Recognizing the finite

We have a simple procedure for recognizing finite sequences. We start at the beginning and go through the sequence one item at a time (e.g., by scanning with our eyes). If we reach the end, we are confident the sequence was finite. This procedure can be relied on if and only if there are no supertasks—i.e., if and only if it is impossible to have an infinite sequence of tasks started and completed.

How do we know that there are no supertasks? Either empirically or a priori. To know it empirically, we would have to know that the various tasks we’ve completed were finite. But how would we know of any tasks we’ve completed that it’s finite if not by the above procedure?

So we have to know it a priori.

And the only story I know of how we could do that is by a priori cognizing some anti-infinity principle like Causal Finitism.

I am not sure how strong the above argument is. It is a little too close to standard sceptical worries for comfort.

Do we have to know that seven is finite to know that three is finite?

Three is a finite number. How do we know this?

Here’s a proof that three is finite:

1. 0 is finite. (Axiom)

2. For all n, if n is finite, then n + 1 is finite. (Axiom)

3. 3=0+1+1+1. (Axiom)

4. So, 0+1 is finite. (By a and b)

5. So, 0+1+1 is finite. (By b and d)

6. So, 0+1+1+1 is finite. (By b and e)

7. So, 3 is finite. (By c and f)

Let’s assume we can answer the difficult question of how we know axioms (a) and (b), and allow that (c) is just true by definition.

I want to raise a different issue. To know that three is finite by means of the above argument, it seems we have to know that the argument is a proof.

One might think this is easy: a proof is a sequence of statements such that each non-axiomatic statement logically follows from the preceding ones, and it’s clear that (d)-(g) each follow from the previous by well-established rules of logic.

One could ask about how we know these rules of logic to be correct—but I won’t do that here. Instead, I want to note that it is false that every sequence of statements such that each non-axiomatic statement logically follows from the preceding ones is a proof. This is the case only for finite sequences of statements. The following infinite sequence of statements is not a proof, even though every statement follows from preceding ones: “…, so I am Napoleon, so I am Napoleon, so I am Napoleon.”

Very well, so to know that (a)-(g) is a proof, I need to know that (a)-(g) are only finitely many statements. OK, let’s count: (a)-(g) are seven statements. So it seems we have to know that seven is finite (or something just as hard to know) in order to use the proof to know that three is finite.

This, of course, would be paradoxical. For to use a proof analogous to (a)-(g) to show that seven is finite, we would need a proof of eleven steps, and so we would need to know that eleven is finite to know that the proof is a proof.

Maybe we can just see that seven is finite? But then we gain nothing by (a)-(g), since the knowledge-by-proof will depend on just seeing that seven is finite, and it would be simpler and more reliable just to directly see that three is finite.

It might be better to say that we can just see that the proof exhibited above, namely (a)-(g), is finite.

It seems that knowledge-by-proof in general depends on recognition of the finite. Or else on causal finitism.

A slightly different causal finitist approach to finitude

The existence of non-standard models of arithmetic makes defining finitude problematic. A finite set is normally defined as one that can be numbered by a natural number, but what is a natural number? The Peano axioms sadly underdetermine the answer: there are non-standard models.

Now, causal finitism is the metaphysical doctrine that nothing can have an infinite causal history. Causal finitism allows for a very neat and pretty intuitive metaphysical account of what a natural number is:

• A natural number is a number one can causally count to starting with zero.

Causal counting is counting where each step is causally dependent on the preceding one. Thus, you say “one” because you remember saying “zero”, and so on. The causal part of causal counting excludes a case where monkeys are typing at random and by chance type up 0, 1, 2, 3, 4. If causal finitism is false, the above account is apt to fail: it may be possible to count to infinite numbers, given infinite causal sequences.

While we can then plug this into the standard definition of a finite set, we can also define finitude directly:

• A finite set or plurality is one whose elements can be causally counted.

One of the reasons we want an account of the finite is so we get an account of proof. Imagine that every day of a past eternity I said: “And thus I am the Queen of England.” Each day my statement followed from what I said before, by reiteration. And trivially all premises were true, since there were no premises. Yet the conclusion is false. How can that be? Well, because what I gave wasn’t a proof, as proofs need to be finite. (I expect we often don’t bother to mention this point explicitly in logic classes.)

The above account of finitude gives an account of the finitude of proof. But interestingly, given causal finitism, we can give an account of proof that doesn’t make use of finitude:

• To causally prove a conclusion from some assumptions is to utter a sequence of steps, where each step’s being uttered is causally dependent on its being in accordance with the rules of the logical system.

• A proof is a sequence of steps that could be uttered in causally proving.

