William M. Briggs

Statistician to the Stars!

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The True Meaning Of Statistical Models

It's catching.

It’s catching. (Image source.)

This came up yesterday (again, as it does often), so I figure one more stab is in order. Because the answer isn’t simple, I had to write a lot, which means it won’t get read, which means I’ll have to write about it again in the future.

Trust your eyes

You’re a doctor (your mother is proud) and have invented a new pill, profitizol, said to cure the screaming willies. You give this pill to 100 volunteer sufferers, and to another 100 you give an identically looking placebo.

Here are the facts, doc: 71 folks in the profiterol group got better, whereas only 60 in the placebo group did.

Now here is what I swear is not a trick question. If you can answer it, you’ll have grasped the true essence of statistical modeling. In what group were there a greater proportion of recoverers?

This is the same question that was asked yesterday, but with respect to the global temperature values. Once we decided what was meant by a “trend”—itself no easy task—the question was: Was there a trend?

May I have a drum roll, please! The answer to today’s question is—isn’t the tension unbearable?—more people in the profitizol group got better. The answer to yesterday’s question was (accepting the definition of trend therein): no.

These answers cause tremendous angst, because people figure it can’t be that easy. It doesn’t sound sciency enough. Well, it is that easy. You can go on national television and trumpet to the world the indisputable inarguable obvious absolute truths that more people in the drug group got better, and that (given our definition of trend) there hasn’t been a trend these twenty years.

Question two: what caused the difference in observed recovery rates? And what caused the temperature to do what it did?

My answer for both: I don’t know. But I do know that some thing or things caused each person in each group to get better or not. And I know that some thing or things caused temperature to take the values it did. I also know that “chance” or “randomness” weren’t the causes. They can’t be, because they are measures of ignorance and not physical objects. Lack of an allele of a certain gene can cause non-recovery, and the sun can cause the temperature to increase, but “chance” is without any power whatsoever.

Results are never due to chance, they are due to real causes, which we may or may not know.

The IPCC claims to know why temperature did what it did. We know the IPCC is wrong, because their model predicted things which did not happen. That means the causes it identified are wrong in some way, either by omission or commission. That’s for them to figure out.

Clever readers will have noticed that, thus far, there was no need for statistical models. But if our goal was only to make the statement which group got better at greater rates or if there was a trend, no model was needed. Why substitute perfectly good reality with a model? That is to commit the Deadly Sin of Reification (alas, an all too common failing).

Enter the models

The classically trained (Bayesian or frequentist) statistician will still want to model, because that is what statisticians do. In the drug trial they will invent for themselves a “null hypothesis”, which is the proposition, “Profitizol and the placebo cause the exact same biological effects”, which they ask us to “accept” or “reject”.

That means, in each patient, profitizol or a placebo would do the same exact thing, i.e. interact with the relevant biological pathways associated with the screaming willies such that no measurement on any system would reveal any difference. But given you are a doctor, aware of biochemistry, genetics, and the various biological manifestations of the screaming willies, it is highly unlikely this “null” proposition holds. Indeed, to insist it does is to abandon or willfully ignore all this knowledge and cast all your attention on only that which can be quantified (the Sin of Scientism).

Of course, you might have made a mistake and created a substance which was (relative to the SW) identical with the placebo. Mistakes happen. How do we tell? Do we have any evidence that profitizol works? That’s the real question, the question everybody wants to know. Well, what does “works” mean?

Uh oh. Now we’re into causality. If by “works” we mean, “Every patient that eats profitizol is cured of the SW” then profitizol does not work, because why? Because not every patient got better. If by “works” we mean, “Some patients that eat profitizol are cured of the SW” then profitizol works, and so does the placebo, because, of course, some patients who ate the drug got better. Defining properly what “works” is not an easy job, as this series of essays on a famous statistical experiment proves. Here we’re stuck with the mixed evidence that patients in both groups got better. Clearly, something other than just interacting with a drug or placebo is going on.

What to do?

Remember the old saw about how the sale of ice cream cones was “correlated” with drownings? Everybody loves to cite—and to scoff at—this example because it is obviously missing any direct causal connection. But it’s a perfectly valid statistical model. Why?

Because a statistical model is only interested in quantifying the uncertainty in some observable, given clearly stated evidence. Thus if we know that ice creams sales are up, it’s a good bet that drownings will rise. We haven’t said why, but this model makes good predictions! (I’m hand-waving, but you (had better) get the idea.)

