William M. Briggs

Statistician to the Stars!

Page 2 of 409

On The Kaya Identity

A valid identity.

A valid identity.

A reader asked about my take on the Kaya wars that are flaming at Anthony Watts’s place.

Here is the form of the Kaya Identity, which is to say, the Kaya non-equation:

Y = X_1 \times \frac{X_2}{X_1} \times \frac{X_3}{X_2} \times \dots \times \frac{X_{n}}{X_{n-1}} \times \frac{Y}{X_n} ,

The Ys and Xis are numbers and a free choice, given the limitations of algebra (no Xi equals 0). Try it and see: for fun, let X_i = i . Works. A perfectly harmless manipulation.

There is no explicit word about causality in the Kaya. The Xis aren’t necessarily causing the Y or each other. If we wanted to know about what caused Y, and we believed that the Xis were in the causal path of Y, we wouldn’t set up an identity but an equation which looked like this:

Y = f(X,\beta) .

Where X is a vector and where the vector of (known and unknown) parameters β may be larger or smaller than X. Notice that Y does not appear on the right hand side. We are solving for Y here. One possibility (and probably a too simple one for most Y)is a linear equation:

Y = \beta_0 +  \beta_1 X_1 + \dots + \beta_1 X_n .

This is not regression. This is a causal model: it says Y will certainly change by \beta_i when X_i increases by 1 unit. Regression is a probability relationship where we first assume Y \sim N(\mu, \sigma) and then substitute μ for Y on the left hand side. Regressions says Y might, not that it certainly will, change.

Anyway, since the Kaya is an identity we can put anything we like in for the Xis and Y. Let’s try.

\mbox{CO}_2 = \mbox{Puppies} \times \frac{\mbox{Cats}}{\mbox{Puppies}} \times \frac{\mbox{Meteors}}{\mbox{Cats}} \times \frac{\mbox{Cosmic rays}}{\mbox{Meteors}} \times \frac{\mbox{CO}_2}{\mbox{Cosmic rays}} ,

where each is a number existent or occurring over a year in some suitable units. Notice that I was careful to put things that we know change over time, but I needn’t have. Everything could be static (more or less), like this:

\mbox{CO}_2 = \mbox{Mountain ranges} \times \frac{\mbox{Continents}}{\mbox{Mountain ranges}} \times \frac{\mbox{Great lakes}}{\mbox{Continents}} \times \frac{\mbox{Gold}}{\mbox{Great lakes}} \times \frac{\mbox{CO}_2}{\mbox{Gold}} .

Some of these quantities are known for sure, and one, the amount of Gold, is not. But whether we know the value of any of these is immaterial to the Kaya. As long as none are null quantities, and Gold isn’t, we’re in business. Also note that the number of entries was up to me. I could have made the list shorter or longer as I pleased.

What do any of these things have to do with CO2? Who said the items had to have anything to do with CO2? Who said I had to use CO2? Insisting that the Xis are causative of Y and using the Kaya and not an equation is doing it, as they say, the hard way.

But we can certainly manufacture cause-like stories. Puppies eat, and their food both requires and releases CO2. Cats, too. Meteorites often have carbon in them, and boy do they disturb the atmosphere; lots of cloud nucleii strewn hither and thither during their journey. Same thing with cosmic rays. Plus, these energetic creatures effect life, and life is important for understanding carbon. This is just off the top of the head; spend some serious time and you can spin this tale out to saga length. Peer reviewed, of course.

The same thing can be done with the fixed Xs. Or with any items you care to put into the Kaya. As long as you stay away from 0, you’re in business.

Here’s what Kaya himself put (adapted to just the USA):

\mbox{CO}_2 = \mbox{Population} \times \frac{\mbox{GDP}}{\mbox{Population}} \times \frac{\mbox{Energy}}{\mbox{GDP}} \times \frac{\mbox{CO}_2}{\mbox{Energy}} .

This is just as valid as the examples above, though this one seems more popular with economists. But then economists are prone to tying everything to GDP, which is a number, and economists love numbers (those without uncertainty, that is), often preferring them over reality. Never mind.

For an example of the Kaya in action, see the Pielke Jr video embedded at this link starting at around 21 minutes. Pielke appears to believe that the Kaya has something to do with causation and that he and fellow economists have captured all they need know about human beings and carbon.

