Imagine Alice claimed she could just see, with reliability, which unprovable large cardinal axioms are true. We would be initially sceptical of her claims, but we could imagine ways in which we could come to be convinced of her having such an ability. For instance, we might later be able to prove a lot of logical connections between these axioms (say that axiom A12 implies axiom A14) and then find that Alice’s oracular pronouncements matched these logical connections (she wouldn’t, for instance, affirm A12 while denying A14) to a degree that would be very hard to explain as just luck.
Suppose, then, that we have come to be convinced that Alice has the intuitive ability to just see which large cardinal axioms are true. This would be some sort of uncanny superpower. The existence of such a superpower would sit poorly with naturalism. An intuition like Ramanujan’s about the sums of series could be explained by naturalism—we could simply suppose that his brain unconsciously sketched proofs of various claims. But an intuition about large cardinal axioms wouldn’t be like that, since these axioms are not provable.
Now as far as we know, there is no one exactly like Alice who just has reliable intuitions about large cardinal axioms. But our confidence in the less abstruse axioms of Zermelo-Fraenkel set theory—intuitive axioms like the axiom of replacement—commits us to thinking that either we in general, or those most expert in the matter, are rather like Alice with respect to these less abstruse axioms. The less abstruse axioms are just as unprovable as the more abstruse ones that Alice could see. Therefore, it seems, if Alice’s reliable intuition provided an argument against naturalism, our own (or our experts’) intuition about the more ordinary axioms, an intuition which we take to be reliable, gives us an argument against naturalism. Seeing the axiom of replacement to be true is just as much a superpower as would be Alice’s seeing that, say, measurable cardinals exist (or that they do not exist).
7 comments:
I think that what mathematicians consider to be "intuitive" is significantly shaped by the history of the subject, and by the complementary faculty which Pascal called "reason".
Frege's original axioms for set theory were intuitive, but they turned out to be inconsistent. (Yes, the Axiom of Replacement in ZF is intuitive, but only because it is a special case of a stonger, also intuitive axiom that turned out to be false.) And before that, a lot of smart people believed that the idea of an actual infinity was incoherent, based on the intuitive (but no longer accepted) common notion that "the whole is greater than the [proper] part".
So I think a better analogy to human set theoretic intuition is if we imagine that Alice keeps making false claims, and changing her mind when other mathematicians find flaws in what she is saying. Human mathematical intuition might be supernatural, but it cannot be taken to be "reliable" in the sense assumed by this parable.
Good points. If we thought like this, I think we would have to assign a pretty low credence to ZF's truth, and maybe even to ZF's consistency. That would be very pessimistic, and would result in a dethronement of modern mathematics as the most certain of the STEM disciplines. I am not saying that this isn't necessarily the right conclusion to draw, just pointing out how radical the consequences are.
I am inclined to be less pessimistic, and to invoke some vague notion of being "close to the truth". Frege's axioms were false, but they were in some sense "close": the typical cases to which comprehension was in fact applied were right. The axiom that the whole is greater than the part is correct for senses of "greater" (say the proper subset sense) and not for others (say the Cantorian one-to-one correspondence sense).
That said, my parable becomes less impressive if all we can say is that Alice is close to the truth. But it still seems to have some force.
Well, it doesn't necessarily lead to pessimism if you think that the process of mathematical dialectic is strongly correlated with truth. But it's hard to see how this gets off the ground unless there is at least some probabilistic validity to mathematical intuition.
Personally, I actually do assign low credence to ZF being true, since I have (Alice-like?) mathematical intuitions which are not compatible with it. In particular, I believe:
(1) The continuum c has cardinality greater than any Aleph,
which is not even compatible with Z (since by invoking Power Set and Specification, one can construct an equivalent to the inconsistent "set of all ordinals").
As for why I believe (1), it seems to follows from the intuitions below:
(2) For any given ordinal O, it is metaphysically possible to flip a coin O times,
such that:
(2a) when O is infinite, the probability of getting any single outcome is zero, &
(2b) the resulting probability measure is O'-additive, where O' is any ordinal (not necessarily equal to O).
