William M. Briggs

Statistician to the Stars!

Uncertainty: The Soul of Models, Probability & Statistics. Chapter Abstracts


The book is chugging along in production. Still the same anticipated publishing date. It has acquired a subtitle, as you see. I was asked to provide short Chapter abstracts, which will only be used on-line for those curious about the book. I give them here as a teaser.

Chapter 1 :: Truth, Argument, Realism

Truth exists and we can know it. The universe (all there is) also exists and we can know it. Further, universals exist and we can know these, too. Any skepticism about truth, reality, or universals is self-refuting. There are two kinds of truth: ontological and epistemological, comprising existence and our understanding of existence. Tremendous disservice has been done by ignoring this distinction. There are two modes of truth: necessary and local or conditional. A necessary truth is proposition that is so based on a chain of reasoning from indubitable axioms or sense impressions. A local truth, and most truths are local, is so based on a set of premises assumed or believed true. From this seemingly trivial observation, everything flows, and is why so-called Gettier problems and the like aren’t problems after all. Science is incapable of answering questions about itself; the belief that it can is called scientism. Faith, belief, and knowledge are differentiated.

Chapter 2 :: Logic

Logical truth is conditional, as are all necessary and local truths, on the premises given or assumed. Logic is the study of the relation between propositions, between premises and conclusion, that is. So too is probability, which is the continuation, fullness, or completion of logic. All arguments use language, and therefore the terms, definitions, and grammar of language are part of the tacit premises in every argument. It is well to bring these tacit premises out when possible. Logic, like mathematics, is not empirical, though observations may inform logic and math, and logic and math may be used on empirical propositions. Probability, because it is part of logic, is also not empirical; and it, too, can be used on empirical propositions. Syllogistic is preferred over symbolic logic for its ease of understanding; syllogisms are an ideal way of grouping evidence. The fundamental principles of logic ultimately are not formal in a sense to be defined. Finally, not all fallacies are what they seem.

Chapter 3 :: Induction & Intellection

There is no knowledge more certain than that provided by induction. Without induction, no argument could, as they say, get off the ground floor. All arguments must trace eventually back to some foundation. This foundational knowledge is first present in the senses; through intellection, i.e. induction, first principles, universals, and essences are discovered. Induction is what accounts for our being certain, after observing only a finite number of instances or even one and sometimes even none, that all flames are hot, that all men are mortal, that for all natural numbers $x$ and $y$, if $x = y$, then $y = x$, and for providing content and characteristics of all other universals and axioms. Induction is analogical; it is of five different kinds, some more and some less reliable. That this multiplicity is generally unknown accounts for a great deal of the controversy over induction. Arguments are not valid because of their form but because of their content.

Chapter 4 :: What Probability Is

Probability is, like logic, an argument. Logic is the study of the relation between propositions, and so is probability. Like logic, probability is not a real or physical thing: it does not exist, it is not ontological. It cannot be measured with any apparatus, like mass or energy can. Like logic, probability is a measure of certainty of some proposition in relation to given or assumed premises—and only on these, and no other, premises, and this includes the tacit premises of language. All probability, without exception, is therefore conditional. Probability is widely misunderstood for two main reasons: the confusion between ontological and epistemological truth, and the conflation of acts or decisions with probability. We know the proposition “Mike is green” is true given “All dragons are green and Mike is a dragon”. This is an epistemological conditional, or local, truth. But we also know the major part of the premise is ontologically false because there are no dragons, green or otherwise. Counterfactuals are always ontologically false; i.e. they begin with premises known observationally to be false. Yet counterfactuals can have meaningful (epistemological) probabilities. Counterfactuals are surely meaningful epistemologically but never ontologically. Not all probabilities are quantifiable; most are not.