The good news from an analytical standpoint is that there’s nothing else. There’s no other levers that you can use out there. This is comprehensive. You may wish there was some rabbit you can pull out of a hat because the good news is also the bad news: this is all you have.

That’s false, as we know from the Puppies example (recall we can add puppies or gold or whatever to the Kaya). And these items—GDP, etc.—are far from all we know about humans and carbon, though Pielke calls it an “extremely powerful tool for policy analysis.” For instance, he says (around 20 min. mark) that, as a lever governments can pull, “Less people, all else equal, equals less emissions” (I believe the guillotine had a similar lever). Another lever the government can yank, he says, is to purposely create poverty, i.e. “Limit generation of wealth” (does that include the wealth of government, one wonders?).

These claims are not quite false, but not quite true, either. More people mean more energy is used, but there’s also a greater chance for more innovation in, say, creating more efficient energy sources. And more people also means more food, and food is a terrific carbon sequestration vehicle, to say it in economic-speak. (Incidentally, one reason that there are more people is that there is more food.)

Now there’s nothing wrong with grappling with crude ratios like Energy/GDP to have some rough, first-blush idea of the amount of energy that is now required to generate such-and-such-a-sized economy, but as for the energy required to drive a future economy, who knows? Nobody in 1990 predicted Google. The Kaya is not a forecasting tool. And since it doesn’t carry any measure of uncertainty, and since every term is mixed up causally with every other term, nobody knows how much credence to give it.

And we can’t bypass the hard work of actually estimating the amount of carbon released and sequestered, both now and in the future. Yet the Kaya is mute on what causes CO2. GDP, after all, doesn’t cause CO2. That’s impossible.

The Kaya should be replaced with a probability model/equation, which can tell us how much change in GDP might be associated with a change in CO2.

That model ought to be under the same constraint as climate models. If it can’t make skillful predictions of future data, we shouldn’t believe it. Right? And how good are economists at forecasting the GDP or energy use one to two decades out?

Update I should have mentioned this above, but it might be in some problem that we know the last ratio Y/X_n (and each other ratio) but that we do not know Y. The Kaya can then be used to calculate Y. But in the case of CO_2, we do not know the last ratio.

Philosophic Issues in Cosmology: II. Relativistic Theories On Universe Origin—Guest Post by Bob Kurland

Robert Millikan, Georges Lamaitre, and A. Einstein (with hat).

Robert Millikan, Georges Lamaitre, and A. Einstein (with hat).

Bob Kurland is a retired, cranky, old physicist, and convert to Catholicism. He shows that there is no contradiction between what science tells us about the world and our Catholic faith.

There was a young lady named Bright,Whose speed was far faster than light;She started one day In a relative way, And returned on the previous night. A.H.R. Buller, Punch

Read Part I.

The usual exposition of Einstein’s General Relativity Field Equations is very forbidding, full of Greek subscripts and tensor notation; a clear, simplified version has been given on the web by John Baez, and is appropriate for considering the Big Bang. The standard general relativity model for cosmology is that given by Friedmann-LeMaitre-Robertson-Walker,1 usually designated by FLRW.

The FLRW model proceeds from the following simplifying assumptions: a) the universe is isotropic (looks the same in every direction, from every point in space); b) there is a constant amount of matter in the universe; c) on a large scale (hundreds of times the distance between galaxies) the universe has a homogeneous matter density (matter is spread evenly throughout space); d) the effects of “pressure” (from radiation or the vacuum) can be neglected.

With these simplifying assumptions, the equation for the “size” of the universe, its radius R, becomes simple, and looks just like the equation of motion for a particle traveling under an inverse square law, like that of gravity. (Note: this is not to say the size of the universe is really given by some value R, but to show how space is expanding; the universe might possibly be infinite—more about that later).

The universe might expand and then contract in a “Big Crunch” (like a ball falling back to earth), corresponding to positively curved spacetime (like a sphere); it might expand with a constant velocity of expansion (like a projectile going into orbit), corresponding to flat space-time (like a plane); or it might expand with an accelerating velocity of expansion (like a projectile achieving escape velocity), corresponding to a saddle-shaped curvature of space-time. It should also be emphasized that the FLRW solution to the Einstein General Relativity equations is by no means unique, nor is it the only solution with a singularity. It is a model, however, that is in accord with measured data (red shift, COBE microwave background radiation).

