Category Archives: Science history

Calculus Made Easy

My way of taking a break from the … everything of everything has been to try my hand at an updated edition of Silvanus P. Thompson’s classic text, Calculus Made Easy (1914). The book deserves its reputation and holds up quite well overall; I have seen folks who aren’t at all “math people” be charmed by his writing style. However, it is outdated in places, with the occasional antiquated turn of phrase or poorly-aged example. Fortunately, Project Gutenberg has a transcribed version including a ZIP file of the LaTeX source code and auxiliary files. So, I went for a minimal update, fixing the archaisms that pose noticeable stumbling blocks. The result is now available as a PDF document and as source code.

The one thing in Thompson’s presentation that I didn’t particularly like is how he introduces derivatives of trig functions. It presumes that the reader has a lot of trig identities in their back pocket, and it makes a simplification that is hard to justify without going into limits, a topic that Thompson doesn’t explicitly teach. I’ve tried my hand at a replacement that appeals to the way he does teach.

Further modifications may come as people apprise me of all the things I missed. I do wish to keep it short and sweet, rather than adding multiple new chapters.

EDIT TO ADD (12 March 2024): I had a Lulu.com account from a print-on-demand project ages ago, and poking around didn’t find any obviously better options, so I ordered some copies from there. I deem them good enough and have made the project available for purchase at cost.

The Lunn–Schrödinger Equation?

Wikipedia claims that Arthur C. Lunn discovered what we now call the Schrödinger equation some years before Schrödinger. I wonder if there is more to say about this than what the references cited there provide (they have the feel of being faithful recollections, but are light on specifics).

In a 1964 interview, the physicist Karl Darrow calls the story “impossible to check”. And in another interview, Robert Mulliken (not to be confused with Robert Millikan) shares the story of Lunn having “sent a paper to the Physical Review which was turned down and which anticipated the quantum mechanics”. Mulliken heard the story from the physical chemist William Draper Harkins. Similarly, Leonard Loeb told Thomas Kuhn that Lunn “was probably a misunderstood genius, and who was completely frustrated, because his one great paper with his one great idea was turned down by a journal”.

Lunn did apparently try to present what sounds like a grandiose paper (“Relativity, quantum theory, and the wave theories of light and gravitation”) at the American Physical Society meeting in April 1923, but his paper was only “read by title”. The abstract ran as follows:

This paper is a preliminary report on a theory originally sought in order to meet the recognized need for a reconciliation between wave theory and quantum phenomena; its scope of adaptation proves to be quite wide. It includes (1) a wave theory of gravitation in quantitative connection with optical, electronic, and radioactivity data; (2) a related general suggestion of a theory connecting molecular properties with properties of matter in bulk; (3) alternatives for some of the current features in the theories of atomic structure; (4) a new interpretation and deduction of formulas for series and band spectra, using in lieu of the quantum condition a substitute directly related to long familiar physical notions; (5) a modification of Lagrangian dynamics which promises to be of service in the study of complex atomic and molecular structures; (6) a non-quantum theory of specific heat and black radiation. Results so far reached deal mostly with problems approachable by elementary methods or approximate computations. A set of formulas has been obtained which yield computation of the electron constants $e$, $h$, $m$ and mass ratios, assuming from observation only the Rydberg constant, velocity of light, gravitation constant, and Faraday constant, with results in each case in practical agreement with measured values.

Darrow says, “I know that in 1924 he wanted to give a twenty or a thirty minute paper before the American Physical Society in Washington, but then authorities of the Society refused him more than ten minutes”.

