Tag Archives: Physics

Re: Paper and slides on indefiniteness of CH

Dear Sy,

In answer to your questions below, it seems to me that my work has philosophical significance in several ways. First, it shows that the reach of Quine’s (and perhaps Putnam’s) indispensability argument is extremely limited (for whatever that’s worth). Secondly, I believe it shows that one can’t sustain the view from Galileo to Tegmark that mathematics (and the continuum in particular) is somehow embedded in nature. Relatedly, it does not sustain the view that the success of analysis in natural science must be due to the independent reality of the real number system.

My results tell us nothing new about physics. And indeed, they do not tell us that physics is somehow conservative over PA. In fact it can’t because if Michael Beeson is right, quantum mechanics is inconsistent with general relativity; see his article, “Constructivity, computability, and the continuum”, in G. Sica (ed.) Essays on the Foundations of Mathematics and Logic, Volume 2 (2005), pp. 23-25. It just tells us that the mathematics used in the different parts of physics is conservative over PA.

Finally, to be “quite happy with ZFC” is not the same as saying that there is a good philosophical justification for it.

Best,
Sol

Re: Paper and slides on indefiniteness of CH

Ok, I agree that we can look at scientists’ actual practice, apart from what they say about meaning and truth, etc., and then draw conclusions about their actual standards on crucial matters such as the meaningfulness (or lack thereof) of statements that are unconfirmable/unfalsifiable in principle, and so forth. If one does this, however, then the picture that emerges is quite at odds with the sort of positivist views suggested in your previous message. Some examples are described in an old paper of mine, “Realist Principles,” Philosophy of Science, 50 (1983): 227-249.

For example, all kinds of statements about gravitational or other field values at various precise locations in the very early universe, say, prior to the decoupling of radiation and matter, are regarded as meaningful despite their being completely inaccessible empirically as a matter of physical principle. In practice, physicists (and even more clearly biologists) adopt realist positions on meaningfulness of statements whose actual truth value may be impossible to determine. (For an insightful critique of logical empiricists’ (positivists’) attempts to frame criteria of cognitive significance, a “must-read” is Hempel’s landmark paper, “Empiricist Criteria of Cognitive Significance: Problems and Changes”, in Aspects of Scientific Explanation (New York: MacMillan, 1965), pp 101-119. Hempel appeals to scientific practice, rather than pronouncements, much as we’re suggesting here.)

The upshot of these studies and reflections is that, contrary to what was suggested in your earlier message, scientific practice, if anything, tends to favor the view that statements like CH, undecidable in current set theory, should be regarded as truth-determinate even if we may never be able to decide them. (I say, “if anything”, because moving from the empirical sciences to pure mathematics is a risky business, although in the future that of course may change.)

PS Nothing in the above diminishes the importance of empirical testing and observation in the process of evaluating hypotheses and theories in the sciences (as true, or approximately probably true, etc.). This–not a narrow standard of meaningfulness–is what has been around in the sciences for a long time. The analogue here in mathematics would be the role of proofs: no one, regardless of whether they favor classicist or constructivist approaches, seriously asserts mathematical claims without proof (available in the community, not necessarily to the person making the assertion); otherwise they are properly put as conjectures.