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> I think your objection is to the fact that most treatments of first order logic informally talk about a "set of variables indexed by the natural numbers" or s

by dls2016 7y ago

> I think your objection is to the fact that most treatments of first order logic informally talk about a "set of variables indexed by the natural numbers" or something of that sort, but that's informal, and it's understood that really what's going on is an algorithm "under the hood." Precisely my point. > an example of a definition of anything that you find is watertight There is no such definition. The existence of anumeric languages suggests that my worrying about this is important for pedagogy, not just philosophy. From my experience, if you want to have students understand the meat of an "intro to proofs" or first real analysis course, then you have to avoid the discussion of what "is" really is. Undergraduates love that crap as it's a distraction from fighting with the new symbols they're attempting to learn. But at the end of the day (or ideally, in the first few lectures), the student must realize three things: there's always a next integer, a set is a bag of distinct objects and a function assigns each object in one bag to an object in another (or possibly the same) bag. Many students are never explicitly told that there a few fundamental, undefinable building blocks of mathematics - the most important being set, function and natural number. And like the person who never learned numeric words, their understanding is hindered.

dwohnitmok 7y ago
I agree that there are undefinable building blocks of mathematics. I don't think the undefinable building blocks of mathematics are set, function, and natural number. My list of fundamental, undefinable building blocks are existence, reference, equality, and memory (the capability to store multiple statements at once), which holds for all definitions ever, not just mathematical ones (this is what I meant by a definition of anything, I meant really a definition of literally anything, mathematical or not). It's important to note that reference is not a relabeling of function, nor is memory a relabeling of natural number and set. It's a bit hard to describe why this is true in English because this is touching on such meta-linguistic foundational issues, but the best example I can give is any formal theorem prover. Those define sets, functions, and natural numbers using existence, reference, (implicit) equality, and memory instead of the other way around. Another example of this is that reference can be used to define both functions and relations. As for memory, although you could use sets to model memory, it's important to note that the notion of memory is an artifact independent of sets, in the same way that ZFC can be modeled by an outer series of sets in an "external" ZFC, but exists independently and could be modeled by something else (see model-theoretic reflection). Taking a page from Wittgenstein and moving away from the ineffable, let's turn our attention to pedagogy. Pedagogically it makes it impossible to explain why first order logic is the logic used for set theory, especially in the face of various model-theoretic problems with first order logic (e.g. Lowenheim-Skolem), instead of second order logic. Namely first-order logic, as I was hoping to show with the BNF syntax, is NOT dependent on a understanding of sets, but second order logic is (which would make using it to define set theory circular). It makes it hard to teach the significance of the Peano axioms. It makes the presentation of logical relations (and functions as a special case thereof) problematic. It makes it difficult to see why basing mathematics on set theory is NOT a trivial, obvious choice and why alternatives might exist. If you're talking about the Piraha language (and other languages like it), now you're talking about very very philosophical issues that also really sit at the cross-section of all epistemology and language, not just mathematics.

dwohnitmok 7y ago

I agree that there are undefinable building blocks of mathematics. I don't think the undefinable building blocks of mathematics are set, function, and natural number. My list of fundamental, undefinable building blocks are existence, reference, equality, and memory (the capability to store multiple statements at once), which holds for all definitions ever, not just mathematical ones (this is what I meant by a definition of anything, I meant really a definition of literally anything, mathematical or not). It's important to note that reference is not a relabeling of function, nor is memory a relabeling of natural number and set. It's a bit hard to describe why this is true in English because this is touching on such meta-linguistic foundational issues, but the best example I can give is any formal theorem prover. Those define sets, functions, and natural numbers using existence, reference, (implicit) equality, and memory instead of the other way around. Another example of this is that reference can be used to define both functions and relations. As for memory, although you could use sets to model memory, it's important to note that the notion of memory is an artifact independent of sets, in the same way that ZFC can be modeled by an outer series of sets in an "external" ZFC, but exists independently and could be modeled by something else (see model-theoretic reflection). Taking a page from Wittgenstein and moving away from the ineffable, let's turn our attention to pedagogy. Pedagogically it makes it impossible to explain why first order logic is the logic used for set theory, especially in the face of various model-theoretic problems with first order logic (e.g. Lowenheim-Skolem), instead of second order logic. Namely first-order logic, as I was hoping to show with the BNF syntax, is NOT dependent on a understanding of sets, but second order logic is (which would make using it to define set theory circular). It makes it hard to teach the significance of the Peano axioms. It makes the presentation of logical relations (and functions as a special case thereof) problematic. It makes it difficult to see why basing mathematics on set theory is NOT a trivial, obvious choice and why alternatives might exist. If you're talking about the Piraha language (and other languages like it), now you're talking about very very philosophical issues that also really sit at the cross-section of all epistemology and language, not just mathematics.