At the Workshop on Infinity and Truth in Singapore last year, we had a special session in which the speakers were asked to imagine that they had been granted an audience with an all-knowing mathematical oracle, given the opportunity to ask a single yes-or-no question, to be truthfully answered. These questions will be gathered together and published in the conference volume. Here is my account.
A question for the mathematics oracle
Joel David Hamkins, The City University of New York
Granted an audience with an all-knowing mathematics oracle, we may ask a single yes-or-no question, to be truthfully answered……
I might mischievously ask the question my six-year-old daughter Hypatia often puts to our visitors: “Answer yes or no. Will you answer `no’?” They stammer, caught in the liar paradox, as she giggles. But my actual question is:
Are we correct in thinking we have an absolute concept of the finite?
An absolute concept of the finite underlies many mathematician’s understanding of the nature of mathematical truth. Most mathematicians, for example, believe that we have an absolute concept of the finite, which determines the natural numbers as a unique mathematical structure—$0,1,2,$ and so on—in which arithmetic assertions have definitive truth values. We can prove after all that the second-order Peano axioms characterize $\langle\mathbb{N},S,0\rangle$ as the unique inductive structure, determined up to isomorphism by the fact that $0$ is not a successor, the successor function $S$ is one-to-one and every set containing $0$ and closed under $S$ is the whole of $\mathbb{N}$. And to be finite means simply to be equinumerous with a proper initial segment of this structure. Doesn’t this categoricity proof therefore settle the matter?
I don’t think so. The categoricity proof, which takes place in set theory, seems to my way of thinking merely to push the absoluteness question for finiteness off to the absoluteness question for sets instead. And surely this is a murkier realm, where already mathematicians do not universally agree that we have a single absolute background concept of set. We know by forcing and other means how to construct alternative set concepts, which seem fully as legitimate and set-theoretic as the set concepts from which they are derived. Thus, we have a plurality of set concepts, and our confidence in a unique absolute set-theoretic background is weakened. How then can we sensibly base our confidence in an absolute concept of the finite on set theory? Perhaps this absoluteness is altogether illusory.
My worries are put to rest if the oracle should answer positively. A negative answer, meanwhile, would raise alarms. A negative answer could indicate, on the one hand, that our understanding of the finite is simply incoherent, a catastrophe, where our cherished mathematical theories are all inconsistent. But, more likely in my view, a negative answer could also mean that there is an undiscovered plurality of concepts of the finite. I imagine technical developments arising that would provide us with tools to modify the arithmetic of a model of set theory, for example, with the same power and flexibility that forcing currently allows us to modify higher-order features, while not providing us with any reason to prefer one arithmetic to another (unlike our current methods with non-standard models). The discovery of such tools would be an amazing development in mathematics and lead to radical changes in our conception of mathematical truth.
Let’s have some fun—please post your question for the oracle in the comment fields below.
A question for the math oracle (pdf) | My talk at the Workshop
Hi Joel,
Very interesting post and great blog!
The question I would ask the mathematics oracle is the following: Cantor claimed that his theory of cardinal numbers (alephs) and ordinal numbers will shed a lot of light on old and new problems in cosmology and in arithmetic. With regards to arithmetic, the claim seems to have been verified by Gentzen, who showed the consistency of arithmetic under the hypothesis that $\epsilon_0$ is well-ordered. Do cardinal numbers, ordinals number and more generally, large cardinals “solve” problems in cosmology? Is Cantor’s claim true?
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I would probably ask the same question you pose about finitude. Pardon my naïveté, but is it fair, for example, to say that any finite ordinal subtracted from itself already supposes a larger set (because n-n=0 –> n+1)? Doesn’t this mean that infinity in some sense precedes finitude — that finite ordinals are derived from a limit ordinal? Or should we be supposing an arithmetic in which it is possible to have no answer instead of zero? (n-n=)
Sorry again if this is a silly question. I’m not much of a mathematician but it does interest me.
Is the Riemann conjecture provable, using current widely-accepted axioms and operations?
Given the most up to date formal mathematical theory of our physical reality, is it possible to mathematically delineate between the mathematical theories and theorems that express facts found in our real physical reality and those that don’t ? (excuses if this is silly).
This isn’t a silly question. There are a lot of such questions on Math StackExchange and MathOverflow, concerning the physical meaning of a wide variety of mathematical structures. Unfortunately, a lot of them are pretty much unanswered, especially since most of the answers given are either misunderstanding the question or not actually answering the question.
Almost all mathematicians believe that first-order PA (perhaps even second-order arithmetic Z2) proves only true facts about natural numbers, in the sense that there is a systematic translation from sentences over PA to sentences in natural language that precisely specify a certain fact that is algorithmically verifiable on a physical medium. For example it seems that HTTPS works (decryption using the private key recovers the original message) not by chance but because PA is sound (under the translation)! It is conceivable that PA is only sound if all quantifiers are restricted to below some large number, but that must be larger than say 2^2048. Perhaps it stops at about 2^(2^64), since the observable universe seems to have size on the order of 2^64.
The same cannot be said for set theories in general. Till today, I see no evidence that set-theoretic statements express true facts about the real world. Naive set theory does express some kind of mental constructions, but is faced with Russell’s paradox. ZFC takes the easy way out by simply restricting the axiom of comprehension, but this doesn’t solve the incongruity with our mental constructions, so ZFC doesn’t even express our mental intuition. Furthermore, there is no evidence that there is anything intrinsically wrong with our mental intuition.
You may also want to look up Reverse Mathematics, where ACA (a predicative subsystem of Z2) is already capable of proving all the basic theorems of real analysis and much much more, and I haven’t seen a theorem that cannot be proven in ACA that has real world meaning under some sound interpretation (as described above). I challenge anyone to give and prove a counter-example!
