A Scientist Claims That We are Living in the Matrix

ACCORDING TO an article in the Huffington Post:

Physicists have devised a new experiment to test if the universe is a computer.

A philosophical thought experiment has long held that it is more likely than not that we're living inside a machine.

The theory basically goes that any civilisation which could evolve to a 'post-human' stage would almost certainly learn to run simulations on the scale of a universe. And that given the size of reality - billions of worlds, around billions of suns - it is fairly likely that if this is possible, it has already happened.

And if it has? Well, then the statistical likelihood is that we're located somewhere in that chain of simulations within simulations.

— from the article Physicists To Test If Universe Is A Computer Simulation, by Michael Rundle.

A group at the University of Washington claims that physics suggests that we are living within a simulation, which “would be a matter of looking for a “signature” in our universe that has an analog in the current small-scale simulations.” They claim that such a signature has been found.

Quantum mechanics is one of the fundamental theories of nature; the other being the theory of relativity. While quantum theory is quite successful at predicting outcomes of experiments on very small scales, it has numerous conceptual and mathematical problems. One rather embarrassing problem is that solutions to its equations tend to blow up to infinity, which requires some method of sweeping the infinities under the rug (albeit in an ad hoc way). Renormalization is one such method, and other is the lattice model, which maps space and time onto a rectangular lattice.

The researchers are using the lattice model, and are using predictions from this model to make the bold claim that we are living in a simulation.

Just the other day, I had some correspondence with a philosophy student, who is writing a thesis on quantum mechanics. Here is part of what I wrote to him:

…I do think that some of the interpretations of QM are motivated from bad philosophical thinking or ideology.

However, I believe that one key to digging deeper into the quantum world is to abandon Euclidean geometry as a framework — it is often assumed — but as we know, both space and time are created things, and not absolute.

Renormalization is one unsatisfying way around the infinities generated by Euclidean geometry, but this directly implies that space is quantized; however, grid lattice formulations are just silly, in my opinion.

Clearly, there has to be something that looks like geometry as we approach classical lengths, but has fewer axioms on the quantum scale. Also, there needs to be something in the formulation which does not strongly differentiate 'here' from 'there', in order to derive a simple explanation of confinement and quantum entanglement. Assuming pure geometric locality has gotten physicists into trouble many times. I haven't thought of it much, but there should be similar nonlocal interactions across time. That would be stunning, but shouldn't that naturally come from the theory somehow?

Perhaps a bit of mystical contemplation is needed, for all these theories — and theology — point to an underlying unity…

Euclidean geometry, despite all of its complexity, derives ultimately from five simple axioms [later expanded, taking into account some assumptions by Euclid]. The fifth axiom, the parallel postulate, was abandoned by Einstein, and from that he derived the general theory of relativity, which posits that space can be warped. On ordinary scales that we experience, space appears to be absolutely flat and so we can use Euclidean geometry without error. Likewise, some other tweaking of the axioms of geometry ought to give us something that is closer to nature, but on the human scale will resemble Euclid.

The grid lattice formulation used by the researchers in their thought experiments is just silly, and the so-called signatures they have discovered could fall out naturally of a better theory, and not be a by-product of computation. Would nature actually work in such a manner? Of course not — but because the physicists use it, leads some of them to believe in the greatest conspiracy ever devised, that we are living in a computer simulation, which was the central premise of the plot of the popular film from 1999, The Matrix.

---

Simulations nested within simulations. This seems rather fishy, as well as familiar to those who know Church history.

Ultimately, this is a modern rehashing of the oldest heresy, that of Gnosticism, said to derive from Simon Magus, who offered to buy Saint Peter's power of healing, and it is from that heretic that we get the sin called simony, the buying and selling of ecclesiastical offices. The Gnostics disbelieved that a good God would create a cosmos with evil within it, and so they posited a series of universes within universes; the final one being created by an evil god, whom they identify with the God of the Old Testament. Jesus, in their system, is simply a god from a higher-order universe, who gives believers true knowledge (gnosis) which will save them. But even a superficial reading of scripture will disprove both ideas.

This contrasts sharply with the orthodox view that the world was created very good, and that evil is a by-product of free will which has been misused, and that Jesus saves through self-sacrifice.

Gnosticism is quite popular today, as seen in the article Rapture and the Gnostic Tendency. We must hope that this computer simulation idea does not gain traction, but knowing the perversity of academia and the media, we may be in trouble, for this type of thinking tends to have an extreme hatred for the material world. The Gnostic tendency is to be extremely violent, for the world in their eyes is evil: and modern believers may think that a computer simulation has no intrinsic value. May God help us if someone proposes that we ought to 'reboot the universe.'

"Curiosity"

One of my most exciting memories from my college years was paying a visit to the Jet Propulsion Laboratory, home of the unmanned space exploration program for the United States. There I watched a live feed of images of the Saturn system photographed during the flyby of Voyager 1 spacecraft. Exciting times! The spontaneous cheers which arose from the crowd — students and scientists alike —  expressed a great joy.

The animation of the landing of the Mars rover "Curiosity,"seen above, is quite dramatic and exciting in itself, and I'm sure that most of us hope that it goes well.

There are some who think that the space program is expensive and wasteful, spending money that is better spent elsewhere, such as on social services — this despite the fact that many Ph.D.s would be homeless were it not for the space program. While the space program often has been wasteful, there have also been crowning moments of awe, like the Apollo 11 moon landing. But what the critics may not realize is that an even greater virtue than liberality is the virtue of magnificence, which exhibits great courage and great honor, found in the best achievements of the space program. I wrote more about magnificence and the space program here.

Faith and Quantum Mechanics

AT THE HEART of all things are problems that are ultimately philosophical and religious. This is despite modernity’s avoidance of philosophy and religion, for these are basics that permeate all of reality whether or not someone likes it. Every intellectual system that makes claims — and certainly all do — is ultimately a metaphysical system, and so is inherently saying something about religion.

See the article Does Quantum Physics Make it Easier to Believe in God? From that article:

Materialism is an atheistic philosophy that says that all of reality is reducible to matter and its interactions. It has gained ground because many people think that it’s supported by science. They think that physics has shown the material world to be a closed system of cause and effect, sealed off from the influence of any non-physical realities --- if any there be. Since our minds and thoughts obviously do affect the physical world, it would follow that they are themselves merely physical phenomena. No room for a spiritual soul or free will: for materialists we are just “machines made of meat.”

And so, I’ve noticed that religion and science tend to be more tightly intertwined than we may suspect, and despite the efforts of modern theorists of science. The materialism noted in the quote above was more found in the older, Newtonian physics than what we find in the newer Quantum mechanics.

During the Reformation, the new religions tended to deny human free will and instead embraced a narrow kind of predestination: that before the beginning of time, God created some humans to be saved and others to be damned. This view became widely accepted among Protestant groups, and even among many Catholics, although this was eventually declared heretical. With this religious view, which became most codified in Deism, it is pretty obvious that a completely deterministic physics would not be far behind, and we eventually found this with Newtonian mechanics.