In defense of (2b), note the following paradox: Suppose that a probability measure were not O'-additive. Without loss of generality, we can consider the case of a unit measure on O', such that there is 0 total probability to select any of the first o ordinals, for each o < O'. Imagine now that the random selection of an ordinal in O' has actually been done, and that as a result we obtain some specific ordinal o_1. This means that, if we do the same process again, we expect to get o_2 > o_1, with probability 1. But by symmetry, it shouldn't matter in which order we do the two processes, so it is equally true that o_1 > o_2 with probability 1. This is a contradiction! Hence valid probability measures must be O'-additive.
[I am aware that in practice, mathematicians usually only assume countable-additivity for their measures, but the point of the previous paragraph is to explain why my own intuitions lead to a stronger requirement, whenever the measure is associated with a possible stochastic process.]
If we accept (2), we can immediately show:
(1') c is not equal to any Aleph,
since we can randomly select an element of c by flipping a coin Aleph_0 times. If this were equal to some Aleph_p, this would not be an omega_p-additive measure.
Without the Axiom of Choice, (1') is a weaker statement than (1), since there might be cardinalities that are incomparable with the Aleph sequence. To demonstrate (1), we need to invoke the full power of (2), by randomly selecting O = Aleph_(p+1) real numbers. The only way this could fail to obtain a map showing that c > Aleph_p, is if all but Aleph_p of these reals are redundant, but such extreme redundancy happens with probability zero.* Hence, with probability 1, we get a map from c to Aleph_(p+1), which obviously implies that such maps exist.
(*Proof of lemma: if such redundancy occured, there would have to be some o < omega_(p+1), such that all of the reals following o are redundant with those before o, but this never actually happens because:
A. For any given ordinal o, this is a probability 0 event. Since there are infinitely many ordinals following o, the only way this could fail, is if we already have a measure 1 subset of c by the time we get to o. But the existence of a nonzero measure on o contradicts omega_p additivity.
B. By using omega_(p+1)-additivity, we can remove the words "For any given o" from the previous statement A.)
I would be curious to know what you think of these arguments.
Absolutly not, as this is a simple argument from ignorance, as a mechanism in the brain used for standard math could be used to shape such intuitions, a very good pattern finding for exemple.
Aron:
That's very interesting. I've grown used to the idea that real-valued probability measures are a crude tool and do not cover all the probability comparisons we would like to have. For a theological example, I doubt that numerical probabilities can be assigned to what God can create, even though I think there could be meaningful comparisons of some sort between different divine creation options. For a non-theological example, I think of relatives of the Banach-Tarski paradox for a countably infinite sequence of coin flips (these require the Axiom of Choice for collections of countable sets of reals, but I think the Axiom of Choice for those is philosophically plausible on probabilistic grounds: http://alexanderpruss.blogspot.com/2014/05/a-metaphysical-argument-for-version-of.html ; I have since gotten a bit suspicious of premise 4, but I can run the Banach-Tarski-style proof in a world where the requisite choice function exists) which show that there are unavoidable nonmeasurable sets.
In your example, off-hand I don't see good reason to think that there would be a probability measure that makes {o&O'} measurable.
I don't see an obvious contradiction with a Bayesian having subjective probability credences concerning what God will do in particular situations. Indeed, if this were not the case, it's hard to see how we'd be justified in having any credences at all, since a credence that e.g. the sun will probably rise tomorrow in the East presupposes that we assign a low credence to God not miraculously causing that not to happen. So in that sense, the predictability of Nature presupposes some amount of divine predictability.
Obviously, my proposed axiom rests on a denial of the Axiom of Choice, since it explicitly asserts that all subsets of the reals are measureable. So you have to try to imagine a state of Edenic innocence before you had to put asterisks about nonmeasurable subsets on every single statement you make about measures.
Your probabilistic argument for Choice is interesting, but I think I would try to evade it by saying that the existence of a probabilistic process for selecting an element of a set S does not always permit one to assert the real possibility of any preselected measure 0 event occuring. For example, if the argument I've mentioned above is correct, it seems that in my ideal form of set theory, for a preselected real number r, one NEVER gets r to come out of a random selection process no matter how many (well orderable # times) you do the process. (But we are entitled to assume that if the process is done, that some measure 0 outcome will occur.)
Furthermore, your premise 2 asserts the ability to have K causally independent copies of a system for any cardinality K, whereas I am only using this for well-orderable K. So your assumption seems a bit stronger than mine is.
I'm not totally sure I understand your last statement. Can you indicate which specific sentence in my comment you are objecting to?
I think the html in my last line of response got mangled. I think I was referring to the event o is less than O'.
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