ONTOLOGY AND PERSONAL IDENTITY, a play in one act and two scenes

ACT I

SCENE I. The Ruskin School of Art.

Enter ARTHUR and MEREOLOGIST

Arthur: I love art. I collect clay sculptures.

Mereologist: Aha, look at all my clever mereological paradoxes involving lumps of clay¹. We are led to the conclusion that two objects can share the exact same matter at the same time (or somesuch).

Enter NIHILIST

Nihilist [in an evil nihilistic voice]: All the purported puzzles of mereology dissolve if we make no existence claims about ordinary objects². When we speak of the statue existing, we are speaking loosely. If a clever mereologist presses the point, we should concede that we really mean “there is matter arranged statue-wise” and not “there is a statue”.

Mereologist [clutching his pearls]: No objects! What an extreme, radical view.

Nihilist: Not so. The view has no practical consequences. In everyday life, we can keep speaking loosely and be prefecly well understood. In ordinary life, no-one cares about this weird, ghostly “existence” property. It’s only in the presence of metaphysicians that we have to retreat to the more cumbersome language to be perfectly exact.

Arthur: Yes, I agree. When I sell a sculpture at an auction, all anyone cares about is that they will get the statue (or the matter arranged statue-wise), and that I will get the money. No one will ask “but does the statue really exist?”. Similarly, if I am admiring the “statue” in my own home, I am perfectly indifferent one way or another to whether it “exists” as a statue. “It” is there for my eyes to look at, and that is all I care about.

Mereologist: But then you live a life of self-deception, for in truth you believe you have no statues in your collection at all!

Arthur: I hate philosophy. I’m going to leave you two and go look at my clay arranged statue-wise.

Exeunt

SCENE II³. All Souls College basement, at night.

Enter ROB and DEREK PARFIT’S EVIL TWIN

Rob: I care a lot about being rational.

Derek Parfit’s evil twin (DPET): You do? Let me try to help you with this goal.

Rob: Sure.

DPET: In ten years I will either torture you, or torture a random stranger. Which would you prefer?

Rob: I would prefer that it be the stranger, obviously.

DPET: What is your reason for that?

Rob: Rob+10 will be identical to me. It’s rational for me to care about people that are me.

DPET: You would say you have a rational reason to care about future people if and only if they are identical to you? (The identity matters (IM) view)

Rob: Yes.

DPET: Take the fission case⁴. You could take multiple occupancy view.⁵

Rob: No, I find it absurd.

DPET: You could take the no-branching view.⁶

Rob: No, it is a silly view. It means that neither Lefty nor Righty is you. They both come into existence when your cerebrum is divided. If both your cerebral hemispheres are transplanted, you cease to exist—though you would survive if only one were transplanted and the other destroyed. Fission is death. You can survive hemispherectomy only if the hemisphere to be removed is first destroyed. This is absurd.

DPET: We have found no plausible defence for the IM view. Hence you must abandon it, and with it your justification to prefer that the random stranger be tortured.

Rob: Oh no. This is a horrible bind, I must accept either torture or irrationality! But wait, could this be a mere sophistical trickery like that of the mereologist? I shall simply replace the IM view with the relation-R-matters ⁷ (RM) view. Since I stand in relation R with Rob+10 and not the stranger, the RM view justifies preferring that the stranger be tortured.

DPET: You are quite right that you should take the RM view. But unlike in the case of mereological nihilism, this has real consequences. Let us say a bit more about relation R. What are the relative contributions of connectedness and continuity to what matters? On one view, connectedness does not matter. Only continuity matters. Suppose I know that, two days from now, my only experience-memories will be of experiences that I shall have tomorrow. On the view just stated, since there shall be continuity of memory, this is all that matters. (Parfit section 100, p. 301)

Rob: I would strongly disagree. Losing all such memories would be something I would deeply regret.

DPET: Hence you must reject the view that only continuity matters. Connectedness matters to some degree.

Rob: I suppose.

DPET: Psychological connectedness comes in degrees. For example, a toddler and an old man are less connected (by shared memories, say) than a toddler and a slightly older toddler. To take another example: immediately after the operation, Lefty and Righty are very connected, over time, they become less so. Psychological connectedness comes in degrees, and hence your reasons to care about future Robs come in degrees.

Rob: The reasons that make it rational for me to care about what happens to Rob+10 also make it rational to care (to a degree) about what happens to strangers. It’s actually irrational to be an pure egoist who nonetheless cares about his future self. This is a big deal, thanks Derek!

