There is no Euler path through a graph of type O with fewer than 100 vertices. This means that there is an alternative explanation of (3) which employs weaker mathematical assumptions:

(1’) The bridges of Konigsberg form a graph of type O with fewer than 100 vertices.
(2’) There is no Euler path through a graph of type O with fewer than 100 vertices.
(3) Therefore, there is no Euler path through the bridges of Konigsberg.

My conclusion, then, is that this puts the burden on the advocates of using IBE to justify mathematical claims to argue that Sensitivity is incorrect or that some other features of these cases have been overlooked.

Inference to the Best Explanation and Sensitivity

Philosophers of science have focused on inference to the best explanation (IBE) as the sort of inference that stands the best chance of ultimately justifying our belief in unobservable entities like atoms and electrons. More recently philosophers of mathematics like Colyvan and Baker have tried to given an explanatory indispensability argument in support some of our mathematical beliefs. The challenge for everyone, though, is to articulate a reasonable form of IBE that accords with scientific practice, but which does not overgenerate beliefs in things which we reject. Despite its clarity, Lipton's discussion of IBE seems to me overly restrictive because he focuses only on causal explanations. Are there plausible principles for the use of IBE which allow non-causal explanations?

Here is one, but it results in problems for any use of IBE to justify our mathematical beliefs. I call it "Sensitivity", although perhaps this not the best label:

Sensitivity: A claim which appears in an explanation can receive support via IBE only when the explanatory contribution tells against some relevant epistemic possibilities.

Here I am imagining an agent who is in doubt about the truth of some competing options A1, A2, A3. Suppose that A1 appears in our best explanation. Sensitivity tells us that this contribution of A1 to the explanation can only license belief in A1 when the way in which A1 contributes makes either A2 or A3 less likely.

This seems to me to be a very weak and plausible restriction on IBE. It is met by the standard atoms and electrons cases, and also by Woodward's non-causal explanation of the stability of planetary orbits. In my next post, I want to outline a case for the claim that sensitivity blocks the use of IBE to support mathematical claims.

Book Project Update

For those few readers tracking my ongoing book project on Mathematics and Scientific Representation some recent good news is that I have signed a contract with Oxford University Press. The delivery date in the contract is Nov. 2010, so over the next six months I will posting some of the key ideas, and eventually the near-final versions of the chapters, for comments and discussion.

The manuscript is projected to be 140 000 words, with twelve chapters:

1. Introduction

Part I: Epistemic Contributions

2. Content and Confirmation
3. Causes
4. Varying Interpretation
5. Scale Matters
6. Constitutive Frameworks
7. Failures

Part II: Other Contributions

8. Discovery
9. Indispensability and Explanation
10. Fictionalism
11. Facades

12. Conclusion: Pure Mathematics

Principia Mathematica at 100 Conference Lineup

2010 marks the 100th year since the publication of the first volume of Russell and Whitehead's Principia Mathematica. A major conference on Principia is scheduled for late May at McMaster University, in conjunction with the yearly meeting of the Bertrand Russell Society. The PM at 100 abstracts were recently posted here. It looks like a great lineup and I expect a great conference!

Pittsburgh Announces Rescher Prize for Systematic Philosophy

Details courtesy of Soul Physics.

From the press release:

Eminent, esteemed, wide-ranging, prolific-these are adjectives that have been aptly used to describe Nicholas Rescher and his contributions to the field of philosophy in a career that spans six decades, with nearly a half century of those years devoted to teaching and research at the University of Pittsburgh. In acknowledgement of his decades-long career at Pitt, Rescher, Distinguished University Professor of Philosophy, is donating his massive collection of materials on philosophy to the University's Hillman Library. In turn, the University is honoring Rescher for his lifetime of achievement and devotion to the University with the establishment of the Dr. Nicholas Rescher Fund for the Advancement of the Department of Philosophy, which will include a prestigious biennial award, the Nicholas Rescher Prize for Contributions to Systematic Philosophy.

[...]