Chapter 5 :: What Probability Is Not

Logic is not an ontological property of things. You cannot, for instance, extract a syllogism from the existence of an object; the imagined syllogism is not somehow buried deep in the folds of the object waiting to be measured by some sophisticated apparatus. Logic is the relation between propositions, and these relations are not physical. A building can be twice as high as another building; the “twice” is the relation, but what exists physically are only the two buildings. Probability is also the relation between sets of propositions, so it too cannot be physical. Once propositions are set, the relation between them is also set and is a deducible consequence, i.e. the relation is not subjective, a matter of opinion. Mathematical equations are lifeless creatures; they do not “come alive” until they are interpreted, so that probability cannot be an equation. Probability is a matter of our understanding. Subjective probability is therefore a fallacy. The most common interpretation of probability, limited relative frequency, also confuses ontology with epistemology and therefore gives rise to many fallacies.

Chapter 6 :: Chance & Randomness

Randomness is not a thing, neither is chance. Both are measures of uncertainty and express ignorance of causes. Because randomness and chance are not ontologically real, they cannot cause anything to happen. Immaterial measures of information are never and can never be physically operative. It is always a mistake, and the cause of vast confusion, to say things like “due to chance”, “games of chance”, “caused by random (chance, spontaneous) mutations”, “these results are significant”, “these results are not explainable by chance”, “random effects”, “random variable”, and the like. All this holds in quantum mechanics, where the evidence for physical chance appears strongest. What also follows, although it is not at first apparent, is that simulations are not needed. This statement will appear striking and even obviously false, until it is understood that the so-called “randomness” driving simulations is anything but random. Coincidences are defined and their relation to cause explained. The ties between information theory and probability are given.

Chapter 7 :: Causality

Cause is analogical. There is not one type, flavor, or aspect of cause, but four. A formal, material, efficient, and final or teleological. Most causation concerns events which occur not separately, as in this before that, but simultaneously, where simultaneous events can be spread through time. Many causal data are embedded in time, and there two types of time series which are often confused: per se and accidental. These should not be mistaken for non-causal data series (the most common) which are all accidental. All causes are activiations of potentials by something actual. A vase is potential a pile of shards. It is made actually a pile of shards by an actual baseball. All four aspects of the cause are there: form of shards, clay fragments, efficient bat, and the pile itself as an end. Deterministic (and probability) models are epistemological; essential causal models are ontological and express true understanding of the nature of a thing. Causes, if they exist and are present, must always be operative, a proposition that has deep consequences for probability modeling. Falsifiability is rarely of interest, and almost never happens in practice. And under-determination, i.e. the possibility of causes other than those under consideration, will always be with us.

Chapter 8 :: Probability Models

A model is an argument. Models are collections of various premises which we assign to an observable proposition, i.e. an observable. Modelling reverses the probability equation: the proposition of interest or conclusion, i.e. the observable Y, is specified first after which premises X thought probative of the observable are sought or discovered. The ultimate goal is to discover just those premises X which cause or which determine Y. Absent these—and there may be many causes of Y—it is hoped to find X which give Y probabilities close to 0 or 1, given X in its various states. Measures of X’s importance are given. A model’s usefulness depends on what decisions are made with it, and how costly and rewarding those decisions are. Proper scores which help define usefulness are given. Probability models can and do have causative elements. Some probability models are even fully causal or deterministic in the sense given last chapter, but which are treated as probabilistic in practice. Tacit premises are added to the predictions from these models which adds uncertainty. Bayes is not all its cracked up to be. The origin and limitations of parameters and parametric models are given.

Chapter 9 :: Statistical & Physical Models

Statistical models are probability models and physical models are causal or deterministic or mixed causal-deterministic-probability models applied to observable propositions. It is observations which turn probability into statistics. Statistical and physical models are thus verifiable, and all use statistics in their verification. All models should be verified, but most aren’t. Classical modeling emphasizes hypothesis or “significance” testing and estimation. No hypothesis test, Bayesian or frequentist, should ever be used. Death to all p-values or Bayes factors! Hypothesis testing does not prove or show cause; therefore, embedded in every test used to claim cause is a fallacy. If cause is known, probability isn’t needed. Neither should parameter-centric (estimation, etc.) methods be used. Instead, use only probability, make probabilistic predictions of observables given observations and other premises, then verify these predictions. Measures of model goodness and observational relevance are given in a language which requires no sophisticated mathematical training to understand. Speak only in terms of observables and match models to measurement. Hypothesis-testing and parameter estimation are responsible for a pandemic of over-certainty in the sciences. Decisions are not probability, a fact with many consequences.

