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...maths is not a formal language by any stretch of the imagination. I think about the best definition one can give is maths is what mathematicians do, and math

...maths is not a formal language by any stretch of the imagination. I think about the best definition one can give is maths is what mathematicians do, and maths notation is what one mathematician writes down to communicate mathematics to another mathematician. It's circular in about a dozen ways but that is as it is IMO.

ukj 7y ago
What mathematicians DO is computation. What a mathematicians WRITE is a formal language used to express computation. Manipulating mathematical expressions has a name. Symbolic Computation. https://en.wikipedia.org/wiki/Computer_algebra https://en.wikipedia.org/wiki/Computer_algebra
- usgroup 7y ago
  I'd say what mathematicians do is "think". We call the sort of thinking mathematicians do "mathematics". I get through quite a bit of grad stats in any given week and I'd say that what I'm not doing reading the computations of other mathematicians. I'm reading words in english, which try to explain concepts which are then sketched formulaically. I'm also reading proofs that are usually completed by a "mathematical" intuition and are very rarely if ever constructivist. Maybe, that's just me busy thinking whilst the rest are busy computing :-)
  - ukj 7y ago
    Once you are done thinking your thought() function returns a result in the form of language, no? Or does thought() fail to halt?
  - ukj 7y ago
    By the way - this is rather circular and it begs questions. > We call the sort of thinking mathematicians do "mathematics" What kind of thinking do mathematicians do? Is it any different to the kind of thinking computer scientists do?
- arkades 7y ago
  Mathematics hasn’t been “computation” since the days of hand-tabulated lists of logarithms.
  - ukj 7y ago
    A lookup table is a particular case of a space-time trade-off. https://en.wikipedia.org/wiki/Space%E2%80%93time_tradeoff https://en.wikipedia.org/wiki/Space%E2%80%93time_tradeoff
- adrianmonk 7y ago
  One can view it as "language" at multiple levels. In a computer language, there is a lexical level, and then there is a slightly deeper syntax level. Beyond that, there are levels of language, too: JSON or XML documents can follow one schema or another, for example. This schema defines a language on top of the JSON or XML language. (And you can do the same thing with any other data structure.) In math, the same sort of levels exist. At the most superficial level is the notation. At that level, I don't think mathematicians use formal language. They're free to invent a notation or even misuse a notation in a proof. There are no absolutely fixed rules, the complete set of rules does not have to communicated up front, and when push comes to shove, anything goes as long as it's clear to another mathematician. But at a deeper level, they deal with formalisms and manipulate information in a formal way. If you want to call that information manipulation language, I think that's reasonable because it can be used (and agreed upon) in an exchange of ideas. For example, do you want to say the Fibonacci sequence starts "0 1 1" or "1 1"? It's just a definition thing, but if you and I agree on one or the other and then start talking about the Nth element of "the Fibonacci sequence", we are communicating with each other in terms of that. (And it's formal because we are basing things on not deviating from that.)
  - ukj 7y ago
    The Chomsky hierarchy covers all of that. It's linguistics for formal languages. Grammar, syntax and semantics together with reflection allows for formalisms and meta-formalisms. It all culminates in Turing Completeness. Chomsky Type 0 grammars. If you find something more powerful than a Chomsky Type 0 grammar - you have made a ground-breaking scientific discovery.
  - ukj 7y ago