Statistical models do not say anything about causality. We’re not learning why people are drowning, or why people are getting better on profitizol, or why the temperature is doing what it’s doing. We are instead quantifying our uncertainty given changes in certain conditions—and that is it.

If we knew all about the causes of a thing, we would not need statistics. We would feed the initial and observed conditions into our causal model, and out would pop what would happen. If we don’t know the causes, or can’t learn them, but still want to quantify uncertainty, we can use a statistical model. But it’s always a mistake to infer (without error; logically infer) causality because some statistical model passes some arbitrary test about the already observed data. The ice cream-drowning model (we assume) would pass the standard tests. But there is no causality.

Penultimate fact: To any given set of data, any number of statistical or causal or combination models can be fit, any number of which fit that observed data arbitrarily well. I can have a model and you can have a rival one, both which “fit” the data. How do we tell which model is better?

Last fact: Every model (causal or statistical or combination) implies (logically implies) a prediction. Since models say what values, or with what probability what values, some observable will take given some conditions, all we do is supply those conditions which indicate new circumstances (usually the future)—voilà! A prediction!

It’s true most people who use statistical models have no idea of this implication (they were likely not taught it). Still, it is true, and even obvious once you give it some thought.

Not knowing this implication is why so many statistical models are meager, petty things. At least the IPCC stuck around and waited to see whether the model they proposed worked. Most users of statistics are content to fit their model to data, announce measures of that fit (and since any number of models will fit as well, this is dull information), and then they run away winking and nudging about the causality which is “obvious.”

Not recognizing this is why we are going through our “reproducibility crisis”, which, you will notice, hits just those fields which rely primarily on statistics.

The IPCC’s And McKitrick’s “Hiatus” Time Series Models

Several readers asked me to look at Ross McKitrick’s paper “HAC-Robust Measurement of the Duration of a Trendless Subsample in a Global Climate Time Series”, which is receiving the usual internet peer-reviewing (here, here, and here).

Before we begin, it is absolutely crucial that you understand the following point: both the IPCC (you know I mean the people and groups which contribute to it) and McKitrick have produced time series models.

Many people and groups have created time series models of the temperature, including the rank amateurs who attended the People’s Climate March. The latter model is, in essence, “The End Is Nigh”. This is simplistic, yes, and stupid certainly, but it is still a time series model.

Now we know, without error, that the IPCC’s time series model stinks. That it should not be trusted. That decisions should not be made based on its forecasts. That it is, somewhere, in error.

How do we know this? Because it has consistently and for many, many years said temperatures would be high when in reality they were low (relative to the predictions). People who refuse to see this are reality deniers.

Because the IPCC’s model said temperatures would be high these past eighteen or so years, when in reality the temperature bounced around but did nothing special, the IPCC has taken to calling reality a “pause” or “hiatus”. Everybody must understand that this “hiatus” is model-relative. It has nothing to do with reality. Reality doesn’t know squat about the IPCC’s model. The reality versus the model-relative “hiatus” is how we know the IPCC’s model stinks.

If the IPCC’s model did not stink, it would have predicted the reality we saw. It did not predict it. Therefore the model stinks. The debate really is over.

Now where the IPCC’s model goes wrong is a mystery. Could be it represents deep ocean circulation badly; could be that cloud parameterizations are poor. Could be a combination of things. It’s not our job to figure that out. The burden is solely on the IPCC to identify and fix what’s busted.

Enter McKitrick, who has his own model (or models; but for shorthand, I’ll speak of one). McKitrick’s model is a standard econometric model, which uses the Dickey-Fuller test (economists are always using the Dickey-Fuller test; I just like to say, “Dickey-Fuller test”; try it).

Is McKitrick’s model any good? There is no reason to think so. (Sorry, Ross.) It’s just a simplistic set of equations which is scarcely likely to capture the complexity of the atmosphere. If McKitrick’s model should be trusted, there is one test it could take to prove it. The same test the IPCC took—and failed.

McKitrick needs to use his creation to predict data he has never before seen. He hasn’t done that; and in fairness, he hasn’t had time. We need to wait a decade or so to see whether his model’s predictions have skill. But in a decade, I predict nobody will care.

The objection will be raised: but McKitrick’s model was built not to make predictions but to measure how long the “hiatus” was.

