Reasoning with one hand tied behind your back

Four examples where hard work pays off

Here are four potentially-unrelated examples with a common structure:

• there are (at least) two ways to achieve some goal
• one is harder than the other
• there is a deeper kind of payoff to taking the harder route

1. Constructive mathematics

Many things in math can be proven without relying on the law of the excluded middle, which says, for any claim \(P\), you can assert that \(P\) or not-\(P\) is true. Using this additional axiom can make things easier; however, proofs using it will not be constructive. For example:

• a constructive proof that \(\sqrt{2}\) is irrational is a mechanism which, for any rational number \(q \in \mathbb{Q}\) it is fed, constructs the nonzero difference \(q - \sqrt{2}\). A non-constructive proof (e.g. assuming \(\sqrt{2}\) is rational, then proving a contradiction, then concluding \(\sqrt{2}\) is irrational) says that \(\sqrt{2}\) is irrational but cannot be used to automatically derive a contradiction from someone asserting that, say, \(\frac{99}{70}\) is equal to \(\sqrt{2}\).

• A constructive proof of the infinitude of prime numbers will take a purported set of all prime numbers (call it \(P\)) and produce a number \(p\prime\) outside that set that is prime \({(p\prime = 1+\underset{p \in P}{\prod} p)}\). A non-constructive proof would tell you that \(P\) is infinite without giving you any concrete witness of this fact.

One might argue that a constructive proof is a stronger form of evidence than a non-constructive proof. Not just definitionally, by using fewer axioms, but in a practical sense: constructive proofs have computational content that witnesses the proof in a concrete way and can be excecuted by a computer.

2. Universal properties

Here, we take aim at a particular attitude towards mathematics:

• to understand an object is to understand its constituent parts, i.e. how it is constructed.

An alternative viewpoint is motivated by category theory:

• to understand an object is to understand how it relates to everything else

The latter approach makes it more work to understand something, for example:

• Suppose we have two sets of things, \(A\) and \(B\). The cartesian product of sets, \(A \times B\) can be defined as the set of pairs \((a,b)\) for all \(a\) in \(A\) and for all \(b\) in \(B\). This is very straightforward. In constrast, the category-theoretic characterization of a product describes how \(A \times B\) relates to everything else rather than describing what \(A \times B\) is in itself, the details of which are left to a footnote.¹

• The set of natural numbers \(\mathbb{N}\) can be defined as the set \(\{0, 1, 2, ...\}\). This is very straightforward. In contrast, the category- theoretic characterization of the natural numbers is very involved because we aren’t allowed to say “what is in the set”, only how the “natural numbers” (whatever that may be) must relate to other things in a given context if it is worthy of being called the “natural numbers” of that context.

What are the upshots of the categorical approach, which forbids us from talking about the constituent parts of the thing we’re trying to describe? Here are two:

• Generality: we define product once and then realize that tons of things that appear in many different contexts are actually the exact categorical definition of product, but specialized to different categories (e.g. multiplying numbers, intersections of sets, the greatest common divisor, logical AND, etc.).
• Essence: the definition of \(\mathbb{N}\) given set-theoretic foundations might mean that it literally is the set \(\{\varnothing, \{\varnothing\},\ \{\{\varnothing\}\},\ ...\}\). Though one could also encode the natural numbers as \(\{\varnothing,\ \{\varnothing\},\ \{\varnothing,\{\varnothing\}\},\ ...\}\) or a very different way, e.g. the number \(3\) defined as the set of all sets with three elements. These can have very different properties that we morally ought to ignore, such as whether or not \(a \subseteq b\) (in their encoding) iff \(a \leq b\) as naturals, because these are “implementation details”. The category-theoretic characterization forbids us from looking at implementation details, which is an advantage in the long run even if it makes things more complicated at the start.²

3. Pragmatist semantics

It’s often important that we clarify the meaning of our statements, such as “The robot is not conscious” or “There do not exist any ghosts.” A theory of semantics is helpful to make explicit how to do this clarification.

We could adopt the “view from nowhere”: we imagine there is a “true model” of the world as it is in itself. Perhaps this lists out all the primitive objects and primitive relationships between them. If one could analyze this “true model”, one could say the meaning of “conscious” is a predicate which holds over objects in the model.³ This is called representatationalism - an ideology that says the meaning of thought and talk should be understood principally in terms of the representational relations that thinkings and sayings stand in to what they (purport to) represent. It gives us an easy template for saying, for any \(\phi\), what \(\phi\) means: namely, that \(\phi\) is only true if the objective world is configured in such-and-such way. It’s so pervasive that it might feel ridiculous to consider any alternative. However, there are issues:

• How are we supposed to achieve the “view from nowhere” in practice? What good is this semantic account if that is in principle impossible?
• We must imagine the “objective model of the world” to be specified in some vocabulary, but it’s not any of our vocabularies (even our scientific ones), so how could our predicates actually hook up to the objects of the objective world?
• Many meaningful concepts (such as mathematics, possibility/necessity, ethics) evade a satisfying descriptive account. This sometimes leads people to conclude those concepts are not what we originally thought (e.g. “evil” turns out to be just a way we talk about things we really dislike, or “necessity” turns out to be just how we feel about stuff that happens really regularly) rather than questioning the semantic model of descriptivism.
• Descriptivism would make it seem like every statement we make crucially hinges on whether or not things ‘actually exist’. Yet many everyday examples (e.g.  witches, Pluto, beauty, and sour acid) show that ordinary statements are not like this.

