Wilczek: Fundamentals

I just finished Fundamentals: Ten Keys to Reality (2021), by Frank Wilczek. It’s yet another book explaining fundamental physics for lay readers, and it does so pretty much entirely within the bounds of mainstream science. I enjoyed reading it, but it’s mainly a review of physics as we know it.

I saw it on the library’s list of new books and put it on hold back on May 14th. It didn’t become available until September 3 — more than a three-month wait. Apparently lots of people wanted to read it.

Bottom line, I recommend it as an easy and enjoyable read, especially for those with a more casual interest in physics.

One thing I like about ebooks is the ability to highlight text. Sadly, the library app doesn’t allow me to copy highlighted text, but at least there’s a listing of the highlights and easy navigation to them. Any quote you see in my posts that’s from a library book I typed in.

[One of the best gifts my parents gave me was forcing me to take a typing class back in high school. At the time, it was actual typewriters, but it sure did set me up for the home computer revolution.]

Anyway, the book was mainstream enough there isn’t much to talk about, but some bits I highlighted as tasty. As often seems to happen, many of those bits come from the Preface or Introduction, when the author is laying out the path they intend to follow in the book.

§

In the Preface, Wilczek points out how many scientific heroes — Galileo, Kepler, Newton, Faraday, Maxwell — were devout Christians studying what they perceived as God’s handiwork. Even Einstein wasn’t immune. One of his more famous quotations is: “Subtle is the lord, but malicious he is not.”

The spirit of their enterprise, and mine here, transcends specific dogmas, whether religious or antireligious. I like to state it this way: In studying how the world works, we are studying how God works, and thereby learning what God is. In that spirit, we can interpret the search for knowledge as a form of worship, and our discoveries as revelations.

Good way to look at it. Under the presumption God created the universe, and that He created a sensible rational universe, it is surely something we can study. After all, if God did create everything, that includes our curious, investigative, inventive minds. Given those minds are what set us apart from the animals, using them to study His handiwork seems almost a calling.

A key aspect of my life’s path has been navigating between my spiritual and scientific views. I’ve never seen them as exclusive or incompatible, but as the Yin and Yang of a complete soul. (“Soul” as in when an aircraft has “124 souls onboard including crew.”)

Wilczek mentions three key themes the book follows:

The first of those themes is abundance. The world is large.

He means this in multiple ways. Compared to the Earth, the Solar System, the Milky Way galaxy, or the universe, we’re increasingly tiny. Compared to cells, atoms, protons, or the Planck Length, we’re increasing huge. There may be 300-billion (or so) stars in our galaxy, but there are about 30-trillion (or so) cells in a human body.

Compared to the 13.8 billion year life of the universe so far, we’re less than the merest instant. Yet for each of us, a lifetime lasts a whole lifetime, and everything we do fits into that span. All of human written history fits into about 300 generations, yet it brought us from campfires to rockets.

While we have only about 86 billion neurons, under 30% of the stars in the galaxy, the number of thoughts those neurons can have far exceeds the number of stars in the visible universe. And those, physically speaking, comparatively tiny minds are capable of attempting to figure out everything from the Planck Length on up to the entire cosmos (or cosmoses as the case may be).

The second theme is that to appreciate the physical universe properly one must be “born again.”

Wilczek writes about his new grandson, Luke, and how that infant began to study and learn about the world he was born into.

In these and many other ways, I could see that Luke was constructing a model of the world. He approached it with insatiable curiosity and few preconceptions. But interacting with the world, he learned the things that nearly all human adults take for granted,…

He compares babies to scientists making experiments and drawing conclusions. As babies we construct a model of reality that allows us to successfully navigate the world. Something as basic and simple as ‘when I put an object down it stays there’ is a property of reality we must all learn.

Crucial is our sense of wonder and curiosity.

§

A bit he mentions in passing caught my eye:

The light has broken up into individual quanta, and quanta cannot be shared. At this fundamental level, we experience separate worlds.

Two people can stand side-by-side, look at the same thing, apparently see the same thing, and yet each is receiving their own personal stream of photons from the scene.

Our experience of reality is one of isolation, both inside the confines of our own brains, and in the stream of information we receive from the physical world. We might as well be brains in a vat — we could certainly never tell the difference!

