Notes on motivation and self-improvement

I’ve recently been thinking about something that strikes me as a remarkable fact about human psychology. Nearly everybody knows of multiple things that consistently bring them significant happiness, and despite this they don’t do these things, or do them much less frequently than they would like.

For example, virtually every time that I meditate I feel better off at the end of it. There’s usually long-term effects too, like that the rest of my day is improved, and sometimes multiple days in the future are improved. There’s even some longer-term benefits; my sense is that consistent meditation allows me to think more clearly and get insight about the structure of my own mind. I’ve meditated hundreds of times and have literally never had a negative experience. The most I can say is I’ve had some neutral experiences, but the vast majority have been positive experiences, and many of them have been extremely positive experiences. And yet, I don’t regularly meditate! If you picked a random moment in the last ten years of my life and asked me “Did you meditate today? What about this week? Or this month?”, the answer to all three would more often than not be “no”.

And it’s not that I just sometimes forget about meditation. It’s actually worse than this! I feel an active resistance towards meditating whenever I think of doing it. One reason I’m saying this is that I used to be very into psychological egoism. The idea is that you can boil down all of our motivations to finding personal happiness. The claim is that if you interrogate your own motivations and keep asking “well why do you want that?”, you’ll eventually find a basic seeking after positive valence experiences at the end of the chain. But when you actually start to introspect on your own mind, it feels like there are these very clear cases, cases that aren’t even very rare, of your mind rejecting happy experiences. One could say “Well really this is a short-term versus long-term type of thing and you’re really rejecting the long-term happiness in exchange for some short term gratification”. But it’s not even that! For me, the benefits I get from meditation happen basically immediately. If you were to try to model me as a simple happiness maximizer, even one that prefers short-term gratification, you would end up making really bad predictions about how I live my life.

So, there’s a whole set of strategies that I just know will effectively bring me happiness. And I can implement these strategies at virtually any time. And yet I don’t do them! Or at least, I do them much less frequently than I could be. And I’m willing to bet that the same is true of you. I want you to think about the following: What are three concrete actions you can take that aren’t very costly, and that pretty much always make you feel happier, healthier, or better in any way. Just think of these three things, maybe even write them down. And then ask yourself: “Well, how often do I actually do these things? Do I do them at a rate that I think makes sense, or do I do them less than that?”

I’ve already given the example of meditation for myself. Another example is exercise. Exercise is a less good example for the psychological egoism point I was making, because there is a good case to be made that it’s a trade off between present suffering and future happiness. But regardless, I know that when I exercise, I feel good for the rest of my day. I have more energy, I’m typically more focused, and I feel better about myself. Long-run consistent exercise makes you look better, which raises self-esteem and makes you more likely to attract social interactions that you might desire, like a romantic partner. And due to the halo effect, you’re probably able to more easily get friends. (Even if you think it’s a bad thing that our society prizes attractiveness so much, it still is a true fact about society. Just because something shouldn’t be the case doesn’t mean you should act as if it isn’t the case!)

Another one for me is taking a walk in the morning. If instead of going on the computer, the first thing I do is just get up and take a walk outside, it consistently makes the rest of my day feel better. I don’t think that this one has longer term effects beyond that single day, but it’s still something I could do which takes maybe 30 minutes and isn’t physically exhausting or anything. And I have a lot of resistance to doing it!

One more example I think I can give is writing. If in the course of a day I write an essay or make progress on an essay, I feel a lot better. This is one which is actually situational, because it’s not the case that at every moment I have an idea for a good essay I can write. If I just sort of sit down and start writing without any idea of something I want to write about, that typically isn’t going to have the same effect. But nonetheless, it’s often the case that there are topics that I want to write about and yet I don’t. So that’s another interesting phenomenon.

So, exercise, meditation, walking in the morning, and writing, these are all concrete things I can put into action literally today. And in fact, I have been doing these for last few weeks and I hope that continues; I’m trying to build them as actual habits rather than just some temporary phase I’m in. But taking an outside view I can see that throughout most of my life I have not had a really consistent practice of any of these things.

Ultimately I just think it’s a good thing to take a moment out of your day and reflect on the things you can do to help yourself, and that you KNOW will help yourself. And possibly just thinking about these things will be motivating to some degree to actually do them! Put a special focus on the fact that you really could just start doing them right now; there’s nothing stopping you. Of course, if one of your choices was something like “every time I vacation in Hawaii it consistently raises my happiness for a period of time”, well… that’s true but not very actionable. On the other hand, things like exercising or meditation can be done in almost any environment! You don’t have to go to the gym, you could just search “high intensity interval training” on Youtube, do ten to fifteen minutes of exercise and you will almost certainly feel better for the rest of your day. So ask yourself the question “why am I not doing that?” And then do it!

First Order Truth Trees

The analytic tableaux (also called truth tree) style of proof system is really nice. The tree structure tends to resemble the actual process of mathematical reasoning better than the Hilbert-style linear proof systems, and I find that the formal proofs are much easier to follow. For first-order logic (without equality), the system involves four new inference rules to deal with quantifiers. There are a few different logically equivalent choices for these rules, but here is my favorite:

Universal instantiation ∀
∀x φ(x)

φ(t)
(where t is any closed term – any term without free variables)

Existential instantiation ∃
∃x φ(x)

φ(a)
(where a is any new constant)

Negated universal ¬∀
¬∀x φ

∃x ¬φ

Negated existential ¬∃
¬∃x φ

∀x ¬φ

The instantiation rules are actually quite interesting. Suppose that you’re working in the language of ZFC. ZFC has no constant or function symbols, only a single binary relation symbol: ∈, interpreted as “is an element of”. This seems to raise a problem for using the instantiation rules. For universal instantiation, we can only apply it to closed terms. But a language with no constants has no closed terms! And looking at existential instantiation, there’s a similar problem: we’re supposed to derive φ(a), where a is a new constant, but our language has no constants!

