Jared Warren's "Infinite Reasoning"

Introduction

This was an exciting paper to read, as it was the first as part of a new reading group on the Philosophy of Maths at the University of Vienna. My thanks to Benjamin Marschall both for organising the group and suggesting the paper. I look forward to reviewing many of the papers that we'll read in this group. Present this week, in alphabetical order, was Julie Lauvsland, Benjamin Marschall, myself and Georg Schiemer.

In the paper, Warren advances the claim that it is possible for us to reason infinitely. What does this mean? A working definition is that inferences are a kind of mental or cognitive act connecting a series of premises to a conclusion. Oxford's dictionary of philosophy says something similar stating: "Any process of drawing a conclusion from a set of premises may be called a process of

reasoning." An inference might be a good (valid) inference or it might be a bad (invalid) one. Typically, though, inferences are finite in the sense that one infers from finitely many premises. A potentially attractive and perhaps orthodox claim is that reasoning can only be finite. More precisely, that every possible inference has only finitely many premises.

It's still possible to reason about infinite domains with this restriction, one simply needs to use tools such as (mathematical) induction or to reason from known universals. All of this is well known in mathematical logic. But the restriction does prevent the use of certain logics and crucially the formulation of certain axioms (for instance Hilbert's "Omega Rule"). If infinite inferences are generally possible and justified, there are numerous useful applications across logic, mathematics and truth theory (see §1 of Warren's paper for examples).

Warren argues that infinite reasoning is possible.

In particular, Warren considers the example of adding Hilbert's omega rule to PA. The omega rule is a possible axiom of arithmetic not stateable in the standard proof theory of classical logic. The reason why it isn't stateable is that it has infinitely premises. The axiom is simply the claim that for some phi with one free variable: phi(0), phi(1), phi(2),...

\vdash

∀x phi(x). Warren does consider some other possible useful cases, but the omega rule is the focus.

My view on Warren's paper is that (1) he is generally successful in responding to criticism of infinite reasoning (2) he is somewhat successful in giving a positive case for infinite reasoning but (3) he fails to demonstrate that the cases of (justified) infinite reasoning go beyond what one can do finitely with tools like induction. In this sense, his conclusion might be true but less interesting than he hopes.

(3) might be an unfair objection to the paper, given that Warren's claim, at least here, is the narrower claim about the possibility of infinite reasoning, not a wider claim about its usefulness. But I think the objection is fair. When evaluating a paper one must consider not just if it's right, but if it's interesting that it's right. (3) targets this latter point.

Arguments against Infinite Reasoning & Warren's responses

Warren considers three arguments against infinite reasoning. As I said above, I'm generally convinced by his responses here. The first two arguments come as a pair and attempt to connect reasoning to notions in mathematical logic, arguing that infinite reasoning clashes with some part of our logic. Warren's response in both cases is to reject the connection. Warren clearly has an interesting and nuanced understanding of the relationship between reasoning and these nearby notions in mathematical logic, which is successfully levied in response to these objections. The final argument against infinite reasoning effectively argues that infinite reasoning would require us to hold infinitely many premises "in our head" at once and that this is implausible. Warren's response here is to rely on a particular account of belief that allows for infinitely many beliefs. Again, the response is convincing

I'll look at each of his arguments and responses in turn and give my view.

The first argument can be called the supertask argument. It argues as follows:

P1: If we can conduct infinite inferences, then we could construct infinitely long proofs.
P2: But we can't conduct infinitely long proofs (this is a supertask)
C: We can't conduct infinite inferences

Supertasks are processes with "more than" omega many steps. For obvious reasons, supertasks likely can't be performed in the real world and, if they can, almost certainly can't be performed by agents like us. Quite simply, one could never reach the "omegath" step and hence never reach the steps after that.

I put the term "more than" in scare quotes just as a matter of taste. Personally, when thinking about different ordinals of the same cardinality, I think of them as the same sized cardinal arranged differently, rather than a bigger ordinal and a smaller one.