Chapter 10 :: Modelling Goals, Strategies, & Mistakes

Here are highlighted only a few of the most egregious and common mistakes made in modeling. Particular models are not emphasized so much as how model results should be communicated. The goal of probability models is to quantify uncertainty in an observable Y given assumptions or observations X. That and nothing more. This, and only this, form of model result should be presented. Regression is of paramount importance. The horrors to thought and clear reasoning committed in its name are legion. Scarcely any user of regression knows its limitations, mainly because of the fallacies of hypothesis testing and the over-certainty of parameter-based reporting. The Deadly Sin of Reification is detailed. The map is not the territory, though this fictional land is unfortunately where many choose to live. When the data do not match a theory, it is often the data that are suspected, not the theory. Models should never take the place of actual data, though they often do, particularly in time series. Risk is nearly always exaggerated. The fallacious belief that we can quantify the unquantifiable is responsible for scientism. “Smoothed” data is often given pride of place over actual observations. Over-certainty rampages across the land and leads to irreproducible results.

Climate Hustle: In Which Yours Truly Becomes A Movie Star. Updates From Last Night


I’m thinking of hiring the same security firm Mark “I’m No Fan Of Guns Unless They’re Used To Protect Me” Zuckerberg. Just like he, I’ll be a big star and in need to protection to keep the adoring throngs at bay.

For the delicious news is that I will become a bona fide Movie Star tonight. At 7 PM (in every time zone). In a theatre near you. Ladies and gentlemen, or reasonable facsimiles thereof, I bring you…

Climate Hustle!

Scorching temperatures. Melting ice caps. Killer hurricanes and tornadoes. Disappearing polar bears. The end of civilization as we know it! Are emissions from our cars, factories, and farms causing catastrophic climate change? Is there a genuine scientific consensus? Or is man-made “global warming” an overheated environmental con job being used to push for increased government regulations and a new “Green” energy agenda?…

CLIMATE HUSTLE, hosted by award-winning investigative journalist Marc Morano, reveals the history of climate scares including global cooling; debunks outrageous claims about temperatures, extreme weather, and the so-called “consensus;” exposes the increasingly shrill calls to “act immediately before it’s too late,” and in perhaps the film’s most important section, profiles key scientists who used to believe in climate alarm but have since converted to skepticism.

Bill Nye Warns Film is ‘Not in Our National Interest’ There may be no better reason to see it than that.

Thomas Richard at Examiner.com has seen the movie and says “Climate Hustle is a smart, energetic global warming documentary.” “…eloquent and viciously clever.”

The film includes a canyon full of frightening climate change statements from the world’s biggest media tycoons, supersized politicians, and vaunted personalities like Leonardo DiCaprio and Mark Ruffalo. From rising seas to extreme weather events to melting ice sheets to famine and disease, an all-too-familiar bombardment of alarming newscasts gets compressed into a few minutes. You’ve heard the debate is over, the science is settled, and the time to act is now. But are all those things really true? Are you being hustled?

Which reminds me: don’t miss this podcast, which has a news clip from 1971 in which an official from the National Science Foundation said mankind was causing global cooling.

Our friend Paul Driessen said (via email; I don’t have a link):

I saw Climate Hustle April 14, at its U.S. premiere on Capitol Hill in Washington. The film is informative and entertaining, pointed and humorous. As meteorologist Anthony Watts says, it is wickedly effective in its using slapstick humor and the words and deeds of climate alarmists to make you laugh at them.

