William M. Briggs

Statistician to the Stars!

Book Update — Uncertainty & Breaking The Law Of Averages


I’ll have the page proofs for Uncertainty mid week and I’ve until 10 June to turn them back in. (I begin teaching on the 13th.) This puts publication in early July. They say.

Probably a good guess, too, because I’m sure they want to have it available for the big statistical meeting held every August, this year in Chicago. One of its main features is a book fair. No, I won’t be there. Costs too much; besides, I’m no longer a member of any organization.

The Lord only knows what typos my enemies have placed in the draft. Besides them, and judging from my re-re-re-…-re-perusal of the manuscript, the corrections should be small stuff. Yet like I said yesterday, I’m already seeing places where I’d like to amplify arguments. I can also see one superfluous example, a distraction, I’d cut.

Also, the editor asked me to begin working on Chapter questions so that the book, in its triumphant second edition, can be used also a textbook. I do not want the book to swell in size to the point at which it is off-putting or intimidating, however.

The math is minimal. On purpose. Those who know the math already know it and don’t need to see more of it, and those who don’t wouldn’t be able to absorb a bunch of math and the new philosophy. And anyway, the philosophy is the point. There is nothing mathematically wrong with frequentism and Bayes, so there is no profit attempting to find mathematical shortcomings of those old philosophies.

Good math is beside the point. Once an equation, proved true to the satisfaction of every mathematician, is in hand, it does not—it most certainly does not—mean the application of the equation is what the mathematician says it is. Mathematicians must take a back seat to philosophers and engineers in deciding useful applications of their work. Your hand-crafted probability space—what a keen sigma field you have!—may sparkle and shine and be worth a theorem or two, but that it’s used in describing a p-value shows it has no practical value.

Law breaker

I received this nice note yesterday.

Dr. Briggs,

I must admit I am a fan of your work in the field of statistics. I myself am a recently graduated high school student who took AP statistics my senior year as I plan on majoring in Financial Engineering. However, I was led to your book “Breaking the Law of Averages” by a PhD in mathematics whom I am also grateful to call a close friend. He showed me your works and allowed me to decide my thoughts on frequentist statistics versus Bayesian statistics and I must admit I am now a convert.

However, while reading through your works I encountered a problem which you have no doubt heard feedback about before. I have been doing the homework that accompanies the book but the lack of answers makes it difficult for me to check my work. I run my answers by my friend who has the PhD in mathematics but I would ideally like your work to see how you solve your problems and to gain insight into your way of thinking (which I have grown accustomed to through reading your blog and listening to your podcasts as well).

I can only imagine how busy you must be and therefore I understand if you are not able to respond in a quick manner I just wanted to reach out to you for your insight and help with an aspiring statistician (to a degree I must concede).

Thank you for all your work and please keep posting, your work and philosophy truly does inspire and allows me to view the world from a new perspective that I appreciate and often agree with,



Yes, I’ve been asked lots about solutions to the book. I don’t have them. I wrote BLOA quickly for use as an Intro text. I’ve used it successfully for many years, and updated only a couple of times, because I’ve spent most of my time on Uncertainty.

Ideally, I’d go back to BLOA and cut out the material on R, which is no longer necessary since it’s all on-line, and I’d redo the questions; plus, I’d do some reorganizing to bring it in line with Uncertainty. Ideally.

Uncertainty is not an introductory book for entering college students.

Way I do it in class is have students go to the board, one by one, and lead the group in an answer. I only kibitz when they stray too far from the farm. I only assign reading for homework—but I’m always asking questions to make sure who knows what.

Point: I never wrote down the answers. And when I’ve gone back to try, I realize I have to redo the whole book. Thanks for the encouragement!

Quantum Mechanics, Potentiality, Ontology, Epistemology, & Probability


What is and what is known about what is are two concepts which are often entangled, especially in quantum mechanics (puns intentional). This is one of the subjects which I cover in my upcoming book, incidentally, but which I’d like to expand in a second edition. (I’m assuming it’ll be a best seller.)

