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This makes no sense to you precisely because you think that mathematics is (only) a language. It's not. A language is invented; mathematics is discovered. In th

This makes no sense to you precisely because you think that mathematics is (only) a language. It's not. A language is invented; mathematics is discovered. In the US it's not called a science, but for all intents and purposes it indeed is.

danbruc 3y ago
A language is invented; mathematics is discovered. I peg to differ. Mathematics is invented when you define the axioms of some structure, after that you discover the consequences of the axioms you picked. First you define the natural numbers and the operations on them, then you discover the primes. So actually invention and discovery are not mutually exclusive here, but invention is more fundamental as it defines what is there to be discovered.
- Koshkin 3y ago
  You define, but the choice is in fact not quite arbitrary; rather, it is guided by something objective, i.e. by something that exists outside of your own mind.
  - danbruc 3y ago
    What are the constraints? I can make up any set of axioms I like and see what they imply, there is no need that they are in any way related to the physical world. Sure, there is a lot of mathematics that was specifically invented in order to deal with the real world, but that is not a general requirement.
    - Koshkin 3y ago
      > I can make up any set of axioms Yeah, but that wouldn't be mathematics.
      - danbruc 3y ago
        Why not? If I decide I want to study the properties of the space of total function from vectors of octonions with prime dimension to the surreal numbers, who is to say that this is not mathematics?
        Koshkin 3y ago
        What you are describing is not "any set of axioms." (Rather, it is a pre-existing concept.)
        danbruc 3y ago
        Yes, but that is a technical detail, it was just easier for me to come up with that than making up a set of axioms. How is an obscure combination of existing axioms still mathematics but not any set of axioms I make up? And how would we ever extend mathematics if coming up with new axioms is not mathematics? But let us just take the integers with the common definitions for addition, subtraction and multiplication, but then redefine them so that every operation first performs the usual operation and then increments the result by one. 1 + 1 = 3 1 * 1 = 2 (1 + 2) * 3 = 13 (a + b) * c = a * c + b * c + c - 4 Still not quite what I had in mind, nothing completely new, but maybe at least different enough from the normal integers to have some weird properties. Not the numbers themselves, they are still just the integers, but the algebraic expressions involving the redefined operations.
        Koshkin 3y ago
        The problem is that the word "any" stands for "random," or "arbitrary." It is not inconceivable that a scientist, say, would mix "random" substances together just to see what happens. But almost all such experiments would result in exactly nothing, nothing interesting anyway. Usually, a scientist first notices something, puts forward a concrete hypothesis, and then conduct an experiment. In mathematics, it is the same: for example, one notices something that certain things have in common and then tries to generalize this common into "axioms." Modern mathematics is highly evolved field full of such examples. Nobody starts with scribbling random doodles on a blank sheet of paper in the hope that something interesting would come out of it. While not impossible, it is extremely unlikely. In the end, one might say that what is mathematics and what is not is merely a matter of definition, but that, as well as calling the "language of doodles" mathematics, would make the word devoid of any useful meaning, IMHO.

danbruc 3y ago

A language is invented; mathematics is discovered. I peg to differ. Mathematics is invented when you define the axioms of some structure, after that you discover the consequences of the axioms you picked. First you define the natural numbers and the operations on them, then you discover the primes. So actually invention and discovery are not mutually exclusive here, but invention is more fundamental as it defines what is there to be discovered.

Koshkin 3y ago
You define, but the choice is in fact not quite arbitrary; rather, it is guided by something objective, i.e. by something that exists outside of your own mind.
- danbruc 3y ago
  What are the constraints? I can make up any set of axioms I like and see what they imply, there is no need that they are in any way related to the physical world. Sure, there is a lot of mathematics that was specifically invented in order to deal with the real world, but that is not a general requirement.
  - Koshkin 3y ago
    > I can make up any set of axioms Yeah, but that wouldn't be mathematics.
    - danbruc 3y ago
      Why not? If I decide I want to study the properties of the space of total function from vectors of octonions with prime dimension to the surreal numbers, who is to say that this is not mathematics?
      - Koshkin 3y ago
        What you are describing is not "any set of axioms." (Rather, it is a pre-existing concept.)
        danbruc 3y ago
        Yes, but that is a technical detail, it was just easier for me to come up with that than making up a set of axioms. How is an obscure combination of existing axioms still mathematics but not any set of axioms I make up? And how would we ever extend mathematics if coming up with new axioms is not mathematics? But let us just take the integers with the common definitions for addition, subtraction and multiplication, but then redefine them so that every operation first performs the usual operation and then increments the result by one. 1 + 1 = 3 1 * 1 = 2 (1 + 2) * 3 = 13 (a + b) * c = a * c + b * c + c - 4 Still not quite what I had in mind, nothing completely new, but maybe at least different enough from the normal integers to have some weird properties. Not the numbers themselves, they are still just the integers, but the algebraic expressions involving the redefined operations.
        Koshkin 3y ago
        The problem is that the word "any" stands for "random," or "arbitrary." It is not inconceivable that a scientist, say, would mix "random" substances together just to see what happens. But almost all such experiments would result in exactly nothing, nothing interesting anyway. Usually, a scientist first notices something, puts forward a concrete hypothesis, and then conduct an experiment. In mathematics, it is the same: for example, one notices something that certain things have in common and then tries to generalize this common into "axioms." Modern mathematics is highly evolved field full of such examples. Nobody starts with scribbling random doodles on a blank sheet of paper in the hope that something interesting would come out of it. While not impossible, it is extremely unlikely. In the end, one might say that what is mathematics and what is not is merely a matter of definition, but that, as well as calling the "language of doodles" mathematics, would make the word devoid of any useful meaning, IMHO.