# The Hidden Aspect of Set Theory [closed]

This question is inspired by a similar question at the beginning of Kunen's new book, "Set Theory".

Many mathematicians believe they are exploring a "real" universe. In such a Platonic point of view they consider the theorems as "real facts" about "real objects" of this "real world" not just a production of a game using meaningless symbols and formal deduction rules.

A part of mathematical Platonism is based on the success of mathematics to explain the patterns of the physical world. In fact many mathematical theories begin with investigating around simple facts about objects and puzzles of the physical world. For example, geometry begins with exploring around calculating areas and volumes of the most natural two and three dimensional objects, the origin of number theory and combinatorics is closely related to daily counting and calculating problems, analysis, calculus and differential equations are describing the nature of some physical phenomenons, many algebraic notions like groups are closely related to very familiar problems like finding symmetries of a shape, etc. Based on this fact one can expect some applications of (elementary/advanced) theorems of usual mathematical fields in physical world.e.g. Even the non-Ecluidean geometries have their own applications in physics.

But the nature of set theory seems quiet different. Set theory begins with infinite number of infinities. It seems too hard to find a correspondence between the basic objects of this particular field to objects and subjects of visible world. The situation becomes more strange when we note the fact that many set theorists believe in a Platonic perspective even much more than usual mathematicians. Based on this view they believe they are talking about "reality" but what kind of reality is set theory talking about when it has no strong connection with the physical world? The answer of this natural question unfolds the hidden aspect of set theory which I came across in these mysterious books.

Amir Aczel, The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity,

Loren Graham, Jean Michel Kantor, Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity, Harvard University Press

Question. Are there more references (book, paper, lecture, media) in the category of mentioned books focused on the historical and sociological researches around possible beliefs of set theorists about the spiritual nature of set theoretic reality?

## closed as off-topic by Andrés E. Caicedo, Stefan Kohl, Yemon Choi, Noah Schweber, user9072 Apr 25 '14 at 17:54

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Andrés E. Caicedo, Stefan Kohl, Noah Schweber
If this question can be reworded to fit the rules in the help center, please edit the question.

• Regarding the reality of mathematical objects I found the chapter "Numbers and Abstraction" in Gowers' "Mathematics: A Very Short Introduction" eye opening. His point is basically: It is not of interest to mathematicians what some mathematical object is but what it does. – Dirk Apr 25 '14 at 11:24
• I don't know if "many" set theorists believe on existence of spiritual interpretations for set theoretic concepts, objects and theorems,but my personal observation confirms this.Quoted from one of my set theorist friends, "The rule of set theory on the other parts of mathematics is divine!" I think there are many isomorphic copies of such a motto amongst set theorists.Furthermore the geographic distribution of set theorists around the world shows some unnatural concentrations on some special cities which could be a sign of popularity of such a spiritual point of view amongst set theorists. – user47697 Apr 25 '14 at 11:46
• I find the distinction drawn in the question between other subjects and set theory quite unconvincing. It would be just as easy, or even easier, to also "link" set theory to daily problems. – user9072 Apr 25 '14 at 11:59
• via Ray Monk "Not really my area, but Rudy Rucker's Infinity and the Mind addresses that kind of issue (spiritual realilty)". – Tom Harris Apr 25 '14 at 12:12
• There seem an awful lot of questionable assumptions in this question. For instance: "analysis, calculus and differential equations are describing the nature of some physical phenomenons" – Yemon Choi Apr 25 '14 at 16:06

I don't know much about this beyond being familiar with the two books you cited, but the following books might be worth looking at:

1. Michael F. Hallett, Cantorian Set Theory and Limitation of Size (1984)

2. Shaughan M. Lavine, Understanding the Infinite (1994)

3. Penelope Jo Maddy, Defending the Axioms. On the Philosophical Foundations of Set Theory (2011)

4. Mary Tiles, The Philosophy of Set Theory: A Historical Introduction to Cantor's Paradise (1989)

Also of possible interest is the following, which I suspect is not very well known:

5. Percy Williams Bridgman, A physicist's second reaction to Mengenlehre, Scripta Mathematica 2 (1933-34), 101-117 and 224-234

For a follow-up (only to Bridgman's comments about the diagonal argument), see

6. William Monroe Rust, An operational statement of Cantor's diagonalverfahren, Scripta Mathematica 2 (1933-34), 334-336

• What about S. Ulam, Combinatorial analysis in infinite sets and some physical theories, SIAM Rev. 6 (1964), 343-355? – bof Apr 25 '14 at 17:48
• @bof: My first few attempts showed that I don't have access (I'm not associated with a college or university), but then I found an apparently bootleg copy. I haven't yet done more than glance at it, but my initial feeling is that Ulam's knowledge of mathematics probably influences his "gut level view" much more than was the case for Bridgman. – Dave L Renfro Apr 25 '14 at 19:07
• I read the Ulam paper this weekend, and probably the most relevant parts for the original poster is the bottom of p. 343 to the top of p. 344, and the top fourth of p. 348. Several of Ulam's speculations (e.g. the universe possibly being like a Cantor set at very great distances and at very small distances) are in other writings by Ulam, such as his autobiography. Reading Ulam's speculations reminded me of another paper that might be of interest: van Vleck's 1915 Bull. AMS paper The role of the point-set theory in geometry and dynamics. – Dave L Renfro Apr 28 '14 at 17:15

Consider for example, in comparison, the law of gravity. Assuming it is true, it is just as spiritual as the natural number 1, or the set of all natural numbers, or the real line, etc. That it predicts something about reality does not make this law itself a part of reality (e.g., you cannot dig it out from anywhere). Thus, its existence is no more physical than that of set theoretic objects.

The main difference is that physical laws, predicting reality, can be tested against reality. Set theoretic objects and facts about them can, often, only be tested against the mathematical reality (known facts about other mathematical objects). Perhaps one can think of this as an incremental (or drilling...) process: Reality $\to$ Physics $\to$ Mathematics $\to$ Set theory.

I have no expertise in any kind of philosophy, so my answer is necessarily naive, but from the little I read, it seems that deep philosophical investigations rarely make things more clear and are mainly good for having more questions. Given that we need more answers than questions, I do not think the philosophical approach would help much here.

It seems too hard to find a correspondence between the basic objects of [set theory] to objects and subjects of visible world.

Yes, pretty much all the mathematics that arises in physical applications, and indeed, the vast majority of mathematics that ordinary mathematicians do, is essentially non-set theoretic in the sense that it can be straightforwardly formalized in systems that are number theoretic in nature. Even most of what set theorists do can be straightforwardly interpreted in number theoretic terms, e.g., as statements about the relative consistency of various formal systems.

I've made similar remarks before but the point seems worth repeating, particularly in light of the comment made above about the divine rule of set theory over the rest of mathematics (an attitude that I really believe is not as prevalent among set theorists as the commenter makes out).

• If your last paragraph refers to my "answer", let me stress that I meant no divinity. Perhaps my diagram is misleading? Diagrams tend to draw too much attention. – Boaz Tsaban Sep 25 '15 at 2:15
• @BoazTsaban: no, not your answer, I was referring to the comment which states that "The rule of set theory on the other parts of mathematics is divine!" – Nik Weaver Sep 25 '15 at 2:50