Here’s a lump of clay. It exists; it is actual. But also existing, in a sense, is the potential for that lump to be an ashtray (attention young readers: the ashtray was a device to catch cigarette ashes. I miss people smoking.) The ashtray is in the lump; it exists in potentia.

Now at the very top is a being of pure act, which is to say, God; a being (being itself!) in which there is absolutely no potency, i.e. no potential to change (we covered this in our Summa Contra Gentiles series). At the bottom (I use these positional terms metaphorically) is prime matter, stuff which is pure potential and which contains no actuality; you can’t measure or collect a bucketful of it. Everything else (including you) is in between, containing a mixture of act and potency. Some objects are weighted more heavily towards act, others towards potency (but there is no claim to form a numerical measure of this “weight”).

All this is preliminary to follow the paper “Werner Heisenberg and Thomas Aquinas on Natural Indeterminism” in New Blackfriars by Ignacio Silva.

Ed Feser in his must-read Scholastic Metaphysics has collected apt quotations from Heisenberg in describing quantum reality. Here’s one (p. 126, from H’s Physics and Philosophy; ellipsis in quotation):

The probability wave of Bohr, Kramers, Slater…was a quantitative version of the old concept of “potentia” in Aristotelian philosophy. It introduced something standing in the middle between the idea of an event and the actual event, a strange kind of reality just in the middle between possibility and reality. (p. 15)

And, more to our point, this:

The probability function combines objective and subjective elements. It contains statements about possibilities or better tendencies (“potentia” in Aristotelian philosophy), and these statements are completely objective, they do not depend on any observer; and it contains statements about our knowledge of the system, which of course are subjective in so far as they may be different for different observers. (p. 27)

Regular readers recognize this is what I insist is the proper definition of “subjective” probability: people having different premises for the same proposition of interest. (The standard Bayesian view I reject because probability is not “feelings.”)

Finally to Silva, who spends most of his time showing Aquinas anticipated quantum mechanics, as it were, at least as far as causes go. He says, for instance, “Thomas explicitly rejects a rigid determinism in nature, i.e. the position that whenever there is a cause a certain determinate effect necessarily follows”, etc. The reasons for Aquinas advances for this are not quite quantum mechanical, however, but have more to do with unaccounted forces and the presence of multiple causes. Granting all this still does not tell us what actuality (God? our non-material intellects, at least sometimes?) reduces the potentia of a QM wave to actuality.

I can’t do the whole paper, but here’s Silva on what’s pertinent to us:

As we descend the degrees of being, the corresponding reduction in actuality correlates with an increase in potentiality, down to the forms of the elements, which are the closest to prime matter, pure potency…

Greater or lesser actuality comes from the participated esse, received by the essence. Essences which are closer to matter would be those that would have lesser actuality, thus, greater potentiality. The farther the substance is from pure actuality, the greater its potentiality, and thus the greater the possibility of an ineffective action. That is why Thomas says that there are three main spheres of action within reality: 1) that being which is only act [God], operates always without defect; 2) that which is only potency, pure matter, needs an act to actualise it; and 3) that which is a mixture of act and potency, every natural being, which acts perfectly most of the times…

We find, then, in every natural being a passive indetermination, which is essentially an imperfection or — more accurately and absolutely speaking – a lack of perfection in relation to the whole of being. According to the hierarchy of being postulated above, we can say that natural things, as they are farther from Pure Act, they participate less in act: they are less determinate.

Now there are some who argue the QM wave exists, i.e. it is ontic (see Ringbauer et al.). This existence is not specified, at least as far as I can tell, in terms of a mixture of actuality and potentiality; instead, the wave just exists. That means all the potential end states exist actually in some “superposition”. But then, since the wave-as-actuality “collapses” to one point, it must exist at least in potential to these end states. And it also exists in potentia in the sense the entire wave can exist potentially in other localities.