Marita Noon also saw the movie: “On Climate, We’re Manipulated By Sleight Of Hand.

Repeatedly throughout the past couple of decades, we’ve been pummeled with dire predictions and told “time is short.” In 1989, the UN predicted “Global warming would destroy entire nations by 2000.” In 2007, we were told: “Scientists believe we have less than ten years to bring emissions under control to prevent a catastrophe.” In 2008, Britain’s Prince Charles said we only had 100 months left to solve the problem. Gore, in 2009, said: “We have to do it this year.” Yet, as the film demonstrates, scientists don’t want to talk about their failed predictions…

Meanwhile, scientists who don’t agree with the “leaders” are accused, by the likes of Robert F. Kennedy Jr., of “treason.” He wants them “in jail.”

Yes, as Climate Hustle makes clear, there are dissenting scientists—but they are marginalized, even called “kooks.” If they speak out, they are insulted, ignored, ridiculed, ostracized, called heretics, hurt professionally, and even terminated for divergent views. This is not the scientific method.

Well, kooks we may be, but we’re at least well dressed. I hope viewers like as much as I do the tie I wore in the film. There is a clear dichotomy of preference for that tie. Many sad people can’t see its beauty.

Yes, I’d rather talk about men’s fashion than global cooling. I mean global warming. That we can’t get actual scientists to realize thirty years of busted forecasts means their theory is wrong is too depressing to contemplate, and if I never ever say another word on this stupid subject it will be too soon (a cliche, but an appropriate one).

I haven’t seen the movie yet, and won’t tonight, either, unfortunately, because I’m in San Francisco, and San Francisco is too enlightened to allow a showing. I’ll rely on you, dear reader, to watch for me. Report back here of your awe and wonder.

Incidentally, I’m in San Francisco. If there are any earthquakes, you’ll know why. If anybody out there wants a talk, seminar, or whatnot, this is the time to ask. Use the Contact Page.

Update Turns out I am going. I’ll be at the Pleasant Hill Century 16 tonight (just north of Walnut Creek). I was invited by a generous reader, who also promises burgers and beers at the Fuddruckers which is apparently near the theater.

Update More reviews. Another.

Update Movie was a great success. About 200 people in the audience in Pleasant Hill, which I find remarkable. The popcorn flowed; bonhomie overflowed. Spontaneous applause at the end. Lot of folks stuck around to chat afterwards. Even better, I got to meet three regular readers! I didn’t ask permission to name them, so I can’t (they may not want to admit it). But heartfelt thanks to them for making the trek. It was deeply appreciated.

Summary Against Modern Thought: God Didn’t Have To Create

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 a lot of words, but they aren’t difficult; it reads quickly. I have cut out extra proofs, which can be had at the links. There’s also not much to say about this chapter, except at the end, mostly because I don’t think it will be controversial.

Chapter 28 How there is anything due in the production of things (alternate translation)

[1] AGAIN. From what has been said it may be shown that God in the creation of things did not work of necessity, as though He brought things into being as a debt of justice.

[2] For justice, according to the Philosopher (5 Ethic.), is towards another person to whom it renders his due. But nothing, to which anything may be due, is presupposed to the universal production of things. Therefore the universal production of things could not result from a debt of justice.

[3] Again. Since the act of justice is to render to each one that which is his own, the act by which a thing becomes one’s own precedes the act of justice, as appears in human affairs: for a man by working has a right to call his own that which, as an act of justice, is rendered to him by the person who pays him. Therefore the act whereby a person first acquires something of his own cannot be an act of justice. Now a created thing begins to have something of its own by creation. Therefore creation does not proceed from a debt of justice.

[4] Further. No one owes something to another except from the fact that in some way he depends on him or receives something either from him or from a third, on whose account he owes something to the other: for thus a son is a debtor to his father, because he receives being from him; a master to his servant, because he receives from him the service he requires; and every man is a debtor to his neighbour for God’s sake, from Whom we have received all good things. But God is dependent on no one, nor needs He to receive anything from another, as is manifestly clear from what has been said. Therefore it was not on account of a debt of justice that God brought things into being.