§

In the Introduction Wilczek says (about applying an existing theory to a new domain):

If it works, then you’ve discovered something useful; if it doesn’t then you’ve learned something important. I’ve called that attitude Radical Conservatism, and to me it’s the essential innovation of the Scientific Revolution.

The third theme of the book is the idea of a “radically conservative approach” — essentially the notion of open exploration bounded by physical evidence and experiment. Newton was a radical.

Wilczek covers the same progression of thought starting with Ptolemy, Copernicus, and Kepler, studying the heavens, Galileo studying how objects fall on Earth, and that radical, Newton, merging them and kicking off the aforementioned Scientific Revolution.

He quotes the radical rascal:

To explain all nature is too difficult a task for any one man or even for any one age. ‘Tis much better to do a little with certainty & leave the rest for others that come after you.

That’s the whole ‘standing on the shoulders of giants’ thing.

I also liked a quote due to John R. Pierce:

We will never again understand nature as well as Greek philosophers did… We know too much.

The more we learn, the less we seem to know. (We owe to Pierce the name transistor.)

§

Regarding the provisional nature of our theories:

Conversely, to the extent that GPS works, its success reinforces our confidence in all the underlying assumptions, including the assumption that Euclidean geometry describes, with good accuracy, the reality of spatial geometry on earthly scales. And so far, GPS has worked flawlessly.

Which should be taken as strong evidence in support of Special and General Relativity as well as in support of the quantum mechanics used in the design of the electronics. Note the crucial “in support of” — our theories are almost never proven.

The fact that Euclidean geometry fails to provide a complete model of reality does not detract from its mathematical consistency nor invalidate its many successes. But it does confirm the wisdom of Gauss’s fact-checking, radically conservative approach.

This bit caught my eye, too:

Yet a nucleus extends less than one-hundred-thousandth of its atom’s radius and — being nearly spherical — occupies less than one part in a million of one part in a billion of its volume. Those are literally astronomical numbers. The way a nucleus is dwarfed by its atom parallels how the Sun is dwarfed by its surrounding interstellar space.

Kind weird how atoms are a tiny, tiny positive seed wrapped in a comparatively vast electron cloud. It took us a while to figure that out.

§

Regarding the apparent meta-law that, in defiance of entropy, grows complexity from simple rules, basic building blocks, energy, and time:

Self-reproduction unleashes the power of exponential growth. Starting with one cell, after ten generations of doubling one has more than a thousand cells, and after forth or so generations one has trillions of cells, which are enough to make a human body.

Giant oaks from tiny acorns grow!

Regarding how every electron is identical to every other electron (true for all fundamental particles — no variation among them):

How does Nature do it? By tracing the common origin of all photons to a common universal electromagnetic field, we come to understand their otherwise baffling sameness. And we are led, by analogy, to introduce a field — call it the electron field — whose excitations are electrons. All electrons have the same properties, because each one is an excitation in the same universal field.

I’d never considered it before, but if “particles” are tiny little things the universe makes, how does it make them so consistently regular? Simple; each type is a manifestation of the same underlying field.

§

Talking about dark matter and dark energy, Wilczek explores other instances of “dark” (i.e. hidden) solutions to observations. For instance, the orbit of Uranus had anomalies that suggested something hidden affecting it. Turned out to be Neptune. The orbit of Mercury had anomalies that suggested something hidden about gravity.

As a little joke, to summarize their historical parallels, we could say that dark matter is from Neptune, while dark energy is from Mercury. The encouraging message from history is that good scientific mysteries often find worthy solutions.

Never underestimate those curious minds! It’s been said that much scientific progresses starts with the phrase, “Huh. That’s weird…”

It’s a neat way to see dark matter and dark energy.

§

Wilczek has a whole chapter, Complementarity Is Mind-Expanding, that speaks to one of the key observed differences between quantum and classical physics: In the quantum world certain properties are mutually exclusive.

The famous Heisenberg Uncertainty Principle is one example: knowing both the precise position and momentum of a particle is impossible because those properties are mutually exclusive.

Wilczek mentions a wonderful analogy (due to a musician friend of his): The contrast between harmony and melody:

Harmony is a local analysis — here monitoring a moment in time, rather than a point in space — while melody is a more global analysis. Harmony is like position, while melody is like velocity.

