Initial development of set theory and determinism in QM

[Mickey1]

I am considering the following question and I want you to agree (but perhaps you don’t):Rutherford wrote a letter to Bohr, as an answer to a previous letter from Bohr containing one of the first of Bohr’s descriptions of the atomic model, saying that he understood the atom model Bohr advocated. He commented further that there was a question he understood that Bohr had no doubt considered himself: when does an electron “know how to jump?”.

From this exchange, it is obvious that the issue of determinism was something that accompanied quantum mechanics from its very first formulation. It was not “discovered” later as a separate entity.

Although not versed in history, I now offer what I believe to be a historical equivalence, regarding set theory and the real numbers. Cantor discussed the fact that real numbers was unaccountable as opposed to rational. The rational was constructible, and easily understood as the solution to simple integer equations such as x*b=a, but the real numbers cannot be defined in a similar simple way, being were more easily defined as a collection of numbers such as (intuitively) all numbers on a number line or all limits of Cauchy sequences.

Playing the role of Rutherford, I would have written to Cantor that I easily understand many mechanisms from rational numbers, but the real numbers cannot be defined as easily. In lieu of an simple general point-wise definition such as a/b above, the question pops naturally up: what can we say about the whole “collection” of the real numbers?

I propose that set theory was a companion of - the recently more rigorously defined - real numbers, just the way as problems of determinism was for quantum physics. Observe that this is not the way it is being portrayed today. (In fact nobody agrees with me on MATH.SE).

 

[micromass]

Although not versed in history, I now offer what I believe to be a historical equivalence, regarding set theory and the real numbers. Cantor discussed the fact that real numbers was unaccountable as opposed to rational.

Uncountable*

The rational was constructible, and easily understood as the solution to simple integer equations such as x*b=a, but the real numbers cannot be defined in a similar simple way, being were more easily defined as a collection of numbers such as (intuitively) all numbers on a number line or all limits of Cauchy sequences.

Can you elaborote? In what way are the rationals constructible that the real numbers are not.

Playing the role of Rutherford, I would have written to Cantor that I easily understand many mechanisms from rational numbers, but the real numbers cannot be defined as easily. In lieu of an simple general point-wise definition such as a/b above

What is your definition of the rationals? You might be surprised that it is not as easy as you think.

 

[Mickey1]

I would certainly like for you to expand on the complexity of rational numbers (being an mathematical amateur in the first place).

However, in my view you haven’t addressed my main question. I will therefore try to restate it more concisely (avoiding also constructability):

The rationals, however strange you find them, have some features the reals have not. One is countability. (Pointwise) definition of real numbers poses problems not found for the rational numbers. See a discussion on
http://en.wikipedia.org/wiki/Definable_real_number

The very fact that this issue is still disputed (mainly by Prof. Hamkins from City University of New York) testifies to a problem. At the time of Cantor and Dedekind, i.e. before the need of an axiom of choice was generally accepted, the idea of choosing a real number was also a problem, although I am not certain they speculated in this direction. I include Dedekind for I consider cantor and Dedekind close on this issue and I don’t want to get into accreditation issues,. Simply view this as something in the air, just as for my example with Rutherford and Bohr. Let me quote a passage from an article on Dedekind”

“As indicated, Dedekind starts by considering the system of rational numbers seen as a whole. Noteworthy here are two aspects: Not only does he accept this system as an “actual infinity”, in the sense of a complete infinite set that is treated as a mathematical object in itself; he also considers it “structurally”, as an example of a linearly ordered set closed under addition and multiplication (an ordered field). In his next step—and proceeding further along set-theoretic and structuralist lines—Dedekind introduces the set of arbitrary cuts on his initial system, thus working essentially with the bigger and more complex infinity of all subsets of the rational numbers (the full power set). It is possible to show that the set of those cuts can, in turn, be endowed with a linear ordering and with operations of addition and multiplication, thus constituting a totally new “number system”.

It is not the cuts Dedekind wants to work with in the end, however. Instead, for each cut—those corresponding to rational numbers, but also those corresponding to irrational quantities—he “creates” a new object, a “real number”, determined by the cut.”

(Reck, Erich, "Dedekind's Contributions to the Foundations of Mathematics", The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), Edward N. Zalta (ed.), URL = http://plato.stanford.edu/archives/win2012/entries/dedekind-foundations/).

I therefore speculate that the real numbers were treated as an initial case by Cantor as a tool to approach some the unique properties of real numbers. According to Reck, “Dedekind was proceeding further along set-theoretic and structuralist lines”. Why should we not be surprised that (a more formal version of) set theory sees the light of day in this environment?

 

[Evo]

Closed pending moderation.

 

[mfb]

@Mickey1: This looks like pure philosophy, I don't see science to talk about. I discussed this with other mentors and we don't think our forums are the right place for such a discussion.