Under Newton’s theory of motion, the initial conditions of a system — even the entire cosmos —determines completely its subsequent motion for all eternity, and this physics corresponded well to the kind of predestination being promulgated by the various reformed religions. Taking this theory seriously led to Deism, where God created the cosmos according to a pre-determined plan, and then did nothing whatsoever afterwards — not even revealing Himself to the world through his Son. Atheistic materialism itself developed from the idea that there was no ‘initial condition’ of the cosmos, that is, from the idea that the cosmos always existed: however, some philosophies of antiquity posited an eternally-existing cosmos along with an equally eternal Ultimate Deity.

While the physicist Albert Einstein (1879–1955) is known more for his Special and General Theories of Relativity, he gained fame initially, in the year 1905, by mathematically demonstrating the quantum hypothesis — the theory that energy is transferred in discrete, and not continually varying, quantities. This theory was a major crack in the Newtonian framework, and eventually led to the theory of Quantum Mechanics.

Einstein developed his theories with scholastic methods — modes of reasoning more associated with Medieval friars such as Saints Bonaventure, Albert the Great, and Thomas Aquinas. But, as Pope Benedict XVI said:

”Yet the human mind invented mathematics in order to understand creation; but if nature is really structured with a mathematical language and mathematics invented by man can manage to understand it, this demonstrates something extraordinary. The objective structure of the universe and the intellectual structure of the human being coincide; the subjective reason and the objectified reason in nature are identical.”

This scholastic reasoning is unreasonably effective, which is something that we ought to keep in mind, especially in light of more modern theories that place all their hope in purely human and skeptical observation of things, and not in things that are larger. But we must be humble enough to realize that human reasoning is much more effective with simple systems, such as physics, and will be less satisfactory when dealing with the complexities of biology and human psychology. But this observation of complexity is in line with the old scholastic notion of the Great Chain of Being, where physics can be found at the far bottom end of the chain, and so is inherently more easily understandable.

Einstein’s Theories of Relativity posit that space and time are not absolute — these being axioms of the old Newtonian physics — but are contingent on something else even more basic, and so are ultimately malleable and changing. But this was something understood even by the ancient rabbis before Christ, that space and time are creatures of God as are we, something made and not perfect unto themselves. Einstein, although a non-practicing Jew, still used the scholastic methods of his forefathers: he knew, ahead of time, that his theories were largely correct, and that the experiments that seemed to contradict his results had to be wrong. Skeptics can doubt as much as they like, but they can never advance science.

Quantum mechanics changes things even more. Like the equations of Newton, quantum theory is deterministic in structure, yet produce results that are only probabilistic in outcome. The theory is rational, it posits true reasons for observations, and yet the future remains for us unknowable. The acceptance of this theory, as you might imagine, led many physicists away from a purely materialistic and atheistic world-view, for the theory shows that transcendence is intrinsic to the laws of nature. Now, this theory in and of itself is not completely convincing, and there are many wildly diverging philosophical opinions on the meaning of quantum mechanics, but it is more consistent with the wider Catholic view of the world than what we found under Newton.

UPDATE: having been in a hurry to get to Mass this morning, I didn’t get to the main point of this article.  The recent claim of the discovery of the Higgs Boson is of major theoretical importance, because it is the final piece required to complete what physicists call the “Standard Model,” a theory which unifies under one consistent umbrella the forces of nature which lead to electricity, magnetism, light, radio waves, nuclear energy, and certain forms of radioactivity. While important, it still leaves out an explanation of gravity. I wrote some more about this here.

Veritas Liberabit Vos

I’VE LONG NOTICED that biology is an attractive field of study for people who like science, but don't understand mathematics. Likewise, physics is attractive to those who are good at math, but don’t understand philosophy.

I got a degree in physics from Caltech, which is an excellent school, but I went for four years without knowing that the world ‘physics’ is from the ancient Greek φύσις (physis) meaning “nature.” Perhaps someone may have mentioned that fact, but as Caltech was akin to trying to get a sip of water from a fire hose, I didn’t explicitly learn it until I started learning philosophy in my adulthood, after my entry into Holy Mother Church. Now, the ancients would have thought this to be a silly order of gaining knowledge — certainly, you have to learn mathematics first, then philosophy, and only after philosophy is a person really ready for professional scientific studies. Sadly, part of the reason for the change is that philosophy these days is often too terribly dry and dead: scholars point out the differences between the various philosophical systems, show how one philosopher influenced another, and so forth. It is not taught as a living and supremely important tool for living the good life. Modern philosophy shows no love for wisdom, nor does it seek out the truth.



This was Caltech’s logo when I started going there. Note the quote from the Gospel, John 8:32: “The truth shall make you free.” The passage in context has Our Lord teaching at the Temple in Jerusalem:

31 Jesus then said to those Jews who believed in him, “If you remain in my word, you will truly be my disciples,
32 and you will know the truth, and the truth will set you free.”
33 They answered him, “We are descendants of Abraham and have never been enslaved to anyone. How can you say, ‘You will become free’?”
34 Jesus answered them, “Amen, amen, I say to you, everyone who commits sin is a slave of sin.
35 A slave does not remain in a household forever, but a son always remains.
36 So if a son frees you, then you will truly be free.”

[New American Bible]

The school’s current logo omits the quote from sacred scripture, and instead adds the year of the school’s founding, 1891. I recall the controversy over the logo change: the quote was removed, and also omitted was the second hand, shown on the right, receiving the torch.

Oddly enough, the removal of the hand was rather controversial. Please recall that the phrase “carrying a torch,” since it is usually is understood to be a symbol of sentimental unrequited love, has quite a different meaning from “passing on the torch,” which is symbolic of the transfer of knowledge, tradition, and responsibility from an older generation to a younger. The powers that be relented on the torch, but not on the quote.

Atheism back in those days was generally more humble than the aggressive, divisive atheism we find today. I do not recall anyone saying that they were offended by the Christian source of the quote, but instead some liberal humanities professors, supporting the change, argued from the point of view of Postmodernism, saying, like Pontius Pilate, that there is no truth. This is not a good philosophy for a school of the sciences and engineering, but sadly this philosophy is now found in public science policy, which seems more intent on the consolidation of power than on finding the truth.

Finding the truth is very difficult, but it is worthwhile, a worthy pursuit for men who truly want to be free. As mentioned, many physicists (and also mathematicians) are very good at finding the truth, but are often bad at philosophy. This is a shame. That we have today a field of inquiry called ‘physics’ is due to the work of the philosopher Aristotle, as well as due to the work of the medieval schoolmen, most specifically the philosophers Saint Albert the Great, Saint Thomas Aquinas, and other Dominicans and Franciscans. ‘Physics’ was the title given to one of Aristotle’s books, and his methodology, filtered and purified through the Church, led to the scientific revolution in the modern period. One of Saint Albert’s considerations was that natural science ought to properly seek out the “immanent causes” found within nature herself, and not conflate the matter with God’s will. Many religions will state that a rock drops because it is God’s direct will that it do so. Catholicism, on the other hand, teaches that the world was created by God in due order as a reflection of His goodness, justice, and wisdom. She teaches that God does not interfere with the natural created order unless a greater good comes from it. An apple will fall from a tree to the ground due to gravity, unless God wills it otherwise — and miracles of this sort are presumably rare. Science therefore can be done without recourse to religion, but science does not exist without a religious and philosophical understanding. Aristotle, although pagan, came from a school of thought that understood the existence of the Supreme Being, the Source of all that is good, just, and wise, and for this reason we understand that there is a rational order that we can investigate via science.

