Are axioms definitions in disguise?

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Some argue that axioms are definitions in disguise. Are they right? Henri Poincaré seems to have though so, but Karl Popper disagreed. If anyone can locate and include their arguments here, please do so, and provide citations.

Search terms: definitions in disguise, axioms are definitions, implicit definitions.

Axioms are definitions in disguise

Pro

Pro A theory providing axioms but no definitions has the axioms serve as definitions. The terms refer to that which meets the axioms. The axioms do not make any claims about anything so long as the terms they use have no definitions, and therefore, no meaning other than the one that axioms give them. Axioms are therefore neither true nor false: before they bind terms to their meaning, the axioms do not mean anything and thus have no truth value.
- Objection To call axioms meaningless would not only deny human nature, it would deny reality itself. Axioms are self-evidently true, and so do not require proof. An axiom is αχιωμα, that which is considered worthy, held as w:self-evident by those who possess common sense and are capable of speaking a natural language.
Pro Even if axioms do not unequivocally pick the referents of the terms, they point to canonical ones, as if located in the shared semantic context. Thus, if one mathematician gives Peano arithmetic to another one, does not tell him what it is supposed to do, does not use the mnemonic symbols 0, s, +, and *, and then asks, what do you think this is, the mathematician should be able to guess these are natural numbers. A guess not worth making is that the referents are the terms themselves since all consistent sets of axioms of first-order logic have such a model, and it has no priority or significance. And when a mathematician sees Peano arithmetic, he sees the natural numbers as he understands them and not the variant where the Gödel sentence is false. This bears witness to natural numbers having a priority in the list of candidate interpretations. Admittedly, this does not hold of all sets of axioms: if one has axioms of the notion of group from algebra, it is hard to guess which group is meant. This idea points to sentences about terms constraining their semantics, even if not completely. Thus, if one has two notions from biology, X and Y, and one says X is part of Y, then X and Y can be leaf and tree, but not the other way around. In this way, the semantic relations of hyponymy and meronymy are part of the definition process, and so is instance of. A set of constraints on a set of terms with unknown or unclear semantics that uses semantic relations and other well understood relations can thereby act as an effective definition helping clarity and precision that a genus-differentia definition often does not reach. Similarly, if one sees axioms of Euclidean geometry that do not use the mnemonic words "point" and "line", one should be able to guess that the referents are points and lines. Of course, one can, in the spirit of modern mathematics, think these are in fact pure sets since everything is pure set, but that is ontologically wrong: points are not sets but rather points.

Con

Con There is another interpretation of axioms: the terms do have meaning, but are undefined. What the axioms do is not define the terms but rather provide true claims that are left without proof or derivation but rather serve as basis for proof or derivation. That axioms do not necessarily provide a definition for terms used by them is clear from Peano arithmetic: while natural numbers are one model (satisfying interpretation of) Peano arithmetic, terms of the arithmetic are another model. Thus, in one model, s(s(0)) is assigned to 2, while in another model, s(s(0)) is assigned to "s(s(0))". The axioms of Peano arithmetic do not pick between the two models, and thus, they do not define the terms "0", "s(0)", etc., (they do not state the genus, to say the least) merely expose their relational properties. Admittedly, once the proof system has axioms at its disposal, it does not need definitions, especially genus-differentia definitions. This gives the meaning to the claim that "mathematicians do not know what they are talking about": systems of axioms used for symbolic or algebraic manipulation do not need definitions and do not need to have any idea of what kind of entities are being referred to: all the entities have to do is have terms be bound to them behave in a way that meets the axioms. Elementary school geometry gets rather far with intuitive definitions by example of "point" and "line", perhaps stressing that the points are "infinitely thin" and that the lines are "straight", but not necessarily wondering about the genus of "point" and its distinguishing characteristics. The definitions rely on children knowing what "infinitely thin" and "straight" means.
- Objection The above interpretation makes sense for Euclidean geometry and natural numbers since they probably correspond to innate human intuitions. But it does not make sense for, say, hyperbolic geometry. If we start with the notion of a line as given but undefined, then we may ask whether it is Euclidean or a hyperbolic geometry that is true; both cannot be true. But if axioms are definitions in disguise, then there is Euclidean line and hyperbolic line, two different notions. As a result, geometries are not true or false. Each geometry is a mathematical object of its own. Without hyperbolic axioms, we would not know what hyperbolic line means. (It seems to have been this problem of incompatible geometries that lead Poincaré to the idea that axioms are definitions in disguise.) See W:Hyperbolic geometry.
  - Objection Good points. However, perhaps we could use some definition process that would explain what hyperbolic lines are without relying on axioms. It would be some kind of definition by pointing. Or we could embed hyperbolic geometry in Euclidean geometry, saying that hyperbolic line is a kind of Euclidean line segment or something of the sort. Axioms are probably not the only way to define hyperbolic line.
  - Objection As a nitpick, a geometry can still be true in some physical sense. Geometry was not originally some kind of abstract mathematical enterprise but rather an attempt to accurately describe the real physical space.
    - Objection Fair point. However, our modern understanding is that mathematical objects live in their own world of abstract objects, and that they merely model the physical world. Thus, a geometry is a better or worse model of the physical world. From the purely mathematical standpoint, whether a particular geometry actually models anything in the real world does not really matter. Mathematics can be interested in more geometries than are relevant to the empirical world, more numbers than are reflected in the empirical world, more modal logics than users in the empirical world find practically relevant, etc.
    - Objection Right, it can be true in the sense that it accurately represents the real physical space. However, in that sense, Euclidean geometry seems false given modern physics, relativity and space curvature, but that is not what mathematicians think. The geometry is consistent, has a model (interpretation meeting the axioms), and that makes it fine. Of course, it is also fine for many practical applications.
    - Objection It is possible that real physical space is so arcane that it will be hard to find a geometry that perfectly accurately represents it. After all, relativity theory is not a theory of everything and not necessarily the final version of physics. Mathematics is not advanced by mathematicians worrying about whether the geometries they are investigating are all false in that physical sense. Thus, the notion that geometries are neither true nor false but rather abstract objects existing in the dedicated abstract world is a good position for a mathematician to take.
Con The proposition commits the w:nominalist fallacy by assuming that the earth (geo) that we measure (meter), exists merely because the geometers said so. See the objection to the first argument pro.

Axioms are definitions in disguise

Pro

Con

Further reading