The assumptions stated above do not apply rigorously. Observations have shown a filament or bubble-like structure to the universe with clusters and meta-clusters of galaxies. (A theoretical picture for this filament structure has been proposed.) In the early stages of the universe radiation pressure was very likely significant.

More recently, measurements have shown that the expansion rate is increasing, which is presumed due to “dark energy”, possibly a pressure due to vacuum energy. Moreover, at some point in the expansion the scale of the universe gets so small that classical physics does not apply and quantum mechanics has to be used for theory. Unfortunately, quantum mechanics and general relativity have not yet been reconciled into one general theory, so there is a fundamental difficulty with this melding of the two theories.

The simple solution above for FLRW models gives an acceleration of R proportional to 1/R^2, which signifies that there is a singularity at R=0, that is to say, if you try to plug in R=0 you’ll get infinity. This would be the same as the infinity at the source for other forces proportional to 1/R^2, coulomb attraction or gravity. Ellis has this to say about the significance and existence of the FLRW singularity:

[T]he universe starts at a space-time singularity…This is not merely a start to matter—it is a start to space, to time, to physics itself. It is the most dramatic event in the history of the universe: it is the start of existence of everything. The underlying physical feature is the non-linear nature of the EFE (Einstein Field Equation): going back into the past, the more the universe contracts, the higher the active gravitational density, causing it to contract even more…a major conclusion is that a Hot Big Bang must have occurred; densities and temperatures must have risen at least to high enough energies that quantum fields were significant, at something like the GUT (Grand Unified Theory) energy. The universe must have reached those extreme temperatures and energies at which classical theory breaks down. [Emphasis in original.]

Ellis is saying that even though we can’t observe the universe at that time when it was so small and temperatures were so high that quantum properties would have been significant, we can infer that this was the case theoretically, that is to say that there was a “Hot Big Bang” at the beginning of the universe with extremely high temperatures (energies)and an extremely small volume.

Thus, given the contracting size of the universe as one goes back to the origin, there will be a time such that quantum effects must come into play. However, there are some basic limitations to using quantum mechanics as a theory for the origin of the universe. As Ellis points out:

The attempt to develop a fully adequate quantum gravity approach to cosmology is of course hampered by the lack of a fully adequate theory of quantum gravity, as well as by the problems at the foundation of quantum theory (the measurement problem, collapse of the wave function, etc.).

The Hawking-Penrose Theorems also show that a class of solutions to the General Relativity equations have a singularity in the solution. The Borde-Guth-Vilenkin Theorem shows that under conditions of universe average expansion, there is a beginning point. Since all such solutions are non-applicable at the singularity because quantum gravity enters the picture, the relevance of such theorems is perhaps questionable.

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1Editor’s note: Georges LeMaitre was a Catholic priest and professor of physics. “He was the first known academic to propose the theory of the expansion of the Universe, widely misattributed to Edwin Hubble.” See also Stigler’s law of eponymy.

Summary Against Modern Thought: The Last Movement

This may be proved in three ways. The first...

This may be proved in three ways. The first…

See the first post in this series for an explanation and guide of our tour of Summa Contra Gentiles.

Previous post.

Recall that the First Way has already been done. Meaning we already have one proof for God’s existence. This is the second, and the good news is that most of our terminology has already been introduced. A lot to do today, but the good news is that this is the last of Chapter 13! But these proofs are all we have. We don’t yet know anything about God except that He’s the Unmoved Mover, the Unchanging Changer. What can that mean? That’s coming.

Chapter 13: Arguments in proof of God’s existence

18 Again, if any two things are found accidentallyi united in a certain subject, and one of them is to be found without the other, it is probable that the latter can be found without the former: thus if white and musical are found in Socrates, and musical without white is found in Plato, it is probable that it is possible to find white without musical in some subject.

Accordingly if mover and moved be united together in some subject accidentally, and it be found that a certain thing is moved without its being a mover, it is probable that a mover is to be found that is not moved. Nor can one urge against this the case of two things one of which depends on the other; because those in question are united not per se but accidentally.