Besides the orthodox Catholic view of Aristotle and science, which took what is good and true in the pagan writings and rejected that which is false, there came about other schools of thought. One became skeptical of all pagan writings, and from this we got the Reformation with its general distrust or rejection of philosophy. Marxism, with its hatred of truth, also derives from this skeptical school of thought. The other school accepted too much of what was in Aristotle, including the notion of an eternal cosmos: but that became the standard understanding of many physicists in the modern era.

The belief that the inductive method is the only valid method of discovering knowledge is also derived from Aristotle, but insisting on this method alone can lead scientists far from the truth in difficult cases where science meets philosophy. But even Aristotle describes at least three methods: deductive, inductive, and abductive reasoning (ever hear of this?), as well as reasoning through analogy. All of these methods are important for good science reasoning.

The book by Aristotle that comes after his book Physics is titled Metaphysics, which simply means “the things after physics.” But metaphysics is a “dirty” word to most contemporary physicists; science, they claim, stops at physics. Metaphysics, in its traditional definition (but not in the contemporary New Age use) is a perfectly respectable field of study, for it asks the questions “what exists?” and “what is it like?” Alas, when the Church largely abandoned the study of metaphysics during the 20^th century, nature abhors a vacuum, and so unreliable religions filled the void. We ought to be grateful that metaphysics has again been taken up by the Church in many of her seminaries, for not only theology but the natural sciences could benefit from that study. Physicists and mathematicians ought to realize that metaphysics is a subject higher and prior to theirs. Debate about the Standard Model of physics is largely a metaphysical problem, as are discussions about the meaning of the otherwise accepted Quantum Mechanics. Contemporary physics has reached a plateau, with progress being slight these days; a rediscovery of proper metaphysics has the potential of greatly expanding the knowledge in those fields, and I think that Catholic philosophers ought to take up again the study of the philosophy of science. See the article Ex Nihilo Nihil Fit for a discussion of some controversies in current physics, where bad philosophy is explored.

One particular thing claimed is that quantum mechanics predicts that matter can be made ex nihilo out of a vacuum. The whole idea of a vacuum was hotly debated during the Middle Ages, with some claiming that the existence of such a thing is impossible, that a vacuum would be a place without the presence of God. But rather we ought to consider that space and time are also made by God, not absolute but created things, and even if space is empty, space itself gets its existence from the Creation. But Quantum Mechanics does not posit the existence of a pure void: ‘empty’ space is truly active and has its own structure and potentialities, as we ought to expect from these philosophical considerations. So the claim of quantum mechanical ex nihilo creation of matter is not to be taken seriously. The other major theory in physics, General Relativity, posits the idea that space and time are contingent and not absolute, which was something believed even by the ancient Rabbis of the Old Testament period. That Jews at one time were typically the leading mathematicians and physicists of their era should not be surprising, nor the fact that Catholics led those fields at other times, likely because they had a strong grounding in philosophy and a better understanding of the role of science.

My own studies at Caltech suffered because I did not have a grounding in philosophy — but that is par for the course in our era. I also had a poor grounding in mathematics — while I knew the advanced stuff, like the calculus, I didn’t know the extreme basics of the field, like the foundations of number theory and geometry as were discovered by the ancient Greeks. This misunderstanding leads to all sorts of mischief and poor mathematical constructs in physics; while not exactly wrong, poor mathematical ideas can be misleading, and prevent scientists from quickly seeing true underlying relationships.

I am pleased to see that Caltech now has a Cardinal Newman Center. Back in my day, many Catholic students fell away from the faith due to the lack of practice and community; although I’ve learned via Facebook that many have come back since then (I fell away from Lutheranism because of the same lack of practice and community). Let us hope that the center will help strengthen their faith, as well as help them become better scientists. The truth shall set you free.

The “proper reason for a certainly established fact"

A QUOTE FROM a book on the liberal arts:

Aristotle had stated the scientific ideal as follows:

We suppose that we possess perfect scientific knowledge of a subject, as contrasted to knowing it in the haphazard way that a sophist knows it, when we think that we know the cause on which the fact depends as the cause of that fact and no other, and further, that the fact could not be other than it is. (Posterior Analytics, 1, c. 2, 71b 8.)

Aristotle knew perfectly well that such a high ideal of knowledge is not easy to obtain and is rarely actually achieved, but we must measure our knowledge by the most perfect type of knowledge possible. Euclid [writing about 30 years after Aristotle] hoped to achieve such knowledge at least in the relatively easy field of mathematics.

You will notice that the difference between science and opinion does not consist in the fact that science is obtained by experiments or the use of some complicated instrument (like a microscope or Geiger counter), while opinion is not. Nor is the difference that science uses measurement, and opinion does not. Nor is it that science is objective, opinion subjective; nor that science is certain, opinion probable. We can have opinions that are certain, objective, acquired by measurement, instruments, and experiment -- and they still remain only opinions. We can have science, on the other band, which involves no instruments, nor experiments, nor measurements (although it must be certain and objective). What makes science to be true science is that it gives us the proper reason for a certainly established fact. Until we have discovered such a proper reason we do not have perfect science, and are still at the level of opinion.

Consequently, mathematicians before Euclid were mainly at the level of opinion, because they had not carefully built [up] the science of mathematics in such a logical and orderly way that they could see the proper reasons for their conclusions.

— from THE ARTS OF LEARNING AND COMMUNICATION, A Handbook of the Liberal Arts, by Benedict M. Ashley, O.P. (The Priory Press, Dubuque - 1958)

In our current age, the practice of what is now called science has decidedly shifted — according to the scheme given above — from true science towards variable opinion. We typically see more political and ideological influence on the sciences these days, and a greater emphasis on the “soft sciences;” this leads, justifiably, to a greater skepticism of the results of state-sponsored research. But this has also led to an unjustifiable skepticism of true science.

Alexis de Tocqueville thought that the arts and sciences in the United States would eventually suffer because of an overemphasis or misplaced emphasis on equality; see the article Catholics and de Tocqueville’s Democracy in America. This is particularly evident in primary and secondary education, which is under considerable direct political control by high levels of government, and is institutionally controlled by ideologues more concerned with power than the truth.