The problem with infinite proofs is that the propositions in the proof are structured such that they contain a limit point. One has to consider infinitely many premises before one reaches the conclusion.

Warren agrees that infinite proofs are impossible to perform but rejects P1. He argues that an inference does not always require a proof. Consider, for instance, students in an introduction to philosophy class who have not yet taken their first logic module. They will not be able to construct proofs, yet they are more than able to perform inferences.

Now one might argue in this case that intro-philosophy students are still able to produce some informal analogue of a proof but there are plausibly agents who are not able to even do that. An inference requires simple first-order beliefs. Animals, for instance, can perform inferences. However, constructing proofs requires a kind of higher-order reflection on the connections between one's beliefs. So there's convincing reason to think these can come apart and that an inference does not require an accompanying proof.

Warren cuts the discussion off here, but there is perhaps a little more to say. In all of these cases, whilst no proof has been performed, it's arguably important that a proof could be performed to explain why the inference was justified. This wouldn't be an argument against infinite inferences but against justified infinite inferences. Now there are finite proofs that are analogue to infinite inferences: these are ones using are ordinary tools such as induction. However, in cases where an infinite inference aims to do more than our finite tools, then there might be a problem as to how these inferences could ever be justified.

I'll remain agnostic-favouring-true about the claim that all valid inferences need to have a possible proof that could be performed in their place. This is at least a plausible enough claim that I think Warren should say something about it.

The second argument is called the computational argument. It argues as follows:

P1: If we can conduct infinite inferences, then we could enumerate the truths of arithmetic
P2: But we can't do that due to Church's thesis
C: We cannot conduct infinite inferences

Understanding this argument requires a brief diversion into some mathematical logic. I will endeavour to keep the explanation as accessible and brief as possible.

Theories in mathematics are sets of sentences. Theories have logical consequences. We can think of these consequences from two perspectives: truth and proof. From the perspective of truth, we might say that a logical consequence is of a theory is something that follows in every "model" (way the world could be) of the theory. From the perspective of proof, we might imagine a computer program that's programmed to apply all the rules of our standard (first-order classical) logic to our theory, and then to keep applying them iteratively to see what follows (I've skipped some nuances here about infinite axiom schemes). Thankfully, in first-order classical logic, these two notions coincide. This is the soundness and the completeness theorem.

A natural question to ask is if there are theories that, in some sense, completely determine the way the (mathematical) world is. One way of formalising that idea is by asking if, for every sentence of the language of the theory, the theory entails that that sentence or its negation. To put this in terms of proof, if one puts a theory into our "logic computer", does the computer eventually churn out every sentence or its negation?

Helpfully, this is also called completeness! We speak of the completeness of a logic in the sense that the two types of entailment match up. We speak of completeness of a theory in the sense of a theory determining the truth value of all sentences.

A common misunderstanding is that there are no complete theories. This is not true, there are many and many of them are infinite. What there aren't, thanks to Gödel's first incompleteness theorem, are complete theories of arithmetic.

What this means is that no theory of arithmetic can be put into the "logic computer" and churn out answers to every statement about the natural numbers. This is Church's thesis.

But Gödel's theorem is restricted to ordinary classical logic without infinite inferences. Hilbert's omega-rule is simply outside its scope. And, it turns out, ordinary arithmetic plus the omega rule (omega arithmetic) is complete, but only in the truth, not proof, theoretic sense. This doesn't get around Church's thesis. It's still the case that finite computers can't handle infinite inferences and this wouldn't amount to a way of enumerating the theorems of the natural numbers. The proofs required to demonstrate all the consequences of omega arithmetic are infinite proofs.

However, if one can perform any infinite inference (in finite time), then this would create a clash with Church's thesis. It would mean that a logic-computer could enumerate all the truth about the natural numbers. This can't be the case. If there's a way of computing all the truths of the natural numbers, then there's a complete theory of arithmetic that does not rely on the omega rule. We know this can't be the case by Gödel's first ICT.