This is why it makes much more sense to go with Heisenberg (and Aquinas) and say the wave exists, like we do, as a mixture of actuality and potentiality. In what portions, I have no idea. A “pure” wave, interacting with nothing, would exist as pure potential, i.e. would be prime matter; though I doubt the QM wave is pure potential. But here is where my ignorance is most glaring. I don’t know enough to see what parts of the QM wave are actual and what part potential. Silva says “it does not sound very implausible to affirm that quantum mechanics is working and describing natural things which are great in potency and low in act.”

Silva also thinks QM waves “cannot be pure potency, as Heisenberg claimed, because they would be primer matter itself, which needs to exist under some kind of formality…For sub-atomic particles to be considered in potency, they need to be under a formal determination, and thus some degree of actuality”

Anyway, the main point (for us) is that our knowledge of the potential end states, which are not actual, is not the wave, even if we have full knowledge of the wave, but is a function of the wave (its “squared modulus”) and other premises. In other words—the exciting conclusion—probability is not actual, nor is it even a potential. It is fully epistemic. Probability doesn’t exist, even if wave functions do as full actualities. Silva (with his spellings):

In particular, matter in these quantum systems can take unpredictable forms, which are only predicted probabilistically by the wave-function included in the Schrödinger equation. Although matter is open to the reception of new forms, it cannot receive any form. The system described by the Schrödinger equation could only receive those forms included probabilistically in that equation.

In other words, not only does probability not exist, epistemically all probability is conditional on the premises fix, here by Schrödinger’s equation. (A QM wave for an electron is potentially, say, a particle and not potentially a Buick.)

Summary Against Modern Thought: Failed Arguments For The Eternity Of The World I

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.

I’m departing from the usual format for this and next week. In Aquinas’s next six chapters, the first three are arguments for the eternal existence of things; the following three are his refutations of the same. I’m doing one of the arguments today; rather, highlights from it. The A reference are those arguments in favor of things existing eternally from the point of view of God, and the B reference are Thomas’s refutations. I’m highlighting only the most interesting arguments on both sides, else this post would run to 4,000 words.

A REFERENCE: Chapter 32 Arguments of those who wish to prove the eternity of the world from God’s side of the question (alternate translation)

B REFERENCE: Chapter 35 Solution of the foregoing arguments, and first of those that were taken from the standpoint of God (alternate translation)

A Every agent that acts not always, is moved either per se or accidentally: per se, as fire which was not always burning, begins to burn, either because it is newly lit, or because it is newly transferred so as to be near the fuel:–accidentally, as the mover of an animal begins anew to move the animal with some movement made in its regard; either from within,–as an animal begins to be moved when it awakes after its digestion is complete,–or from without, as when there newly arise actions that lead to the beginning of a new action. Now God is not moved, neither per se nor accidentally, as we proved in the First Book. Therefore God always acts in the same way. But created things are established in being by His action. Therefore creatures always have been.

B For it does not follow that God is moved either per se or accidentally if His effect begin to be anew; as the first argument pretended. Because newness of effect may argue change of the agent in so far as it proves newness of action: since it is impossible for a new action to be in the agent, unless the latter be in some way moved, at least from inaction to action. But newness of effect does not prove newness of action in God, since His action is His essence, as we have proved above. Neither therefore can newness of effect argue change in God the agent.

A Again. The effect proceeds from the active cause by the latter’s action. But God’s action is eternal: else He would become an actual agent from being an agent in potentiality: and it would be necessary for Him to be reduced to actuality by some previous agent, which is impossible. Therefore the things created by God have been from eternity.

B And yet it does not follow, if the action of the first agent is eternal, that His effect is eternal, as the second argument inferred. For it has been shown above, that in producing things God acts voluntarily. Not, however, as though there were an intermediate action of His,–as in us the action of the motive power intervenes between the act of the will and the effect,–as we have proved in a foregoing chapter: but His act of understanding and willing must be His act of making.