Notes Hence, at the very least, Honor your father and mother, love God, and love your neighbor.

[5] Moreover. In every genus that which is on account of itself precedes that which is on account of another. Consequently that which is simply first of all causes, is a cause on its own account only: whereas that which acts by reason of a debt of justice does not act on its own account only, for it acts on account of the thing to which the debt is due. Therefore God, since He is the first cause and the first agent, did not bring things into being from a debt of justice…

[7] Hereby is refuted the error of some who strive to prove that God cannot do save what He does, because He cannot do except what He ought to do. For He does not produce things from a debt of justice, as we have proved.

[8] Nevertheless, although nothing to which anything can be due precedes the universal creation of things, something uncreated precedes it, and this is the principle of creation. This may be considered in two ways. For the divine goodness precedes as the end and first motive of creation, according to Augustine, who says: Because God is good we exist. Also His knowledge and will precede, as by them things are brought into being.

[9] Accordingly if we consider the divine goodness absolutely, we find nothing due in the creation of things. For in one way a thing is said to be due to someone on account of another person being referred to him, in that it is his duty to refer to himself that which he has received from that person: thus it is due to a benefactor that he be thanked for his kindness, inasmuch as he who has received the kindness owes this to him. But this kind of due has no place in the creation of things: since there is nothing pre-existent to which it can be competent to owe anything to God, nor does any favour of His pre-exist. In another way something is said to be due to a thing in itself: since that which is required for a thing’s perfection is necessarily due to it: thus it is due to a man to have hands or strength, since without these he cannot be perfect. Now God’s goodness needs nothing outside Him for its perfection. Therefore the production of creatures is not due to Him by way of necessity…

[11] Moreover. It has been proved that God brings things into being neither by necessity of His nature, nor by necessity of His knowledge, nor by necessity of His will, nor by necessity of His justice. Therefore by no manner of necessity is it due to the divine goodness that things be brought into being…

[14] If, however, we consider the divine ordinance whereby God decided by His intellect and will to bring things into being, then the production of things proceeds from the necessity of the divine ordinance: for it is impossible that God should decide to do a certain thing which afterwards He did not, otherwise His decision would be either changeable or weak. It is therefore necessarily due to His ordinance that it be fulfilled. And yet this due is not enough for the notion of justice properly so called in the creation of things, wherein we can consider nothing but the action of God in creating: and there is no justice properly speaking between one same person and himself, as the Philosopher says (5 Ethic.). Therefore it cannot be said properly that God brought things into being from a debt of justice, for the reason that He ordained by His knowledge and will to produce them.

[15] If, however, we consider the production of a particular creature, it will be possible to find therein a debt of justice by comparing a subsequent creature to a preceding one. And I say preceding, not only in time but also in nature.

[16] Accordingly in those divine effects which were to be produced first, we find no due: but in the subsequent production we find a due, yet in a different order. For if those things that are first naturally, are also first in being, those which follow become due on account of those which precede: for given the causes, it is due that they should have actions whereby to produce their effects. On the other hand if those which are first naturally are subsequent in being, then those which are first become due on account of those which come afterwards; thus it is due that medicine precede in order that health may follow. And in either case there is this in common,–that what is due or necessary is claimed by that which is naturally first from that which is naturally subsequent.

[17] Now the necessity that arises from that which is subsequent in being, and yet is first by nature, is not absolute but conditional necessity: namely, if this must be done, then that must precede. Accordingly with regard to this necessity, a due is found in the production of creatures in three ways.

First, so that the conditional due is on the part of the whole universe of things in relation to each part thereof that is necessary for the perfection of the universe. For if God willed such a universe to be made, it was due that He should make the sun and moon, and suchlike things without which the universe cannot be.

(Notes Don’t get stuck on “sun and moon”. Thomas means the physical things that are without which no universe can be. This is a sort of anthropic argument.)