Simultaneous notes in a given moment create a harmony, but a sequence of notes over time create a melody. The two are mutually exclusive. There is no melody in a chord, and there is no harmony in a melody (or rather there are many many different harmonies along the way).

§

Wilczek makes an important point regarding competing views and debates about them:

Of course, trying to understand different ways of thinking does not necessarily mean you must agree with them, much less adopt them as your own.

Just about every Thinker I’ve ever known or read about engaged in passionate, sometimes even fractious, debate. It seems to go with the territory of an intellectual analytical life. It tends to grow strong worldviews.

That said, science has a special status. It has earned enormous credibility, both as a body of understanding and as an approach to analyzing physical reality, through its impressive success in many applications. Scientists who define themselves narrowly fail to enrich their minds, but people who avoid science impoverish theirs.

Indeed. Life is both Yin and Yang.

To quote Einstein again: “But science can only be created by those who are thoroughly imbued with the aspiration toward truth and understanding. This source of feeling, however, springs from the sphere of religion. To this there also belongs the faith in the possibility that the regulations valid for the world of existence are rational, that is, comprehensible to reason. I cannot conceive of a genuine scientist without that profound faith. The situation may be expressed by an image: science without religion is lame, religion without science is blind.

Stay balanced, my friends! Go forth and spread beauty and light.

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

21 responses to “Wilczek: Fundamentals

  • Sai Sundar S

    Anoraniyaan mahitho maheeyaan is a sanskrit verse to describe God. GOD is a conscious Upanishaths clearly explain the science and God. The duality is clearly explained by Adi Sankara by his principle called Adwaitha . Dwaitha is the sanskrit word of duality . Brahma sathyam Jagadh midhya. The universe is may a ie illusion we are calling it as holographic. .If you get time please do the treasure hunt….

  • Peter Morgan

    Regarding Harmony and Melody, one path is to use a wavelet basis. There’s an engaging article about Daubechies, who is the eponym of the “Daubechies wavelet”, here, https://www.nytimes.com/2021/09/14/magazine/ingrid-daubechies.html, which also engages with the math at a nice level.

    • Wyrd Smythe

      I don’t understand,… how does that simultaneously resolve harmony and melody?

      • Peter Morgan

        Wavelets give us a complete and orthogonal basis that is “between” the basis elements of another orthogonal basis. A given wavelet is, for example, neither at a single frequency nor at a single point. Insofar as a “Harmony basis” is complete and orthogonal, a given wavelet will not be just a single harmony; same for a “Melody basis”, but now that I’m thinking it through more carefully because of your reply, I’m not aware of very formal mathematical statements about the properties of either, so my comment seems to me now to be somewhat at cross-purposes with your post. Anyway, I don’t think a wavelet basis “resolves” harmony and melody, it’s more that it gives a more formal and in some ways less informative way to encode the information in a piece of music.

      • Wyrd Smythe

        I don’t know about cross purposes. More an addition to or extension of? Wavelets, as I understand them, straddle between frequency and time localization, and it takes many sine waves to comprise them.

        For a moment there I thought maybe you had a way around the Heisenberg Uncertainty Principle! 🙂

      • Peter Morgan

        I think I’d say that the mathematics of the HUP is not something to find a way around, it’s something to get used to. x and d/dx do not and cannot commute, and, because of that, if we transform between the basis sets of eigenstates of x and d/dx —if we use Fourier transforms— then there will be measurement incompatibility and there will be the HUP. The scale of the HUP is about the amplitude of the noise that affects something that is being measured, which can be small or large, but if Planck’s constant is the noise amplitude concerned, we haven’t found a way to reduce it the way we can reduce thermal noise by refrigeration.

        Crucially, I think, classical physics could have or ought to have introduced measurement incompatibility in the 19th Century, when it was already known that two probability distributions sometimes are not marginal probability distributions of a single JOINT measurement distribution.

        Classically, this is just called “contextuality”: that measurements typically cannot be performed in the same context, although we usually focus on measurements that are measurable in the same context. One way of thinking about experiments that I find helps a lot is to notice that all actual measurement results both happened in one context, the past, and, by applying appropriate post-selections, happened in multiple contexts, separate parts of the past: the latter structure introduces many complications, but it also introduces many useful opportunities.
        2¢.