This kind of control, as well as the current emphasis on “teaching the test,” has led to the unfortunate result that youths very often see the sciences as the mere opinion or ideology of the teacher. Many youth seem to think that all they have to do in a class is mindlessly regurgitate the lessons back to the teacher and then quickly forget the lessons learned, since they aren’t facts and have no relevance to real life. This is a great tragedy, for a true scientific understanding is one of the pillars of virtue.

As mentioned by Fr. Ashley, mathematics is the most certain as well as the simplest of the sciences. Ancient and medieval mathematicians may not have had a broad knowledge of their field, but it was certainly deep, and they had a better understanding of mathematics’ place in the realm of reality, which is something that many contemporary mathematicians struggle with. Mathematics sometimes goes too far into metaphysics, and these philosophical issues aren’t handled well by many mathematicians. Problematic fields include Probability Theory and Set Theory, which can give us absurd and contradictory results due to poor assumptions.

I’ve been reading up on some of the basics of arithmetic and geometry, both according to the ancients and also modern views. As it turns out, many mathematical problems I used to struggle with when I was studying physics would have been much clearer if I had a more thorough background in the basics.

The basics of number theory and geometry are also of great use to artists: premodern art is intimately tied to a deep understanding of mathematics. If I have the time and make the effort, I hope to write a bit about mathematics and how it ought to apply to the arts.

Springtime and Man’s Relationship with Nature

LENT, in the Northern Hemisphere, is associated with the lengthening of days (whence we get the name of this liturgical season), and is determined by the vernal equinox, which happens to be today. Of course, in the southern hemisphere, today marks the autumnal equinox. Today at the equator the sun at local noon is directly overhead, and were it not for atmospheric distortion and the large disk of the sun, it would be precisely 12 hours from sunrise to sundown worldwide.

[When I was a child, I had a public school textbook which illustrated the globe of the Earth: it properly showed that when it was summer in the northern hemisphere, it was winter in the southern. Absurdly, it also (falsely) showed that when it was spring in the western hemisphere, it was autumn in the eastern hemisphere.]

Nowadays, the equinox is around March 20^th or the 21^st, but it was not always so: in the ancient world it was on the 25^th, coinciding with the Feast of the Annunciation. An imprecise calculation of leap years found in Julius Caesar’s reformed calendar caused a drift in the date. You can read more about this in the article The Vernal Equinox.

Meteorologists define the start of spring as the date of the Vernal Equinox, although this appears to be a bit late according to local weather patterns hereabouts. Older European traditions mark the start of spring on Saint Bridget’s Day or Candlemas, February 1^st or 2^nd.

---

Catholics, other Apostolic Christians, and the Jews have annual liturgical cycles that attempt to precisely coincide with the seasons of the year. These ancient faiths have a unique tie with nature that is usually not found in newer religious groups. Fundamentalist and liberal Christians will attack Catholicism because of this relationship, calling it pagan, although there are real pagans nowadays who ought to be of much more concern to these Christians.  Fundamentalist Christians inadvertently cut themselves off from sacred tradition when they attempt to divorce the faith from nature, ultimately leading to a gnostic-like attitude towards the material world, seeing it as evil; likewise, liberal Christians do the same when they fail to distinguish faith from nature, and slide towards pantheism.

The seasons of the year, controlled by the relationship between the earth and sun, are an undeniable influence on human life, even though technology and systems of control attempt to moderate these influences. But the seasons and natural cycles of the heavens make us, in many ways, what we are. What are we to make of it? Pagans worship these things as gods, while modernists consider them to be of little to no importance, who are more intent on imposing human will on nature.

Neo-pagans are usually passionate environmentalists, as is fitting for someone with a nature religion. Generally speaking, the environmentalist movement is highly anti-Christian, for many reasons. One particular claim is this biblical passage, Genesis 1:26 (NAB):

Then God said: Let us make human beings in our image, after our likeness. Let them have dominion over the fish of the sea, the birds of the air, the tame animals, all the wild animals, and all the creatures that crawl on the earth.

They claim that this dominion of man over nature has lead to the ruination of nature, because of industrialization, strip-mining, pollution, etc. But what does dominion mean? Is their idea of dominion a totalitarian, absolutist State, led by an all-powerful great leader or by a vanguard political party? Do they think that property rights are absolute, that each individual is master of his own destiny? But these are all heresies. The Catholic idea of dominion is neither totalitarian nor is it radically individualistic. Catholic good government is rather humble and seeks harmony rather than overwhelming command and control. Dominion becomes tyranny as orthodoxy is lost, and we find this loss of orthodoxy in modern forms of Christianity, and in the forms of political governance preferred by these sects.

Rather, dominion is a fact, not a license. Man names and classifies each creature into individual, species, and genera, but no animals are scientists. Man can kill any animal he desires, and animals run in fear from him. Man eats whatever plant or animal he considers good, and is not limited to an instinctive diet. Plants and animals are limited to specific ecological niches, whereas man lives wherever he wants. Man’s dominion over nature does not necessarily imply what the environmentalists claim.

Catholicism sees the things of nature as fellow creatures, worthy of being treated well, being also made by God. This attitude did not start with Saint Francis of Assisi (seen in his Canticle of the Sun), but can be found in numerous places in the Old Testament; here is Psalm 148 (Douay):

1 Praise ye the Lord from the heavens: praise ye him in the high places.
2 Praise ye him , all his angels: praise ye him, all his hosts.
3 Praise ye him, O sun and moon: praise him, all ye stars and light.
4 Praise him, ye heavens of heavens: and let all the waters that are above the heavens
5 praise the name of the Lord. For he spoke, and they were made: he commanded, and they were created.
6 He hath established them for ever, and for ages of ages: he hath made a decree, and it shall not pass away.
7 Praise the Lord from the earth, ye dragons, and all ye deeps:
8 Fire, hail, snow, ice, stormy winds which fulfil his word:
9 Mountains and all hills, fruitful trees and all cedars:
10 Beasts and all cattle: serpents and feathered fowls:
11 Kings of the earth and all people: princes and all judges of the earth:
12 Young men and maidens: let the old with the younger, praise the name of the Lord:
13 for his name alone is exalted.
14 The praise of him is above heaven and earth: and he hath exalted the horn of his people. A hymn to all his saints: to the children of Israel, a people approaching to him. Alleluia.

We shouldn’t be surprised that many neo-pagans come from heterodox Christian groups, and found them lacking, perhaps since these groups had no notion of harmony with nature.

Happy New Year and Most Popular of 2011

MAY EVERYONE HAVE a happy new year, in this the year of Our Lord 2012, and may this year be one of peace, prosperity, and joy.

In remembrance of the year past, here are some of the most popular postings and photographs from this website during 2010:



This is billboard, advertising the Latin Mass at Saint Francis de Sales Oratory, showing a monstrance on the left, and the shield of the Institute of Christ the King Sovereign Priest on the right. The Flickr photo-sharing website, which hosts most of my photos, tells me that many people view this image every day. I'm not sure that I'd call this my best photo of the year, since I hurredly took it while driving, but it is certainly the most viewed.  My second-most viewed image is of the monstrance itself:



This fine piece of liturgical art was made for the church itself. A crop of this photo is seen on the billboard above.



