6 ms·

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 ju

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.
    - Koshkin 3y ago
      But unlike the transistor, and like the electron, the natural integers do exist in nature (i.e. have "natural existence"). To quote my other comment here, [number 3] exists as the thing that is common between three apples and the three horses you might want to feed these three apples to; and the fact of the (almost) physical existence of that common (i.e., the number) can be proven by you being bitten by the third horse if you only happen to have two apples instead of three. This common exists, objectively (no sane person can say that it does not), so does the number 3 (because that's what it is called). Now, you can say that we invented the infinity, but that's like saying "we invented the idea that jumping off a cliff is dangerous" - because in both cases the "invention" is unavoidable, as it is dictated by the objective reality which tends to harshly punish those who fail to "invent" the facts.
      - deleted 3y ago

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.
  - Koshkin 3y ago
    But unlike the transistor, and like the electron, the natural integers do exist in nature (i.e. have "natural existence"). To quote my other comment here, [number 3] exists as the thing that is common between three apples and the three horses you might want to feed these three apples to; and the fact of the (almost) physical existence of that common (i.e., the number) can be proven by you being bitten by the third horse if you only happen to have two apples instead of three. This common exists, objectively (no sane person can say that it does not), so does the number 3 (because that's what it is called). Now, you can say that we invented the infinity, but that's like saying "we invented the idea that jumping off a cliff is dangerous" - because in both cases the "invention" is unavoidable, as it is dictated by the objective reality which tends to harshly punish those who fail to "invent" the facts.
    - deleted 3y ago