I somehow never realized that the puzzling fact that infinite sets could be in bijection with proper subsets was as old as Galileo:

Simplicio: Here a difficulty presents itself which appears to me insoluble. Since it is clear that we may have one line greater than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity, because the infinity of points in the long line is greater than the infinity of points in the short line. This assigning to an infinite quantity a value greater than infinity is quite beyond my comprehension.

Salviati: This is one of the difficulties which arise when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this I think is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another. To prove this I have in mind an argument which, for the sake of clearness, I shall put in the form of questions to Simplicio who raised this difficulty.

- I take it for granted that you know which of the numbers are squares and which are not.

Simplicio: I am quite aware that a squared number is one which results from the multiplication of another number by itself; this 4, 9, etc., are squared numbers which come from multiplying 2, 3, etc., by themselves.

Salviati: Very well; and you also know that just as the products are called squares so the factors are called sides or roots; while on the other hand those numbers which do not consist of two equal factors are not squares. Therefore if I assert that all numbers, including both squares and non-squares, are more than the squares alone, I shall speak the truth, shall I not?

Simplicio: Most certainly.

Salviati: If I should ask further how many squares there are one might reply truly that there are as many as the corresponding number of roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square.

Simplicio: Precisely so.

Salviati: But if I inquire how many roots there are, it cannot be denied that there are as many as the numbers because every number is the root of some square. This being granted, we must say that there are as many squares as there are numbers because they are just as numerous as their roots, and all the numbers are roots. Yet at the outset we said that there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers, Thus up to 100 we have 10 squares, that is, the squares constitute 1/10 part of all the numbers; up to 10000, we find only 1/100 part to be squares; and up to a million only 1/1000 part; on the other hand in an infinite number, if one could conceive of such a thing, he would be forced to admit that there are as many squares as there are numbers taken all together.

Sagredo: What then must one conclude under these circumstances?

Salviati: So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all the numbers, nor the latter greater than the former; and finally the attributes “equal,” “greater,” and “less,” are not applicable to infinite, but only to finite, quantities. When therefore Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number.

This came to my attention because I’m procrastinating by looking at the August 1951 issue of *The Times Review of the Progress of Science*, a quarterly supplement to the British newspaper. This issue features an article by Schrödinger which lays out the modern theory of transfinite cardinals, including Cantor’s diagonal argument, the bijection between the line segment and the square, and the hierarchy of infinities obtained by iterating “power set.” Can you imagine a mathematical exposition of similar depth appearing in the *Science Times* today?

Schrödinger’s closing paragraph is striking:

While these higher infinities have not hitherto acquired half the importance of the two that we have been studying here, the physicist is keenly interested in the probable bearing of the startling properties of the continuous infinite on the theories of atoms and energy quanta. I consider these theories a weapon of self-defence, contrived by the mind against the “mysterious continuum.” This does not mean that these theories are pure invention, not founded on experiment. It is, however, fairly obvious that in their historical development some part was played by the desire to replace the continuous by the countable infinite, which is easier to handle. To disentangle the influence of this mental urge on the interpretation of experimental evidence is a task for the future.”

Can someone with more knowledge of quantum theory than me explain what Schrödinger might have meant?

How did you get your hands on this article from August 1951? Is it archived on-line somewhere? Is it in one of our libraries?

I am working in Memorial Library. Whenever I come in here I like to take a volume or two off the shelves that I feel are otherwise in danger of never being seen again.

I think Schrodinger is referring to how old school quantum mechanics (not the modern formalism due to, well, Schrodinger and Heisenberg, but the older quantum mechanics of Bohr and Summerfeld, which was based on restricting energy levels to a discrete (i.e. “quantized”) set) managed to avoid divergences such as the ultraviolet catastrophe that were predicted by classical mechanics. So, for a brief period, quantum mechanics was actually viewed as having fewer (or at least more countable) infinities in it than classical mechanics. The situation is of course completely different now, what with path integrals, string theories, and whatnot…

I would like to be able to give an inspired explanation of Schrodinger’s thought but I don’t know that I can, but since your question caught me when I was reviewing the history and structure of quantum mechanics I’ll make a guess: in quantum mechanics the values of a physical observable are taken from the eigen values of a linear operator. The relevant operators in the cases which drove the development of quantum mechanics all have discrete spectra. Schrodinger may not have been aware of the instances (most glaringly in condensed matter physics) where the operators you encounter have fully continuous spectra.

When I was an undergraduate I tried studying in the Memorial Library and found it distracting for that reason. The library has all sorts of interesting material that one generally doesn’t even imagine exists. Don’t get squished in the moveable stacks!

I also tried studying at the old medical library on Linden Dr., and wandered the stacks there. That’s how I discovered that dermatology books are filled with trophy photographs. Ack!

I think Schrödinger would have known many instances of continuous spectra in quantum mechanics, e.g., in scattering or in beta-radiation, so it’s not clear that this is what he had in mind. (The position operator also typically has a continuous spectrum.)

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Thanks for the post Jordan.

Schrödinger had very broad interests I think. He wrote a book on biology, touching on the topics of life and consciousness (if I remember well), that is often cited. He is also famous for claiming to have generalized general relativity (again if I remember well), and saying something like… “I shall look an awful fool if I am wrong.” (ok I searched the quote now). I guess this was with a view to integrating it to quantum mechanics, or more probably subsuming both -since he didn’t like QM all that much.

All this to say that he had far-reaching ideas and was not afraid of grandiose statements (I sympathise, I too indulge in mania more than I should).

If we suppose as Terence does that Schrödinger refers to QM by “these theories” and not to the higher infinities, set theory, then as Terence says QM is “more finite” than classical mechanics -though he says this view has changed and I would rather disagree. The hydrogen atom has a lowest energy state, and continuous spectrum for position or momentum is not relevant here. Classically the electron spiraled to the proton/nucleus in the Coulomb potential.

This was one of the major attracts of QM, as is often mentioned in textbooks, and this is probably what Schrödinger refers to by “the desire to replace the continuous by the countable infinite”, and since he doubts QM he closes his text mentioning “the influence of this mental urge on the interpretation of experimental evidence”.