We needed a model for that? No, sir. We did not. We could just use our eyes. We need no model of any kind. We just take reality as she comes. To show you how easy it is to fool yourself with time series, here’s Figure 1 from McKitrick’s paper:

McKitrick's Figure 1.

McKitrick’s Figure 1.

It shows “Globally-averaged HadCRUT4 surface temperature anomalies, January 1850 to April 2014. Dark line is lowess smoothed with bandwidth parameter = 0.09.” Let’s don’t argue about the dots, i.e. the temperature, a.k.a. reality, which really should have accompanying error bounds. Let’s just assume that the dots were the reality, full stop.

The black line is a chimera, a distraction, put there to fool the eye into believing the author has discovered some underlying “signal” in the reality. Well, he might have done. But if he has, he should be able pass the reality test mentioned above. Unfortunately, you can’t make forecasts with that kind of black line. The black line is not what happened! To say it is is to commit the Deadly Sin of Reification.

We must take reality as she is. All we need is a working definition of trend. Easy, right? No, sir. Not really. See this post. But skip all that and call a trend, “Over any ten year period, the temperature increased more than it decreased.” That’s one possible definition of trend.

Accepting that definition (but feel free to make up your own, using the post as a guide), there is no trend in the last two decades. But then there are many other periods sine 1850 without trends. So maybe bump up the time window to 20 years. Still no trend in the latter years.

And so on. No model is needed. None. We just look. There is no need for “statistical significance”, or any other pseudo-quantification.

Listen: make sure you get this. It doesn’t even matter if the IPCC or McKitrick perfectly predicted reality. We still do not need their models to see whether there was a trend. A trend only depends on (1) its definition, and (2) reality.

Update Ken below discovered this gem, which shows Richard Feynman destroying the IPCC’s global warming models.

Update See this on the true meaning of statistical models.

The Rise Of Bayes

The man himself.

Thanks to reader Frank Kristeller we learn that the far left New York Times yesterday ran an article by F.D. Flam praising the rise of Bayesian statistics: The Odds, Continually Updated.

The replacement of frequentist statistics is, if true, moderately cheering news. And Bayes is the next step in the removal of magical and loose thinking from statistics. But far from the destination. That, I argue, is logical probability, which you can think of as Bayes sans scientism and subjectivism.

However, baby steps:

Bayesian statistics are rippling through everything from physics to cancer research, ecology to psychology. Enthusiasts say they are allowing scientists to solve problems that would have been considered impossible just 20 years ago. And lately, they have been thrust into an intense debate over the reliability of research results.

Nothing like a little hyperbole, eh? I don’t think our frequentist friends would agree they couldn’t solve the same problems as Bayesians. And of course they can. But so can storefront psychics solve problems. What we’re after is good solutions.

Flam got this right:

But the current debate is about how scientists turn data into knowledge, evidence and predictions. Concern has been growing in recent years that some fields are not doing a very good job at this sort of inference. In 2012, for example, a team at the biotech company Amgen announced that they’d analyzed 53 cancer studies and found it could not replicate 47 of them.

This is what happens when you base your decisions on p-values, little mystical numbers which remove the responsibility of thinking. P-values aren’t the only scourge, of course, willful transgressive thinking (especially in fields like sociology) and false quantification are just as, and probably even more, degrading.

False quantification? That’s when numbers are put to non-numerical things, just so statistics can have a go at them. Express your agreement with that statement on a Likert scale from 1 to 5.

Again:

“Statistics sounds like this dry, technical subject, but it draws on deep philosophical debates about the nature of reality,” said the Princeton University astrophysicist Edwin Turner, who has witnessed a widespread conversion to Bayesian thinking in his field over the last 15 years.

This is true. But just try to get people to believe it! Most academics, even their Bayesian variety, feel the foundations are fixed, that most or all that need be known about our primary premises is already known. Not true. Philosophy in a statistician’s education is put last, if at all. The error here is to assume probability is only a branch of mathematics.

One downside of Bayesian statistics is that it requires prior information — and often scientists need to start with a guess or estimate. Assigning numbers to subjective judgments is “like fingernails on a chalkboard,” said physicist Kyle Cranmer, who helped develop a frequentist technique to identify the latest new subatomic particle — the Higgs boson.