Income from the Rescher Fund will be used to achieve key initiatives of the Department of Philosophy and to establish the Nicholas Rescher Prize. Awarded biennially, the prize will recognize an individual “for distinguished contributions to philosophical systematization” and include a gold medal, a $25,000 award, and an invitation to the University to deliver a lecture. Currently there is no major recognition in the field of philosophy, says Rescher, that is even remotely akin to the Field Medal in mathematics; the Pulitzer Prize in journalism, letters, and the arts; or the Nobel Prize in the sciences, medicine, economics, and literature.

The prize-to be awarded for the first time in the fall of 2010-reflects the seriousness of Pitt's commitment to philosophy. “It is our aspiration that the new Rescher Prize will become recognized as the most prestigious award in the field of philosophy, emphasizing the life's work and contributions to philosophy by a preeminent, world-renowned figure,” Maher said.

The Disunity of Climate Science

While there has been a lot of misleading coverage of the stolen e-mails from East Anglia, the Guardian offers an intriguing look inside the fallout from the more significant retraction of the 2007 IPCC report claims about the Himalayan icepack:

Speaking on condition of anonymity, several lead authors of the working group one (WG1) report, which produced the high-profile scientific conclusions that global warming was unequivocal and very likely down to human activity, told the Guardian they were dismayed by the actions of their colleagues.

"Naturally the public and policy makers link all three reports together," one said. "And the blunder over the glaciers detracts from the very carefully peer-reviewed science used exclusively in the WG1 report."

Another author said: "There is no doubt that the inclusion of the glacier statement was sloppy. I find it embarrassing that working group two (WG2) would have the Himalaya statement referred to in the way it was."

Another said: "I am annoyed about this and I do think that WG1, the physical basis for climate change, should be distinguished from WG2 and WG3. The latter deal with impacts, mitigation and socioeconomics and it seems to me they might be better placed in another arm of the United Nations, or another organisation altogether."

The scientists were particularly unhappy that the flawed glacier prediction contradicted statements already published in their own report. "WG1 made a proper assessment of the state of glaciers and this should have been the source cited by the impacts people in WG2," one said. "In the final stages of finishing our own report, we as WG1 authors simply had no time to also start double-checking WG2 draft chapters."

Another said the mistake was made "not by climate scientists, but rather the social and biological scientists in WG2 ... Clearly that WWF report was an inappropriate source, [as] any glaciologist would have stumbled over that number."

As I understand the science, the climate models used to support the central claims of the report are unequivocal. But they don't always give information relevant to policy makers such as exactly how much hotter it is going to get in Indiana or what year the Himalayan icepack will melt. This creates a temptation to leap in and provide more precise predictions than the models support. What is interesting here is that the "hard scientists" are blaming the "social and biological scientists" for giving in to this temptation.

New Book: Nasim, Bertrand Russell and the Edwardian Philosophers: Constructing the World

NDPR has an instructive review by Bernard Linsky of Omar Nasim's 2008 book on Russell and his 'Edwardian' philosophical contemporaries like Stout, Nunn and Alexander. I haven't read this book yet, but it seems to mark a new level of scholarship on Russell's external world program and its relationship to Russell's intellectual context. As Linsky summarizes things,

Nasim argues that Russell took ideas that were being debated and made them precise to formulate his own views on sense data and matter. Most importantly, Russell replaced what Nasim describes as a "socio-psychological" notion of construction with the precise method of "logical construction" modeled on the construction of numbers as equivalence classes, which he brought to the "Controversy" from his work on logic and the foundations of mathematics. Both the origins of some of the unusual aspects of Russell's theory of sense data as being non-mental, but also not material, are found in the Edwardian controversy. We also learn what new ideas Russell brought to the debate to make it his own and to come up with his distinctive project of constructing matter from sense data.

This is a very helpful contribution to our understanding of the history. Linsky raises some points about the amount of detail which Nasim is able to go into about the philosophers he discusses. For example, Linsky explains how Alexander influenced the distinctive form of realism which later flourished in Australia.

A larger question about Nasim's project concerns the extent to which we can reconstruct Russell's views by focusing on philosophers alone. It seems that we may need to look beyond the philosophical context to the scientific debates, especially in psychology and physics, concerning space and our representation of space. Gary Hatfield has made some progress in this direction and my understanding is that Alexander Klein is also pursuing some research into the links between Russell's constructions and the psychology of his day. This sort of work will hopefully complement Nasim's story by expanding what counts as Russell's intellectual context.