Warren's response to this is to reject that we could always perform infinite inferences. The argument only works if one could perform every infinite inference required to enumerate all the consequences of omega-arithmetic. It's not generally the case that in order to perform a type of inference one must accept every consequence of it.

This is a successful response to the objection, it rejects P1, but is potentially a pyrrhic victory. What Warren has to say is that we can perform some but not all infinite inferences. The question, therefore, is which infinite inferences can we perform and if these offer anything more than our current tools such as induction. In fending off this objection, Warren opens the door to my worry: that whilst infinite inferences might be possible, they might only be possible, or at least justified, in cases where we can do without them.

The final objection is called the inference argument. It argues as follows:

P1: If we can engage in infinite reasoning, then we can perform infinite inferences.
P2: But we can't perform infinite inferences
C: we can't engage in infinite reasoning

So far I've been using the terms "reasoning" and "inference" fairly interchangeably. Clearly, that's not appropriate here. This argument is not trivial, it's making an important and subtle distinction between reasoning and inferences.

Inferences are cognitive processes connecting some premises to a conclusion. Because the process is a real cognitive process, the agent performing the inference needs to stand in some appropriate cognitive relation to the premises. Typically, this relation will be belief but sometimes it might be other relations such as supposition.

But, so says the argument, beliefs (or other relevant propositional attitudes) take up "space in heads" and our heads are only finite. This means that we can only have finitely many appropriate propositional attitudes at any one time. Infinite inferences, however, require infinitely many of these propositional attitudes.

Putting this together into a full argument:

P1: To engage in an inference one must stand in an appropriate propositional attitude towards each of the premises.
P2: We can only have finitely many propositional attitudes at any one time.
P3: Infinite inferences have infinitely any premises
C: Infinite inferences are impossible.

Warren rejects P2.

He does this by adopting a kind of dispositionalism about beliefs. The idea is that to believe a proposition is to have certain behavioural dispositions relating to that proposition (chiefly, but not entirely, to assent to it if heard). This sort of view is well known and is motivated by cases such as our apparent beliefs that 175631 is an odd number, despite the fact that we've likely never had a to consciously form that belief before in our lives.

Similar views are likely available for other kinds of propositional attitudes relevant for inferences.

But it's possible to have infinitely many dispositions at once (these are just counterfactual facts about some agent), so P2 is false.

My worry here is similar to the supertask argument. It's plausibly true that we can have infinitely many beliefs and hence perform infinite inferences, but it seems like the only time we can have infinitely many beliefs (or other propositional attitudes) is when they are schematic, in some sense. I can believe of every odd number that it is odd because I know a rule about looking at the last character in the name.

But if this is true of infinite sets of beliefs, that they must always be generated "schematically", then there's a worry that the infinite inferences we can perform are just those we could already do using the sorts of finite tools that we're familiar with.

In summary, this covers §2-4 in the paper. I'm convinced that Warren has successfully responded to these objections to infinite reasoning (point (1) in the intro). However, the responses do involve giving up some ground at crucial points and it's unclear that the sorts of infinite inferences one's left with are able to do more than the current finite tools already available.

Examples of Infinite Reasoning

In §5 Warren gives a basic attempt at a constructive case for his conclusion. I don't think Warren intends this as a definitive argument for the conclusion. He says as much, opening the section with the claim that "Completely uncontroversial examples of actual infinite reasoning are not to be had". So, at best, what we can expect is a series of cases that can plausibly be understood as cases of infinite reasoning but don't have to be. At worst, it's implausible that these are cases of infinite reasoning. My view on this is that Warren's right that these cases could plausibly involve infinite reasoning but that the cases fail to establish that there are interesting cases of infinite reasoning. I.e. if there are cases of infinite reasoning that go beyond what one could do with other tools such as mathematical induction.

