# Axioms, meaning, and notation

According to at least one philosophy of mathematics, the axioms determine the meaning of the primitive symbols that are used in the axioms. The phrase "used in" is somewhat imprecise, so please see the note at the bottom.

Could it be worthwhile to conduct actual experiments to see whether there is evidence of less confusion or better understanding of written materials when -- for example -- set theory conjectures that are to be deduced without relying upon the axiom of choice are written using some mathematical symbol other than "∈" to represent a primitive, binary membership relation for which we refrain from assuming the axiom of choice?

Given that we build a hierarchy of defined terms upon the primitives, we would need many new mathematical symbols to conduct such experiments, and the need to learn those symbols would put a burden on the memory. So we face the question of how to conduct a fair experiment.

Of course, an inventory of logical fallacies includes the fallacy of equivocation. If axioms determine the meaning of the primitive symbols that are "used in" the axioms, then using the same primitive symbol in connection with different systems of axioms would be an example of equivocation. Are there published examples of errors discovered in mathematical literature that have been traced back to that kind of (alleged) equivocation?

Regarding the phrase "used in": The primitive symbols may be used directly, in the sense that they explicitly appear in the axioms, as the axioms are ordinarily written ... or they may appear indirectly. Given that definitions in mathematics -- in contrast with ordinary language definition in dictionaries -- provide a way to rewrite discourse to eliminate defined symbols, it is in principle possible to rewrite the axioms so that all of the symbols that appear are either symbols of logic or primitive symbols of mathematics, and so that there is no need for definitions of mathematical symbols. Of course, what is possible in principle is not necessarily practical in terms of conciseness, clarity, avoiding human error, or other related considerations.

• What philosophy are you talking about? The meaning of { } is not defined axiomatically, see the top answer here: math.stackexchange.com/questions/1452425/…. Oct 9, 2023 at 4:36
• @J Kusin I think that you will find it quite explicitly in the mathematical philosophy of Carl G. Hempel and/or Afred Jules Ayer. Oct 9, 2023 at 4:46
• For equivocation, the late David McCarty seems to touch on this "the classical mathematician can do all the mathematics in the world done classically, not just the logical rules you lay down, but contentual things you prove and say; all of that will not determine that the connectives that you use name the connectives that you think you are using name in fact classical negation rather than intuitionistic negation" youtu.be/…. Similar argument about functions youtu.be/cLpHvZH6avg?t=690 by him Oct 9, 2023 at 4:53
• You can use ZF set theory instead of ZFC if you intend to exclude axiom of choice. There're also well known constructive set theories (CST, Bishop CST, CZF, etc) further excluding LEM whose rigorous construction requirement for any proof is perhaps nothing but your intended 'experiments'. There're plenty of symbols to be pumped for math so equivocation fallacy is very rare in math, even you have to use other system axioms instead of their foundational set theory symbols... Oct 9, 2023 at 5:44
• Re note: "appear indirectly"? See a "typical" example: Zermelo-Fraenkel Set Theory (ZF). There is only one specific primitive symbol: the binary predicate "in" () and it is used in all set axioms. Other symbols: union, intersection, subset are all defined explicitly in terms of it. Oct 9, 2023 at 7:27

The reference in your comment to Ayer and Hempel is on point. The logical positivist approach to philosophy and mathematics involves identifying the meaning of a sentence with its justification conditions. On this view, the meaning of a mathematical sentence is nothing more than the role it plays within proofs. Mathematical sentences cannot make statements about verification-transcendent propositions. They do not have a meaning that relates to some platonic reality. This position is no longer popular, particularly within philosophy generally. Even within the philosophy of mathematics, it mainly survives in the form of intuitionism.

In terms of how mathematicians typically work, I think you will find they are aware of when they doing things like relying on the axiom of choice. ZF and ZFC are different theories, and there are important theorems of ZFC that are not theorems of ZF. This does not mean that the set membership relation is different. The difference lies mainly with the conditions under which one may assert the existence of a set.

In the case of the underlying logic, most mathematicians work with classical logic, but some use intuitionistic logic, and a few use linear logic, or relevance logic or other paraconsistent logics. Again, as long as one is clear which logic is being used, it should not lead to equivocation.

In practice, most mathematicians probably work with an intuitive understanding of mathematical concepts, rather than referring constantly to axioms provided by logic and set theory. I suspect that if you surprised a working mathematician and asked them to state all of the axioms of ZFC from memory, most would struggle.

Primitive symbols and definitions are not arbitrary. Their job is usually to take some naive, pre-theoretic concepts and make them formal. We start with a pre-theoretic understanding of a collection of things and we progress to a formal set theory. We start with a pre-theoretic understanding of mechanical calculation and we progress to a formal theory of computation. We will always need some underlying understanding of mathematical concepts to get us started. Definitions can formalise these, but not eliminate them.

