4 ms·

> precise definition of each term I’m not a mathematician, but aren’t some fundamental terms in mathematics NOT clearly defined? For example a set?

manifestsilence 7y ago
I'm not either, but I think it depends on which system you're working from within. There's the ugly, beautiful thing from Godel's second incompleteness theorem saying a formal system cannot prove its own consistency from within. With set theory, it looks like there was some reformulating of the fundamental terms to prevent paradox, but I'm not quite clear on whether that leaves us with a consistent formal system in the end: https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...
- dwohnitmok 7y ago
  It leaves us with a system that we strongly suspect is consistent, but which we cannot prove is such (at least not without resorting to a stronger system whose consistency is again unproven). Note that even were ZFC be proven to be inconsistent it would probably not be a huge issue practically speaking for most mathematicians. This Math Overflow answer describes it best: https://mathoverflow.net/a/41030 https://mathoverflow.net/a/41030
  - manifestsilence 7y ago
    Awesome, thank you for clarifying!
dls2016 7y ago
The definitions of set, function and integers will always have a bit of Justice Stewart "know it when I see it" quality.
- dwohnitmok 7y ago
  The major achievement of early modern mathematics was precisely to remove this ambiguity. Sets, functions, and integers all have precise definitions. They are respectively provided by ZFC set theory, ZFC set theory again, and the Peano axioms (with your chosen method of extending the natural numbers to the integers, the easiest being an algebraic extension, or you could use equivalence classes of pairs).
  - dls2016 7y ago
    Ok, then produce a definition of the syntax of the logic in which you interpret set theory without referring to sets, functions or integers.
    - dwohnitmok 7y ago
      Are you talking about a definition of the syntax of first-order logic? Would a BNF grammar do? Here's one of the basics of FOL. Formula = PrimitiveFormula | Formula Connective Formula | not Formula | Quantifier Variable Formula VariableList = Variable | Variable,VariableList PrimitiveFormula = Predicate(VariableList) Term = Constant | Variable Connective = implies | and | or Quantifier = forall | exists Constant = c | c Constant Variable = v | v Constant Predicate = p | p Predicate It's adapted and formalized from https://web.engr.oregonstate.edu/~afern/classes/cs532/notes/fo-ss.pdf https://web.engr.oregonstate.edu/~afern/classes/cs532/notes/.... Note I've left out function symbols so as to minimize confusion. Function symbols are not functions themselves, but they can be represented by predicates with an additional argument so I've left them out for clarity. It's kind of restricted in that variables are now called v, vv, vvv, etc., predicates are called p, pp, ppp, etc., but that seems minor and I was being lazy. 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." Crucially I'm using recursion here, but a recursive definition has no need to appeal to sets, functions, or integers. In fact since we're on HN it's worth pointing out all that recursion is is a while loop, and I could've done that instead. In that case though I wouldn't have provided a BNF and I instead would need to provide a full program (which is definitely possible, this is what tools like bison do for you automatically). If what you're concerned about is the "interpret" part (i.e. the model part of model theory), that's a far deeper philosophical question that touches on whether it's possible to define anything at all. Modern mathematics can be thought of as simply a syntactic game. That's why it can be formalized by computers. Philosophically this may be troubling because it doesn't explain why certain results are interesting or why an application of math to real-world objects "works," but like I said, that's starting to touch on the broader issue of whether it's possible to define anything at all and what exactly language is and that's a deep deep deep philosophical hole that affects the entirety of human language. This is where you have to start closely scrutinizing what the word "is" really, well, is. Perhaps it's first useful to sort out an example of a definition of anything that you find is watertight, compare and contrast, and discuss from there?
dwohnitmok 7y ago
Sets are clearly defined. The usual formalism for defining a set is ZFC set theory. In fact to a first approximation all of modern mathematics is sufficiently well-defined to be entirely formalized in a computer. This is an approximation primarily because currently the burden to do so is immensely high. With today's technology formalizing a proof can easily take several orders of magnitude more time (if not more!) than writing the informal, but commonly accepted as rigorous, proof. There are philosophical questions around what it means to "define" something. But if you want to go there, well then nothing is clearly defined.
- ganzuul 7y ago
  Isn't that definition just consistency, both in theory and in practice?
Someone 7y ago
You can’t rigidly define your first term out of nothing, so in the end, all of mathematics is built on _some_ common understanding that isn’t defined at all. For example, if mathematicians say (https://www.britannica.com/science/Peano-axioms https://www.britannica.com/science/Peano-axioms): “Zero is a natural number.” they don’t say what is a means. Even before that, the statement there’s a concept called ‘natural number’” builds on shared understanding of what there’s* and concept mean. Mathematicians have worked hard on minimizing the amount of common understanding needed, though. Certainly, the fairly complex concept “set” gets a clear, unambiguous definition.