      • Wyrd Smythe

        Yes, that’s my view of the HUP also. It’s a direct consequence of a wave-based mechanics wherein conjugate properties are Fourier transforms of each other. (Although I do know a guy who insists the HUP is strictly an epistemological issue.) That’s why I was startled by what I misunderstood you to suggest about wavelets! Confusion resolved.

        I think you might Lee Smolin’s thinking interesting. He speaks of a view-based physics that limits thinking to light cones, and he’s speculated about what surprises extremely short-scale physics might hold. He’s also talked about how a (very successful) Newtonian view created a ‘physics in a box’ way of thinking that would explicitly denies noise on the presumption it doesn’t belong in the box. There would be no noise in the abstraction of the system being studied.

        So classical physics qua Newton was a view that explicitly excluded noise. Maybe that was the mistake that led to the blind alley were in? An intriguing idea!

      • Peter Morgan

        If we add noise into classical physics, with one way to do that being to use probabilities, then one way to push into the consequences is to model measurements using operator algebras, which leads to Hilbert spaces, which allows us to model anything that quantum physics can model. The big question seems then to be how to understand the measurement problem.

        I’ve described an approach that I think works very well to you before, but I’m definitely still trying to find a better way to say it, so to say it again, whether better or not: quantum states are not absolute things, but only relative to what operators we choose to model the measurements we perform, and we can always make choices of operators in which there is no collapse of the quantum state. There’s a measurement problem only if we choose operators that do not commute for measurements that for a different choice behave straightforwardly.

        Sadly, that still seems like word salad. I know what I’m saying, but I will have to forgive most everybody else for thinking it’s nonsense.

        Lee Smolin is always interesting but he’s also always on a course slightly different from mine, for a multitude of reasons.

      • Wyrd Smythe

        I guess a question I have is what the value is of adding randomness to classical physics. What are we saying about physics if we do that? Is the randomness of quantum physics just the way it is, then? Is this part of merging classical and quantum?

        My question about choosing operators (an idea I’m still not clear on) is what kind of freedom I actually have. My impression is operators depend on what we want to know about a quantum state, so what happens when we want to know orthogonal things?

      • Peter Morgan

        In classical physics, we can either take the random noise to be caused by some process that happens at a smaller scale and that interacts in a some way with our measurements, like Brownian motion, or we can just model the noise carefully without worrying too much about what causes it, if anything. For now, any model we might introduce for quantum noise would be a wild, uneducated guess, so my feeling is that it’s best to characterize the noise with increasing care and not to fuss too much about what our descendants will have better data for. Not to give up about it, but not to be stupid about guessing. Quantum theory is the same, as far as I’m concerned, in that quantum noise could be described by a lower level dynamics, at 10^{-40} meters or 10^{-200} meters or whatever, that uses a different-from-Planck constant as the scale of its noise, but all still within the quantum physics formalism; or quantum noise could truly be completely irreducible, just the way the Universe is, was and will be.

        When we use operators and states to model measurement and their results, we have ρ(A) as the expected value for the measurement A’s results in the state ρ and we can construct ρ(δ(A-x)) as a probability density for A’s results, et cetera. There’s a duality between transformations of the measurements and the states, so that we can always find a ρ’ and an A’ for which ρ'(A’)=ρ(A) [and also, ρ'(δ(A’-x))=ρ(δ(A-x))]. This freedom is what makes possible the so-called Schrödinger and Heisenberg and other “pictures”. Empirically, we can choose either ρ'(A’) or ρ(A) because they represent the same expected measurement results. The choice of A’ or of A might make a difference to our intuitions about the experiment, and switching to ρ’ means that we also have to switch to B’, C’ for all measurements, so there is overall consistency and more to consider, but we can choose either the Schrödinger, the Heisenberg, or another “picture” quite freely for whatever mathematical, intuitive, or other benefits we get from making one or the other choice, without the measurement results changing.