This isn’t really so. The problem here is blind parameterization, which is the assigning of probability models for the sake of convenience without understanding where the parameters of those models arise. This is an area of research that most statisticians are completely unaware of, so used are they to taking the parameters as a given. Logical probability removes the subjectivism and arbitrary quantification here, so that the true state of knowledge at the beginning of a problem is optimally stated.

Others say that in confronting the so-called replication crisis, the best cure for misleading findings is not Bayesian statistics, but good frequentist ones. It was frequentist statistics that allowed people to uncover all the problems with irreproducible research in the first place, said Deborah Mayo, a philosopher of science at Virginia Tech. The technique was developed to distinguish real effects from chance, and to prevent scientists from fooling themselves.

Mayo (our friend) is wrong. It was the discordance between scientists’ commonsensical knowledge of causality and the official statistical results that allowed us to see the mistakes. Statisticians do causality very, very badly. Indeed, frequentism is based on a fallacy of mixing up ontology (what is) with epistemology (our knowledge of what might be). Bayes does slightly better, but errs but introducing arbitrary subjective opinion.

Uri Simonsohn…exposed common statistical shenanigans in his field — logical leaps, unjustified conclusions, and various forms of unconscious and conscious cheating.

He said he had looked into Bayesian statistics and concluded that if people misused or misunderstood one system, they would do just as badly with the other. Bayesian statistics, in short, can’t save us from bad science.

Simonsohn (whom I don’t know) is right, mostly. The problems are deep. But you notice he left out p-values.

Flam missed that resistance to Bayes is still strong in many traditional fields, like medicine, where p-values are demanded. Still, that Bayes is becoming more available is good. But since we’re at the start and let’s try and do it right, and not, say, re-introduce old notions (like p-values!) into new theory.

A Wandering Mind Is An Unhappy Mind?

ransom8

The gentleman who runs Shadow To Light asked me to take a look at a paper which Sam Harris approvingly quoted. The 2010 peer-reviewed one-page paper shares today’s title (sans question mark) and was written by Matthew A. Killingsworth and Daniel T. Gilbert, appearing in Science.

The pair are from Harvard which allowed them, it appears, to garner national attention for their project, which asked people to log onto the website TrackYourHappiness.org. The website boasts itself as “a new scientific research project that investigates what makes life worth living.”

Which is an immediate failure in the narrow sense that science must remain forever mute on what makes life worth living. That is the task of religion, philosophy, literature, and other arts. Saying science can tell us means the billions of people who lived before (say) 1500 had no clue why they were happy or sad. But never mind.

The 5,000 or so participants had to have a (surprise) iPhone, which was in part given over to alerting holders, via text message or email, from 1 to 3 times a day, to answer several questions, including in what activity were they engaging, whether their “minds” were “wandering”, and how numerically happy they were.

How long after receiving these messages it took participants to answer I couldn’t discover—perhaps their minds were wandering when they were received?—but since some people reported engaging in sexual intercourse and others in driving, it might have been appreciable. But perhaps iPhone users are more dedicated to their hand machines than I suspect?

Anyway, everybody was contacted from between 1 and 39, average 8, times; compliance was about 83%, meaning not everybody responded to every message.

The authors say, “Unlike other animals, human beings spend a lot of time thinking about what is not going on around them,” which is true. That is because humans are rational beings, which implies having wander-capable minds, and other animals do not. Yet somehow from this the authors conclude:

Although this ability is a remarkable evolutionary achievement that allows people to learn, reason, and plan, it may have an emotional cost. Many philosophical and religious traditions teach that happiness is to be found by living in the moment, and practitioners are trained to resist mind wandering and “to be here now.” These traditions suggest that a wandering mind is an unhappy mind. Are they right?

Living in the moment? If your mind is always “in the moment”, how does it escape into the next moment and have more than one thought? Skip it.

The authors claim to to have “solved” the problem of sampling people’s thoughts with their iPhone app. And that was to ask participants “How are you feeling right now?” (from 0 to 100) and “Are you thinking about something other than what you’re currently doing?” Such as letting the traffic before you dissolve and instead think about designing an internet survey? Or not paying attention to the television commercial (several participants claimed to be watching TV) and thinking about something more pleasant?

Now comes the wee p-values. “[M]ultilevel regression revealed that people were less happy when their minds were wandering than when they were not”, confirmed, as said, by a wee p-value. Further, and in the category of Who Knew?: “people’s minds were more likely to wander to pleasant topics (42.5% of samples) than to unpleasant topics (26.5% of samples) or neutral topics (31% of samples)”.