A mosaic of the Crucifixion of Our Lord, at Resurrection Cemetery, in Shrewsbury, Missouri. The following mosaic of Our Lady is located nearby:





A bridge and railroad tracks in the snow, at Carondelet Park, in Saint Louis, Missouri.

A Snowy Night in Saint Louis shows photos that I took on one bright night in the snow. It was so rare and beautiful that I was up until past 4 o'clock in the morning snapping photos. Several of the photos I took that night, as well as more than a hundred others taken at other times and places will be featured — full resolution and optimized for print — in a forthcoming book to be published by Reedy Press.



Clifton Heights Park, in Saint Louis, at dusk.



Cardinal Burke, with seminarians, at Saint Francis de Sales Oratory.

Half a Billion is an article which discussed the architecture and philosophy behind the Gateway Arch, the great symbol of the City of Saint Louis, which is one on the most-visited tourist attractions in the world. The title of the article is the estimated cost of renovating and expanding the Arch grounds, which intends to correct some of the flaws of its existing unrelenting Modernist design.



I didn't take this photo of Archbishop Ritter, but it was one of the most popular on this website. Here His Grace leaves for Rome on an aircraft named "Star of Rome of the West"; he came back a Cardinal.

A Warning to Schools that are "Catholic In Name Only" sounds the alarm that the Federal Government now arrogantly decides for itself which schools are Catholic and which are merely Catholic in name only. The secularization of our schools must reverse.



Detail of a monument, at Calvary Cemetery, in Saint Louis.





Palm Sunday procession at the Oratory.



Mosaic at Saint Elizabeth of Hungary Church, in Crestwood, Missouri.



Young Conservation Area, in Jefferson County, Missouri. Photo taken by the light of the full moon. I've been inspired to take photographs at night ever since I've seen these photos of Cambridge University.



Forest Park at night.



Flora and fauna are always popular. See the posts Spring Flowers at Shaw's Garden, Spring Wildflowers, Mushrooms and Moss, and Birds, Beasts, and Bugs.



















The articles Photos of Fog and Morning Fog shows pictures taken while enveloped by mist. Fog obscures our sight, and so is a natural symbol of mystery and divinity. Fog is a good photographic subject: by breaking the rules of Photographic Modernism, fog forces the artist to fall back on the older and more delicate principles of Pictorialism.







Tower Rock, located in the Mississippi River.



Dawn over the Meramec River.



Archbishop Carlson at the Oratory.



Sculpted bust of Saint Francis de Sales, at the Cathedral Basilica of Saint Louis.



Altar at Saint Peter Church, in Jefferson City, Missouri. This photograph has a muted color inspired by the old Autochrome process, the first practical method of color photography. See my posting Autochrome for more attempted examples of recreating the subtle color used by this French process . If you are technically minded and interested in this process, see my article Using ICC Profiles for Creative Color Control, which is located on my photography blog, The Refracted Light.



I was trained in the sciences, so I occasionally like to write on the subject. See Unsolved Problems and Higgs Boson Not Found. I got a degree in physics, but a close second interest of mine was theology; as a young man, I also wanted to be an architect and design cathedrals (and secretly I still do)! See my article Build Your Own Gothic Cathedral and New Geometric Patterned Art.

My interest in ecclesiastical architecture led me to dig more deeply into art theory. The subject of Catholic Art Theory is problematical, because not too much magisterial guidance is available, that is, we don't have too many authoritative sources; however, the great Catholic arts tradition speaks for itself, and as these churches and works of art were commissioned by our bishops, this tradition is authoritatively magisterial. There are a number of ancient and more modern philosophical works that proved to be greatly influential on this tradition. Also see the article Catholic Art, which goes into some detail about the great Tradition. Also see the article On the Sublime, which describes a very important idea in the arts, which tends to be sadly neglected or confused these days.

Altar at Saint Elizabeth Church, in Marine, Illinois.



Stained glass window at Saint Nicholas Roman Church, in Pocahontas, Illinois.

Saint Paul Church, in Highland, Illinois.



Silence! At the Cathedral Basilica of Saint Louis.



Immaculate Conception, in Augusta, Missouri.



Statue of Saint Francis of Assisi, at Saint Joseph Church, in Apple Creek, Missouri.



Holy Cross, in Cuba, Missouri.

Many of my readers come to this website come for the church photos. These posts proved popular this past year:

Photos of Saint Maurus Church, in Biehle, Missouri
Photos of Saint Joseph Church, in Highland, Missouri
Some Photos of Saint Mary's Church, in Alton, Illinois
Photos of Saint Elizabeth Church, in Marine, Illinois
Photos of Saint Joseph Church, in Louisiana, Missouri

And sadly, this posting, from a few days ago:

Saint Mary's Church, in Brussels, Illinois, Destroyed by Fire

Now as it so happens I've taken photos of many more churches this past year but have been too busy/lazy to post them. Perhaps during these upcoming cold days of winter I'll work on them.

Higgs Boson Not Found

HIGH ENERGY PHYSICISTS are disappointed at the lack of evidence for the elusive Higgs boson. Fermilab, a laboratory in the western suburbs of Chicago, Illinois, recently discovered an unexplained signal, which some had hoped to be the long-sought-after Higgs boson, but subsequent analysis disproves this. See the article Tevatron's DZero Cannot Find New Particle.

The Higgs boson is a theoretical consequence of the Standard Model of physics, a theory which unifies three of the four known fundamental forces in the cosmos: the strong nuclear force (which explains nuclear energy), the electromagnetic force (which explains electricity, magnetism, light, and the basics of chemistry), and the weak force (which explains a particular kind of radioactivity). Fundamental particles predicted by the Standard Model include the electron (familiar as the carrier of electricity) and the photon, which is an irreducibly small particle of light (The existence of photons leads to noise in digital photography). Other predicted particles include the quarks, which make up the familiar protons and neutrons found in the nuclei of atoms. All of the particles predicted by the Standard Model have been discovered — except the Higgs boson. As far as I know, no subatomic particle has ever been found which cannot be explained by the Standard Model.

The Standard Model does not contain an explanation of gravitation: the best current theory is Einstein's General Theory of Relativity; however, unlike the Standard Model, general relativity is not compatible with quantum mechanics. A logical extension of the ordinary classical or Newtonian mechanics on a very small scale, quantum mechanics posits that energy is not infinitely divisible, but rather is bound up in quanta, or small packets. [Back when I was at Caltech, I proposed doing some research in the interaction of relativity and quantum mechanics, but was told not to bother with it, because my proposal  threw out lots of the accepted theory of that era.]

The Standard Model was developed in the 1960s and is based on symmetry arguments. From the 1920s through the 1960s, experimenters had discovered a wild variety of subatomic particles, and many despaired of the fundamental idea that nature is ultimately simple. Eventually attempts were made to put these particles in a rational order. In chemistry, the development of the Periodic Table of the Elements immediately led to the prediction of previously unknown elements, which were indeed discovered either in nature or subsequently synthesized. Likewise, the development of the Standard Model led to the prediction of various subatomic particles — and multiple researchers quickly proposed the existence of what is now called the Higgs boson.