    >when push comes to shove, anything goes as long as it's clear to another mathematician. Well, yes! That's not inherent to Mathematicians. That's how all human communication works. That's what we use "natural" languages for. What we use formal languages for is to minimise the ambiguity, imprecision and non-determinism (entropy) in natural languages. Perhaps one way to put it: the purpose of natural languages is communication and context-setting. The purpose of formal languages is precision. That line gets blurred in computer science (and in my own mind) because of Homoiconicity. >For example, do you want to say the Fibonacci sequence starts "0 1 1" or "1 1"? It's just a definition thing I don't want to say absolutely anything about Fibonacci in a vacuum. The problem I am solving WITH Fibonacci will dictate whether it needs to start at "0 1 1", "1 1", or whether the the difference is immaterial. Off-by-1 errors are just a fact of programmer's life. We make those decisions on case-by-case basis. > if you and I agree on one or the other and then start talking about the Nth element of "the Fibonacci sequence", we are communicating with each other in terms of that. What you are pointing out is problems of convention and consensus. Off the top of my head I can think of a handful of strategies for coming to agree on the 0th value of the Fibonacci function in English. I bet you do too - all those things are intuitive to humans and we take them for granted. In computer science many of those problems have already been formalised, and some have been solved. https://en.wikipedia.org/wiki/Consensus_(computer_science) https://en.wikipedia.org/wiki/Consensus_(computer_science) P.S You have been using the concepts of "communication" and "information". Shannon formalised those! P.P.S Fibonacci is an numerical formalisation of the Golden spiral. In a nutshell, what I am pointing out is succinctly expressed by Donald Knuth. Science is what we understand well enough to explain to a computer. Art is everything else we do.
olooney 7y ago
I'm fairly certain that Hilbert would disagree with you: https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathe... In particular, most modern mathematics can be formalized within first-order logic with the ZFA axioms: https://en.wikipedia.org/wiki/First-order_logic https://en.wikipedia.org/wiki/First-order_logic https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... As a concrete example, the metamath project has formalized (and formally verified) huge chunks of mathematics: http://us.metamath.org/mpeuni/mmset.html http://us.metamath.org/mpeuni/mmset.html Certainly there are aspects of mathematics which are not formal (beauty, parsimony, elegance, clarity, geometric intuition, etc.) but saying math isn't a formal language is like saying chess doesn't have rules.
- jcranberry 7y ago
  He's not saying mathematics doesn't have rigorous foundations. Betrand Russell's Principia Mathematica is computation. But math textbooks/papers are not like this. They share insights, opinions, and metaphors, and frequently explain things in non-rigorous terms, although these are not substitutes for proofs even though proofs for something given a non-rigorous explanation may not appear. In less heavily proof-read contexts, such as in discussions or a class room, it's not uncommon for mathematicians to be pretty free-wheeling with their notation, and leave it up to context and intuition to figure out what an individual symbol means at a given time. And it wouldn't be a faux pas, or sign of a lesser mathematician or anything like that. These things can be valuable and in some sense are a part of mathematics. Although the end results of mathematics can be distilled into formal computations, it's far too reductive to say that's all mathematics is.
  - ukj 7y ago
    The exact same thing can be said about any human dynamic where the objective is to teach. The end-goal is to make the material accessible to the learner by connecting it some pre-existing human intuition or experience, thus providing the learner with a vocabulary to express their thoughts. The important question is how are the vocabularies (both formal and informal) produced by Mathematics different from the vocabularies produced by Computer Science when both fields are working on the same real-world problems? And if the two vocabularies are of equal utility - do we really need two different tools for the same job?