Secondly, so that the conditional due be in one creature in relation to another: for instance, if God willed the existence of plants and animals, it was due that He should make the heavenly bodies, whereby those things are preserved; and if He willed the existence of man, it behoved Him to make plants and animals and the like, which man needs for perfect existence: although God made both these and other things of His mere will.

Thirdly, so that the conditional due be in each creature in relation to its parts, properties, and accidents, on which the creature depends either for its being, or for some one of its perfections: thus, given that God willed to make man, it was due, on this supposition, that He should unite in him soul and body, and furnish him with senses and other like aids, both within and without. In all of which, if we consider the matter rightly, God is said to be a debtor not to the creature, but to the fulfilment of His purpose. There is also in the universe another kind of necessity whereby a thing is said to be necessary absolutely.

(Notes Analogy: If God wanted to make a triangle, and He first created two sides, He was bound to create the third. As Thomas says next. Remember cause has four components: formal, material, efficient, and final.)

This necessity depends on causes which precede in being, for instance on essential principles, and on efficient or moving causes. But this kind of necessity cannot find place in the first creation of things, as regards efficient causes. For there God alone was the efficient cause, since to create belongs to Him alone, as we have proved above; while in creating, He works not by a necessity of His nature, but by His will, as we have shown above; and those things which are done by the will cannot be necessitated, except only by the supposition of the end, on account of which supposition it is due to the end that those things should be whereby the end is obtained.

On the other hand, with regard to formal and material causes, nothing hinders us from finding absolute necessity even in the first creation of things. For from the very fact that certain bodies were composed of the elements, it was necessary for them to be hot or cold: and from the very fact that a superficies was drawn in the shape of a triangle, it was necessary that it should have three angles equal to two right angles. Now this necessity results from the relation of an effect to its material or formal cause. Wherefore on this account God cannot be said to be a debtor, but rather does the debt of necessity affect the creature.

But in the propagation of things, where the creature is already an efficient cause, an absolute necessity can arise from the created efficient cause: thus the lower bodies are necessarily influenced by the movement of the sun…

Notes Hence science!

Worlds Without End: The Conformal Cyclic Cosmology Of Roger Penrose — Guest Post by Bob Kurland


An extended version of this article is available at Kurland’s place.

Perhaps the best argument in favour of the thesis that the Big Bang supports theism is the obvious unease with which it is greeted by some atheistic physicists. At times this has led to scientific ideas, such as continuous creation or an oscillating universe, being advanced with a tenacity which so exceeds their intrinsic worth that one can only suspect the operation of psychological forces lying very much deeper than the usual desire of a theorist to support his/her theory. –Chris Isham, “Creation of the Universe as Quantum Process” in Physics, Philosophy and Theology–A Common Quest for Understanding.


Elsewhere, I wrote that the conformal cyclic cosmology (CCC) proposed by Roger Penrose deserves an article, “nay, a chapter” on its own. Although there are several cosmology theories that propose multiple universes (see here and here), the CCC is perhaps the only one that might be subject to experimental tests. And if such tests were to confirm Penrose’s hypothesis, what then would be the consequences for Catholic teaching? We’ll answer that question below.

This is only a summary of Penrose’s thesis; for a fuller account go to his 2005 paper, Before the Big Bang: an Outrageous New Perspective, his 2013 paper, On the Gravitization of Quantum Mechanics 2: Conformal Cyclic Cosmology, or his book, Cycles of Time.

The starting point for this theory is the puzzle of the Second Law of Thermodynamics. Why does entropy increase for the universe? In one of his other books, The Road to Reality, Penrose poses this question, which appears to contradict a prime principle of physics, time reversibility of physical processes. Why is the Universe so ordered at the moment of the Big Bang? (He estimates the relative volume of phase space into which the universe expands as 10^(10^123), a number greater than all the particles in the universe.)