But wasn’t it just the case that when faced with dull or familiar topics, participants’ minds would wander? And wouldn’t whether their minds wandered into happy or sad places depend on the (unsampled) nature of what urgent matters were pressing down on participants’ minds, and don’t negative matters cause us to consider them more urgently than positive ones?

They say no. “[T]ime-lag analyses strongly suggested that mind wandering in our sample was generally the cause, and not merely the consequence, of unhappiness.” Time-lag analysis? In supplementary material, they say: “We used multilevel regression to determine whether there was a relationship between happiness in given sample (T) and mind-wandering in the previous sample (T-1) and/or the next sample (T+1).” The conclusion of which was

…we found a strong negative relationship between mind-wandering at T-1 and happiness at T, but no relationship between mind-wandering at T+1 and happiness at T. In other words, a person’s happiness was strongly related to whether they had been mind-wandering in the previous sample, but was unrelated to whether they were mind-wandering in the next sample.

This is a silly statistical procedure, of course. The times were not constant, the things thought about where not constant or controlled for, and then consider some samples came from previous days. And also that quantifying happiness on a numerical scale, as often as it is done, is absurd. Is my “50” the same as yours? Is my “50” the same as my “50” yesterday?

The problem with this study is the same as that with “big data”, incidentally. The ability to collect massive amounts of “data”, most of it highly suspect, does not bring about an increase in intelligence.

Update On religions not wanting your mind to wander, listen to this speech by Peter Kreeft, starting about here.

Summary Against Modern Thought: God Is Not A Body. Part III

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. All posts are under the category SAMT.

Previous post.

This is the last part proving God is not a body (more proofs, that is; we have already had several), a proposition which in unfamiliar to moderns and therefore not under much dispute, except for the steady stream of demiurges put forth by modern atheists as misconceptions of who or what God is. Yet I see that we’re growing weary of this subtopic, and so we’ll finish it today, and in a circumscribed fashion. I’ll also keep my footnotes to a minimum. Don’t forget we already know the Unmoved Mover, God, is outside time, i.e. is eternal. There are a lot of infinities involving God, and today we meet some of them. Next week, we start on a new and essential topic, that God is His own essence.

Chapter 20: That God is not a body

13That the power of the first mover is infinite is proved thus. No finite power can cause movement in an infinite time. Now the power of the first mover causes movement in an infinite time, since the first movement is eternal. Therefore the power of the first mover is infinite.i

The first proposition is proved thus. If any finite power of a body causes movement in infinite time, a part of that body having a part of that power, will cause movement during less time, since the greater power a thing has, for so much the longer time will it be able to continue a movement, and thus the aforesaid part will cause movement in finite time, and a greater part will be able to cause movement during more time. And thus always according as we increase the power of the mover, we increase the time in the same proportion. But if this increase be made a certain number of times we shall come to the quantity of the whole or even go beyond it. Therefore the increase also on the part of the time will reach the quantity of time wherein the whole causes movement. And yet the time wherein the whole causes movement was supposed to be infinite. Consequently a finite time will measure an infinite time: which is impossible…ii

16The second objection is that, although a body be divided, it is possible for a power of a body not to be divided when the body is divided, thus the rational soul is not divided when the body is divided.iii

17To this we reply that by the above argument it is not proved that God is not united to the body as the rational soul is united to the human body, but that He is not a power residing in a body, as a material power which is divided when the body is divided. Wherefore it is also said of the human intellect that it is neither a body nor a power in a body.[11] That God is not united to the body as its soul, is another question.[12]

18The third objection is that if the power of every body is finite, as is proved in the above process; and if a finite power cannot make its effect to endure an infinite time; it will follow that no body can endure an infinite time: and consequently that a heavenly body will be necessarily corrupted. Some reply to this that a heavenly body in respect of its own power is defectible, but acquires everlastingness from another that has infinite power. Apparently Plato approves of this solution, for he represents God as speaking of the heavenly bodies as follows: By your nature ye are corruptible, but by My will incorruptible, because My will is greater than your necessity.[13]iv

19But the Commentator refutes this solution in 11 Metaph. For it is impossible, according to him, that what in itself may possibly not be, should acquire everlastingness of being from another: since it would follow that the corruptible is changed into incorruptibility; and this, in his opinion, is impossible. Wherefore he replies after this fashion: that in a heavenly body whatever power there is, is finite, and yet it does not follow that it has all power; for, according to Aristotle (8 Metaph.)[14] the potentiality to (be) somewhere is in a heavenly body, but not the potentiality to be. And thus it does not follow that it has a potentiality to not-be.

