Hi Richard,
Sorry this response is so late; I have been busy. You asked me,
*****Are you saying that there are no axioms in set theory? You said:
***There are no axioms whatever in Dick's theory, additional or otherwise***
And then you implied that his theory is based on set theory....
I never heard of mathematics without axioms.*****
Sorry about the confusion. There is nothing magic or mysterious going on here. It is simply a little confusion over the boundaries around some concepts like, formal systems, mathematics, analysis, set theory, theorems, Dick's discovery, etc. I had hoped to explain my position with the metaphor of mathematics as a tree, but I see that I didn't make myself clear.
In that metaphor, the roots are the axioms and primitive terms, the trunk comprises the basic definitions and theorems, branches are formed by introducing new definitions and developing theorems involving these definitions, and I guess, leaves and twigs would be individual theorems and lemmas. Returning to that metaphor, your questions become, "Are you saying that branches have no roots? You said, 'There are no roots whatever in Dick's branch, additional or otherwise' And then you implied that his theory is based on a thicker branch further down the tree." I hope you can see from this metaphor that there was nothing inconsistent in what I said.
Trying to give credit where credit is due, I give Dick credit for developing only what he developed in the formalism of his paper. In other words, I give him credit only for the branch that he has grafted onto the already-existing tree of mathematics. Just as there are no roots anywhere on a new branch that grows on a tree, there are no axioms on any new branch of mathematics that is deduced from the pre-existing set of theorems that make up the body of mathematics up to that point.
I probably confused the issue by mentioning the historical development of mathematics. Let me try to clear that up by using the tree metaphor again. Euclid planted a sturdy thriving tree of Geometry which stood for some two thousand years on the roots of the Euclidean Axioms. Then Hilbert sawed it off at ground level and grafted it onto a new trunk that he had gotten to take root. Hilbert's root system consists of Axiomatic Set Theory.
It gets more complicated. Set Theory itself was "sawed off" from its roots consisting of Zermelo's axioms, and grafted onto a more primitive set of axioms of Propositional Calculus. Mathematicians have shown that there are many optional ways of choosing the primitive terms and axioms and, from them, deducing the same body of theorems. The game, for a while, was attempting to find the very most primitive starting point that would lead to all of mathematics. Goedel, of course, pulled the rug out from under that effort and proved that no complete, consistent set of axioms could be found that would provide the basis for any mathematical system robust enough to include Arithmetic.
So this leaves us with two questions that seem to have confused our conversation: On what axiomatic basis does modern mathematics rest? and, How shall we divide up, and apportion credit, for the various piece-parts of mathematics?
The first question is not interesting at all to physicists, engineers, most mathematicians, or anyone else who simply wants to trust and use mathematics. It is interesting only to mathematicians involved in the Foundations of Mathematics, and to historians of mathematical development.
The second question is at the heart of your misunderstanding of what I previously wrote. It seems reasonable to me to divide mathematics up into non-overlapping sets in order assign people's names to theorems, or sets of axioms, or even to whole branches, as in The Galois Theory, as I described earlier. In this kind of division and attribution, most contributions to mathematics do not include new axioms. But, of course, at the root of any mathematical theorem, if you go back to the foundations, you will find axioms of some sort or another. But these are not part of the new contribution and may not even be clearly identifiable.
For example, if someone discovered a new theorem of Euclidean Geometry, it would be an arbitrary choice to say that the theorem rested on Euclid's axioms or that it rested on Hilbert's axioms. The choice would make no difference to the validity or usefulness of the new theorem.
Similarly, Dick's discovery begins very high up in the mathematical tree, and as I said, it introduces no axioms whatever. He simply assigns names to some sets of numbers he wishes to consider, and then proceeds to deduce what I consider to be a theorem about some of the properties of those sets. In particular, he deduces a set of constraints which apply to any function which can possibly produce the probability amplitudes for predicting some patterns of some of these sets.
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****Are you saying that you can define fields in set theory that have no relationship to each other. Now that is something I would like to see in a journal.****
No, I don't think I said that, and I'm not sure exactly what you think I might have said.
Let me take your question piece by piece to try to make sure we understand one another.
"Are you saying that you can define fields in set theory[?]"
Yes, fields are defined in set theory.
But you have to be careful here. The term 'fields' looks like it is plural and therefore you might think this means that there are, or can be, multiple definitions for fields. For example, you might think that we can define five different kinds of fields, and if so, then it would make sense to ask about the relationships among these five kinds of fields.
Of course, you could define five different kinds of fields, but that is not done in mathematics. Instead, the term field is singular, and there is typically one definition given for a field. (It is defined as a ring with a couple specific constraints).
Now the fact that there is only a single definition for 'field', does not mean that there are not many examples of sets of various objects that happen to qualify as fields according to the definition. There are indeed many such examples. Some of them involve sets of numbers, others involve sets of elements that have nothing to do with numbers.
It might be that these various examples of fields are what you were referring to in your question. If so, it should have read,
"Are you saying that you can [find examples of] fields in set theory that have no relationship to each other[?]"
The answer to this is "no". All such examples have the relationship that they share the property of meeting the conditions of the definition of a field. This is sort of a trivial relationship, though, and I doubt that that is what you were getting at.
Let's take a couple of specific examples: Rational numbers form a field, and five-dimensional vectors form a field. Is there a relationship between them? Well, yes. They both happen to involve tuples of numbers. But that is just coincidental. There are examples of fields that have nothing to do with numbers.
So, if you disregard the fact that all fields meet the conditions of the definition, the answer is that there are examples of fields that have no relationship to each other. I think there is no profound revelation here, so I doubt that you will find this conclusion written up in the journals.
Sorry for being so late with this; I hope it answers your questions.
Warm regards,
Paul |