Wouldn’t the mere existence of an oracle that answers every finite question truthfully already settle the question of definiteness of truth for finitary statements? 🙂
I find that position reasonable, but since we are in the land of imagination here, I would also find it reasonable to say that when asked about whether P=NP, say, she would reply, “It depends on which concept of finite number you use.”
For example, think about how you would answer a nineteenth-century question about whether the angle sum of every triangle is 180 degrees.
I’m not convinced by your argument. Either there is a notion of a proof over some formal system, or there isn’t. In the former case, we already have the standard natural numbers in the meta-system, so the question is moot. In the latter case we don’t even have proofs, so there is no mathematics. “P=NP” refers to a specific conjecture that can be stated over any formal system that can perform sufficient reasoning about the execution of a program (in a fixed recursive Turing-complete language). Both the formal system and the program execution can be purely syntactically described in any meta-system with string manipulation capability, which of course is essentially the same as having natural numbers. So this question has a well-defined answer unless one insists that the standard properties of strings are false, but I don’t see how one can even assert this viewpoint precisely without already assuming the standard notion of strings. Another alternative is to insist on rejecting the law of excluded middle even in the topmost meta-system, so that the answer might be “Don’t know”. But I don’t think any logically consistent philosophy of the world is non-classical at the intrinsic level (observables might very well be non-classical because they are merely extrinsic effects).
The other question about the sum of angles in a triangle is different, because at the time it was stated it was not at all precise. Euclid did not, contrary to popular opinion, specify proper axioms for Euclidean geometry. If you read his actual work, not only are his axioms utterly useless, even with very forgiving interpretations he uses principles (preservation of lengths and angles under rigid motion, for instance) that are not stated as axioms! So one can justifiably say that without proper specification of the axiom system, the question is not even well-defined. But once the system has been specified formally, then the question is as well-defined as “P=NP”, and perhaps more so because Euclidean geometry is decidable.
In case it is still not clear, consider whether you believe that the halting problem is meaningful or not, in whatever sense you wish. In most reasonable senses you will find that the provability of every well-defined mathematical statement reduces to whether a program halts or not. So unless you believe that there is no such thing as idealized programs (namely string manipulating processes), you would by LEM agree that any program either halts or does not halt, and hence you would have to accept that provability of any mathematical statement has a well-defined answer. This nevertheless is different from asking about the truth of the statement, which is my next point.
The original comment about oracles is indeed crucial to this discussion. Is your oracle’s answer merely her opinion? If so then that’s not an oracle. If not, then it is absolute and hence there is an absolute truth and a suitable translation from some mathematical sentences to well-defined notions, each of which are hence absolutely true or absolutely false. There may be mathematical sentences that cannot be translated, which just means they are meaningless. For example, “Eucnfdixlwmqpz.” is a meaningless statement and has no truth-value. Similarly I think most mathematical sentences that cannot be stated in the language of PA are meaningless and hence asking the oracle about them would get you the answer “Meaningless question”.
The bottom line is, if you meet such an oracle you’d better not ask any set-theoretic question as a true-false question unless you also provide your own interpretation of it in the real world, though you could ask about its provability rather than truth. Provability and consistency have nothing to do with truth.
I would like to know what an “absolute concept of the finite” is (not necessarily of the mathematics oracle–of her I would like to know whether ordinary mathematicians doing ordinary mathematics are actually working in a [standard] countable transitive model of $ZFC$, since the model in question cannot ‘say of itself’ that it is countable). I have been reading Levy’s paper “The independence of various definitions of finiteness” which states that ‘standard definitions of finiteness are all equivalent assuming Choice’. As far as ‘concepts of the infinite’ are concerned, are the definitions stated in Levy’s paper ‘concepts of the infinite’ (if they are, then in ‘choiceless’ models off $ZF$ there would not be ‘absolute’ concepts of finiteness, as Levy seems to show….)?
the phrase ‘concepts of the infinite’ should, of course, read, ‘concepts of the finite’–sorry for the typo, but if there is no absolute ‘concept of the finite, can it be said that there is an absolute ‘concept of the infinite’?
The “ordinary mathematicians” aren’t working in any Model of ZFC, that would exist in some stronger Theory, etc.; they’re actually working syntactically in ZFC itself! 😊
One small way to approach the start of an attempt to do such things: per Frege, one introduction of the empty set is as {x: x doesn’t equal x}. But we can generalize and say that a set world’s empty set is “always” the set of things that violate the axioms of that world. So in ZFC, it is neater to have {x: x isn’t well-founded}.
Anyway, different worlds will have different nonexistent-element conditions as such, but then what is an empty set in one world, with one parcel of such conditions, will be overflowing with elements in another world, with another parcel of such conditions. The only “version” of an empty set that would be strictly empty “everywhere” would be one introduced extensionally, i.e. just “defined as” empty, not on account of some peculiar Fregean analysis.
So, maybe we can start off with relativizing the concept of 0 (as the numerical carrier of the empty set’s value), and proceed from there?
Thanks for the comment. To my way of thinking, the difficulty of giving a successful account of the finite has mainly to do with the “proceeding from there” part. Frege’s idea, similar to Dedekind, is to start with some concept of 0, as you have described, and then have a process of going from a number to it successor. But how do we describe the totality of numbers that we get by iterating? Of course, Dedekind solves this with the second-order induction axiom, and in my view Frege’s solution amounts to the same. But my trouble with this approach is that it seems to define the finite only by reference to the set-theoretic realm in which arbitrary sets are part of the ontology. But if we were worried about the definiteness of finite, I don’t see how that approach can succeed, since it relies on concepts even less secure as far as definiteness is concerned.