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.[1] Reality is a causal fabric, a collection of variables ("stuff") that interact with each other.[2] 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.[3] 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.[4] That is, when you take a true mathematical statement and plug in any acceptable inputs,[5] 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.[6]
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,[7] 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:[8] 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[9], 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.[10]
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.[11] This is closely related to topos theory; the internal logic of a topos is at least[12] intuitionistic. As I understand it, many topoi can be considered as foundations for mathematics,[13] 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,[14] 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.
ADDENDUM
[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.
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[1] See The Useful Idea of Truth.
[2] See The Fabric of Real Things and Stuff that Makes Stuff Happen.
[3] See The Useful Idea of Truth and The Fabric of Real Things.
[4] See Proofs, Implications, and Models and Logical Pinpointing.
[5] 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.
[6] A visual example using modus ponens and cute cuddly kittens is found in Proofs, Implications, and Models.
[7] See The Useful Idea of Truth.
[8] See this paper by biologist E O Wilson.
[9] What I mean is that I would not claim that it "isn't true," which usually makes the debate stagnate.
[10] For concreteness, read these examples of non-constructive proofs.
[11] See here, paragraph two.
[12] Given certain further restrictions, a topos is Boolean and its internal logic is classical.
[13] This is an amusing and vague-as-advertised summary by John Baez.
[14] 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.
A Pragmatic Epistemology
For the past three thousand years epistemology has been about the truth, the whole truth, and nothing but the truth. Philosophers and scientists have continuously attempted to pinpoint the nature of truth, to find general logico-syntactic criteria for generating justified inferences, and to discover the true nature of reality. I happen to think that truth is overrated. And by that I don't mean that I'm a stereotypical postmodernist, prepared to say that all views are on equal footing (because after all, who can really say what's true and what isn't?). Instead I mean that I don't even think that the truth is a useful or coherent concept when stretched to accommodate what philosophers have tried to make it accommodate. It's not a malleable enough concept to have the generality that philosophers are asking of it. We need something else in its place.
A view similar to this is reservationism, which was first introduced[1] by Moldbug in A Reservationist Epistemology. If you haven't read it, I suggest at least skimming it before reading the rest of this post, but the basic idea is that you can try to cram reason into an explicit General Theory of Reason for as long as you like, but at best it will always be a special case of "common sense." I have mixed feelings about Moldbug's post. On the one hand, it's delightfully witty and I agree with the general thrust of the argument. On the other hand, I think you can go a bit farther to explain his "common sense" notion than he lets on, and the abrasiveness and vagueness of his writing are likely to cloak some of the finer points. And despite giving (likely unintentional) hints about what we might replace "truth" with, he never does criticise the concept of truth, although he obviously criticises general theories of truth.
Since I do depart from Moldbug, I'll call myself a pragmatist rather than a reservationist. I'll also give my pragmatism a slogan: "It's just a model."[2] What's just a model? Bayesianism, falsificationism, positivism, naturalism, physicalism, panpsychism, quantum mechanics, operant conditioning, phlogistic chemistry, Catholicism, atheism, Hinduism, category theory, number theory, constructive analysis ... we could go all day with obvious examples. Here are some other examples: "Bayesian reasoners are optimal," "loop quantum gravity will give us a theory of everything," "a sentence is meaningful iff it, by itself or in conjunction with further premises, entails some observation statement not entailed by those other premises alone,"[3] and more mundane examples like "It's raining outside," "My mother is 52," "Common sense," and "It's just a model." Here's another, an example central to my position: models are conceptual tools that help us think about some aspect of our experience and achieve our goals. I italicised "conceptual tools" because I want to emphasise their role as tools rather than their role as theories or propositions, and I want to emphasise the utility of model-tools rather than their truth.
Lots of other models have been called pragmatism. Charles Peirce and William James came up with pragmatic "theories of truth." Richard Rorty and Ludwig Wittgenstein advanced pragmatic "theories of meaning." Instead of pragmatically explaining truth and related concepts, I'm giving it a rest. There are plenty of theories of truth already, and truth-focused epistemologies have their shortcomings. After all, what have the correspondence theory and Quinean naturalism given us in the philosophy of math except Platonism and confusion?[4] Of course, these shortcomings shouldn't come as a surprise under the models-as-tools theory. Tools are built and tested with specific domains of application in mind by agents with limited imagination, and when we try to apply the tools to other domains we run the risk that they could be utterly worthless.
Of course, to provide a working alternative I need to convince others that it's worth trying, so let me try. Under this paradigm, where we judge models by their utility, there is no need to fret over whether the continuum hypothesis is "true" or not, whatever that might mean: we just note that as far as we can tell it's neither here nor there and move on.[5] And suddenly the famous fact/value distinction looks very silly: of course facts[6] inform us about how we should act; "facts" are just another model-tool in our system of model-tools, and the whole point of building our model-tools is to use them. These benefits should be enough to get your attention, at the very least. Another is that we don't have to use awkward, gross-feeling terms like "common sense." Common sense, in Moldbug's usage, is just the process that leads us to justify using models. So instead of common sense being the standard, we have our goals and instrumental rationality. Model building and model use are special cases of tool building and tool use, and agent-like goal-directed behaviour in general.
