4 ms·

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
    Sure, randomly generating axioms would probably not be very productive, most of the time you will continue from known territory, generalize something, add additional constraints, whatever. And if that is what you mean with discovery, identifying things that might be interesting to change in one way or another, fine, that is not an unreasonable description of the process. But you are still inventing a new structure when you decide what to change and write down the new rules. If you have the naturals, you might wonder what would happen if there was an inverse of addition or of multiplication for each natural, if you pick the former you will invent the integers, if you pick the later the non-negative rationals. You can also call that a discovery if you like, but it is not a discovery in the sense of say discovering the electron, you did not discover something that has always been there, you actually brought it into existence by the choices you made.
    - Koshkin 3y ago
      > you will invent the integers You will discover them, not invent. Albeit subtle, there is a difference between these notions, and you will indeed discover the integers in (almost) exactly the same sense as electron was discovered - what you do is, you put forward an idea, play with it, test it, correct mistakes, and in the end you find what you have been looking for. What you can invent in mathematics, is a proof; but even in this case the path may lead to a series of discoveries. Whether it does, depends on the attitude towards it (Grothendieck's analogy between building a proof and opening a nut).
      - danbruc 3y ago
        I would say it is exactly the opposite - you invent the integers, you discover a proof. Discovery for me means finding something that already exists, inventing means bringing something into existence. You discover the electron, you invent the transistor. Before you write down the axioms for integers, they do not exist, you invent them. That there are infinitely many primes is a consequence of the axioms, you just have to notice those special numbers and give them a name. Here I find it actually quite tempting to say you invent the primes, or maybe better the idea of primes, but I still think it is more correct to say you discover them, given the axioms they were always there even if you have not yet noticed, named and described them. The same for the proof that there are infinitely many of them, given the axioms and the laws of logic, the truth of that statement and the possible proofs for it are fixed, you just have to find one. Maybe back to the integers. Yes, you may play around with different ideas, tweak definitions and so on. But what you are actually doing is inventing a whole bunch of similar structures and then you pick the one that you like the most, that works the best or whatever. Some of your attempts might be inconsistent, some might not do what you want, but you all invented them. And among them you might discover one that works just like wanted it to work.

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
  Sure, randomly generating axioms would probably not be very productive, most of the time you will continue from known territory, generalize something, add additional constraints, whatever. And if that is what you mean with discovery, identifying things that might be interesting to change in one way or another, fine, that is not an unreasonable description of the process. But you are still inventing a new structure when you decide what to change and write down the new rules. If you have the naturals, you might wonder what would happen if there was an inverse of addition or of multiplication for each natural, if you pick the former you will invent the integers, if you pick the later the non-negative rationals. You can also call that a discovery if you like, but it is not a discovery in the sense of say discovering the electron, you did not discover something that has always been there, you actually brought it into existence by the choices you made.
  - Koshkin 3y ago
    > you will invent the integers You will discover them, not invent. Albeit subtle, there is a difference between these notions, and you will indeed discover the integers in (almost) exactly the same sense as electron was discovered - what you do is, you put forward an idea, play with it, test it, correct mistakes, and in the end you find what you have been looking for. What you can invent in mathematics, is a proof; but even in this case the path may lead to a series of discoveries. Whether it does, depends on the attitude towards it (Grothendieck's analogy between building a proof and opening a nut).
    - danbruc 3y ago
      I would say it is exactly the opposite - you invent the integers, you discover a proof. Discovery for me means finding something that already exists, inventing means bringing something into existence. You discover the electron, you invent the transistor. Before you write down the axioms for integers, they do not exist, you invent them. That there are infinitely many primes is a consequence of the axioms, you just have to notice those special numbers and give them a name. Here I find it actually quite tempting to say you invent the primes, or maybe better the idea of primes, but I still think it is more correct to say you discover them, given the axioms they were always there even if you have not yet noticed, named and described them. The same for the proof that there are infinitely many of them, given the axioms and the laws of logic, the truth of that statement and the possible proofs for it are fixed, you just have to find one. Maybe back to the integers. Yes, you may play around with different ideas, tweak definitions and so on. But what you are actually doing is inventing a whole bunch of similar structures and then you pick the one that you like the most, that works the best or whatever. Some of your attempts might be inconsistent, some might not do what you want, but you all invented them. And among them you might discover one that works just like wanted it to work.