        There is more freedom than this, however, because for JOINT measurements, we can use operators that commute with each other OR we can use operators that do not commute with each other, with, loosely, “collapse” of the quantum state in between. That “collapse” of the state, however, can instead be thought of as a constraint on the following measurement, so that we can write ρ(A.collapse(B)), or we can write ρ'(AB’), with [A,B’]=0. We want to preserve the empirical results exactly, which can be done, but free up our intuition to work in different ways, insofar as that’s useful. Now, the previous paragraph was easy, because it’s been well-known practically forever, but this paragraph is tricky, because as far as I know it’s only hinted at a little wildly and very vaguely in two papers, one from the 90’s and one from the 00’s, there’s another better known paper from the 10’s that’s less wild but only waves in the general direction, and there’s a mathematical construction called the sequential product from the 90’s in the quantum measurement theory literature. So I’m trying to make those ideas clear as physics in an elementary, slightly less wild, and slightly less vague way, because I’m morally certain, as they say, that this very much reduces the distance between how we think of classical and quantum physics. Not to nothing, but by a lot. Doing that well enough may be beyond me, but so far I’m keeping at it. The math is in the paper I’ve linked to here before, as you know, https://arxiv.org/abs/2101.10931, but part of my process is to cause ripples and hope for caustics in people’s understanding. Simplicity, and writing compellingly, is hard!

  • SelfAwarePatterns

    I actually learned how to type on a typewriter too. I think we’re seriously dating ourselves. In my case, the first year was on an old mechanical typewriter at my middle school, and the second was on an electrical one in the first year of high school. I remember needing a few days to get used to the sensitive electrical model. I took typewriting primarily because I hoped to be a writer someday. But it turned out to be handy because that was right about the time I was discovering programming.

    I listened to a couple interviews of Wilczek’s when his book came out. He struck me as the most conventional of conventional physicists. So it doesn’t surprise me he talks up complementarity. Have to admit I still don’t understand that one. It seems like a simple acknowledgment of wave-particle duality, but Bohr and a lot of his fans seem to see it as a profound realization about reality.

    • Wyrd Smythe

      Likewise we had an old mechanical at home. There were periods I had access to an IBM Selectric, which to this day I think is an awesome machine. Typing was almost like firing bullets! My dad bought electronic typesetting machines for his print shop, and they had electronic keyboards. There is indeed a whole different touch to each type. (I’ve played keyboards since I was a kid, and the same ‘whole different touch’ thing exists there, too.)

      Yeah, this is a pretty conventional book. 😀

      Various aspects of complementarity do pervade quantum mechanics far beyond the wave-particle duality. It’s the root of the Heisenberg Uncertainty Principle, and a fundamental issue in any wave-based mechanics. The complementarity of harmony and melody is an evocative metaphor, but a more accurate one is frequency and location in time, which is a direct analogue for the HUP.

      It’s formalized in the mathematical notation of a commutator:

      \displaystyle[A,B]\equiv(AB)-(BA)

      Which gives the difference between A-then-B versus B-then-A. With classical measurements, [A,B]=0, but quantum measurements of conjugate pair properties result in non-zero values. For instance, the HUP is expressed with the canonical commutation relation:

      \displaystyle[\hat{x},\hat{p}]=(\hat{x}\hat{p})\!-\!(\hat{p}\hat{x})={i}\hbar

      Where x-hat and p-hat are the position and momentum operators. The non-zero value is the uncertainty of precision, which is ultimately due to the wave-based description of QM.

      Bottom line, it’s ubiquitous in QM, and it seems to be a characteristic that sets QM apart from classical mechanics. Understanding it could be a major key.

  • diotimasladder

    Nice write-up! There’s definitely an advantage to physical books when it comes to referencing them later, at least that’s what I find. Then again, I don’t mind writing all over my books, whereas others would consider it sacrilege.

    It’s funny how actual working scientists seem far less antagonistic to religion than those who, for lack of a better word, “follow” science. If you know what I mean.

    One point I take issue with:

    “Our experience of reality is one of isolation, both inside the confines of our own brains, and in the stream of information we receive from the physical world. We might as well be brains in a vat — we could certainly never tell the difference!”

    Well, I think you know what I’m gonna say. I think this sounds like a scientific assumption about experience, but it’s not really what we experience.

    Actually, the quote you gave doesn’t go that far:

    “The light has broken up into individual quanta, and quanta cannot be shared. At this fundamental level, we experience separate worlds.”

    Key words—at this fundamental level. But I don’t think we consciously experience the fundamental level of quanta, except as a theory, so there’s that. A tedious point, I’m sure.

    • Wyrd Smythe

      Thanks! I was brought up to revere books but not to take them as sacred. I’ve never had a problem writing or highlighting in them. OTOH, I seem to have the ability to read a paperback without cracking the spine. In that sense I do take care of my books.