Now the effect follows from the intellect and the will according to the determination of the intellect and the command of the will. And just as every other condition of the thing made is determined by the intellect, so is time appointed to it: for art determines not only that this thing is to be such and such, but that it is to be at this particular time, even as a physician determines that a draught is to be taken at such and such a time. Wherefore, if his willing were per se efficacious for producing the effect, the effect would follow anew from his former will, without any new action on his part. Therefore nothing prevents our saying that God’s action was from eternity, whereas His effect was not from eternity, but then when from eternity He appointed.

Notes Don’t forget “eternity” means outside of time, not existing on a time line indefinitely. Time is change; if everything were always actual and changeless, like God, there’d be no potentiality, and no time. So God exists necessarily and is outside of time, but His creations are in time, and so there can be a beginning of time. Some of A’s premises are true (the implied one about God not being in potential).

A Moreover. Given a sufficient cause, its effect must necessarily be granted. For if, given the cause, it were still unnecessary to grant its effect, it would be therefore possible that, given the cause, the effect would be or not be. Therefore the sequence of the effect to its cause would only be possible: and what is possible, requires something to reduce it to actuality. Hence it will be necessary to suppose some cause whereby it comes about that the effect is made actual, and thus the first cause was not sufficient. But God is the sufficient cause of creatures being produced: else He would not be a cause; rather would He be in potentiality to a cause: since He would become a cause by the addition of something: which is impossible. Therefore it would seem necessary, since God is from eternity, that the creature was also from eternity.

B Hence it is also clear that, although God is the sufficient cause of bringing things into being, it is not necessary to suppose that because he is eternal His effect is eternal; as the third argument contended. For if we suppose a sufficient cause, we suppose its effect, but not an effect outside the cause: for this would be through insufficiency of the cause, as if for instance a hot thing failed to give heat. Now the proper effect of the will is for that thing to be which the will wills: and if something else were to be than what the will wills, this would be an effect that is not proper to the cause but foreign thereto. But just as the will, as we have said, wills this thing to be such and such, so does it will it to be at such and such a time.

Wherefore, for the will to be a sufficient cause, it is not necessary for the effect to be when the will is, but when the will has appointed the effect to be. On the other hand, it is different with things which proceed from a cause acting naturally: because the action of nature is according as nature is; wherefore the effect must necessarily follow if the cause exist. Whereas the will acts, not according to the mode of its being, but according to the mode of its purpose. And consequently, just as the effect of a natural agent follows the being of the agent, so the effect of a voluntary agent follows the mode of his purpose.

Notes A is a clever counter-argument and it ought to be studied for its form, to show how easy it is to be confused when discussing infinities/eternities and chains of causes, and in particular the tacit arguments about the primary and all secondary causes. Now just how (and why) God, who is timeless, causes things to be in time I have no idea.

I left off four other arguments, which you can and should read. I don’t include them because I think the point is already well made, and because next week we have another group of similar arguments and rebuttals (from a different perspective).

Think You Can Simulate A Brain? Think Again

A technician communicates with the future futurist Ray Kurzweil.

A technician communicates with the future futurist Ray Kurzweil.

Since is Silly Saturday, a few fun back-of-the-envelope calculations on simulating a brain. I’m drawing from the marvelous, must-read (go do it now) essay “The empty brain: Your brain does not process information, retrieve knowledge or store memories. In short: your brain is not a computer” by Robert Epstein.

(Update: About what I mean by “simulating”, see the exchange with DAV below.)

Lots to mine from this article, many fascinating implications, which we’ll come back to in the future. For now, what about the idea that we can “simulate” a brain in the Ray Kurzweil sense of being able to “download” a man onto a chip. Can we quantify the scope of the problem?