Lunn’s abstract in the 1924 proceedings has a similar explain-everything atmosphere:

Relativity, the quantum phenomena, and a kinematic geometry of matter and radiation. A. C. LUNN, University of Chicago. The theory indicated in an earlier paper (Phys. Rev. 21, 711, 1923), has since been developed, extended in scope, and so ordered as to permit of treatment as a deductive space-time geometry. It unites the treatment of the quantum phenomena with the rest of physical theory in a way that yields to illustration by familiar physical images. It resolves into matters of choice a number of hitherto controversial alternatives in the interpretation of phenomena, and allows freedom of use of a range of concrete types of representation including many other concepts commonly discarded. Among special topics more recently found to affiliate with the scheme may be mentioned the Stark and Zeeman effects and fine structure, resonance potentials, and the intensity and distribution of general x-radiation. Improvements have been made in the setting of the formulas connecting $e$, $h$, and $m$ with pre-electron data. A program has emerged for the foundation of a trial mathematical chemistry by determination of types of atoms, valence, number of isotopes, atomic weights, and spectrum levels.

I can easily imagine a paper with that attempted scope being incomprehensible to whoever had the task of evaluating it, and so any really good morsels within it would have been lost.

UPDATE (4 November): I wrote to the Physical Review offices on the chance that they had more information and received this reply from Robert Garisto, the Managing Editor of Physical Review Letters.

Thank you for your query. Our records from the early 20th century are fragmentary. I am not sure if we have any from before 1930, much less a complete set that could answer your question.

But I see that Arthur C. Lunn published 7 papers in the Physical Review from 1912-1922. So he was a known author to the editors. Those were different times, and while it is possible that he submitted a paper that was rejected and never published elsewhere, for what it’s worth, it strikes me as unlikely.

Friends Don’t Let Friends Learn Physics From Yudkowsky

With the demise of Reddit, we have lost /r/SneerClub, the Internet’s hot spot for mocking those who proclaim allegiance to capital-R Rationality and related ideologies like longtermism. Somewhere in between the discussions of heavy stuff like sexual harassment in Effective Altruism culture and total frivolity were the rambles about science. I thought I would pull a couple such comments out of the archives and edit them into something shaped like a blog post. So, consider this your Attention Conservation Notice: if you’d rather not work through a self-admittedly rough explanation of how Eliezer Yudkowsky’s claims about quantum physics are just silly, exit now.

Yudkowsky clearly intends to argue that the scientific community is broken and his brand of Rationalism(TM) is superior, but what he’s actually done is take all the weaknesses that physicists have when discussing quantum foundations and present them in a more concentrated form. There’s the accepting whatever mathematical formulation you learn first as the ultimate truth, the reliance upon oversimplified labels and third-hand accounts rather than studying what the pioneers themselves wrote, the general unwillingness to get out of the armchair and go even so far as the library…
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Saturday Thought

One thing I just don’t get is people proclaiming “the End of Physics”. Like “the End of History”, it’s a very mockable phrase! Folks will be going, “Oh, our giant colliders haven’t found any surprises in years, and we never figured out an experiment to test string theory, so everyone’s drifting into quantum information and exotic condensed-matter physics, truly this is the sunset of an era.”

And I’m all, “So, instead of testing one effective field theory by putting matter into extreme conditions, you’re testing … multiple … effective field theories … by putting matter into extreme conditions.” I’m making my astonished face, can’t you tell?

More on Bohr

This post carries further on in the vein of my earlier writings on how the way most physicists talk about “the Copenhagen interpretation of quantum mechanics” is largely ahistorical.

It’s common to present “the Copenhagen interpretation” as a kind of dynamical collapse model, in which wavefunctions are ontic entities (like a sophomore’s picture of the electromagnetic field) that evolve according to the Schrödinger equation, except in moments of “measurement” that take place in unspecified conditions. This portrayal is typically intended to make “the Copenhagen interpretation” sound like a mutant form of Newtonian mechanics where $F = ma$ almost always, except at peculiar instants when $F$ suddenly becomes $ma/2$ and then switches back again. Of course, this is abhorrent and pathological.