• "Primitive symbols and definitions are not arbitrary. Their job is usually to take some naive, pre-theoretic concepts and make them formal." There's a lot that could be said to support that point of view. However, if Ayer, Hempel, etc. never challenge it directly, then they are tactically sophisticated in refraining from providing an opportunity -- via debate against their claims -- to support it. Ideally, it would be possible to remain agnostic on the big picture in order to provide an answer with persuasive power or traction, regardless of the views of the individual reading your answer. Oct 10, 2023 at 16:38
• Consider the claim that the project of deciphering Egyptian hieroglyphics remains incomplete. In particular, it could be claimed that until somebody finds an ancient Egyptian tablet that lists the mathematical axioms they accepted, we don't have enough information to determine the meaning they assigned to the mathematical vocabulary written in their hieroglyphic writing system. Is that observation enough to hint that the mathematical philosophy of Ayer, Hempel, (etc.) is incompatible with a full appreciation for the historical development of human cultures? Oct 10, 2023 at 17:08
• I wouldn't say that the logical positivist philosophy is incompatible with human culture. It does tend toward holding that any question must have an answer, otherwise the question is without meaning. That seems to me and many others an unreasonable assumption. Even if our knowledge of Egyptian hieroglyphs is incomplete, and even if there is insufficient information to complete it, we can speculate and perhaps understand the significance of the hieroglyphs. Oct 11, 2023 at 14:35

The ambiguity involved in using the ∈ symbol as well as the choice of the background logic is frequently voiced by constructive mathematicians such as Bishop and Richman. Namely, classically-trained mathematicians frequently formulate theorems without specifying whether or not they depend on the Law of Excluded Middle and the Axiom of Choice. The constructivists frequently call on them to be more explicit about the background requirements of their theorems.

There is an area of classical mathematics where researchers are more careful about this sort of thing: Reverse Mathematics. Here great importance is attached to the precise logical system (typically, systems of second order arithmetic) being used. Some recent work has explored (more explicitly than is usually done) the role of the axiom of choice in infinitesimal analysis; see this page for an introduction.

• Incidentally, I have never seen the notation "∈<sub>**AC**</sub>" used in set theory papers. In fact, it would be more appropriate to place the subscript "AC" on the existence quantifier rather than on the membership relation itself, since it is the meaning of the quantifier that is being modified (in constructive mathematics). Oct 9, 2023 at 13:29
• As soon as it is maths, it should be straightforward to read the proof and make the list of the axioms used.
– Plop
Oct 9, 2023 at 14:37
• @Plop, perhaps this should be the case as you say, but in fact it is seldom the case. The problem of determining a minimal axiom system that would produce the result being thought is highly nontrivial, and in fact is one of the motivations of the program of Reverse Mathematics, with numerous publications on record. Moreover, the answer is not always unique; constructive mathematicians would favor a different set of hypotheses than would classically-trained mathematicians. Oct 9, 2023 at 14:40
• That's not what I meant. In the proof "let x be a real number, such that x=2; then, by the axiom of choice, x^2 = 4", the axiom of choice is used. Whether or not it can be removed is indeed much more subtle than the mere fact that it appears in the proof. Therefore, it's no big deal if someone doesn't write, in his/her algebra book, "be careful, the axiom of choice is needed in the following proof of Krull's theorem", because the constructivist reader can probably just notice it by reading the proof (while the other readers may not care).
– Plop
Oct 10, 2023 at 10:06
• @Plop, things are even more subtle than you suggest. Many proofs appearing in the literature rely on the axiom of choice without realizing it. Thus, Halmos in his famous measure theory book proves the countable additivity of Lebesgue measure apparently without using the axiom of choice. However, there is a gap in his proof: there are models of ZF where the Lebesgue measure is not countably additive; see for example this publication. Oct 10, 2023 at 14:32

A (semi-)classical counterexample to the "axiomatics fix semiotics" viewpoint is the Gulf of TONK incident (forgive the pun!). Arthur Prior, of temporal-logic fame, invented a random logical connective whose introduction and elimination rules allowed spontaneous inferences of whatever conclusions, something like methodological trivialism (if you will).

As for the effect on efficiency (zounds, another pun!) of distinguishing between, say, ∈AC and ∈AD (for the axiom of determinacy, that is), well, I don't doubt that it would be somewhat worthwhile to look into this, but I also don't doubt that set theorists already do such things. I can't remember an extremely clear example of them doing so off the top of my head except that they are extremely fastidious about equivalence relations, but having read through an inordinate amount of set-theoretic texts over the last few years, I'm confident enough to tell you that the semiotic-efficiency problem has been faithfully approached in many and varied ways, here (for instance, just try out the acronym AOC instead of AC on your average set theorist and watch them squirm and sputter!).