As an independent scholar, I am not the kind of scholar who can get her work peer reviewed.
From my previous post: “People that work for the journal or the book publisher provide an initial filter to see if a work should be accepted for peer review. They evaluate if the person is a reasonable candidate to be regarded well by others in the field and they consider the abstract to see if it seems reasonable and offers something new or interesting.” I don’t fit the criteria since I do not have a doctorate from an esteemed school with an esteemed advisor.
That’s just the way it is.
Maybe it is relevant to share some examples of attempts to get my work reviewed by people in the academic community; although, without knowing the quality of my work, it’s hard to know how justified my brush-offs were.
I developed a paper that captured some defects I have observed in Set Theory — 2 problems with formal set theory and 2 problems with informal set theory. There’s at least one well-known problem with Set Theory associated with a certain type of paradox — so it’s not like Set Theory has an unblemished reputation. However in the absence of another theory in the first half of the 20th century to fill its role as the theoretical foundation of Mathematics, scholars have moved forward with the idea that everything is okay with Set Theory as long as practitioners abandon formalism and avoid the special cases leading to the paradoxes. (1)
If you are up for the challenge, you can judge for yourself if I support my claim in this paper: “Why Set Theory is Not an Acceptable Theoretical Foundation for Mathematics.”
[http://www.placeofunderstanding.us/pdf_files/WhySetTheory.pdf]
I am interested to know if my arguments can be refuted.
I submitted this paper to “The Journal of the American Mathematical Society” (under 2000 Mathematics Subject Classifications 03A05 and 03E99) in September 2002. I was prepared that it would not be published, but I was hoping for some review and comments on my work. I only received a short email saying “I am sorry to say that we will not be able to publish your paper.” No comments, no feedback.
In another case, I found a post-doc in Mathematics at MIT who was willing to read my paper on Set Theory and look at my paper proposing a new foundation for Mathematics. He obtained copies of my references. He seemed to give it a serious go. But he did not deliver any observations on problems with my reasoning.
He consulted with the professor that he took his one course on set theory with.
The professor indicated that formal Set Theory and the Predicate Calculus of formal Mathematical Logic are 2 distinct theoretical systems. This pertains to the first problem with formal set theory.
Per the textbooks I read, formal Set Theory is a formal theory defined within the theory of formal theories, the Predicate Calculus (of Mathematical Logic). Formal Set Theory is formally used to define numbers. So there is a problem with circular definition. Numbers are used to define the Predicate Calculus (supposedly because counting numbers are part of natural language), and Predicate Calculus is the theoretical setting in which formal Set Theory is defined, and formal Set Theory is the theory used to formally define numbers (via successive sets). Natural numbers define the Predicate Calculus, the Predicate Calculus defines Set Theory, and Set Theory defines natural numbers.
Doesn’t it seem on the face of it that the attempt to understand the theoretical foundations of mathematics should not start with using numbers as primitive concepts?
I don’t agree with the position put forward by the professor that formal Set Theory and the Predicate Calculus are independent. Consider this. If Set Theory is defined outside of the Predicate Calculus, is it a formal theory? What makes a theory a formal theory according to the tenets of academic mathematics? I’m interested to know what other scholars think.
The post-doc in a sense agreed with one of the problems of informal set theory, because it is involved with the known paradoxes. He seems to think that the axioms of Zermelo-Frankel Set Theory correct the problem, but ZF Set Theory is a case of formal set theory, not informal set theory. He felt I was wasting people’s time to include this known problem in my paper. I include this issue because it is not some weirdness that can be fixed by avoidance; it is a serious flaw in which “the whole” and “a part, not all, of the whole” are NOT distincly different; they can be the same.
In the end, the post-doc confessed that he believed the ideas and work of the respected giants of the field must be more correct than the ideas of a lone independent female scholar. I asked, “What about the new numbers defined in my paper on the new foundation for mathematics.” He said, “What are they useful for?”
I have to tell you that I was shocked to get this reaction from a mathematician. Yes, the “imaginary” number also got this reaction; I just thought that a modern mathematician would understand from the example of i (the square root of -1) that numbers are objects worthy of analysis and most likely a practical purpose will show up over time. We agreed to disagree and went our separate ways.
So one of the changes that I would love to see in academic science — working together on knowledge — is a way for independent scholars and scholars from lesser schools to get their work reviewed. Consider that if the current system of academic review existed in 1905, that Albert Einstein would NOT have had his 3 seminal articles published — he was an independent scholar working as a patent clerk.