When I was a child, my parents bought me a magnet from a museum gift shop. It had a long handle, likely made deliberately to resemble a magic wand, and as educational toys go, it served its function, since I went around poking all sorts of things to see if the magnet would grab them. I suspect this is a common enough type of learning experience. One discovers, for example, that it will pick up paperclips but not pennies. Having calibrated one’s understanding of the magnet, one can then use it as a tool — say, by telling which of two matchboxes is filled with paperclips, or that something is different about a wire coil connected to a battery versus one that is not.

What concerned Bohr himself was that this transition — between the calibration phase, when an object is under scrutiny, and its later use as a laboratory instrument — is conceptually nontrivial. First a lens is a strangely curved block of glass we must work to comprehend, and then it is a means to overthrow Aristotle. There are not two different dynamical laws, but two different languages.

Here’s how John Wheeler put it:

“Bohr stresses […] that the stick we hold can itself be an object of investigation, as when we run our fingers over its surface. The same stick, when grasped firmly and used to explore something else, becomes an extension of the observer or—when we depersonalize—a part of the measuring equipment. As we withdraw the stick from the one role, and recast it in the other role, we transpose the line of demarcation from one end of it to the other. The distinction between the probed and the probe, so evident at this scale of the everyday, is the without-which-nothing of every elementary phenomenon, of every closed quantum process.”

[From “Law Without Law”, in the Wheeler–Zurek collection, p. 206]

The commonalities and contrasts with QBism should be evident enough. Extension of the observer, yes; depersonalize to mere dead “equipment”, no, for it is the latter move that gets one into trouble with Wigner’s Friend. And, on a perhaps more practical level where the choice of research problems is concerned, Bohr takes the quantum formalism pretty much as given and leaves “the quantum principle” not explicitly defined.

It may also be illustrative to consider how Rovelli’s “Relational Quantum Mechanics” treats this point. I tentatively infer that Rovelli thinks giving a special role to an agent means imposing two different dynamical laws, one for systems of agent-type and another for all nonagent physical entities. Even if he doesn’t spell it out, that seems to be the mindset he operates with, and the background he relies upon. Of course, he balks at that dichotomy. I would, too!

What’s Wrong with this Sting Operation?

To the extent that academic peer review is good for anything, it is optimized to catch honest mistakes. It is weaker against deliberate fraud and stubborn denial. Science has a presumption of fair play, a sense that the natural world isn’t a cheater. If you want to explain how a “psychic” operates, you’re better off asking a magician than a physicist.

Nearly two decades ago now, there was a dust-up when a couple French TV personalities got a clutch of physics and mathematics papers published, and even received PhD’s, and their “work” turned out to be nonsense. (The Wikipedia article on l’affaire Bogdanov is currently not terrible, and it contains more pointers to details than almost anyone could honestly desire.) The news stories about the incident really played up the “even the physicists can’t tell if the papers are nonsense or not” angle. That rather oversells the case, though. I read the Bogdanovs’ “Topological field theory of the initial singularity of spacetime” when I was a first-year grad student, and I could see through it. If you know what a Lagrangian is, and the fog doesn’t intimidate you, then you can tell something is wrong. If you don’t know what a Lagrangian is, you’re probably not reading theoretical physics papers yet.

So, what went wrong?
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Underappreciated

Some time ago, I had one of those odd little thoughts that could be the spark of an essay. But in this particular case, the point I wanted to make felt like it could be made most clearly by demonstration, rather than explication. So, I wrote a concise report on “An Underappreciated Exchange in the Bohr–Einstein Debate.” Judging by the modest splash of positive e-mail that I received after posting it, I think I layered the whimsy and the serious point adequately well.

My 2019 in Science

First, of course, there was the doubt and the pain.

But we’ve already covered that.

Let’s talk about the papers I managed to get out the door and into public view. In retrospect, the list is pleasingly not insubstantial:

There was also From Gender to Gleason, my review of Adam Becker’s book What is Real? (2018). By the time I was done, it was as lengthy as a paper, but the arXiv isn’t really a host for book reviews, so I just posted it here at Sunclipse and moved on.