My model is also compatible with the conception of rationality as winning. There is no holy reason juice in the universe[7] that stops us from picking a winning but decidedly not reason-juice-flavoured strategy[8]; the standard for picking a strategy is that it helps us achieve our goals, and strategies that make us sit in the corner don't pass. But my model is not compatible with the division between instrumental and epistemic rationality. Since the correspondence theory (and the map-territory metaphor) is just another tool in the toolbox, epistemic rationality is just a tool in the toolbox too, whereas instrumental rationality is the process we use to choose which tools we want to use and when (and why) we want to us them. In this model, instrumental rationality just is rationality, that "common sense" thing that Moldbug claimed subsumed everything else as a special case.
And before I'm accused of being a relativist, let me say that not all tools are created equal, and we do have reason to use some in certain situations as opposed to others; namely, we have reason to use tools in certain situations when they produce outcomes we like better relative to other tools at our disposal. So when it comes to a models of physics, we use Aristotelian physics for simple everyday situations[9], classical mechanics for many engineering projects and pedagogical functions, and quantum mechanics for many other engineering projects and current research.[10] Now, often people will read this transition through different models as evidence for their favourite epistemology, and I won't disappoint you there: this transition shows us that as people began encountering new problems, old tools often didn't cut it. Go figure. After all, they weren't built with those future problems in mind, and foreseeing every possible roadblock that a tool could face would require another very powerful tool!
Which brings me to the problem of induction. Traditionally the problem is to find a general justification for the truth of universal claims on the basis of particular cases. We can translate this into my pragmatic framework fairly easily: construct the one tool to rule them all, a tool so awesome that we can use to achieve any achievable goal and that has provisions for any pesky roadblocks. The traditional statement reads easily as "carry out a foundationalist programme like Descartes," or in other words create a bedrock of certainty. It's generally agreed that this is impossible. My reformulation can be reread in a similar way: "carry out a reductionist programme like a theory of everything." Since the problem of induction is unsolvable, I strongly doubt that a reductionist theory of everything is on the menu. And if such a theory is ever announced I suspect the pragmatic slogan will still apply: It's just a model. A model with a fancy name, sure, but nonetheless with a limited domain of applicability and its own set of weaknesses.
That all having been said, my views aren't as alien to the general LW memecluster as you might expect. My position assumes consequentalism, and it's Quinean in that it's continuous with science rather than "prior" to it. I think that the results of science are some of the best tools we've developed, that physicalism is a good model for conceptualising and solving many problems, and that the correspondence theory of truth is a good tool in certain contexts. My goal here is not really to be a contrarian, as fun as that is. Rather, one of my goals is to find a better way to conceptualise a broader class of epistemological and scientific problems than current frameworks comfortably allow.
If this post receives favourable feedback, I plan to write more posts expanding on these ideas. Specifically:
- The extent to which I am kind of sort of a relativist after all, but still not really.
- Foundational issues in math as seen through a pragmatic lens (potentially featuring a mysterious co-author).
- An epistemological analogue to the orthogonality thesis in ethics.
- The interface theory of perception and an evolutionary perspective on my model.
- The relationship between my pragmatism and probability theory.
- Criticism and commentary on recent MIRI research.
- Criticism and commentary on key posts in the Sequences.
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[1] First introduced under that name, anyhow. For similar ideas, see Richard Rorty's Consequences of Pragmatism and Paul Feyerabend's Against Method.
[2] "My" is meant a bit loosely; I owe a lot to people I've discussed these ideas with, and to the reading material I've consumed. I'll elaborate on any of those contributions by request.
[3] This is a paraphrase of A. J. Ayer from Language, Truth and Logic, Dover ed. pp. 38-39.
[4] Quine gives us Platonism.
[5] It's been begrudgingly agreed that we can't decide on whether the continuum hypothesis is true since CH and its negation are independent of ZFC, but many people still argue about whether it is, ultimately, true or not. A pragmatic take on this debate is that since CH and ¬CH are both consistent with ZFC, we can strategically add either one of them as axioms for the purposes of making proofs easier if we like.
[6] I doubt the usefulness and coherence of "fact" as much as I do "truth," but conventional language is conventional language.
[7] "Universe" being another example of a model.
[8] Despite the arguments of champions of causal and evidential decision theories.
[9] See Induction by Holland, Holyoak, et al., pp. 203-9 and 224-5, and A Function for Thought Experiments by Thomas Kuhn.
[10] Obviously these are meant as examples of uses, not an exhaustive list.
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