Since the Higgs boson has the interesting property of giving mass to the other particles in the Standard Model, it is quite important to our theoretical understanding of the cosmos. The existence of the Higgs boson explains a lot; the non-existence of the Higgs boson means that some other theory, such as a quantum gravity theory, is needed to explain the existence of mass.

---

The Higgs boson is often referred to in the press as the ‘God particle’, an unfortunate name taken from a popular science book. That name comes from the theory of materialistic monism, a theory that states that the observable cosmos is all that exists; and this monism is at the core of much modern scientific and secularistic thought — even American public school pedagogy is based on materialistic monism.

The rational consideration of existence goes by the name metaphysics. Nowadays that word has an irrational New Age connotation, and this is partially due to Catholic universities denigrating the study of metaphysics in favor of more worldly and materialistic philosophies; nature abhors a vacuum, and so the word metaphysics was taken over by others.  However, metaphysics is a good solid subject: it merely asks the questions ‘what exists?’ and ‘what is it like?’ The existence of the Higgs boson is a currently a metaphysical question, as is the question of the validity of the Standard Model.

The standard history of science ends with Democritus and restarts again with Galileo, ignoring everything from Plato through the Medieval schoolmen. But we can still learn things from these ancient thinkers. In particular, we can consider the Catholic notion of the Great Chain of Being, which posits — via an argument of fittingness — a hierarchical structure of all that exists, which starts from God and continues down through inanimate matter.

The Chain of Being ends in fundamental matter, which is quite remote from the Source of all being. With matter being so remote, it is fitting to assume that the properties of fundamental matter are very simple, and get simpler the closer we look. So the wild variety of chemistry devolves to a simpler set of chemical elements, which are made up of simpler, well-ordered particles; these particles are vast in number, but are reduced to a small collection as described by the Standard Model. The existence of the Periodic Table of Elements is not in dispute, while the existence of the Standard Model remains a metaphysical problem.

As we go all the way down the chain, we ought to find simpler models with simpler interactions, which display greater symmetry — models that mathematical physicists call beautiful. Beauty is a great organizing principle in fundamental physics, with the more beautiful models having a greater likelihood of being true. Beauty and simplicity was one of the factors which led to the acceptance of the Standard Model.

As we move up the chain of being, these symmetries are broken, and things get messier. The periodic table of elements gets messier as the elements get heavier, chemicals have more complexity as they get larger, and living things get more complicated as they go up the chain: protozoa are more more complicated than viruses; fungi are more complicated than protozoa; plants are more complicated than fungi; higher animals are more complicated than lower; and rational animals (which as far as we know only consists of Man) are the most complicated beings in existence. Consider the existence of sexuality — what is pretty simple in lower orders becomes highly messy in human beings.

Platonic and Catholic philosophy also posits the existence of a higher order of being beyond the material world — a metaphysical, intellectual, or spiritual world. It is beyond and above mere matter, and is outside of space and time. Rational animals are seen as straddling these two worlds, being both material and simultaneously spiritual.

Part of the order of being in the metaphysical realm is mathematics. According to Plato, an understanding of mathematics is a great step towards enlightenment — for example, being able to see with the mind's eye, a true, mathematically perfect triangle, beyond the mere material drawn copies of triangles. Because math exists outside of space, time, and matter, it is a reliably true guide to understanding the material world; this is implicit in the modern sciences, especially in physics.

In his parables of the divided line and the cave, Plato asks us to go beyond mere opinion to more solid reality, which includes science of the material world, and mathematics in the metaphysical world — and even higher. Sadly, so many people live and die because of mere changeable opinion, with the most opinion-bound and power-hungry people being the worst of humanity. We can break the chains of this bondage to opinion by rising to contemplate higher things.

As we ascend into the higher realms, things get simpler. Angels and demons, having intellect but lacking bodies, are simpler than humans. Mathematics is extremely complicated, but less complicated than human psychology, and the logical foundations of math are much simpler than say, non-linear mathematical analysis. We reach ultimate simplicity when we rise to the Source, or God. Spiritual directors say that living an excessively complicated life is problematic, and that simplifying your life (in the upwards direction) is key to ultimate blessedness.

God is not a particle, but the existence of the Higgs boson is still an important metaphysical question.

Unsolved Problems

AN INTELLECTUAL HISTORY of the world, according to contemporary scholars:

• Evolving from bonobo-like apes, early hominids slowly developed language skills, starting with basic grunts and gestures, to pidgin-like vocalizations, and finally to languages with full grammar and syntax.
• Increased travel and leisure led to an intellectual interest in the natural world. The Greek philosopher Democritus (ca. 460 - ca. 370 B.C.) posited the existence of atoms, indivisible primary constituents of matter.
• ...nothing interesting happens for 2000 years...
• Galileo (1564-1642) confronts the backwards Catholic Church by declaring that the sun — and not the earth — is at the center of the cosmos, and also that the moon has no influence on the ocean tides. Because he is a scientist, Galileo is harshly persecuted, worse than any other man in history. Scientists subsequently return the favor many times over.
• Progress in the sciences is made through the work of Descartes, Pascal, Newton, Leibniz, Linnaeus, Lavoisier, Gauss, Faraday, Darwin, Kelvin, Maxwell, Mendeleev, Lorenz, Freud, Planck, Curie, Jung, Einstein, Bohr, Heisenberg, von Neumann, Skinner, Levi-Strauss, Turing, Feynman, Hawking, and many others. Each generation of scientists, building on the work of those before, increases and perfects Man's knowledge of the world. Progress is inevitable.
• The accelerating increase in the rate of change of scientific progress means that soon we will pass through a technological singularity, after which there will be no more want, war, or suffering (most likely because everyone will be dead). Until that time, more money is needed for research.

One of my goals for Lent is to read more. Recently I was looking at lists of unsolved problems in science, and one such list can be found here. What I find fascinating is that while many of these problems are quite complex and involved, such as superstring theory in physics, other unsolved problems — like why we sleep and dream — are commonplace. Just for fun, let's look at some of these unsolved problems:

Paradox of the plankton

Evolution seems to demand that when multiple species compete against each other for limited resources, one species will become dominant, wiping out other species, until evolutionary progress creates a new, more fit species, which in turn will wipe out its former rival. We can extend this evolutionary analogy to other cases, such as men versus women, capitalists versus proletariats, Walmart versus main street, and so forth, all of which will struggle for supremacy. But wide varieties of plankton species coexist, and we do not see a struggle for mastery.

I don't see this as a paradox. Rather, I see winner-take-all competition as the exception and not the rule: certainly among humans, competition is something that is very often imposed, and is not the normal state of affairs. In nature, a fairly harmonious balance is what we almost always see, and that is as it must be.