Predator/prey or Perish

Looking at academic publishing from the perspective of Fully Automated Luxury Gay Space Communism is an interesting experience.

Consider, for example, the term predatory publisher for shady outfits that will accept anything for the right fee and put it on a website that calls itself a “journal”. Scummy behavior, right? But is it really “predatory”? What fraction, exactly, of their customers are being conned, and how many are walking into the deal with their eyes wide open? A used-car salesman might be a sleaze, but if you’re going to his dealership to pay cash for a getaway car, the relationship is more of a symbiosis.

I’m sure it’s convenient for the legacy institutions to present the situation as saintly scholars being exploited by deceptive newcomers. [cough]

Suppose the Web came to be, but there never were any respectable Open Access journals. No “Open Letter to Scientific Publishers” in 2001, so no Public Library of Science; no Budapest Initiative in 2002 or Berlin Declaration in 2003. Would the morass of “predatory” OA really look all that different? Perhaps not. Websites are cheap, calling yourself a journal is easy, and as we just noted, there’s a ready market.

But without the cover of PLOS and the like, would “predatory” OA have a veneer of respectability to offer its customers? Well, consider that paying to attend conferences is a thing that academia finds universally respectable. So, a “predator” could do what outfits like WASET do now: offer “conferences” with no standards, no dedicated space, perhaps not even a physical event. And if you’ve got a paper, great! For only a modest additional fee, it can go in the conference proceedings, which will conveniently be available online.

Quality is always the hard course of action. Legitimate OA journals were optional; only the pay-to-play racket was inevitable.

As We Would Not Actually think

There’s an aspect of Vannevar Bush’s “memex” that, I think, would still be difficult to achieve with current software, and that is its intensely personal character. The memex that his 1945 essay “As We May Think” imagined was to be “an enlarged intimate supplement to [the user’s] memory.”

A modern analogue would have to be something like a personal wiki, hybridized with a social-media platform. Every post you make is intended to be retrievable: cross-indexed, hyperlinked. Like, if every time you posted to your Mastodon instance, it was also added as a page to your own MediaWiki setup. And you could share pages from your MediaWiki with just a few clicks, sending any set of them you wish to another Mastodon user. Instead of just sharing a news story, you could pull up every news story you ever shared, along with whatever comments you made about them, and all the ways that you had decided to tag them.

It’s not beyond what software can do, but we don’t generally seem to have worked toward what Vannevar Bush had in mind. There wasn’t supposed to be just one Memex for everybody.

The bits and pieces are present, but there hasn’t been the drive to put them together in a way that makes the package readily usable. We have software for sharing personal records and observations (social media), and we have platforms for making association trails (e.g., Wikipedia, TV Tropes, etc.). But the Memex that VB envisioned was an individual possession that facilitated social exchanges. In slogan form: The memex was like building your own Wikipedia, with adjustable privacy settings, one blog or microblog post at a time.

From Gender to Gleason

… or, The Case of Adam Becker’s What Is Real? (2018).

It is easy to argue that the founders of quantum mechanics made statements which are opaque and confusing. It is fair to say that their philosophical takes on the subject are not infrequently unsatisfying. We can all use reminders that human flaws and passions are a part of physics. So, it would be nice to have a popular book on these themes, one that makes no vital omissions, represents its sources accurately and lives up to its own ideals.

Sadly, we’re still waiting.

Full review (PDF).

In Re “CopenHagen” and “COLLAPSE”

I was having an e-mail conversation the other day with a friend from olden days — another MIT student who made it out with a physics degree the same year I did — and that led me to set down some thoughts about history and terminology that may be useful to share here.

My primary claim is the following:

We should really expunge the term “the Copenhagen interpretation” from our vocabularies.

What Bohr thought was not what Heisenberg thought, nor was it what Pauli thought; there was no single unified “Copenhagen interpretation” worthy of the name. Indeed, the term does not enter the written literature until the 1950s, and that was mostly due to Heisenberg acting like he and Bohr were more in agreement back in the 1920s than they actually had been.

