Tuesday, December 26, 2006

proof, mathematics and theology

I don’t know enough about math to judge the accuracy of this article in American Scientist on the nature of proof, but it makes for an interesting read. I would have thought that when it comes to proof, the fields of math and theology are about as far apart as you can possibly get, but apparently that is not exactly the case –

Davis and Kline both wrote as mathematical insiders—as members of the club, albeit iconoclastic ones. In contrast, John Horgan positioned himself as a defiant outsider when he wrote a Scientific Americanessay titled "The Death of Proof" in 1993. "The doubts riddling modern human thought have finally infected mathematics," he said. "Mathematicians may at last be forced to accept what many scientists and philosophers already have admitted: their assertions are, at best, only provisionally true, true until proved false."

…In the 17th century, when algebraic methods began intruding into geometry, the heirs of the Euclidean tradition cried foul. (Hobbes was one of them.) At the end of the 19th century, when David Hilbert introduced nonconstructive proofs—saying, in effect, "I know x exists, but I can't tell you where to look for it"—there was another
rebellion. Said one critic: "This is not mathematics. This is theology."

Richard Feynman reportedly commented regarding proof, "A great deal more is known than has been proved."

To return to a discussion from a few days ago, I wonder what modern day fans of evidentialism will make of these statements. Recall Annie Besant’s argument in Why I Do Not Believe in G-d - "It is not for me to prove that no such beings exist before my non-belief is justified, but for him to prove that they do exist before my belief can be fairly claimed." (emphasis mine) It is wonderful to speak in such rhetorical platitudes of rationalism about adherence to proof as the sole basis for knowledge, but the practical reality is that not just theology, but even "pure" sciences like math and physics cannot meet such a standard.

8 comments:

  1. Oh no, proof is not dead yet. What is happenning is that the concept of proof is being gradualy widened reflecting the growing diversity of mathematical practice. In case of Hilbert's nonconstructive proofs, they were much more abstract than conventional proofs of his time. But they were accepted. Now more and more mathematicians accept very concrete machine computations as proof. So what?

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  2. I am not an expert in math (as it seems you are) so I cannot really comment other than to say my impression from the article is that the notion of absolute proof is vanishing. To quote, the author expresses fear "that proof cannot be trusted to lead us to eternal and indubitable truth." To say proof suffices to lead us to what might be true or what is probably true is not at all the same thing as saying proof establishes what must be true (which is how I think most non-mathematicians like myself understand the concept of proof).

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  3. > many scientists and philosophers already have admitted: their assertions are, at best, only provisionally true, true until proved false."

    You just disproved your own arguments. Of course Science, is provisional, if better evidence (or 'proof') turns up, it will change. However religion, which doesn't hardly have any good 'proof' at all, claims it can never change, and every one of its beliefs are 100% true always. Thats EXACTLY the problem. Religion needs to have more humility.

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  4. I suggest you read McGrath's book, referenced last post, who discusses the development of religion and the problem it poses for atheists who focus on denying a specific static set of beliefs, ignoring the fact that religion does indeed evolve and change. For a radical formulation to that same effect, see Tzidkas haTzadik of R' Tzadok haKohen #90 which I have discussed here as well. The notion that science instantly changes in response to "better" evidence is incorrect. Thomas Kuhn explains in "The Structure of Scientific Revolutions" that science for the most part is "directed to the articulation of those phenomena and theories that the paradigm already supplies", and any evidence to the contrary is dismissed as an anamoly. The professionalization of science leads to "immense restriction of the scientists' vision, rigid science, and resistance to paradigm change." The thrust of Kuhn's book is that it is only a rare crisis (see the book for his definition of what can lead this to occur) which can lead to a paradigm shift, but under normal conditions science conservatively reinforces accepted doctrines and sweeps evidence to the contrary under the rug.

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  5. > The thrust of Kuhn's book is that it is only a rare crisis (see the book for his definition of what can lead this to occur) which can lead to a paradigm shift, but under normal conditions science conservatively reinforces accepted doctrines and sweeps evidence to the contrary under the rug.

    I'm not debating Kuhn, mostly because I haven't read him. But you don't need to read Kuhn to see that science changes a LOT, whereas orthodox religion hardly changes at all. Its obvious.

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  6. Kuhn writes that paradigm shifts are exceedingly rare, and in the interim science basically reinforces its own set of preconceived doctrines and dismisses contrary evidence as anamolies. See latest posting on mishmar
    http://mishmar.blogspot.com/2006/12/science-religion-and-dispassionate.html

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  7. kishnevi9:56 PM

    At heart, all mathematics assumes that logic is valid without being able to prove that logic is valid. (Since to do so would require using logic, and therefore committing the fallacy of question begging.) Just as it assumes that the empirical correspondence of mathematics to the real world is actually an ontological reality, without actually being able to prove it. (IOW, why does the square of the hypotenuse always equal the sum of the squares of the sides in the real world, and not just as a piece of logical abstraction?)

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  8. >>>Just as it assumes that the empirical correspondence of mathematics to the real world is actually an ontological reality, without actually being able to prove it.

    Which is why Kuhn takes the logical leap of denying that science can ever make a statement regarding ontology. "Newton's mechanics improves on Aristotle's and ... Einstein's improves on Newton's as instruments for puzzle-solving. But I can see in their succession no coherent direction of ontological development."

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