Penrose’s answer in “Before the Big Bang” is that gravitational degrees of freedom are not activated in the very early universe, and it’s gravitational effects that give rise to the increase in entropy:

The answer lies in the fact that the high entropy of the microwave background refers only to the matter content of the universe and not to the gravitation field, as would be encoded in its space-time geometry in accordance with Einstein’s general relativity. What we find…is an extraordinary uniformity, … interpreted as gravitational degrees of freedom…being not excited at all…the entropy rises as the initially uniform distribution of matter begins to clump, as the gravitational degrees of freedom begin to be taken up. This allows stars to be formed, which become much hotter than their surroundings…and finally this gravitational clumping leads to the presence of black holes (particularly the huge ones in galactic centres), which represent an enormous increase in entropy

So, where does “conformal” enter into the picture? By means of a mathematical transformation of spacetime coordinates, the zero—the singularity at the origin—can be transformed into a finite boundary, as shown in the diagram above. Further, the infinity—the singularity at the end of the universe—can also be transformed into a finite boundary. Penrose’s “outrageous perspective” is to assume that these transformations are more than mathematical devices, but portray reality. Moreover, not only are these mathematically transformed beginnings and ends real, they are conjoined: the end of one universe is the beginning of the next.

An important way in which Penrose’s cosmology differs from other proposals is that it can (in principle) be tested experimentally. Symmetry requirements enable particles with integral spin (which include photons) to cross the boundary between the nth and (n+1)th universe, but not particles with half-integral spin (mass particles such as electrons and protons).

Accordingly there might be signals from a preceding universe shown in the microwave background radiation, COBE. Penrose and Gurzadyan claimed to have found such by concentric circles in the background radiation, presumed to be due to gravity waves formed by collisions between massive black holes in the preceding Aeon.

However this “evidence” has been challenged by several authors who claim that intrinsic structure that can be deduced by statistical manipulation is part of the natural variation of the COBE radiation. In fact, one group showed that statistically significant equilateral triangles could also be deduced by appropriate manipulation.

Other objections have been raised to the CCC thesis: for the conformal mapping at the end of the universe to be physically significant, it is required that no matter be left in the universe. Protons have a finite lifetime (albeit very long—their decay has not yet been observed), but electrons do not. So, unless all electrons in the universe were to be annihilated by positron-electron collisions (or some other interaction), the massless condition would not hold, nor would it hold if neutrinos continued to exist.

Theological Significance

Catholic theologians breathed a huge sigh of relief (presumed) when Abbe LeMaitre’s primordial atom thesis (“The Big Bang”) was confirmed by galactic red shift measurements and became scientifically acceptable. Indeed, Pope Pius XII argued that this scientific theory validated Catholic teaching. LeMaitre himself was not inclined to such a strong view; rather he disassociated the science from theology with this statement (from “The Primeval Atom Hypothesis and the Problem of Clusters of Galaxies”):

As far as I can see, such a theory remains entirely outside any metaphysical or religious question. It leaves the materialist free to deny any transcendental Being…For the believer, it removes any attempt at familiarity with God…It is consonant with Isaiah speaking of the hidden God, hidden even in the beginning of the universe.

St. Thomas did not say that reason could prove a universe beginning in time; although reason could prove a First Cause—God—for the origin of the universe (possibly as an eternal entity, since God is eternal); it is an article of faith to believe a universe created in time. The Big Bang notion is consistent with Creation at t=0, but does not prove it. If we believe God is the author of all, a First Cause, then He can create an infinity of universes, as in the bubble universe hypothesis of Lande or in the parallel worlds given by some interpretations of quantum theory. Economy of effort is not required of God.

Accordingly, faith trumps science in this. Revelation gives us reason to believe, and despite the efforts of non-believing scientists to trash a strong confirmation of God’s creative power, the Big Bang, we still can remain confident in that power as the source of being. Such a position is not “fideism”, an adherence to Catholic teaching in the face of empirical evidence to the contrary. It is a recognition of the limits of science, and the authority of theology and philosophy in final verdicts on truth.

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