It must be observed, however, that this reply of the Commentator is insufficient.v Because, although it be granted that in a heavenly body there is no quasi-potentiality to be, which potentiality is that of matter, there is nevertheless in it a quasi-active potentiality, which is the power of being: since Aristotle says explicitly in 1 Coeli et Mundi,[15] that the heaven has the power to be always. Hence it is better to reply that since power implies relation to act, we should judge of power according to the mode of the act. Now movement by its very nature has quantity and extension, wherefore its infinite duration requires that the moving power should be infinite. On the other hand being has no quantitative extension, especially in a thing whose being is invariable, such as the heaven. Hence it does not follow that the power of being a finite body is infinite though its duration be infinite: because it matters not whether that power make a thing to last for an instant or for an infinite time, since that invariable being is not affected by time except accidentally…

28Again. No movement that tends towards an end which passes from potentiality to actuality, can be perpetual: since, when it arrives at actuality, the movement ceases. If therefore the first movement is perpetual, it must be towards an end which is always and in every way actual. Now such is neither a body nor a power residing in a body; because these are all movable either per se or accidentally. Therefore the end of the first movement is not a body nor a power residing in a body. Now the end of the first movement is the first mover, which moves as the object of desire:[21] and that is God. Therefore God is neither a body nor a power residing in a body…vi

31Hereby is refuted the error of the early natural philosophers,[23] who admitted none but material causes, such as fire, water and the like, and consequently asserted that the first principles of things were bodies, and called them gods. Among these also there were some who held that the causes of movement were sympathy and antipathy: and these again are refuted by the above arguments. For since according to them sympathy and antipathy are in bodies, it would follow that the first principles of movement are forces residing in a body. They also asserted that God was composed of the four elements and sympathy: from which we gather that they held God to be a heavenly body. Among the ancients Anaxagoras alone came near to the truth, since he affirmed that all things are moved by an intellect.

32By this truth, moreover, those heathens are refuted who maintained that the very elements of the world, and the forces residing in them, are gods; for instance the sun, moon, earth, water and so forth, being led astray by the errors of the philosophers mentioned above.vii

——————————————————-

i[Comment updated to fix stupid typo.] Don’t forget to review what these terms mean. The first movement is not some movement that caused the universe to start on its way in the dim dark past. It is the movement that starts all other movements, and we have seen that it must take no time. Again, I ask you to review Chapter 13. And Chapter 17, which proves God is not made of matter, and Chapter 15, which proves God is eternal. These are all premises here.

iiThis is not as bad as it looks when you first scan it. Read it. Two objections answered about conditionals and divided bodies are skipped.

iiiThe first time we hear the soul is immaterial! See also the next point, at that word “united.” This is only a hint of what is to come in other books of STG. Book One is all God all the time. Do not become a Descartesian over this one small word.

iv“For the sword outwears its sheath…And the soul wears out the breast.”

vSo much for slavishly following his predecessor!

viThis is pretty, but you must have Chapter 13 assimilated before you understand what he’s talking about. For instance, movement is actualization of a potential, and some actual power must actualize the potential. Potentials are powerless. (There’s a new business slogan for you.)

viiThe two-paragraph passage has interest in itself, as philosophical history, but is also proof that progress can be and is had in philosophy and theology, just like in science. Decay and distraction, again just in like science, also happen. Theology is thus, in the same sense as science, self correcting.

[11] Cf. Bk. II., ch. lvi.
[12] Cf. Ch. xxvii.
[13] Timaeus xli.
[14] D. 7, iv. 6.
[15] Ch. iii. 4; xii. 3.
[16] See above: But the Commentator…p. 46.
[17] 3, iv. 11; 6, ii. 8.
[18] Averroës, 12 Metaph. t. c. 41.
[19] See above: To this we reply…p. 46.
[20] Ch. vii. seqq.
[21] Cf. ch. xiii.: Since, however,…p. 31.
[22] Bk. IV., ch. xcvii.
[23] Cf. 1 Phys. ii.

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