Astray with the Truth: Logic and Math
LessWrong has one of the strongest and most compelling presentations of a correspondence theory of truth on the internet, but as I said in A Pragmatic Epistemology, it has some deficiencies. This post delves into one example: its treatment of math and logic. First, though, I’ll summarise the epistemology of the sequences (especially as presented in High Advanced Epistemology 101 for Beginners).
Truth is the correspondence between beliefs and reality, between the map and the territory. Reality is a causal fabric, a collection of variables (“stuff”) that interact with each other. True beliefs mirror reality in some way. If I believe that most maps skew the relative size of Ellesmere Island, it’s true when I compare accurate measurements of Ellesmere Island to accurate measurements of other places, and find that the differences aren’t preserved in the scaling of most maps. That is an example of a truth-condition, which is a reality that the belief can correspond to. My belief about world maps is true when that scaling doesn’t match up in reality. All meaningful beliefs have truth-conditions; they trace out paths in a causal fabric. Another way to define truth, then, is that a belief is true when it traces a path which is found in the causal fabric the believer inhabits.
Beliefs come in many forms. You can have beliefs about your experiences past, present and future; about what you ought to do; and, relevant to our purposes, about abstractions like mathematical objects. Mathematical statements are true when they are truth-preserving, or valid. They’re also conditional: they’re about all possible causal fabrics rather than any one in particular. That is, when you take a true mathematical statement and plug in any acceptable inputs, you will end up with a true conditional statement about the inputs. Let’s illustrate this with the disjunctive syllogism:
((A∨B) ∧ ¬A) ⇒ B
Letting A be “All penguins ski in December” and B be “Martians have been decimated,” this reads “If all penguins ski in December or Martians have been decimated, and some penguins don’t ski in December, then Martians have been decimated.” And if the hypothesis obtains (if it’s true that (A∨B) ∧ ¬A), then the conclusion (B) is claimed to follow.
That’s it for review, now for the substance.
Summary. First, from examining the truth-conditions of beliefs about validity, we see that our sense of what is obvious plays a suspicious role in which statements we consider valid. Second, a major failure mode in following obviousness is that we sacrifice other goals by separating the pursuit of truth from other pursuits. This elevation of the truth via the epistemic/instrumental rationality distinction prevents us from seeing it as one instrumental goal among many which may sometimes be irrelevant.
What are the truth-conditions of a belief that a certain logical form is valid or not?
A property of valid statements is being able to plug any proposition you like into the propositional variables of the statement without disturbing the outcome (the conditional statement will still be true). Literally any proposition; valid forms about everything that can be articulated by means of propositions. So part of the truth-conditions of a belief about validity is that if a sentence is valid, everything is a model of it. In that case, causal fabrics, which we investigate by means of propositions, can’t help but be constrained by what is logically valid. We would never expect to see some universe where inputting propositions into the disjunctive syllogism can output false without being in error. Call this the logical law view. This suggests that we could check a bunch of inputs and universes constructions until we feel satisfied that the sentence will not fail to output true.
It happens that sentences which people agree are valid are usually sentences that people agree are obviously true. There is something about the structure of our thought that makes us very willing to accept their validity. Perhaps you might say that because reality is constrained by valid sentences, sapient chunks of reality are going to be predisposed to recognising validity …
But what separates that hypothesis from this alternative: “valid sentences are rules that have been applied successfully in many cases so far”? That is, after all, the very process that we use to check the truth-conditions of our beliefs about validity. We consider hypothetical universes and we apply the rules in reasoning. Why should we go further and claim that all possible realities are constrained by these rules? In the end we are very dependent on our intuitions about what is obvious, which might just as well be due to flaws in our thought as logical laws. And our insistence of correctness is no excuse. In that regard we may be no different than certain ants that mistake living members of the colony for dead when their body is covered in a certain pheromone: prone to a reaction that is just as obviously astray to other minds as it is obviously right to us.
In light of that, I see no reason to be confident that we can distinguish between success in our limited applications and necessary constraint on all possible causal fabrics.
