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In what way is 'the definition of $x$ is $y$' ($x:=y$) not the same as '$x$ is equivalent to $y$' ($x=y$)? I can find no justification for making the distinction aside from 'it feels right'. It seems that every property of $:=$ is also a property of $=$ and that neither possesses a property that the other does not.

Furthermore, trying to casually force a distinction either renders one relation meaningless, with the other assuming the meaning of 'equality', or leads to glaringly obvious contradictions (ex $1:=S(0)\land S(0):\neq1\implies 1\in\Bbb{N}\land 1\notin\Bbb{N}$ using the Peano axioms).

Is there any objective distinction between a term and its definition (without explicit typing)?

I know its a bit (okay, extremely) pedantic to insist on a formal distinction between the definition of an object and the equivalence class to which that object belongs, but making this distinction effectively sets the limits on the definability of objects within a theory.

Note: the definition of an object is an equivalence class in the sense that a definition is specified via the conjunction of a sequence of formulae $(\Phi_i)_{i\in I}$, and the collection of objects $X=\{x\mid\bigwedge_{i\in I}\Phi_i(x)\}$ is an equivalence class since the relation $x\sim y\iff \left(\bigwedge_{i\in I}\Phi_i(x)\right)\land\left(\bigwedge_{i\in I}\Phi_i(y)\right) \iff x\in X\land y\in X$ is an equivalence relation. There are more technical details regarding the nature of this relation depending on whether or not a universe of discourse has been specified, but that's another thing entirely.

R. Burton
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  • Basically, an equiv class is an "object" while with a definition you introduce a new "name", i.e. a linguistic entity. – Mauro ALLEGRANZA May 24 '19 at 14:56
  • Aren't these used interchangeably though? As long as we aren't introducing physical objects (and even then), can't the name be substituted for the object and vice-versa? – R. Burton May 24 '19 at 14:58
  • Related: https://math.stackexchange.com/questions/182101/appropriate-notation-equiv-versus Also: https://math.stackexchange.com/questions/63553/difference-between-and – JMoravitz May 24 '19 at 15:00
  • @R.Burton You're right. At a philosophical level, it's hard to separate a (reference to) a mathematical object from its name (because these are probably the same thing, although I am not a philosopher). This is more complicated by the suggestion that $:=$ is a "definition" where we all know that the same mathematical structure may admit multiple different yet equivalent definitions. The problem is that when building things up axiomatically, we are sort of always bounded by the problem that we have to assume one of those definitions to be the definition and that the rest are merely equivalent – Jack Crawford May 24 '19 at 15:08
  • Category theorists have a nice way of arguing that we shouldn't really care which is the canonical definition and which is the "equivalent" definition though, via their means of defining things up to isomorphism. Although we are still forced to choose a particular definition as the "canonical" one whenever we want to construct an object, though. The category theory/type theory just lets us swap between the constructions interchangeably. Homotopy type theory does this at an even deeper level. – Jack Crawford May 24 '19 at 15:10
  • @JackCrawford Why should there be a the definition though, if all definitions identify the same referent? – R. Burton May 24 '19 at 15:11
  • Tthere's a bit of a running conflict between constructivists and platonic idealists at the heart of mathematics. The platonic idealists sort of like to think that a lot of mathematical structures are only grasping at some universal platonic form, whereas the constructivists like to believe that there is no particular universal canonical form, but that we can demonstrate isomorphisms to show that the many separate definitions are in some sense interchangeable. It's just that when talking about a particular object constructively in terms of data, we need to pick a particular representation/defn. – Jack Crawford May 24 '19 at 15:15
  • It's kind of like how if we're trying to represent coordinates on $\mathbb{R}^2$, we could, for example, interchange between $(x,y)$, $(y,x)$ or even polar coordinates $(r,\theta)$. Yes, they are all somewhat interchangeable, but in order to even examine that they are interchangeable we sort of have to pick just one of the definitions to start with and then demonstrate its interchangeability with the others. From the perspective of (computational) theorem proving/verification, this usually means picking a particular definition to be the canonical one and defining the others as theorems. – Jack Crawford May 24 '19 at 15:19

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Equality, in many system and particularly for first-order logic with equality, is a relation of the logic. Most logical systems don't have a mechanism for making a definition. It simply makes no sense to say $(1:=S(0))\land P$ in these systems. A definition isn't a claim. It's not a statement that has a truth value.