My infinite “proof” that I am the Queen of England cannot be causally given if causal finitism is true, because then each day’s utterance will be causally dependent on the previous day’s utterance, in violation of causal finitism. However, interestingly, the above account of proof does not guarantee that a proof is finite. A proof could contain an infinite number of steps. For instance, uttering an axiom or stating a premise does not need to causally depend on previous steps, but only on one’s knowledge of what the axioms and premises are, and so causal finitism does not preclude having written down an infinite number of axioms or premises. However, what causal finitism does guarantee is that the conclusion will only depend on a finite number of the steps—and that’s all we need to make the proof be a good one.

What is particularly nice about this approach is that the restriction of proofs to being finite can sound ad hoc. But it is very natural to think of the process of proving as a causal process, and of proofs as abstractions from the process of proving. And given causal finitism, that’s all we need.

From the finite to the countable

Causal finitism lets you give a metaphysical definition of the finite. Here’s something I just noticed. This yields a metaphysical definition of the countable (phrased in terms of pluralities rather than sets):

1. The xs are countable provided that it is possible to have a total ordering on the xs such if a is any of the xs, then there are only finitely many xs smaller (in that ordering) than x.

Here’s an intuitive argument that this definition fits with the usual mathematical one if we have an independently adequate notion of nautral numbers. Let N be the natural numbers. Then if the xs are countable, for any a among the xs, define f(a) to be the number of xs smaller than a. Since all finite pluralities are numbered by the natural numbers, f(a) is a natural number. Moreover, f is one-to-one. For suppose that a ≠ b are both xs. By total ordering, either a is less than b or b is less than a. If a is less than b, there will be fewer things less than a than there are less than b, since (a) anything less than a is less than b but not conversely, and (b) if you take something away from a finite collection, you get a smaller collection. Thus, if a is less than b, then f(a)<f(b). Conversely, if b is less than a, then f(b)<f(a). In either case, f(a)≠f(b), and so f is one-to-one. Since there is a one-to-one map from the xs to the natural numbers, there are only countably many xs.

This means that if causal finitism can solve the problem of how to define the finite, we get a solution to the problem of defining the countable as a bonus.

One of the big picture things I’ve lately been thinking about is that, more generally, the concept of the finite is foundationally important and prior to mathematics. Descartes realized this, and he thought that we needed the concept of God to get the concept of the infinite in order to get the concept of the finite in turn. I am not sure we need the concept of God for this purpose.

Infinite proofs

Consider this fun “proof” that 0=1:

      …

• So, 3=4

• So, 2=3

• So, 1=2

• So, 0=1.

What’s wrong with the proof? Each step follows from the preceding one, after all, and the only axiom used is an uncontroversial axiom of arithmetic that if x + 1 = y + 1 then x = y (by definition, 2 = 1 + 1, 3 = 1 + 1 + 1, 4 = 1 + 1 + 1 + 1 and so on).

Well, one problem is that intuitively a proof should have a beginning and an end. This one has an end, but no beginning. But that’s easily fixed. Prefix the above infinite proof with this infinite number of repetitions of “0=0”, to get:

• 0=0

• So, 0=0

• So, 0=0

• So, 0=0

      …

      …

• So, 3=4

• So, 2=3

• So, 1=2

• So, 0=1.

Now, there is a beginning and an end. Every step in the proof follows from a step before it (in fact, from the step immediately before it). But the conclusion is false. So what’s wrong?

The answer is that there is a condition on proofs that we may not actually bother to mention explicitly when we teach logic: a proof needs to have a finite number of steps. (We implicitly indicate this by numbering lines with natural numbers. In the above proof, we can’t do that: the “second half” of the proof would have infinite line numbers.)

So, our systems of proof depend on the notion of finitude. This is disquieting. The concept of finitude is connected to arithmetic (the standard definition of a finite set is one that can be numbered by a natural number). So is arithmetic conceptually prior to proof? That would be a kind of Platonism.

Interestingly, though, causal finitism—the doctrine that nothing can have an infinite causal history—gives us a metaphysical verificationist account of proof that does not presuppose Platonism:

• A proof is a sequence of steps such that it is metaphysically possible for an agent to verify that each one followed by the rules from the preceding steps and/or the axioms by observation of each step.

For, given causal finitism, only a finite number of steps can be in the causal history of an act of verification of a proposition. (God can know all the steps in an infinite chain, but God isn’t an observer: an observer’s observational state is caused by the observations.)

A causal finitist definition of the finite

Causal finitism says that nothing can have infinitely many causes. Interestingly, we can turn causal finitism around into a definition of the finite.