manifestsilence 7y ago

I'm not either, but I think it depends on which system you're working from within. There's the ugly, beautiful thing from Godel's second incompleteness theorem saying a formal system cannot prove its own consistency from within. With set theory, it looks like there was some reformulating of the fundamental terms to prevent paradox, but I'm not quite clear on whether that leaves us with a consistent formal system in the end: https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...

dwohnitmok 7y ago
It leaves us with a system that we strongly suspect is consistent, but which we cannot prove is such (at least not without resorting to a stronger system whose consistency is again unproven). Note that even were ZFC be proven to be inconsistent it would probably not be a huge issue practically speaking for most mathematicians. This Math Overflow answer describes it best: https://mathoverflow.net/a/41030 https://mathoverflow.net/a/41030
- manifestsilence 7y ago
  Awesome, thank you for clarifying!

dls2016 7y ago

The definitions of set, function and integers will always have a bit of Justice Stewart "know it when I see it" quality.

dwohnitmok 7y ago
The major achievement of early modern mathematics was precisely to remove this ambiguity. Sets, functions, and integers all have precise definitions. They are respectively provided by ZFC set theory, ZFC set theory again, and the Peano axioms (with your chosen method of extending the natural numbers to the integers, the easiest being an algebraic extension, or you could use equivalence classes of pairs).
- dls2016 7y ago
  Ok, then produce a definition of the syntax of the logic in which you interpret set theory without referring to sets, functions or integers.
  - dwohnitmok 7y ago
    Are you talking about a definition of the syntax of first-order logic? Would a BNF grammar do? Here's one of the basics of FOL. Formula = PrimitiveFormula | Formula Connective Formula | not Formula | Quantifier Variable Formula VariableList = Variable | Variable,VariableList PrimitiveFormula = Predicate(VariableList) Term = Constant | Variable Connective = implies | and | or Quantifier = forall | exists Constant = c | c Constant Variable = v | v Constant Predicate = p | p Predicate It's adapted and formalized from https://web.engr.oregonstate.edu/~afern/classes/cs532/notes/fo-ss.pdf https://web.engr.oregonstate.edu/~afern/classes/cs532/notes/.... Note I've left out function symbols so as to minimize confusion. Function symbols are not functions themselves, but they can be represented by predicates with an additional argument so I've left them out for clarity. It's kind of restricted in that variables are now called v, vv, vvv, etc., predicates are called p, pp, ppp, etc., but that seems minor and I was being lazy. 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." Crucially I'm using recursion here, but a recursive definition has no need to appeal to sets, functions, or integers. In fact since we're on HN it's worth pointing out all that recursion is is a while loop, and I could've done that instead. In that case though I wouldn't have provided a BNF and I instead would need to provide a full program (which is definitely possible, this is what tools like bison do for you automatically). If what you're concerned about is the "interpret" part (i.e. the model part of model theory), that's a far deeper philosophical question that touches on whether it's possible to define anything at all. Modern mathematics can be thought of as simply a syntactic game. That's why it can be formalized by computers. Philosophically this may be troubling because it doesn't explain why certain results are interesting or why an application of math to real-world objects "works," but like I said, that's starting to touch on the broader issue of whether it's possible to define anything at all and what exactly language is and that's a deep deep deep philosophical hole that affects the entirety of human language. This is where you have to start closely scrutinizing what the word "is" really, well, is. Perhaps it's first useful to sort out an example of a definition of anything that you find is watertight, compare and contrast, and discuss from there?

dwohnitmok 7y ago

Sets are clearly defined. The usual formalism for defining a set is ZFC set theory. In fact to a first approximation all of modern mathematics is sufficiently well-defined to be entirely formalized in a computer. This is an approximation primarily because currently the burden to do so is immensely high. With today's technology formalizing a proof can easily take several orders of magnitude more time (if not more!) than writing the informal, but commonly accepted as rigorous, proof. There are philosophical questions around what it means to "define" something. But if you want to go there, well then nothing is clearly defined.

ganzuul 7y ago
Isn't that definition just consistency, both in theory and in practice?

Someone 7y ago

You can’t rigidly define your first term out of nothing, so in the end, all of mathematics is built on _some_ common understanding that isn’t defined at all. For example, if mathematicians say (https://www.britannica.com/science/Peano-axioms https://www.britannica.com/science/Peano-axioms): “Zero is a natural number.” they don’t say what is a means. Even before that, the statement there’s a concept called ‘natural number’” builds on shared understanding of what there’s* and concept mean. Mathematicians have worked hard on minimizing the amount of common understanding needed, though. Certainly, the fairly complex concept “set” gets a clear, unambiguous definition.