      I think a lot of scientists retain a child-like sense of wonder about reality. It’s very much about discovering how things tick. A passion for the “how” and “why” questions. Had I not gotten into the arts in high school, I was headed for a science career. I took German, rather than the probably much more useful Spanish, expressly with the hope of reading scientific papers in their original German.

      You raise a very good point. Our experience of reality is many (many!) layers above fundamental physics. There’s the quantum level, the atomic level, the chemical level, the organic chemistry level, the biological level, the species level, and the conscious intelligence level. And it’s perhaps only at that top level we even have the tools to recognize our shared experiences. (A key thing I love about literature and art is the shared humanity of it.)

      That said, it still remains true, pedantically, maybe even fancifully (but then I’m definitely one for fancies), that we are in some sense alone in our experience of reality. Long ago someone said to me something I’ve remembered ever since: “No matter how close you are to someone, no matter for how long, in that last moment when you fall asleep, you are alone.” Depressing, but true.

      There is also that, while we might experience different streams of photons, as an ontological realist, I see those different streams as coming from the same shared external (real) reality. We’re all listening to the same record album. 🙂

      • Peter Morgan

        Wyrd, sorry to pick up on just one sentence, but “A passion for the “how” and “why” questions.” happens to hit me where I’m thinking at the moment. It seems to me very close to the passion for knowing what causes what happens that has been all over philosophy of physics for a couple of decades now.

        Whaddya think of the idea that we can also ask what happens between and beyond and inside what happens? It being physics I’m talking about, and as empiricist as I am, by “what happens” I only mean whatever records have been kept of an experiment. Being able to say something about what happens between and … the records we actually have is then to use what we now about the apparatus and the situation —encoded as theoretical knowledge— to interpolate and extrapolate (and, not a word, intrapolate) from the records we have to the records we might have had or would have liked to have had from the past or will in future have.

        Because of the low standards I keep, I even wonder whether being able to say a lot about what happens between and beyond and inside (and deep down and …) is as much as to know the “how”. It being physics I’m talking about, I don’t suppose this much touches on “why”.

        Keep up the good words, Wyrd:-)

      • Wyrd Smythe

        Thanks, Peter; I’ll try!

        I think science (and human curiosity in general) is very much about what’s “between and beyond and inside”. It’s kind of what I was trying to express before when we’ve talked about experimental data and the possibility of contributions far below the Planck level.

        I see us experiencing/observing/measuring reality at many levels, from our crude senses to our best single “particle” detectors — a spectrum of data we use as input to our thinking. That data also has many types: position, distance, mass, duration, charge, etc. There is also meta-data that comes from persistent and consistent results (within some margin of error). Over time a consensus view of the reality behind that data develops (that view is always contingent on new data).

        I see that view as a “wireframe” version of “the real thing” (whatever that may be, but as a Realist I do believe such exists). As you likely know, a wireframe is a collection of (spatial) data points, and it’s only in the rendering that lines connecting them are drawn. Those lines are already extrapolations — approximations — based on just the factual data we’ve gathered. The more sparse that data, the worse the approximation. (Imagine a wireframe of a sphere and how not-spherical it becomes with fewer and fewer data points.)

        Our imagination (our theories) can fill in the wireframe even more with what we think reality might be like between or beyond or inside of those data points. Theories like LQG or ST do exactly that — both are far beyond any data points we have.

        Science is the process of making a better and better wireframe model. 😀

      • diotimasladder

        Interesting quote. Actually, I can see that applying more to death than sleeping. In the last moments when I fall asleep, I do so with a kind of security and comfort that comes from not being alone, even when I’m not thinking about it. But death and even dying seems to be very much a solo experience.

        How much German do you remember? I probably should have taken that instead of French. Maybe I would have if there had been a good German instructor at the time. Oh well, c’est la vie.

      • Wyrd Smythe

        Fast nichts. I took it for all four years in high school, but I don’t seem to have much facility for spoken languages. (Ironically, I’m fluent in a dozen programming languages and conversant in many more.) The first two years it was taught by a Yugoslavian woman who’d been a tennis star when she was younger. The last two years were taught by an American who’d been a Mormon missionary to Vienna for several years. Budget cuts forced everyone into a single class, and we advanced students were left mostly to self-study, and that’s not an effective way to learn a language.

And what do you think?

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: