Professor Richard Zach of Calgary University (http://www.ucalgary.ca/~rzach/) kindly wrote to me to correct me on what I’d said about Milton Friedman’s interpretation of Gödel (also mentioning Eric Schliesser’s reply). I found it beneficial, so maybe others will also. He agreed to let me post the exchange we had. So here it is (him in blue, me in brown). Be warned that I haven’t done much editing.
Dear Alex (if I may) and Eric,
I read your post about Milton Friedman’s Gödel remark
and Eric’s reply to it. I think you’re too harsh on Friedman: Gödel’s theorem does show the impossibility of a comprehensive and self-contained logic if you understand this the way it was widely understood at the time Friedman was writing. The prototypical example of what was hoped to be such a system, and which is even mentioned in the title of Gödel’s paper, was, after all, Principia Mathematica. It’s only a few decades later, after the influence of Quine’s criticism of second-order logic as “set theory in sheep’s clothing” took hold, that second-order logic no longer counted as logic — and, that people started to think of the predicate calculus as an example of a “comprehensive and self-contained logic.” For instance, Carnap — who was Friedman’s colleague at Chicago — called predicate logic “the lower functional calculus” at around the same time.
It’s a mystery to me as well what Gödel’s theorem should have to do with observer-dependence in physics.
Thank you for this. One thing I hope to gain from my blog is a chance to learn from others’ expertise. In this case the hope has been very much fulfilled.
What you say sounds quite convincing. I owe the memory of Friedman an apology. Perhaps the problem is that I had read his ‘comprehensive’ as meaning something like ‘sound’, whereas what he may have meant by ‘a comprehensive logic’ was more like ‘a logic that contains a proof theory for second-order statements’. In that case, you are right to remind me that second-order logic was commonly regarded as an essential part of any ‘comprehensive’ logic until the time of Quine, and Gödel did indeed prove that no such logic can be ‘self-contained’.
I suppose I was more puzzled about what Friedman meant by connecting Gödel’s theorems to observer-independence in physics. If I’m right that it has something to do with the notion (not one I’m particularly inclined to hold) that one implication of the first theorem is that ‘true’ must mean only ‘true in the intended model’, then it seems more like a point specific to the system of the Principia Mathematica than a general point about second-order logic. The idea would be (very roughly) that the truth of some statement that can be formulated within that system might depend not only on how things are independently of us but also on our intending some particular model for the system. This is in some ways analogous to the case in which the truth of some statement about the state of a physical system might depend not only on how things are independently of us but also on whether we have made a certain kind of observation of that system.
But maybe this makes no sense, and then it’s simply a mystery what connection Friedman imagined between the Gödel result and observer-dependence.
I’d think “comprehensive” in this case means “a logic that includes everything”, i.e., not a fragment. And “self-contained” means “complete”.
I too am at a loss why he thought observer-dependence in physics has anything to do with Gödel. Of course whether a sentence is true depends on what it is intended to be about, or which model we intend it to be evaluated in. But whether any sentence is true depends on what we mean by it, and we usually don’t take this pedestrian fact to have anything to do with Gödelian incompleteness. So I hope that’s not what he means.
Thanks again! Yes, I think you’re right about what ‘comprehensive’ and ‘self-contained’ mean, and I’m convinced by your argument that a ‘comprehensive’ logic in this sense, as it seemed to somebody of Friedman’s time, would include second order logic.
In the days of the Principia I suppose the hope was that statements about arithmetic could be taken to be true regardless of what model we intend them to be evaluated in, since provable statements are true in every model. And so Gödel’s demonstration that at least one sentence that can be formulated in the system of the Principia can’t be true in every model (since it’s unprovable) isn’t so pedestrian a fact. It shows that our intended model plays a role in determining the truth of statements about arithmetic in a way that would not have been the case had the Principia programme been successful. Does this seem right?
That seems right. If you have a complete system (ie, one that either proves or refutes every sentence) the question of which model is intended need not arise, and that certainly was the hope for foundational systems like type theory, analysis, set theory.
Good – thanks! So, to put it crudely, the hope of the Principia programme was that our mathematical statements could be true/false regardless of what we were talking about; Gödel ruined this hope. And somehow there seems a very weak analogy between the way in which our intentions to talk about one model rather than another determines the truth of mathematical statements (which had previously been taken to be independent of such intentions) and the way in which what we observe determines the TRUTH (not just the justifiability) of statements in natural science (which had previously been taken to be independent of our observation). I’m not saying any of this is correct, but maybe it explains how Friedman thought about it?
Anyway thanks so much for your help on this! Can I post this discussion?
I wouldn’t go as far as saying that this was the hope of the Principia
programme, or that before Gödel people took the truth of mathematical
claims as being independent of which model we intended them to be about
— just that had mathematics/logic turned out to be complete, it
wouldn’t have mattered.
It’s not clear to me how good the analogy can be made. If we have
arithmetic in mind, then the independent sentences are true or false in
the natural numbers. Sure if we had intended a different model, they
might have different truth values. But that’s unlike the case in
physics. For instance, we don’t take “Particle x has velocity v” to be
observer-dependent because it might have a different truth value if we
intend to refer to a different particle. Incompleteness in higher-order
logic or set theory may be a better fit, where we have examples of
independent statements such as the Continuum Hypothesis which, according
to some at least, are genuinely indeterminate in truth value. But it’s
it’s not observation which makes them determinate: they require stipulation.
Well… the analogy is VERY weak. We have to do *something* in either case to give the relevant statements determinate truth-values (on some views), though for very different reasons in each case (I agree that higher-order logic and set theory provide better examples). And this is, in each case, something we may not have expected; indeed it is in each case disputed.
Maybe I go too far in speaking of the *hope* of the Principia programme. Maybe I can call it an *advantage* of that programme, if only it had worked. It’s easy enough to say that the natural numbers are what arithmetic is about. But then must the natural numbers exist for statements about them to be true? Better if the question of what arithmetic is about doesn’t have to arise at all.