For Bohr, the “collapse of the wavefunction” (or the “reduction of the wave packet”, or whatever you wish to call it) was not a singular concept tacked on to the dynamics, but an essential part of what the quantum theory meant. He considered any description of an experiment as necessarily beginning and ending in “classical language”. So, for him, there was no problem with ending up with a measurement outcome that is just a classical fact: You introduce “classical information” when you specify the problem, so you end up with “classical information” as a result. “Collapse” is not a matter of the Hamiltonian changing stochastically or anything like that, as caricatures of Bohr would have it, but instead, it’s a question of what writing a Hamiltonian means. For example, suppose you are writing the Schrödinger equation for an electron in a potential well. The potential function $V(x)$ that you choose depends upon your experimental arrangement — the voltages you put on your capacitor plates, etc. In the Bohrian view, the description of how you arrange your laboratory apparatus is in “classical language”, or perhaps he’d say “ordinary language, suitably amended by the concepts of classical physics”. Getting a classical fact at your detector is just the necessary flipside of starting with a classical account of your source.

(Yes, Bohr was the kind of guy who would choose the yin-yang symbol as his coat of arms.)

To me, the clearest expression of all this from the man himself is a lecture titled “The causality problem in atomic physics”, given in Warsaw in 1938 and published in the proceedings, New Theories in Physics, the following year. This conference is notable for several reasons, among them the fact that Hans Kramers, speaking both for himself and on behalf of Heisenberg, suggested that quantum mechanics could break down at high energies. More than a decade after what we today consider the establishment of the quantum theory, the pioneers of it did not all trust it in their bones; we tend to forget that nowadays.

As to how Heisenberg disagreed with Bohr, and what all this has to do with decoherence, I refer to Camilleri and Schlosshauer.

Do I find the Bohrian position that I outlined above satisfactory? No, I do not. Perhaps the most important reason why, the reason that emotionally cuts the most deeply, is rather like the concern which Rudolf Haag raised while debating Bohr in the early 1950s:

I tried to argue that we did not understand the status of the superposition principle. Why are pure states described as [rays] in a complex linear space? Approximation or deep principle? Niels Bohr did not understand why I should worry about this. Aage Bohr tried to explain to his father that I hoped to get inspiration about the direction for the development of the theory by analyzing the existing formal structure. Niels Bohr retorted: “But this is very foolish. There is no inspiration besides the results of the experiments.” I guess he did not mean that so absolutely but he was just annoyed. […] Five years later I met Niels Bohr in Princeton at a dinner in the house of Eugene Wigner. When I drove him afterwards to his hotel I apologized for my precocious behaviour in Copenhagen. He just waved it away saying: “We all have our opinions.”

Why rays? Why complex linear space? I want to know too.

QBism and the Ithaca Desiderata

Time again for the New Paper Dance!

B. C. Stacey, “QBism and the Ithaca Desiderata” [arXiv:1812.05549].

In 1996, N. David Mermin proposed a set of desiderata for an understanding of quantum mechanics, the “Ithaca Interpretation”. In 2012, Mermin became a public advocate of QBism, an interpretation due to Christopher Fuchs and Ruediger Schack. Here, we evaluate QBism with respect to the Ithaca Interpretation’s six desiderata, in the process also evaluating those desiderata themselves. This analysis reveals a genuine distinction between QBism and the IIQM, but also a natural progression from one to the other.

To Thems That Have

Occasionally, I think of burning my opportunities of advancing in the physics profession — or, more likely, just burning my bridges with Geek Culture(TM) — by writing a paper entitled, “Richard Feynman’s Greatest Mistake”.

I did start drafting an essay I call “To Thems That Have, Shall Be Given More”. There are a sizable number of examples where Feynman gets credit for an idea that somebody else discovered first. It’s the rich-get-richer of science.
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