I think it is relevant to this ongoing monologue to share some of my experience so that you have more understanding about personal context related to the subject. Also, it allows you to get to know me a little bit.
Footnote
“Since all mathematics can be reduced to set theory, all one need consider is the foundation of set theory. However, it was Russell himself who discovered that the seemingly transparent notion of set contained unexpected traps.
“…
“The Russell paradox and the other antinomies showed that intuitive logic, far from being more secure than classical mathematics, was actually much riskier, for it could lead to contradictions in a way that never happens in arithmetic or geometry.
“This was the ‘crisis in foundations,’ the central issue in the famous controversies of the first quarter of this century. Three principal remedies were proposed.
“…
“… In 1930, Gödel’s incompleteness theorems showed that the Hilbert program was unattainable — that any consistent formal system strong enough to contain elementary arithmetic would be unable to prove its own consistency. The search for secure foundations has never recovered from this defeat.”
Page 344
“In recent years, a reaction against formalism has been growing. In recent mathematical research, there is a turn toward the concrete and the applicable.”
Page 347-348
“Thus, Lakatos applied his epistemological analysis, not to formalized mathematics, but to informal mathematics, mathematics in process of growth and discovery, which is of course mathematics as it is known to mathematicians and students of mathematics. Formalized mathematics, to which most philosophizing has been devoted in recent years, is in fact hardly to be found anywhere on earth or in heaven outside the texts and journals of symbolic logic.”
Foundations of Science 
With free information on the internet and independent scholars who’s business it is to share analyze and forward this information I agree. The predicate calculus and set theory are powerful tools but they are lacking in that our intuition is often more powerful than the definitions we develop. Yes, I have known of no way to develop the predicate calculus without having an intuition of natural numbers. I remember one definition using the power set of the empty set, since the “empty” element is a member of every set. Even then this leads me to question the inductive principle and infinity and even the axiom of choice. New and original ideas in this field will certainly contribute to out side fields like Ai and even mathematical biology.
I think, especially for independent scholars, there is fertile grounds in researching these topics regardless of peer review. I just hope that such scholars are not disheartened by the illusive “doctorate” status and continue to share and forward their original ideas until the environment is mature for their work.
Your comment provided nice encouragement; thank you.
The Predicate Calculus does not only require an intuition about natural numbers. It requires a mathematical understanding of what it means to have a one-to-one and onto mapping relationship (as well as the notions of “constants”, “variables”, “function variables” and “predicate variables”). The notion of a 1-to-1 map is not a primitive meaning from natural language, it is defined by mathematicians in the context of math. I think this makes it off-limits as a starting place for a formal theory of formal theories, especially where one of the formal theories is Math.
In the attempt to have clear-cut definitions the creators of the Predicate Calculus drew from vocabulary and meanings from their training as mathematicians. But what they did was put forth a form of circular reference.
I am so glad to learn that someone else has intuited that math does not have the desired rigor when it comes to infinity.
And whether it is the power set of the empty set, or the successor of the null set, the corresponding definition of natural numbers is problematic. In my paper, Why Set Theory is Not an Acceptable Foundation for Mathematics, I set up a model for this set-theoretic approach to the definition of natural numbers. I correlate a set to a list in the English language (could be another language). For example, the set {1, 2, 3} would be modeled by the list “1, 2 and 3”. So in this model, start with an empty list, then try to form another list that contains the empty list. This is still an empty list. In the world of English, all empty lists are the same, the (semantic) meaning of 1 empty list is the same as any other empty list.
And this exercise, I believe, points out an inconsistency; because a set is supposed to be determined by its members.
I think that the people who embrace the empty-set approach have in mind that the set which contains the empty set as its only member, is like making a concept out of “nothing.” In other words, conceive of nothing. Now consider that in your mind there is a concept of nothing — and this concept is a something, a meaning in your brain. But the problem with this is that the identity of a set [according to the founders] is supposed to be determined solely by its members. The ability to create something out of nothing confers extra capability beyond the idea of group membership — a capability that is not explicitly defined.
On another note, the reason I have not pursued the usual academic path involving a doctorate is because most likely it is impossible to change the foundation of a major structure by working within an institution that grades people on how well they embrace, understand, and work with the structure — essentially trying to change the foundation of a structure while working inside the structure. I think a person has to go at it from the outside. There are other reasons too, but this is a key one.