Len Adleman: Quantum Mechanics and Mathematical Logic

Today I’m pleased to announce that we have a guest post from a very distinguished colleague of mine, Len Adleman. Len is best known as the “A” in RSA and the inventor of DNA-computing. He is a Turing Award laureate. However, he considers himself “a rank amateur” (his words!) as a physicist. len_adleman He’s one of my colleagues on whom I can always rely for a fun and interesting conversation, even if it is just for a fleeting moment during a chance encounter in an elevator. The other day he told me he’d been thinking a lot about quantum mechanics, and it seemed like it would be fun to share his thoughts with others here on the blog. So join in using the comment form if you’ve some thoughts of your own in response.

Here’s Len.

-cvj

_________________________________________________________________________________

For a long time, physicists have struggled with perplexing “meta-questions” (my phrase): Does God play dice with the universe? Does a theory of everything exist? Do parallel universes exist? As the physics community is acutely aware, these are extremely difficult questions and one may despair of ever finding meaningful answers. The mathematical community has had its own meta-questions that are no less daunting: What is “truth”? Do infinitesimals exist? Is there a single set of axioms from which all of mathematics can be derived? In what many consider to be on the short list of great intellectual achievements, Frege, Russell, Tarski, Turing, Godel, and other logicians were able to clear away the fog and sort these questions out. The framework they created, mathematical logic, has put a foundation under mathematics, provided great insights and profound results. After many years of consideration, I have come to believe that mathematical logic, suitably extended and modified (perhaps to include complexity theoretic ideas), has the potential to provide the same benefits to physics. In the following remarks, I will explore this possibility.

But, be warned: I am not a physicist and these ideas are embryonic. At best they indicate a possible direction; a fully functional theoretical framework, if possible at all, would be the work of lifetimes.

For most of my academic life, my primary topic of research (and affection) has been number theory. Number theory is the study of “the standard model of arithmetic”: the set {0,1,2,…} together with the operations of addition and multiplication. Where by addition and multiplication, I mean those functions computed by the algorithms we learned when we were little.

Despite how simple the standard model may seem, finding out what is true about it is sometimes remarkably difficult. For example, only by building on over three centuries of prior research was Andrew Wiles able to establish that Fermat’s Last Theorem is true in the standard model.

In the 1930′s, Godel proved the famous incompleteness theorem. Godel’s theorem shows that if you “know” a set K of statements (e.g., Giuseppe Peano’s axioms) which are true in the standard model, there must exist a statement S which cannot be proven true and cannot be proven false using K as axioms. Further, there is an infinite collection of “parallel models” (my phrase), in all of which the statements in K are true, but in infinitely many of which S is true, and in infinitely many of which S is false. Such an S is said to be independent of K, and you cannot use K to figure out whether the standard model is one of those parallel models where S is true or one of those parallel models where S is false.

Now, consider the following story of a physicist and a mathematician (no, they do not go into a bar).

Our physicist has designed an experiment during which the spin of an electron will be measured. He attempts to predict whether the outcome of the measurement will be “up” or the outcome of the measurement will be “down”. He knows many laws of our universe (e.g., quantum mechanics, relativity, etc.). The physicist may be able to use his “known laws” to deduce many things about the outcome of the experiment (e.g., perhaps he can use quantum mechanics to deduce that the probability of the experimental measurement producing an “up” is 0.5 and the probability of the experimental measurement producing a “down” is 0.5). But, unfortunately, the known laws do not allow him to deduce whether the outcome of his measurement will be “up” or the outcome of his measurement will be “down”.

Though it may seem strange, it is very important to keep in mind that “known laws” and “laws” mean different things. Special relativity has (presumably) always been a law of our universe; however, in 1905, it became a “known law” of our universe.

Our mathematician has written an algorithm. He attempts to determine whether the algorithm halts on all inputs. He knows many true statements about the standard model (e.g., Peano’s axioms). The mathematician may be able to use his known truths to deduce many things about the algorithm. But, unfortunately, the known truths do not allow him to deduce whether his algorithm halts on all inputs.

How does our mathematician interpret his situation?

The mathematician considers the fact that the known truths do not allow him to deduce whether the algorithm halts on all inputs. He concludes that the statement “the algorithm halts on all inputs” is independent of the known truths. He realizes that there is an infinite collection of parallel models, in all of which all of the known truths are true, but in infinitely many of which the algorithm halts on all inputs, and in infinitely many of which there exists an input on which the algorithm does not halt. The mathematician’s difficulty is that despite all he knows, he does not know whether the standard model is one of the infinitely many in which the algorithm halts on all inputs or the infinitely many in which there is an input on which it does not halt.

The mathematician accepts that while he is having difficulty, the standard model is not. In the standard model either the algorithm halts on all inputs or there exists an input on which the algorithm does not halt. Which ever it is, it does not change over time and does not change because our mathematician has wondered about the question or even run the algorithm.

The mathematician views his difficulty as stemming from the inadequacy of the truths he knows. He is aware that Godel’s results establish that neither he nor any future mathematician can ever know all of the truths about the standard model. Even if future generations of mathematicians are equipped with a bigger set of known truths, algorithms will still exist about which these future mathematicians will be unable to deduce whether or not they halt on all inputs. Essentially, the mathematician has come to understand that the object of his study is simply too complex to yield all of its truths to one with only the tools he has available.

What have mathematicians gained from this interpretation? A great deal, for example, it has provided them with answers to many of their meta-questions – including all those asked at the beginning of this note.

Now how does our physicist interpret his situation? The Copenhagen interpretation would have the physicist view the electron as existing in a superposition of “up”/“down” until the moment of measurement when its wavefunction would collapse and the result of the measurement would be determined. This interpretation places the physicist’s difficulty within our universe rather than in the inadequacy of the known laws.

Many other interpretations have been proposed, notably the Everett many-worlds interpretation, but I will reserve a discussion of them for another time.

But, what about a new interpretation based upon the mathematical one given above? I suspect that such an interpretation is possible, that it is distinct from those previously proposed, and that it would have profound implications. With such an interpretation, the physicist, like the mathematician before him, must accept that the tools he possesses for understanding the object of his study are insufficient to unlock all of its truths. With such an interpretation, the physicist’s lack of knowledge is not imposed upon the universe itself. The electron probability cloud is no longer seen as a picture from our universe, but rather as a diagram of our lack of knowledge.

-Len Adleman

Acknowledgment: I want to thank David Deutsch – a real physicist – for acting as a sounding board for some of these ideas. The remarks above are partially derived from a series of emails between us a few years ago. I also want to thank my students for helping me compose this note. Finally, thanks to Clifford Johnson for providing a forum.

_________________________________________________________________________________

Bookmark the permalink.