If, however, the aforesaid proposition is true in itself, again there follows something impossible or unfitting. For the mover must needs be moved either by the same kind of movement or by another kind. If by the same kind, it follows that whatever causes alteration must itself be altered, and furthermore that the healer must be healed, that the teacher must be taught, and in respect of the same science. But this is impossible: for the teacher must needs have science, while the learner must needs not have it, and thus the same will be both possessed and not possessed by the same, which is impossible.ii

And if it be moved by another kind of movement, so that, to wit, that which causes alteration be moved in respect of place, and that which moves in respect of place be increased, and so on, it will follow that we cannot go on indefinitely, since the genera and species of movement are finite in number.iii And thus there will be some first mover that is not moved by another. Unless, perchance, someone say that a recurrence takes place, in this way, that when all the genera and species of movement have been exhausted, a return must be made to the first; for instance, if that which moves in respect of place be altered, and that which causes alteration be increased, then again that which is increased be moved in respect of place. But the consequence of this will be the same as before; namely, that which moves by one kind of movement is itself moved by the same kind, not immediately indeed but mediately.iv It remains therefore that we must needs postulate some first mover that is not moved by anything outside itself.

19 Since however, given that there is a first mover that is not moved by anything outside itself,v it does not follow that it is absolutely immovable, Aristotle proceeds further, saying that this may happen in two ways. First, so that this first mover is absolutely immovable. And if this be granted, our point is established, namely that there is a first immovable mover. Secondly, that this first mover is moved by itself. And this seems probable: because what is of itself is always prior to what is of another: wherefore also in things moved, it is logical that what is moved first is moved by itself and not by another.

20 But, if this be granted, the same consequence follows.[18] For it cannot be said that the whole of that which moves itself is moved by its whole self, because then the absurd consequences mentioned above would follow, namely that a person might teach and be taught at the same time, and in like manner as to other kinds of movement; and again that a thing would be at the same time in act and in potentiality, since a mover, as such, is in act, while that which is moved is in potentiality.vi It remains, therefore, that one part thereof is mover only, and the other part moved. And thus we have the same conclusion as before, namely that there is something that moves and is itself immovable.

21 And it cannot be said that both parts are moved, so that one is moved by the other; nor that one part moves both itself and the other; nor that the whole moves a part; nor that part moves the whole, since the above absurdities would follow, namely that something would both move and be moved by the same kind of movement, and that it would be at the same time in potentiality and in act, and moreover that the whole would move itself not primarily but by reason of its part. It remains, therefore, that in that which moves itself, one part must be immovable, and must move the other part.

22 Since, however, in those things among us which move themselves, namely animals, the part which moves, namely the soul, though immovable of itself, is nevertheless moved accidentally, he goes on to show that in the first mover, the part which moves is not moved neither of itself nor accidentally.[19]vii

23 For in those things which among us move themselves, namely animals, since they are corruptible, the part which moves is moved accidentally. Now those corruptible things which move themselves must needs be reducible to some first self-mover that is everlasting. Therefore that which moves itself must have a mover, which is moved neither of itself nor accidentally.

24 It is clear that, in accordance with his hypothesis, some self-mover must be everlasting. For if, as he supposes, movement is everlasting, the production of these self-movers that are subject to generation and corruption must be everlasting. But no one of these self-movers, since it does not always exist, can be the cause of this everlastingness. Nor can all of them together, both because they would be infinite, and because they do not exist all together. It follows therefore that there must be an everlasting self-mover, that causes the everlastingness of generation in these lower self-movers. And thus its mover is not moved, neither of itself nor accidentally.

Again, we observe that in self-movers some begin to be moved anew on account of some movement whereby the animal is not moved by itself, for instance by the digestion of food or a change in the atmosphere: by which movement the mover that moves itself is moved accidentally. Whence we may gather that no self-mover, whose mover is moved per se or accidentally, is always moved. But the first self-mover is always in motion, else movement could not be everlasting, since every other movement is caused by the movement of the first self-mover. It follows therefore that the first self-mover is moved by a mover who is not moved, neither per se nor accidentally…viii

30 The Philosopher proceeds in a different way in 2 Metaph. to show that it is impossible to proceed to infinity in efficient causes, and that we must come to one first cause, and this we call God. This is how he proceeds. In all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate, whether the intermediate be one or several. Now if the cause be removed, that which it causes is removed. Therefore if we remove the first the intermediate cannot be a cause. But if we go on to infinity in efficient causes, no cause will be first. Therefore all the others which are intermediate will be removed. Now this is clearly false. Therefore we must suppose the existence of a first efficient cause: and this is God.ix

