February 13, 2021 davidWJ

Cats Are Not Annoying; Except, Perhaps, Schrödinger’s

We’ve all heard people refer to “Schrödinger’s cat,” and sometimes that usage is spot-on, and useful. In my experience, however, more often than not the poor cat is deployed as a metaphor which (ahem) fails to land on its feet. That’s likely because the speaker has strayed from what Erwin Schrödinger actually wrote and, more to the point, what he meant. The best place to start, then, is with his original text from 1935:

“A cat is penned up in a steel chamber, along with the following diabolical device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small that perhaps in the course of one hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The ψ-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.”(1)

The classical theory at the time held that a radioactive atom either does, or does not, shed a particle during the stated one hour timeframe. Quantum theory, being posited by Einstein at the time, accepted that such a system operates in superposition, meaning both possibilities co-exist at once. Thus the two possible states of the particle must be deemed “smeared out in equal parts” – unless and until the probability wave function collapses the particle’s position into one real-world state, or another.

Schrödinger used the cat paradox precisely because he believed it was a “ridiculous case” that could more clearly show the paradox of quantum theory. Indeed, this paragraph was written in direct response to Einstein at the time superposition was being posited. To explicate the limitation of the binary model, in which the particle must be in either one state or the other, he simply connected the state of the particle to the state of a cat-at-risk-of-instant-death, substituting the cat for unpredictable atomic decay, surely the more abstract idea.

This puts the cat in superposition. It means the cat-at-risk exists in the unfortunate state of being at once partly alive and partly dead – a state that bivalence abhors but multivalence (cf., fuzzy set theory) comfortably manages by allowing us to see the cat as holding partial and simultaneous membership in set X (Alive) and set not-X (Dead). This is a nice example of what fuzzy set theorist Lotfi Zadeh later dubbed the Law of the Included Middle.

Fifty years after Schrödinger, Stephen Hawking and Roger Penrose wrote:

“The wave function of the system is a superposition of these two possibilities. But why does our perception not allow us to perceive macroscopic superposition, of states such as these, and not just the macroscopic alternatives ‘cat is dead’ and ‘cat is alive’?” (2)

The answer is that we have all been taught the theories, rules and explanations formed by the Law of the Excluded Middle, which are intended to provide us with certainty in our lives. And this is a good thing, in many contexts, because it allows us to have certainty about things in the world. But rules of certainty are not universally applicable. They can become troublesome when applied haphazardly to contexts for which they are not well-suited. And we are increasingly finding such context in both our physical and non-physical realities: 

“Uncertainty is . . . so ubiquitous that scholars for centuries have studied how to escape it rather than to work with it. They have preferred the solace of certainty to a grasp of uncertainty.”(3) 

We must, then, adopt a mindset that acknowledges the limitations of our ingrained knowledge, so as to sharpen our awareness to the possibility that we may not be seeing all of the uncertainty extant in a problem domain.(4) Design-thinking, fuzzy sets and interdisciplinary training can serve as worthwhile ‘antidotes,’ in the pragmatist sense, to the conclusion that the only useful model is the binary, set X or set not-X, comfort in the face of the uncertainties all around us, every day.

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(1) (emphasis in the original). Erwin Schrodinger, “Die gegenwartige Situation in der Quantenmechanik,” Die Naturwissenschaften 23: (1935), translated by Trimmer, John D., “The Present Situation in Quantum Mechanics: A Translation of Schrödinger’s ‘Cat Paradox’ Paper.” Proceedings of the American Philosophical Society 124(5) 1980, p. 323, 328.  JSTOR, www.jstor.org/stable/986572 (accessed 28 Apr. 2020).

(2) Stephen W. Hawking and Roger Penrose “The Nature of Space and Time,” Scientific American July 1996 p. 63 (emphasis added).

(3) P.N. Nikiforuk quoted in Fuzzy Automata and Decision Processes, Gupta, Saridis, and Gaines, eds., at p. ix (Elsevier North, 1977), at p. 175. “[M]ost of us are aware that there is much uncertainty in the world, and one of our most basic choices is whether we will accept that uncertainty as a fact or try to run away from it.”  Robyn M. Dawes, Rational Choice in an Uncertain World (Harcourt, 1988) p. 273.

(4) Cf., Lera Boroditsky, “Does Language Shape Thought?: Mandarin and English Speakers’ Conceptions of Time” Cognitive Psychology 43(1): 1-22 (2001).

 

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