And despite what I said about “success so far,” there are clear cases where sticking to our strong intuition to take the logical law view leads us astray on goals apart from truth-seeking. I give two examples where obsessive focus on truth-seeking consumes valuable resources that could be used toward a host of other worthy goals.
The Law of Non-Contradiction. The is law is probably the most obvious thing in the world. A proposition can’t be truth and false, or ¬(P ∧ ¬P). If it were both, then you would have a model of any proposition you could dream of. This is an extremely scary prospect if you hold the logical law view; it means that if you have a true contradiction, reality doesn’t have to make sense. Causality and your expectations are meaningless. That is the principle of explosion: (P ∧ ¬P) ⇒ Q, for arbitrary Q. Suppose that pink is my favourite colour, and that it isn’t. Then pink is my favourite colour or causality is meaningless. Except pink isn’t my favourite colour, so causality is meaningless. Except it is, because either pink is my favourite colour or causality is meaningful, but pink isn’t. Therefore pixies by a similar argument.
Is (P ∧ ¬P) ⇒ Q valid? Most people think it is. If you hypnotised me into forgetting that I find that sort of question suspect, I would agree. I can *feel* the pull toward assenting its validity. If ¬(P ∧ ¬P) is true it would be hard to say why not. But there are nonetheless very good reasons for ditching the law of non-contradiction and the principle of explosion. Despite its intuitive truth and general obviousness, it’s extremely inconvenient. Solving the problem of the consistency of various PA and ZFC, which are central to mathematics, has proved very difficult. But of course part of the motivation is that if there were an inconsistency, the principle of explosion would render the entire system useless. This undesirable effect has led some to develop paraconsistent logics which do not explode with the discovery of a contradiction.
Setting aside whether the law of non-contradiction is really truly true and the principle of explosion really truly valid, wouldn’t we be better off with foundational systems that don’t buckle over and die at the merest whiff of a contradiction? In any case, it would be nice to alter the debate so that the truth of these statements didn’t eclipse their utility toward other goals.
The Law of Excluded Middle. P∨¬P: if a proposition isn’t true, then it’s false; if it isn’t false, then it’s true. In terms of the LessWrong epistemology, this means that a proposition either obtains in the causal fabric you’re embedded in, or it doesn’t. Like the previous example this has a strong intuitive pull. If that pull is correct, all sentences Q ⇒ (P∨¬P) must be valid since everything models true sentences. And yet, though doubting it can seem ridiculous, and though I would not doubt it on its own terms, there are very good reasons for using systems where it doesn’t hold.
The use of the law of excluded middle in proofs severely inhibits the construction of programmes based on proofs. The barrier is that the law is used in existence proofs, which show that some mathematical object must exist but give no method of constructing it.
Removing the law, on the other hand, gives us intuitionistic logic. Via a mapping called the Curry-Howard isomorphism all proofs in intuitionistic logic are translatable into programmes in the lambda calculus, and vice versa. The lambda calculus itself, assuming the Church-Turing thesis, gives us all effectively computable functions. This creates a deep connection between proof theory in constructive mathematics and computability theory, facilitating automatic theorem proving and proof verification and rendering everything we do more computationally tractable.
Even if we the above weren’t tempting and we decided not to restrict ourselves to constructive proofs, we would be stuck with intuitionistic logic. Just as classical logic is associated with Boolean algebras, intuitionistic logic is associated with Heyting algebras. And it happens that the open set lattice of a topological space is a complete Heyting algebra even in classical topology. This is closely related to topos theory; the internal logic of a topos is at least intuitionistic. As I understand it, many topoi can be considered as foundations for mathematics, and so again we see a classical theory pointing at constructivism suggestively. The moral of the story: in classical mathematics where the law of excluded middle holds, objects in which it fails arise naturally.
Work in the foundations of mathematics suggests that constructive mathematics is at least worth looking into, setting aside whether the law of excluded middle is too obvious to doubt. Letting its truth hold us back from investigating the merits of living without it cripples the capabilities of our mathematical projects.