So, not only are equality and definition not the same, they aren't even the same sort of thing. The most common way to handle definitions is meta-logically and informally. A definition like $1:=S(0)$ is an instruction to you, as the reader, to mentally replace $1$ with $S(0)$ everywhere you see it. You then trivially get $1=S(0)$ be reflexivity, because this really means $S(0)=S(0)$.

There are several ways of formalizing the notion of definitions. The typical way would be an extension by definition. For this perspective, a (meta-logical) definition like $1:=S(0)$ means: "Add the constant $1$ and the axiom $1=S(0)$ to the current theory, then continue with this new theory." Note how operational this is. It's instruction to create a new theory. "Add the constant $1$ and the axiom $1=S(0)$ to the current theory, then continue with this new theory," is not a statement that is true or false.

Some logical systems have an internal mechanism for making definitions. One example is the system used by Coq. The $\delta$-reductions do a formal version of the "mental replacement" described earlier. To over-simplify, it basically states that when you are in a context with a definition like $1:=S(0)$, then $1$ reduces to $S(0)$. The typing judgment is (supposed to be) sensitive only to normal forms of the reduction relation. There are constraints on the rules that introduce definitions to avoid (or clearly define the behavior of) multiple, potentially inconsistent definitions.

Derek Elkins left SE
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  • So, if we were being completely literal, would it be correct to say that we can't prove that $1$ is a natural number from the Peano axioms? – R. Burton May 27 '19 at 13:06
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    @R.Burton If we're being "completely literal", $1$ is not a term of Peano Arithmetic (PA) so any formula involving it would be ill-formed. Also, PA doesn't prove that things are or are not naturals. We simply call the individuals of PA "natural numbers". Being a single-sorted theory that means every term is a "natural number"; there's nothing else a term could be so that you would need to prove it was a "natural number". That said, there are models of (the first-order form of) PA that aren't the (standard) natural numbers. – Derek Elkins left SE May 27 '19 at 13:19
  • This was one of the biggest issues I had with a particular professor of number theory, who simultaneously insisted on strict adherence to PA, and a doctrine of "absolute rigor" - often in a seemingly contradictory fashion. It's good to know that I wasn't mistaken by thinking that, in an "absolutely rigorous" sense, PA doesn't provide a means of proving most of their claims. This is also why I think the words 'literally' and 'absolutely' should probably be banned from academic literature. – R. Burton May 27 '19 at 13:31
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The notion of equality in mathematics that you've touched upon is something very deep.

Every axiomatic system in mathematics (that I am aware of) has some way of imbuing the canonical $=$ sign (if there even is a canonical notion of equality, which there sometimes is not) with some definitional value that is distinguished from just-any-other equivalence class defined over a structure.

In most (all?) versions of set theory, two sets are equal in the purest sense if they contain the same elements, and the elements of a set themselves are also just composed out of smaller sets (where axiomatically we only assume the existence of the empty set and a few ways of constructing new sets out of that) so all of the equality follows this same definition (in some sense, it reduces to being able to compare the empty set to any other set), whereas the same cannot be said of just any arbitrary equivalence relation over a set (although sometimes the equivalence of cosets under a quotient can confuse this).

If you're partial to constructive mathematics, then in certain forms of the calculus of inductive constructions, equality is just another inductive data type with nothing particularly special about it. Depending on the exact implementation of your type system though, you could end up with a definition of equality that basically boils down to two things having the same data representation on the computer. Or you could just as well come up with a very different kind of equality.

The real answer is that there are many different takes on this topic under many different systems, and to understand some of the variety involved I can only recommend checking out some type theory, axiomatic set theory, and category theory. This is a very good question, though, and one I wish I had a better answer for.

Jack Crawford
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