The first case involves Malament-Hogarth machines. These are theoretically possible entities that involve two parts. First, a theoretical feature of spacetime known as a Malament-Hogarth event. This is a theoretical feature of certain spacetimes wherein spacetime is curved in such a way that there's a path that "subjectively" experiences an infinite period of time despite this only happening in a finite amount of time from the reference frame of some external point p (strictly, the term Malament-Hogarth event refers to this point p). The second part of a Malament-Hogarth machine is setting a turning machine along this path. The machine "subjectively" experiences an infinite amount of time but an individual standing at p could receive information from the machine relating to the completion, or lack thereof, of a process on the machine.

I don't know how possible MH-machines are or the technical details relating to General Relativity. I'll grant Warren the assumption that they're possible but look forward to any more informed comments than I can offer. Later in the paper, Warren points to an interesting fact that MH-machines don't need to be possible for his example to work, it just needs to be possible for someone to believe that they're possible. This move is interesting a probably correct but feeds more oxygen to the claim that infinite reasoning might only be possible in (in some sense) uninteresting cases.

The example Warren uses is of a computer falling towards an event horizon of a black hole. As the computer approaches, time speeds up from its perspective and, theoretically, it's able to complete an omega-task. The computer is set to check every natural number to find if it's a counterexample to Goldbach's conjecture. If such an example is found, the machine sends a signal and an observer at the Malament-Hogarth event can infer that Goldbach's conjecture is false. What happens if no such signal is received?

Surprisingly, this would be insufficient in standard Peano Arithmetic to conclude that Goldbach's conjecture is true. To understand why one needs to understand the difference between standard and non-standard models of (first-order) PA. The standard model of PA is what one thinks of when one thinks of integers. It's a sequence starting at 0, containing every and only the successors but with a limit omega. But this is not the only model of PA. PA has many non-standard models that extend beyond omega, typically by packing particular structures of Z-chains (sequences that look like the integers) after omega. Simply from the perspective of PA, one doesn't know which model "one is in". This means that one can't infer from something being true of all the finite numbers to something being true of all numbers, simpliciter. One can't claim, merely with the resources of PA, that the MH-machine has checked every number.

By adding the omega rule to PA one can bypass the difficulties created by non-standard models. In fact, the omega rule just is the claim that if something is true of all the finite numbers, then it's true of all the numbers. But this isn't the only way of bypassing this. Either by working embedded in some sufficiently strong Set Theory or by moving to second-order PA, one can specify that one is talking specifically about the standard model of arithmetic N, not about other non-standard models and consequently claim that the MH-machine checks every number. One could then just use standard reasoning from universals to conclude that Goldbach's conjecture is true.

Note that none of this rules out the possibility of infinite reasoning. One could understand this case as a case of infinite reasoning and, given Warren's convincing reply to objections to the contrary, I'm inclined to agree with him that infinite reasoning is possible in this case.

Nevertheless, I'm unconvinced that there are "interesting" cases of infinite reasoning, in the sense of a case of infinite reasoning where one can use infinite reasoning to do something one couldn't ordinarily do with our finite tools. This means that whilst I think Warren has given us good reason to believe his conclusion, we shouldn't overstate how much that's achieved. It still remains unclear, at least from this paper, if infinite reasoning can be used as a basis for more interesting philosophical moves relating to truth or mathematics.

The paper then finishes with a couple of objections and responses to those objections. I won't unpack them here, but suffice to say I find Warren's responses reasonable.

Closing remarks

I'll certainly keep myself aware of any papers Warren writes on this subject in the future. At the least, this paper opened the door for interesting infinite reasoning and Warren's certainly earned the right to be heard-out on this line of enquiry. I remain sceptical, but he could certainly change my mind on this.

For anyone wanting to read the article it can be found via these links:

https://onlinelibrary.wiley.com/doi/10.1111/phpr.12694
https://philpapers.org/rec/WARIR

Jared Warren is an Assistant Professor at Stanford. https://philosophy.stanford.edu/people/jared-warren

I welcome comments or friendly criticism below, or you're welcome to email me at gareth.pearce@univie.ac.at

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Photo credit: Jonas Lekevicius, CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons

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