Think how difficult this problem is. To understand even the basics of how the brain maintains the human intellect, we might need to know not just the current state of all 86 billion neurons and their 100 trillion interconnections, not just the varying strengths with which they are connected, and not just the states of more than 1,000 proteins that exist at each connection point, but how the moment-to-moment activity of the brain contributes to the integrity of the system. Add to this the uniqueness of each brain, brought about in part because of the uniqueness of each person’s life history, and Kandel’s prediction starts to sound overly optimistic.

Okay, that’s 100 trillion interconnections, or 1013, times the number of active proteins (103) at each connection, and we’re at 1016 degrees of freedom at a minimum. For each “moment” of action. (The basic “step time” unit is microseconds or smaller.)

And this is only in the brain itself, and doesn’t include the rest of the nervous system (which, in our metaphor, makes it sound like a separate entity) and it’s connections. Then add That Which We Do Not Yet Know about workings we should be modeling but aren’t, and can’t because, by definition, we don’t know what they are, and we’re probably at 1020 (see inter aliaBlood exerts a powerful influence on the brain“). At the least. I’m only guessing. You can make your own guess. I’m doing all this on one cup of coffee. Mistakes will be made.

All this is happening in three-dimensions. Proteins move. Chemicals swap electrons at the connections between synapses and between nerve cells and other cells in the body. And so on. This adds several more orders of magnitude. A wild guess here, which I’m happy to disdain upon cogent criticism, but I’d say, for fun, about 1,000 degrees per protein, though maybe up to a million. We’re up to 1023~1026. And this on on the low end. Think of it as A Very Best Case Scenario.

Now computers are (at this point still) made of transistors. How many transistors does it take to model the actions of one protein? Well, an Intell Quad-core + GPU Core i7, an everyday processor, has 1.4 x 10 9, and this is enough to do one protein. Not very speedily, but it can do it with some power left over. Is one i7 enough to do two proteins interacting? I’m not an expert.

What we’re after is the number of processors it takes to simulate those 1023+ degrees of freedom. Say a billion for each degree of freedom. That puts us in need of 1031+ transistors, at a raw minimum, to fully simulate the organism which is a brain (and its connections). This simulate ignores vast areas of a human being, of course. But let’s pretend those areas don’t matter.

I’ve mixed up the time component in there surreptitiously by speaking of modeling a protein. Probably this is another underestimation. Probably a laughable underestimation. It must be, because those proteins are made of finer stuff, all of which has to be taken into account. I wouldn’t be shocked if 10100 (or more) is the right answer.

If we believe Moore’s “Law”, the number of transistors on a chip doubles every two years. We have a billion now and want to arrive at 1031+. I make it something less than a century (73 years). Maybe less if “quantum” computers fulfill any of their promises, maybe more depending on how badly I’ve botched the above calculations. All assuming Moore doesn’t break down and become logarithmic, which every single innovation in human history has done.

(Incidentally, I’m betting Moore lasts only twenty years more, or even fewer, before the vigor is gone.)

Of course, that’s on one “chip”. We can string processors together and reach the goal faster. Right now we’d need 1022 i7 computers linked up. That’s a big number.

So, plus or minus, it’s a century from now (or even two or three) and we might have the computer muscle, and the intelligence enough to figure out how to program such a monstrosity. No small thing, either, because many of the interactions we’ll have to model are quantum mechanical, and nobody in the world has any idea—as in NONE, even if the computers are quantum computers—of what actualizes potential states of QM objects. Are these potentia actualized one-by-one? Or is there coordination, which is to say a sort of entanglement, between some, a few, all of the elements? Not only do we not know this, I think we cannot know this.

Anyway, forget the insurmountable difficulty. We’ve got the thing. We switch it on and…

It still won’t work. “Brains”, which is to say the organisms (in their entirety) which are us, are not solely material. Our intellects are not just the physical stuff which makes us up. We are more than animated dust.

This sad finding destroys some science fictional concepts, but it invites new ones.

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