Unfortunately, not all constructivists or dialetheists (as proponents of paraconsistent logic are called) would agree how I framed the situation. I have blamed the tendency to stick to discussions of truth for our inability to move forward in both cases, but they might blame the inability of their opponents to see that the laws in question are false. They might urge that if we take the success of these laws as evidence of their truth, then failures or shortcomings should be evidence against them and we should simply revise our views accordingly.
That is how the problem looks when we wear our epistemic rationality cap and focus on the truth of sentences: we consider which experiences could tip us off about which rules govern causal fabrics, and we organise our beliefs about causal fabrics around them.
This framing of the problem is counterproductive. So long as we are discussing these abstract principles under the constraints of our own minds, I will find any discussion of their truth or falsity highly suspect for the reasons highlighted above. And beyond that, the psychological pull toward the respective positions is too forceful for this mode of debate to make progress on reasonable timescales. In the interests of actually achieving some of our goals I favour dropping that debate entirely.
Instead, we should put on our instrumental rationality cap and consider whether these concepts are working for us. We should think hard about what we want to achieve with our mathematical systems and tailor them to perform better in that regard. We should recognise when a path is moot and trace a different one.
When we wear our instrumental rationality cap, mathematical systems are not attempts at creating images of reality that we can use for other things if we like. They are tools that we use to achieve potentially any goal, and potentially none. If after careful consideration we decide that creating images of reality is a fruitful goal relative to the other goals we can think of for our systems, fine. But that should by no means be the default, and if it weren’t mathematics would be headed elsewhere.
[Added due to expressions of confusion in the comments. I have also altered the original conclusion above.]
I gave two broad weaknesses in the LessWrong epistemology with respect to math.
The first concerned its ontological commitments. Thinking of validity as a property of logical laws constraining causal fabrics is indistinguishable in practical purposes from thinking of validity as a property of sentences relative to some axioms or according to strong intuition. Since our formulation and use of these sentences have been in familiar conditions, and since it is very difficult (perhaps impossible) to determine whether their psychological weight is a bias, inferring any of them as logical laws above and beyond their usefulness as tools is spurious.
The second concerned cases where the logical law view can hold us back from achieving goals other than discovering true things. The law of non-contradiction and the law of excluded middle are as old as they are obvious, yet they prevent us from strengthening our mathematical systems and making their use considerably easier.
One diagnosis of this problem might be that sometimes it’s best to set our epistemology aside in the interests of practical pursuits, that sometimes our epistemology isn’t relevant to our goals. Under this diagnosis, we can take the LessWrong epistemology literally and believe it is true, but temporarily ignore it in order to solve certain problems. This is a step forward, but I would make a stronger diagnosis: we should have a background epistemology guided by instrumental reason, in which the epistemology of LessWrong and epistemic reason are tools that we can use if we find them convenient, but which we are not committed to taking literally.
I prescribe an epistemology that a) sees theories as no different from hammers, b) doesn’t take the content of theories literally, and c) lets instrumental reason guide the decision of which theory to adopt when. I claim that this is the best framework to use for achieving our goals, and I call this a pragmatic epistemology.
 See The Useful Idea of Truth.
 See The Useful Idea of Truth and The Fabric of Real Things.
 Acceptable inputs being given by the universe of discourse (also known as the universe or the domain of discourse), which is discussed on any text covering the semantics of classical logic, or classical model theory in general.
 A visual example using modus ponens and cute cuddly kittens is found in Proofs, Implications, and Models.
 See The Useful Idea of Truth.
 See this paper by biologist E O Wilson.
 What I mean is that I would not claim that it “isn’t true,” which usually makes the debate stagnate.
 For concreteness, read these examples of non-constructive proofs.
 See here, paragraph two.
 Given certain further restrictions, a topos is Boolean and its internal logic is classical.
 This is an amusing and vague-as-advertised summary by John Baez.
 Communication with very different agents might be a way to circumvent this. Receiving advice from an AI, for instance. Still, I have reasons to find this fishy as well, which I will explore in later posts.