Say that a plurality of objects, the xs, is finite if and only if it possible for there to be a plurality of beings, the ys, such that (a) it is possible for the ys to have a common effect, and (b) it is possible for there to be a relation R such that whenever x₀ one of the xs, then there is exactly one of a y₀ among the xs such that Rx₀y₀.

Here's a way to make it plausible that the definition is extensionally correct if causal finitism is true. First, if the definition holds, then clearly there are no more of the xs than of the ys, and causal finitism together with (a) ensures that there are finitely many of the ys, so anything that the definition rules to be finite is indeed finite. Conversely, suppose the xs are a finite plurality. Then it should be possible for there to be a finite plurality of persons each of which thinks about a different one of the xs in such a way that each of the xs is thought about by one of the ys. Taking being thought about as the relation R makes the definition be satisfied.

Of course, on this account of finitude, causal finitism is trivial, for if a plurality of objects has an effect, then they satisfy the above definition if we take R to be identity. But what then becomes non-trivial is that our usual platitudes about the finite are correct.

Mathematical Platonist Universalism, consistency, and causal finitism

Mathematical Platonists say that sets and numbers exist. But there is a standard epistemological problem: How do we have epistemic access to the sets to the extent of knowing some of the axioms they satisfy? There is a solution to this epistemological problem, mathematical Platonist universalism (MPU): for any consistent collection of mathematical axioms, there are Platonic objects that satisfy these axioms. MPU looks to be a great solution to the epistemological problems surrounding mathematical Platonism. How did evolved creatures like us get lucky enough to have axioms of set theory or arithmetic that are actually true of the sets? It didn’t take much luck: As soon as we had consistent axioms, it was guaranteed that there would be a plurality of objects that satisfied them, and if the axioms fit with our “set intuitions”, we could call the members of any such plurality “sets” while if they fit with our “number intuitions”, we could call them “natural numbers”. And the difficult questions about whether things like the Axiom of Choice are true are also easily resolved: the Axiom of Choice is true of some pluralities of Platonic objects and is false of others, and unless we settle the matter by stipulation, no one of these pluralities is the sets. (The story here is somewhat similar to Joel Hamkins’ set theoretic multiverse, but I don’t know if Hamkins has the kind of far-reaching epistemological application in mind that I am thinking about.)

This story has a serious problem. It is surely only the consistent axioms that are satisfied by a plurality of objects. Axioms are consistent, by definition, provided that there is no proof of a contradiction from them. But proofs are themselves mathematical objects. In fact, we’ve learned from Goedel that proofs can be thought of as just numbers. (Just write your proof in ASCII, and encode it as a binary number.) Hence, a plurality of axioms is consistent if and only if there does not exist a number with a certain property, namely the property of encoding a proof of a contradiction from these axioms. But on MPU there is no unique plurality of mathematical objects deserving to be called “the numbers”. So now MPU faces a very serious problem. It said that any consistent plurality of axioms is true of some plurality of Platonic objects, and there are no privileged pluralities of “numbers” or “sets”. But consistency is itself defined by means of “the numbers”. And the old epistemological problems for Platonism resurface at this level. How do we have access to “the numbers” and the axioms they satisfy so as to have reason to think that the facts about consistency of axioms are as we think they are?

One could try making the same move again. There is no privileged notion of consistency. There are many notions of consistency, and for any axioms that are consistent with respect to any notion of consistency there exists a plurality of Platonic satisfiers. But now this literally threatens incoherence. But unless we specify some boundaries on the notion of consistency, this is going to literally let square circles into Platonic universalism. And if we specify the boundaries, then epistemological problems that MPU was trying to solve will come back.

At my dissertation defense, Robert Brandom offered a very clever suggestion for how to use my causal powers account of modality to account for provability: q can be proved from p provided that it is causally possible for someone to write down a proof of q from p. This can be used to account for consistency: axioms are consistent provided that it is not causally possible to write down a proof of a contradiction from them. There is a bit of a problem here, in that proofs must be finite strings of symbols, so one needs an account of the finite, and a plurality is finite if and only if its count is a natural number, and so this account seems to get us back to needing privileged numbers.

But if one adds causal finitism (the doctrine that only finite pluralities can together cause something) to the mix, we get a cool account of proof and consistency. Add the stipulation that the parts of a “written proof” need to have causal powers such that they are capable of together causing something (e.g., causing someone to understand the proof). Causal finitism then guarantees that any plurality of things that can work together to cause an effect is finite.

So, causal finitism together with the causal powers account of modality gives us a metaphysical account of consistency: axioms are consistent provided that it is not causally possible for someone to produce a written proof of a contradiction from them.