ukj 7y ago

What mathematicians DO is computation. What a mathematicians WRITE is a formal language used to express computation. Manipulating mathematical expressions has a name. Symbolic Computation. https://en.wikipedia.org/wiki/Computer_algebra https://en.wikipedia.org/wiki/Computer_algebra

usgroup 7y ago
I'd say what mathematicians do is "think". We call the sort of thinking mathematicians do "mathematics". I get through quite a bit of grad stats in any given week and I'd say that what I'm not doing reading the computations of other mathematicians. I'm reading words in english, which try to explain concepts which are then sketched formulaically. I'm also reading proofs that are usually completed by a "mathematical" intuition and are very rarely if ever constructivist. Maybe, that's just me busy thinking whilst the rest are busy computing :-)
- ukj 7y ago
  Once you are done thinking your thought() function returns a result in the form of language, no? Or does thought() fail to halt?
- ukj 7y ago
  By the way - this is rather circular and it begs questions. > We call the sort of thinking mathematicians do "mathematics" What kind of thinking do mathematicians do? Is it any different to the kind of thinking computer scientists do?
arkades 7y ago
Mathematics hasn’t been “computation” since the days of hand-tabulated lists of logarithms.
- ukj 7y ago
  A lookup table is a particular case of a space-time trade-off. https://en.wikipedia.org/wiki/Space%E2%80%93time_tradeoff https://en.wikipedia.org/wiki/Space%E2%80%93time_tradeoff
adrianmonk 7y ago
One can view it as "language" at multiple levels. In a computer language, there is a lexical level, and then there is a slightly deeper syntax level. Beyond that, there are levels of language, too: JSON or XML documents can follow one schema or another, for example. This schema defines a language on top of the JSON or XML language. (And you can do the same thing with any other data structure.) In math, the same sort of levels exist. At the most superficial level is the notation. At that level, I don't think mathematicians use formal language. They're free to invent a notation or even misuse a notation in a proof. There are no absolutely fixed rules, the complete set of rules does not have to communicated up front, and when push comes to shove, anything goes as long as it's clear to another mathematician. But at a deeper level, they deal with formalisms and manipulate information in a formal way. If you want to call that information manipulation language, I think that's reasonable because it can be used (and agreed upon) in an exchange of ideas. For example, do you want to say the Fibonacci sequence starts "0 1 1" or "1 1"? It's just a definition thing, but if you and I agree on one or the other and then start talking about the Nth element of "the Fibonacci sequence", we are communicating with each other in terms of that. (And it's formal because we are basing things on not deviating from that.)
- ukj 7y ago
  The Chomsky hierarchy covers all of that. It's linguistics for formal languages. Grammar, syntax and semantics together with reflection allows for formalisms and meta-formalisms. It all culminates in Turing Completeness. Chomsky Type 0 grammars. If you find something more powerful than a Chomsky Type 0 grammar - you have made a ground-breaking scientific discovery.
- ukj 7y ago
  >when push comes to shove, anything goes as long as it's clear to another mathematician. Well, yes! That's not inherent to Mathematicians. That's how all human communication works. That's what we use "natural" languages for. What we use formal languages for is to minimise the ambiguity, imprecision and non-determinism (entropy) in natural languages. Perhaps one way to put it: the purpose of natural languages is communication and context-setting. The purpose of formal languages is precision. That line gets blurred in computer science (and in my own mind) because of Homoiconicity. >For example, do you want to say the Fibonacci sequence starts "0 1 1" or "1 1"? It's just a definition thing I don't want to say absolutely anything about Fibonacci in a vacuum. The problem I am solving WITH Fibonacci will dictate whether it needs to start at "0 1 1", "1 1", or whether the the difference is immaterial. Off-by-1 errors are just a fact of programmer's life. We make those decisions on case-by-case basis. > if you and I agree on one or the other and then start talking about the Nth element of "the Fibonacci sequence", we are communicating with each other in terms of that. What you are pointing out is problems of convention and consensus. Off the top of my head I can think of a handful of strategies for coming to agree on the 0th value of the Fibonacci function in English. I bet you do too - all those things are intuitive to humans and we take them for granted. In computer science many of those problems have already been formalised, and some have been solved. https://en.wikipedia.org/wiki/Consensus_(computer_science) https://en.wikipedia.org/wiki/Consensus_(computer_science) P.S You have been using the concepts of "communication" and "information". Shannon formalised those! P.P.S Fibonacci is an numerical formalisation of the Golden spiral. In a nutshell, what I am pointing out is succinctly expressed by Donald Knuth. Science is what we understand well enough to explain to a computer. Art is everything else we do.

olooney 7y ago

I'm fairly certain that Hilbert would disagree with you: https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathe... In particular, most modern mathematics can be formalized within first-order logic with the ZFA axioms: https://en.wikipedia.org/wiki/First-order_logic https://en.wikipedia.org/wiki/First-order_logic https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... As a concrete example, the metamath project has formalized (and formally verified) huge chunks of mathematics: http://us.metamath.org/mpeuni/mmset.html http://us.metamath.org/mpeuni/mmset.html Certainly there are aspects of mathematics which are not formal (beauty, parsimony, elegance, clarity, geometric intuition, etc.) but saying math isn't a formal language is like saying chess doesn't have rules.

jcranberry 7y ago
He's not saying mathematics doesn't have rigorous foundations. Betrand Russell's Principia Mathematica is computation. But math textbooks/papers are not like this. They share insights, opinions, and metaphors, and frequently explain things in non-rigorous terms, although these are not substitutes for proofs even though proofs for something given a non-rigorous explanation may not appear. In less heavily proof-read contexts, such as in discussions or a class room, it's not uncommon for mathematicians to be pretty free-wheeling with their notation, and leave it up to context and intuition to figure out what an individual symbol means at a given time. And it wouldn't be a faux pas, or sign of a lesser mathematician or anything like that. These things can be valuable and in some sense are a part of mathematics. Although the end results of mathematics can be distilled into formal computations, it's far too reductive to say that's all mathematics is.
- ukj 7y ago
  The exact same thing can be said about any human dynamic where the objective is to teach. The end-goal is to make the material accessible to the learner by connecting it some pre-existing human intuition or experience, thus providing the learner with a vocabulary to express their thoughts. The important question is how are the vocabularies (both formal and informal) produced by Mathematics different from the vocabularies produced by Computer Science when both fields are working on the same real-world problems? And if the two vocabularies are of equal utility - do we really need two different tools for the same job?