Exeunt

THE END

Praise for ONTOLOGY AND PERSONAL IDENTITY

A work of great literary subtlety and not at all a hackneyed morality play.

— Oxford University Press

ONTOLOGY AND PERSONAL IDENTITY is of uncommon moral depth, […] a play that pushes the boundaries of the genre with vivid, believeable characters.

— Cherwell

My new favourite book.

— My grandmum

Dies ist zu meiner Kenntnis die einzige Kopie dieses Briefes, die es auf dem Internet gibt. Ich habe den Scan persönlich beantragt, von dem Original bei der Staatsbibliothek zu Berlin, Preußischer Kulturbesitz.

Valerio Filoso (2013) writes:

Most econometrics textbooks limit themselves to providing the formula for the $\beta$ vector of the type

$$\beta = (X′X)^{-1} X'Y.$$

Although compact and easy to remember, this formulation is a sort black box, since it hardly reveals anything about what really happens during the estimation of a multivariate OLS model. Furthermore, the link between the $\beta$ and the moments of the data distribution disappear buried in the intricacies of matrix algebra. Luckily, an enlightening interpretation of the $\beta$s in the multivariate case exists and has relevant interpreting power. It was originally formulated more than seventy years ago by Frisch and Waugh (1933), revived by Lovell (1963), and recently brought to a new life by Angrist and Pischke (2009) under the catchy phrase regression anatomy. According to this result, given a model with K independent variables, the coefficient $\beta$ for the k-th variable can be written as

$$\beta_k = \frac{cov(y,\tilde{x}_k)}{var(\tilde x_k)}$$

where $\tilde x_k$ is the residual obtained by regressing $x_k$ on all remaining $K − 1$ independent variables.

The result is striking since it establishes the possibility of breaking a multivariate model with $K$ independent variables into $K$ bivariate models and also sheds light into the machinery of multivariate OLS. This property of OLS does not depend on the underlying Data Generating Process or on its causal interpretation: it is a mechanical property of the estimator which holds because of the algebra behind it.

From, $\beta_k = \frac{cov(y,\tilde{x}_k)}{var(\tilde x_k)}$, it’s easy to also show that

I’ll stick to the first expression in what follows. (See Filoso sections 2-4 for a discussion of the two options. The second is the Frisch-Waugh-Lovell theorem, the first is what Angrist and Pischke call regression anatomy).

Multiple regression with $K\geq3$ (a constant and two or more variables) can feel a bit like voodoo at first. It is shrouded in phrases like “holding constant the effect of”, “controlling for”, which are veiled metaphors for the underlying mathematics. In particular, it’s hard to see what “holding constant” has to do with minimising a loss function. On the other hand, a simple $K=2$ regression has an appealingly intuitive 2D graphical representation, and the coefficients are ratios of familiar covariances.

This is why it’s nice that you can break a model with $K$ variables into $K$ bivariate models involving the residuals $\tilde x_k$. This is easiest to see in a model with $K=3$: $\tilde x_k$ is the residual from a simple $K=2$ regression. Hence a sequence of three simple regressions is sufficient to obtain the exact coefficients of the $K=3$ regression (see figure 2 below, yellow boxes).

Similarly, it’s possible to arrive at the coefficients of a $K>3$ regression by starting with only simple pairwise regressions of the original $K$ independent variables. I do this for $K=4$ in figure 1. From these pairwise regressions (in black and grey¹), we work our way up to three $K=3$ regressions of one $X$-variable on the two others (orange boxes), by regressing each $X$-variable on the residuals obtained in the first step. We obtain expressions for each of the $\tilde x_k$, ($g,f,q$ in my notation). We regress $Y$ on these (yellow box). Figure 1 also nicely shows that the number of pairwise regressions needed to compute multivariate regression coefficients grows with the square of $K$. According to this StackExchange answer, the total time complexity is $O(K^2n)$, for $n$ observations.

Figure 1:

Judd et al. (2017) have a nice detailed walk-through of the $K=3$ case, pp.107-116. Unfortunately, they use the more complicated Frisch-Waugh-Lovell theorem method of regressing residuals on residuals. I show this method here (in green) and the method we’ve been using (in yellow), for $K=3$. As you can see, the former method needs two superfluous base-level regressions (in dark blue). For this reason, that method becomes quickly intractable at $K\geq 4$. But they should be equivalent, hence I use the same $\theta$ coefficients in the yellow and green boxes.

Figure 2:

I made this is PowerPoint, not knowing how to do it better. Here is the file.