The philosophical school of pragmatism offers an alternative target vocabulary for explaining semantics, one that does not have to resort to the “view from nowhere.” Rather, one explains meanings in terms of social practices (sometimes thought of as “language games”), rather than some sort of direct relationship between the expression and the world.

• Think of the difference between: “Rojo in Spanish and rot in German mean the same thing because they pick out the same objects in the world” versus “…because rojo plays the same role in Spanish that rot plays in German.”

One pragmatist analysis of ‘sentience’ might conclude that sentience is not a property of agents (nor the presence of “mind stuff”). Rather, a sentient agent is (to first approximation) one whose verbal noises can be given the social significance of being judgments.

• Making a judgment both makes us responsible for something as well as entitles us to making other inferences.
• Think of the difference of a parrot squacking “The building is on fire” and an adult saying the same. Our difference in how we initially react (and how we treat the speaker if it turns out to be false) reflects the different social status of the parrot and human (the difference between mere words and a judgment).
• Sentience can be explained in terms of this social status by pragmatists, without referring to some “true model” of the world (beyond the ordinary and accessible world of speakers and talking).

4. Anti-authoritarianism

We have opinions on how the world should work. For example, the world might be better if everyone stopped drinking sugary soda, or spanking their kids, or eating meat. Politics involves proposing ways to bring the world closer to how it should work, and the dead-simple way to characterize a system where people don’t do X is to say: “X is illegal”. Any alternative, i.e. working towards a system where people do X less yet X is still not illegal, takes more creativity and likely more work. For example, in the case of reducing meat eating:

• We could subsidize meat alternatives or urban agriculture.
• We can create public educational resources (e.g. inform people of the environmental consequences, show visceral imagery of the slaughterhouse, health benefits of vegetarianism, teach high schoolers how to cook)
• We could fund fundamental research about how to grow artificial meat.

What are the upshots in this scenario? For some, there are only practical reasons to not use coercion: it’s hard to get the bill passed, or it will cost one’s political party too much political capital, or it would be hard to enforce. For others, it feels like there is something immediately negative about coercion, and while it may be good to use coercion to stop something even worse, it’s still a case of picking the lesser of two evils. In the past, people were coercive for (what they thought) were the right reasons, but in retrospect they were wrong. This gives us pause when considering the use of force, to the extent that we are not absolutely sure that we are right in our belief of how the world should work.

Conclusion

I started writing this without knowing what the common theme would be, but now I think of it as a cautionary tale about adopting a “God’s-eye view” (even hypothetically!) because it is such a seductive idea that we inadvertently buy into it to some extent.

1. A God’s-eye view can see whether a statement is true or false.
   • there is no further need for a mechanism that (in a finite amount of time) demonstrates why.
   • The favoring of constructive over non-constructive proofs mirrors the pragmatist tendency to have more interest in justification over truth.
2. A God’s-eye view is does not have to observe things by observation but can simply know objects directly.
   • There is no worry that one has picked an arbitrary encoding that might not cohere with other design choices and need to be changed.
   • Our implementation details may be imperfect, while God’s are perfect and
     immutable (so us mortals alone need to worry about dependencies on implementation details which may change).
3. A God’s-eye view in semantics ignores the inherent subjectivity of our concepts. It forces one to engage in ontology and metaphysics.
   • Category theory likewise can be thought of as filtering out ontology.
4. A God’s-eye view of politics would allow one to use one’s omnipotence to alter anything as one sees fit to achieve greater good.
   • There is no worry about this good actually being mistaken or an illusion, thus, without any humility, the ends will always justify the means.
   • Rorty interprets Platonists (which he uses to characterize the god’s-eye-view) as seeking to end the conversation, while pragmatism seeks to continue the conversation. The former is more useful in practice for authoritarians.

Although it can feel like one is reasoning with one hand tied behind one’s back when one refuses to invoke a God’s-eye view, the short-term advantages of invoking it are outweighed by longer-term friction from the fact that we are not actually omniscient and omnipotent. Phrased another way, I believe there are practical and moral benefits to showing this kind of restraint.^{4 5}

Footnotes

1. You can pick whatever representation you want for \(A \times B\), but for it to act as a product, it must be able to take any pair of functions like in the figure below and derive a unique function which first maps into it and then applies the projection maps (e.g. get rank or get suit) to recover the original pair of functions. In the case of the \((a,b)\) encoding, this unique map sends “low” to \((1, \spadesuit)\) and “high” to \((A, \clubsuit)\).

2. When programming, we can also take advantage of implementation details to do some things easier - it is likewise not a good idea to do this, in the long run.

3. Think of a ‘predicate’ as a function which takes in the identity of the object and evaluates it to tell you whether the property holds or not.

4. This is also a theme of minimizing ontology talk.

5. Another possible analogue to add to the list: “Jaynes is very careful to always work with conditional probability. Rather than positing some big measure space \(\Omega\) and assuming there is a single distribution on it that all conditional distributions are a result of, he works with conditional distributions natively, which gives everything a more local feel.”