Why are there two sexes?

Undoubtably, there are some people who would prefer to reproduce themselves asexually via cloning, and I claim that these people are likely the worst candidates for reproduction. From an evolutionary standpoint, sexual reproduction is seemingly poor because only half the individuals in a population bear offspring, and each individual can only pass on half of its genes at one time. And so, a supremely fit individual cannot fully reproduce itself and must deign to mix its genes with a genetic inferior (which is why some misguided people want cloning). However, sexual reproduction is quite robust in providing healthy genetic material to individuals, because the process of genetic recombination usually eliminates damaged genes.

Usually it is asserted that sexual reproduction increases variation in the population, but this may not be true when you consider whole populations. Although controversial, some think that sexual reproduction instead acts as a brake on evolution, since mutations tend to be largely eliminated. Forced breeding, as we see with domestic animals, does generate a great variety of individuals, but only at the risk of very poor fitness. Tabbies and mutts are usually healthier than purebreds. My thoughts on forced breeding can be found here.

Evolution is often seen by theorists as ‘whatever’; but rather all organisms by necessity follow the laws of chemistry, physics, and mathematics. Some people go into biology because they like science but aren't good at math. Scaling, for example, is often considered to be a problem in biology: small organisms simply don't operate like larger ones. But this is a problem only when you ignore the math. Problems in biology become certainties in mathematics. Scale is also a major problem with fiction writers.

Why we have sexual reproduction is considered by biologists to be an unsolved problem, although there are some odd and largely unsupported theories to its origin. But let us consider the possibility that sexuality is somehow inherent in the cosmos or is a basic reality. A bird cannot fly unless its wings correspond to the laws of aerodynamics; might it also be possible that sexuality is somehow a law of nature? Chinese philosophy certainly thinks so — consider the concept of yin and yang. We find similar ideas in traditional Western philosophy also, although they aren't popular these days. Alas, many people go into science because they like the truth, but aren't good at philosophy: this in my opinion is a huge problem.

The problem of consciousness

Why is there a subjective component to experience? Although this is a core fact of theology, scientists have made no progress on this problem. For various reasons, scientists self-limit themselves in this inquiry, and usually seek purely materialistic solutions, but this doesn't work. Some hold to a theory of panpsychism or hylopathism (such as Phillip Pullman's series of anti-Christian novels) where consciousness is somehow inherent in matter or in some particles. This seems rather arbitrary and reductionistic, and its First Cause is a lesser god indeed. Quantum-mechanical solutions to the problem are problematical since we don't know the meaning of quantum mechanics.

Others hold that consciousness is an illusion, but to whom is this illusion presented? Undoubtably there may be people who believe that they are the only ones who are conscious, with the rest of us being unthinking sheep. Certainly you can see the side-effect of this kind of solipsism when it comes to public policy: if our laws tell us what to do in minute detail, then undoubtably our leaders must think that we are in fact unthinking machines. This attitude likely is derived from the Puritan theory of double-predestination, where some people can know for a fact they are saved, while others are certainly damned. I've written more about this problem here.

Although the mind-body dualism of Descartes seems to be a popular opinion, scientists are sharply divided on philosophy of the mind. Catholic anthropology posits the existence of a world of matter and a world of spirit, with the human person straddling the two worlds; the human brain acts as the sense organ for the spiritual human intellect, and this distinguishes us from pure matter which has no intellect, and from the spiritual angels who are pure intellect. This anthropology has a bit in common with Cartesian dualism, but it avoids certain existential problems inherent in dualism, since it does not set up a divorce between the body and the spirit.

The Problem of Free Will

This is closely related to the problem of consciousness. Scientists are mixed in their opinion of free will, and adherents to one side or the other will often embrace the extreme political philosophies of libertarianism or totalitarianism. Some Protestants deny free will, whereas free will is a core doctrine in Catholicism, Orthodoxy, and Judaism. The determinism inherent in reform theologies led to determinism in the sciences and politics: the discovery of the apparently non-deterministic quantum mechanics did much to change opinions among scientists. This apparent non-determinism awoken a new interest in spirituality among scientists, but they notably reject Catholic spirituality in favor of philosophies of the far east or the New Age. Ultimately we can hope that scientists will recognize the Catholic roots of the scientific method, as well as the reason why science is even important.

Many think that free will and determinism are contradictory. Catholic theology instead posits the greatness, infinitude, and transcendence of God as the solution to the paradox. A lesser god cannot both allow freedom while knowing the future, while the God who transcends space and time can. Unlike the libertarians, Catholics believe that free will in humans is limited by the damage caused by Original Sin.

Why do we dream?

It was horrible. I was flat on my back for hours, unable to move and unconscious of everything going on around me. During this time I suffered extreme hallucinations that I thought were real. It is hard to believe, but people call this sleeping ‘normal’.

Why do we dream? Nobody knows. We don't even know why we sleep. Maybe the brain is always active, so we interpret this activity as dreams. Perhaps dreams have something to do with sorting out our thoughts and experiences of the day, saving some as memories, and discarding others as junk; this seems quite plausible. Maybe dreams can include the intersection of our mind with the spiritual world: most scientists, but not all, would discard this hypothesis. I wrote about dreaming here.

Nature versus Nurture

What is the relative importance of our genetics compared to our environment? Even relatively lately, your worth as an individual was seen as being largely dependent on one or the other: Nazis judged genetics, while Communists discriminated on environment.

Obviously genetics plays an important role. Take a human baby and a rock, raise them in the same environment, and the rock will show hardly any personality development at all. But environment also plays a role: raise a baby in Alaska, and raise its identical twin in Kenya, and they will likely be quite different in adulthood. This relationship is now seen to be particularly complex, with great interdependence of factors. Quality schooling is more predominant among certain genetic groups, but do certain genes cause a predilection for good education? Or if you send children from various groups to the same school, can we expect a similar outcome? But unlike the 20th century, scientists now mainly agree that this is not an either/or issue of nature versus nurture, but rather is nature and nurture. This does not mean that we understand how the factors interrelate, and so studies on personality development still have difficulty.

This is still considered a major unsolved problem, and is mirrored by the controversies on grace versus nature in theology. In fundamentalism grace predominates, in liberalism nature is dominant, while Catholicism marries grace and nature together.

We ought not think that a reasonable understanding of nature and nurture will be found in the political sphere. Some groups organize themselves, or are distinguished, by what is claimed to be purely genetic factors — such as ethnicity and sex — while others groups emphasize nurture, such as religion and ideology. The modern trend is to invert nature and nurture: sex, for example, is claimed by some to be a purely cultural phenomenon, while some have proposed that religious belief is caused by a 'god gene'; however I see this as being more of a political strategy rather than a careful investigation of the truth.

Catholic anthropology emphasizes the sameness of human beings, while modernity emphasizes, and strongly acts on even minor differences among humans. This variation in belief has extreme consequences, including the rise of slavery, oppression, and genocide in the post-Reformation world.