So what’s up? Does the standard first-order analytical tableaux approach not work for simple languages like ZFC? Insofar as ZFC is a shining beacon of what first-order logic can accomplish, this is a little disconcerting.

Actually, the instantiation rules work exactly as stated for ZFC. The subtlety in their interpretation is that when running proofs, you are allowed to (and in fact, often must) expand the language. So when you do existential instantiation, you don’t choose a constant symbol that already exists inside your language. Instead, you add a totally new constant symbol to your language, and then declare that φ holds of this symbol. Similarly, you can do universal instantiation in a language with no closed terms, simply by expanding the language to include some constants.

Now, if we’re expanding the language in the course of the proof, then who’s to say that in the end our proof will still be valid for our original language? Well, suppose we’re working in some minimal language L that has no constants. Let T be our background theory (the set of non-tautologies that we’re taking as assumptions). When we do existential instantiation, we always create a new constant. Let’s call it a. We then expand the language into L’ = L ⋃ {a}, and we expand our assumptions to T’ = T ⋃ {φ(a)}. When we do universal instantiation, we either use an existing closed term or create a new one. In the second case, we create a new constant b and form a new language L’ = L ⋃ {b}. We also expand our assumptions to a new set T’ = T ⋃ {φ(b)}.

The important thing to note is that we haven’t done anything that invalidates any models of T! If T originally contained a sentence of the form ∀x φ(x), then adding c and declaring φ(c) doesn’t conflict with this. And if T originally contained a sentence of the form ∃x φ(x), then in every model of T at least one thing satisfies φ. So when we add a new constant c, that constant can just refer back to any of these φ-satisfiers.

You might think: “But hold on! How can we be sure that it’s safe to just add new constants? Couldn’t we expand the domain too much, in a way that’s inconsistent with our original theory?” The answer to this is that the domain doesn’t have to expand to accommodate new constants! These constants can refer to existing elements of the domain. For instance, suppose T contains six existential statements and a sentence that says there are exactly five objects. Each time we run existential instantiation on one of the six existential sentences, we create a new constant. So we’ll get six new constants. But these constants can refer to the same value! And since our theory already says that there are five objects, the models of our expanded theory will also contain exactly five objects, meaning that in every model of our original theory, the new constants will refer to elements of the domain that already exist. No domain expansion!

Ok, but how do we know that there isn’t some very complicated sentence ψ that was originally true in every model of T, but becomes false in some models of T’? To fully prove that this never happens, we could do induction over all sentences in the language, showing that any sentence that is entailed by T is also entailed by T’. But considering the details of the expansion process, and trying to come up with ways that the expansion might fail, is a good way to get an intuition for why this proof system works.

I’ll close with a few examples of using analytic tableaux to prove basic results in PA and ZFC. Think of this as a proof of concept! (Each of these proofs also uses some rules regarding equality, which are obvious enough despite that I haven’t explicitly defined them).

First, we show that PA proves ∀x (0 + x = x). This is nontrivial, despite that ∀x (x + 0 = x) is an axiom!

Next, a proof from ZFC that the empty set exists and is unique:

Now a proof from PA that addition is commutative. This one is shockingly complicated for such a simple concept, and requires three uses of induction, but this goes to show how basic the axioms of PA really are! If you look closely, you’ll also notice that this proof replicates the proof of ∀x (0 + x = x) from above!

Final example, here’s a PA proof that no number is its own successor. This rules out nonstandard models of arithmetic with one nonstandard number!

One nonstandard is worth infinitely many standards

Suppose that M is a nonstandard model of true arithmetic (the set of all first-order sentences in the language of PA that are true in the standard model of arithmetic, ℕ). Now, take any sentence φ(x) with one free variable. Suppose that there’s some nonstandard number k in M such that φ holds of k. Since k is larger than every standard natural, the following infinite set of sentences are all true:

∃x (x > 0 ∧ φ(x))
∃x (x > 1 ∧ φ(x))
∃x (x > 2 ∧ φ(x))

∃x (x > 1000000 ∧ φ(x))

Since these sentences are true in M and M is a model of true arithmetic, these sentences must also be true in the standard model ℕ. So it must be true for every standard natural that there’s a larger standard natural that satisfies φ. In other words, you can guarantee that there are infinitely many standard naturals that satisfy a property φ just by finding a single nonstandard number k that satisfies φ in a model of true arithmetic!

Furthermore, since in ℕ it is true that every standard natural has a larger standard natural satisfying φ, the sentence ∀x ∃y (y > x ∧ φ(y)) is true in ℕ. So this sentence must be true in every model of true arithmetic, including M! This means that just by finding a single nonstandard satisfying φ, you can immediately be sure that there are infinitely many standard numbers AND infinitely many nonstandard numbers (in every nonstandard model of TA!) satisfying φ. This is pretty dramatic!

As an example, consider the twin prime conjecture. We can construct the predicate isPrime(x) in first-order PA with the formula ∀y (∃z (y⋅z = x) → (y=1 ∨ y=x)). Then the predicate isTwinPrime(x) is just: isPrime(x) ∧ isPrime(x+2). Now the twin prime conjecture just says that ∀x ∃y (y > x ∧ isTwinPrime(y)), which is exactly the form we saw in the last paragraph! So to prove the twin prime conjecture, it suffices to demonstrate a single nonstandard twin prime in a model of true arithmetic.