31 Another reason can be drawn from the words of Aristotle. For in 2 Metaph.[22] he shows that those things which excel as true excel as beings: and in 4 Metaph.[23] he shows that there is something supremely true, from the fact that we see that of two false things one is falser than the other, wherefore it follows that one also is truer than the other. Now this is by reason of approximation to that which is simply and supremely true. Wherefore we may further conclude that there is something that is supremely being. And this we call God.x

32 Another argument in support of this conclusion is adduced by Damascene[24] from the government of things: and the same reasoning is indicated by the Commentator in 2 Phys.[25] It runs as follows. It is impossible for contrary and discordant things to accord in one order always or frequently except by someone’s governance, whereby each and all are made to tend to a definite end. Now we see that in the world things of different natures accord in one order, not seldom and fortuitously, but always or for the most part. Therefore it follows that there is someone by whose providence the world is governed. And this we God.xi

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iA circle can be accidentally red or yellow and still be a circle, etc. A thing is not defined by its accidents.

iiAquinas created this argument in ignorance of our modern educational system, of course. But he’s still right.

iiiYou can change quantity, mass, color, place, and the like, but there are not an infinite number of changeable qualities.

ivDon’t loose sight of the main example, the stone moved by the stick moved by the arm moved by the muscles, etc., all in the here and now. This movement cannot be circularly caused.

vWe’re starting to hone in on (but have not reached) the idea that there can be only one Unmoved Mover, and not an assemblage of them. Not gods, but God.

viYou cannot potentially and actually be in Cleveland simultaneously. Aquinas is speaking ontologically, not epistemologically. And this reminds me that we have to, sometime soon, introduce Jaki’s criticisms of complementarity.

viiWe’ll also do this more later. But, for now, the soul of the animal is its form. Things are composed of forms and substances or matter. Clay, a substance, can be in the form of an ashtray (eek!) or (say) a model car. Speaking loosely, and later correctly, the soul animates the substance. The souls of plants are lesser than the souls of animals, which in turn are lesser than the souls of men. Like I said, this is only a crude introduction. Let’s stick here to the argument of movement, else we will be arguing that of which we have incomplete information.

viiiYou may feel buried after these last two paragraphs. Slow reading is in order. Again, Aquinas is leading us to why there can be only one Unmoved Mover. There’s no getting around that this last proof is rather dry—I even leave out some discussion about movement of heavenly bodies—but it dots all the Js and crosses all the Xs.

ixHere it is in a compact form! This is the elevator proof. Pretty, too.

xI don’t think Aquinas expected this snippet to be convincing in itself, but included it as an augment to main argument. Here’s a good article from Swinburne on the argument from the beauty of design, and another by Williams on beauty.

xiAnd this sketch is cruder still. It’s doubtful that it, in this telegraphic form, would convince any moderns. I’m not expert enough in Medieval history to know why Aquinas included it, which audience he had in mind.

Hooray! We are finally done with Chapter 13! But not entirely done with motion.

[18] 8 Phys., l.c.
[19] 8 Phys. vi.
[22] D. 1a. i. 5.
[23] D. 3. iv. 27, 28.
[24] De Fide Orth. i. 3.
[25] Text 75.

Readers: Pick Your Theme Song

It’s summer and everybody’s on vacation, except, it seems, Yours Truly. And whoever is left is over at Anthony’s place fighting the Kaya wars. (My side? Take a look at the guy who introduces Pielke Jr in this video. Would you believe any equation/identity introduced from a man who purposely spikes his hair? Fashion tip: Never do that.)

So with all this inactivity on our hands, it’s about time we settled on theme songs, which is to say, leitmotifs. A tune or snippet of a melody that, when heard, tells listeners Briggs has done it again.

Long ago, it had been suggested to me that this offering by Mac Davis would fit best:

But that was then. And since this is now a more humble and self-effacing self speaking to you, something softer and more enlightened is in order.

Note carefully: a theme song is not your favorite pop song. If it were, I might nominate Where Oh Where Are You Tonight? (apologies for not linking to a better version, but the rights holders keep having YouTube delete the good ones).

No, a theme song tells people who are you, what you are about, and what you are going to do. It should accomplish this task in as few as notes as possible.