“Nurture” was once assumed to be mainly family life, and was particularly focused on the mother. Nowadays, environmental factors such as peers, schooling, and the media are seen to have a greater effect on personality than family life, which is a strong argument for homeschooling.

But notice that this debate is based purely on determinism. We must also take free will into account, but most contemporary scientists are uncomfortable with this.

Does extraterrestrial life exist?

The idea of life in outer space was rare until the modern age, particularly after it was discovered that Earth does not physically occupy the center of the cosmos. The common scenario presented by fiction writers is that extraterrestrial life is quite common, and that sometime in the near future we will be able to travel to these exotic locales and interact with these lifeforms, if they haven't visited us already. But we ought to contrast this fictional view with the apparent fact that space, as far as we can tell, is lifeless, and there is no practical way we can travel much beyond earth (of course, this reality doesn't have much dramatic potential). I wrote more about space travel here.

What is the structure of water?

The composition of water, we are told, is H₂O, but that isn't quite right, for water molecules connect to their neighbors in complicated and unknown ways. Water just isn't like other chemicals.

Do one-way functions exist?

In computer science, a function is a process that takes an input, transforms it in some way, producing an output. For example, take a function that simply multiplies the input number by two. Given an output, you can easily determine the input by dividing by two. But some functions appear to be vastly more complex; given an output, you can't quickly determine the input with a simple formula, even if you know precisely how the function works. These are called one-way functions.

Here is another simple function: input two prime numbers and multiply them together, producing an output.  Given an output, is there any quick way to determine the two numbers? (A prime number is indivisible by any positive whole number other than 1 and itself: 2, 3, 5, 7, and 11 are examples of small prime numbers.) OK, if the function returns 6, you know that the input numbers must be 2 and 3. But what if the number is really, really big? At present time, the only method of solving this problem involves trial-and-error: is the number evenly divisible by 2? No, well how about 3? What about 5 then? 7? And so on. A 232 digit number was successfully broken in the year 2009 using hundreds of computers that worked over the course of two years. Just adding a few digits makes the problem vastly harder to solve; no matter how fast computers get, you can easily overwhelm them with little effort on your part.

One way functions are used in cryptography, which is not only important to spy agencies, but also to ordinary folks who use wireless technology. The common WiFi networking standard, found in innumerable Internet cafes, uses one-way functions to ensure privacy of data transmitted over the air.

But do one-way functions actually exist? Is there some simple formula, currently unknown, which will quickly solve them? This is closely related to NP-complete problems, for which no fast solution is known.

Valuation problems in economics.

Investors are more likely to highly value companies that pay dividends, contrary to utilitarian models. This dividend puzzle ought to be easily explainable. Economists tend to define all money as being equal, but a bird in hand — a dividend — is worth two in the bush — that is, owning a stock. Cash in hand is yours, while cash tied up in companies is not quite so easy to extract, and getting that cash is contingent on constantly fluctuating stock prices.

Options allow an investor to manage risk, but economists are puzzled since they appear to be overpriced. There are formulas which calculate the expected price of options, but some of these severely underestimate risk, and so give a wrong value for options. These formulae also incorporate something called a “risk free interest rate” — typically defined as the U.S. Treasury bond rate — but that concept is unreasonable; I can come up with all sorts of plausible and likely scenarios which will drive any investment to zero, including treasury bonds. Are theorists really that optimistic? Should we be listening to them? Reality is riskier than theory, and so it is plausible that options ought to cost more.

Is there an objective measure for the quality of translation?

This is highly debated with the new translation of the Roman Missal into English, due to come into use at Advent this year.

In theory, the dynamic equivalence translation found in the current missal is valid, since it theoretically converted idiomatic Latin into good idiomatic American English. However, the translators sometimes seemed to go out of their way to change the meaning of the texts, which is something else altogether. The translation is also rather flat and unpoetical, unlike the original Latin.

A literal translation, which translates a text word-for-word, will usually produce a stilted text. The presence of idioms and unusual word orders will make texts hard to understand. Changing word order and substituting some idioms in the translation can make texts more readable while still being largely literal. But perhaps we should keep a certain amount of Latinity in our translations in the Latin Church; Catholics ought not be ignorant about their liturgy and Rite.

A literary translation will attempt to preserve the meaning and style of a text. This is the most difficult to do well, and requires great expertise. How do you translate puns, rhyming, and alliteration? The old Anglican Book of Common Prayer is well-regarded as a largely elegant translation of Latin texts into English.

The most extreme philosophies will state that even primary texts themselves have no meaning, and so there is no objective component to translation, and so translation is arbitrary. But this is nonsense; language is more than just random utterances. We can't live if language is meaningless, or if it means only what we want it to mean. Words are not mere vocal utterances, they instead ought to correspond to real things, and we must attempt to keep these real-world references intact. Never, ever allow a postmodernist near your texts.

The Yang–Mills Existence and Mass Gap.

“Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on R4 and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in [45, 35].”

Um. I don't think I have anything to offer here. Math is hard.

What is randomness?

That there are several schools of thought in probability theory — utterly unlike most other branches of mathematics — leads me to believe that the theory of randomness is a great unsolved problem. Can we state with certainty that something is utterly, irreducibly random, or is there order behind the chaos?

Probability and statistics is the bane of most university students, largely because much of it is counter-intuitive, and (I think) that some of it may be — well, not strictly wrong, but problematic. Mathematics can be seen as logic taken to an extreme degree, but there seems to be an attitude that it is all a complex word game, not relating to anything real. But if one of your axioms is false, has no being, has no reality, then your theorems based on that axiom will fall apart, and that is precisely what we see in probability theory: people can come to differing conclusions based on the same axiom of randomness. We see this when dealing with imaginary numbers — they are useful, but need to be handled according to convention, and so we can conclude that imaginary numbers are not real, but are truly fabricated. (There are better mathematical entities than imaginary numbers, which have the same usefulness without the need to handle them according to arbitrary convention.)

Is a random variable something that exists ontologically? Or is it just a useful convention based on our ignorance? I tend to think that you can't be both Catholic and think that random variables have ultimate existence, since you'd be denying God's omniscience, and you would be positing randomness as a First Cause. One of the best purely secular explanations of this is found in Volume 1 of the book The Art of Computer Programming, by Donald Knuth, a book which also helped me think of art as being something more than just what is found in museums. Here Knuth, who is a practicing Lutheran, states that we ought to consider randomness to be merely a measure of naive unpredictability, even if the process used to generate these random numbers is completely deterministic (see the section above on one-way functions). The problem of randomness figures into debates on the interpretation of quantum mechanics and free will.

Some of my thoughts on randomness and the scientific method are found in the article How Random?

What is Art?

This is an easy one, but too often we are given narrow explanations by specialists, which isn't helpful and leads to controversy. According to Saint Thomas Aquinas, art is “right reason with regard to external productions” (recta ratio factibilium). It isn't just something that hangs in museums, or something purely aesthetic. Everything habitually made by humans according to right reason is art. I wrote more about it here.