"Finite"

In conversation last week, I said to my father that my laptop battery has a “finite number of charge cycles”.

Now, if someone said to me that a battery had fewer than a billion charge cycles, I’d take the speaker to be implicating that it has quite a lot of them, probably between half a billion and a billion. And even besides that implicature, if all my information were that the battery has fewer than a billion charge cycles, then it would seem natural to take a uniform distribution from 0 to 999,999,999 and think that it is extremely likely that it has at least a million charge cycles.

One might think something similar would be the case with saying that the battery has a finite number of charge cycles. After all, that statement is logically equivalent to the statement that it has fewer than ℵ₀ charge cycles, which by analogy should implicate that it has quite a lot of them, or at least give rise to a uniform distribution between 0, inclusive, and ℵ₀, exclusive. But no! To say that it has a finite number of charge cycles seems to implicate something quite different: it implicates that the number is sufficiently limited that running into the limit is a serious possibility.

Actually, this may go beyond implicature. Perhaps outside of specialized domains like mathematics and philosophy, “finite” typically means something like not practically infinite, where “practically infinite” means beyond all practical limitations (e.g., the amount of energy in the sun is practically infinite). Thus, the finite is what has practical limits. (But see also this aberrant usage.)

A reason why voting methods are compromises

Voting involves compromise on two levels. On the ground level, a vote involves coming to a compromise decision. But on the meta level, a voting system embodies compromise between different desiderata. Arrow's Theorem is a famous way of seeing the latter point. But there is also another way of seeing it, which in one way goes beyond Arrow's Theorem: while Arrow's Theorem only applies where there are three or more options, what I say applies even in binary cases.

We suffer from both epistemic and moral limitations. Good voting systems are a way of overcoming these, by combining the information offered by us in such a way that no small group of individuals, suffering as it may from epistemic or moral shortcomings, has too much of a say. It is interesting to see that there is an inherent tension between overcoming epistemic and moral limitations.

Consider one of two models. On both models, a collection of options is offered to a population.

1. Model 1: Each voter comes up with her honest best estimate of the total utility of each option, and offers a report of her estimate.
2. Model 2: Each voter comes up with her honest best estimate of the utility for her of each option, and offers a report of her estimate.

On the assumption that (a) the voters' errors in their estimations are independent Gaussians with mean zero and we have no information as to who has bigger variances, and that we want to maximize total expected utility (which will be approximately true) and (b) the voters accurately report their estimates, there is provably an optimal voting system under both models: we simply arithmetically average the voters' estimates and select the option with the highest average utility estimate (see my earlier post on this for some computer simulation data). Any voting system whose departs from this will be inoptimal under these circumstances.

Assuming that whatever people are going to say in a vote is going to be somehow based on their estimates of utility on the whole or utility to them, this averaging system is the best way to leverage the information scattered in the population. Unfortunately, while this is a good way to overcome our epistemic limitations, it does terribly with regard to our moral limitations. If one lies boldly enough, namely comes up with utility estimates that are far more inflated than anybody else's, one controls the outcome of the vote. Let's say that option 2 is the best one for me. Then I simply specify that the utility for option 2 is 10^100000000 and for option 1 is −10^100000000. And of course, there will be an arms race in the population to specify big numbers if there is more than one dishonest member of the population. But in any case, the dishonest will win.

In other words, the optimal system in the case of honest utility estimates is pretty much the worst system where honesty does not generally hold. A good voting system for morally imperfect voters must cap the effect each voter has. But in capping the effect each voter has, information can will in general be lost.

This is most clear in Model 2. We can imagine that an option moderately benefits a significant majority but horrendously harms a minority. Given honest utility reports from everyone and the averaging system, the option is likely to be defeated, since the members of the minority will report enormously negative utilities that will overcome the moderate positive utilities reported by members of the majority. But as soon as one caps the effects of each voter, the information about the enormously negative utilities to the minority will be lost. Model 1 is more helpful (presumably, civic education is how we might get most people to vote according to Model 1), but information will still be lost due to the differences in epistemic access to the total utility. On Model 1, capping will lose us the case where one individual genuinely has information about an enormous negative effect but is unable to convince others of this information. But capping of some sort is necessary because of moral imperfection.

(The optimal method of utility estimation also faces the problem that we are better at rank orderings than at absolute utilities. This can in principle be overcome to some degree by giving people additional hypothetical options to rank-order and then recovering utility estimates from these.)

A brief way to make the point is this. The more trusting a voting system is, the more information it brings to the table; but the more trusting a voting system is, the worse it does with regard to moral imperfection. A compromise is needed in this regard. And not just in voting.