Yet must brevity be all? The most memorable television theme songs used to last a full minute; now they are five to ten seconds. But that’s because production companies have to make a profit. Ideally, Yours Truly would, too, but we can’t have everything.

There is no end to this post. I don’t have a candidate replacement in mind. I’ll work on it.

Meanwhile, I’d like to hear your theme songs, especially those of regular contributors. If any are at their computers, that is.

Note that links to YouTube are sometimes quarantined by WordPress. Best to cut and paste the URL directly. If yours seems lost, don’t worry, I’ll un-quarantine it in time.

Disbelief In Free Will Causes Disbelief In Free Will

"I have no choice but to love thee, Theory."

“I have no choice but to love thee, Theory.”

Never was the West’s wholesale flight from philosophy and a classic education more evident than in the title of this peer-reviewed paper: Prosocial Benefits of Feeling Free: Disbelief in Free Will Increases Aggression and Reduces Helpfulness.

How could we have forgotten that it is impossible—not unlikely, impossible—to “disbelieve in free will”? Answer: scientism, the curious belief that science and only science is fit to answer all questions. In order to believe you must have free will because to believe is an act of will, and to believe in one proposition is to disbelieve in its contrary; therefore, in order to disbelieve you must have free will.

The paper appears in Personality and Social Psychology Bulletin and it’s by Roy Baumeister, E.J. Masicampo, and C. Nathan DeWall.

The abstract begins with the words, “Laypersons’ belief in free will may foster a sense of thoughtful reflection and willingness to exert energy, thereby promoting helpfulness and reducing aggression, and so disbelief in free will may make behavior more reliant on selfish, automatic impulses and therefore less socially desirable.”

Laypersons. Laypersons are those unfortunate souls who are not trained in the Ways of Science and who cling to superstitions like they are rational beings.

To paraphrase the abstract: since people believe they can make free choices, the free choices they make are better than the free choices they make when they disbelieve they can make free choices.

Preposterous isn’t in it. Yet there it is. And here is more. The opening words, worth paying attention to:

Belief in free will seems widespread and intuitive. Almost every person every day has the subjective impression of making a choice in which more than one outcome is possible. The most influential religious beliefs in Western culture give prominent emphasis to doctrines of free will, assuming that human individuals can freely choose whether to perform virtuous or sinful actions and even stating that eternal judgment of individual souls rests on the choices they make. Likewise, the legal system allocates guilt and punishment differentially based on whether the rule breaker could have acted differently such that perceived reductions in the capacity for free choice (including external pressures, lack of awareness, mental illness, or intense emotion) constitute valid reasons for reduced punishment or even acquittal.

We could spend a week on this. Almost every person? Phffag. Every person. And why pick on religion, why single out jurisprudence? I’ll tell you why, because these are the areas, religion and the common law, which intellectuals are most keen on dismantling. Now that’s a judgement of psychology and not philosophy, but it’s not made lightly. Here are the authors’ next words:

Intellectuals and scientists, however, seem rather less uniformly comfortable with the idea of free will than the general public. Many scientists regard the belief in free will as untenable if not downright absurd…Although not explicitly siding with them, Wegner (2002) summarized the opposition to free will as embodying the assumption that only “bad scientists” could believe such a thing.

Do the authors consider themselves good scientists?

How in the holy heckfire do you congratulate yourself for believing you cannot believe! The only possible answer is insanity. Scientists are driven mad by love of their pure and perfect theories. They have become Pygmalion.

Authors: “To be sure, the impossibility of free will cannot be proven either empirically or conceptually.”

The reason it cannot be proven impossible is that it exists. You cannot prove that which exists does not exist, though you might conduce somebody a few slices short of a loaf to believe that which exists does not—you might even convince them that what does not or cannot exist does, like bigfoot or Utopia.

The rest of the paper is given over to “experiments” where groups of college kiddies are exposed to the researchers’ particularities and then the kiddies fill out questionnaires. The questionnaires are given numerical answers and technologically sounding names. This is what makes it science. There are statistics and wee p-values. None of this is of the slightest interest.

The end the paper with this:

The broader implication is that many people in Western culture share a belief in human freedom of action and that, moreover, human society benefits from such a belief. (Indeed, we suspect that most cultures will have found beliefs in free will to be socially beneficial and hence will tend to favor and promote those beliefs.)

I despair, I despair.

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Thanks to Mangan (@Mangan150) where I first learned of this paper.

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