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> 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

> 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.
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  - danbruc 3y ago
    What then about mathematical structures that do not exist in the real world? Do you discover some and invent others? Or do you think all mathematical structures are realized? You say that no sane person will deny the existence of three, I am actually willing to go even further, I am willing to deny the natual existence of your apples and horses. The idea of cutting the universe into pieces is arguably an invention. It is of course an extremely useful idea for comprehending the universe, but if you think about it, you are somewhat arbitrarily drawing borders around collections of atoms. There is one forest. And a hundred trees. And millions of cells. Is the water in a tree part of the tree? What about the water inside of the cells? When does it become part of the tree? What about a water molecule just evaporating from a leave, when does it become part of the atmosphere? But even if there are things that can be counted, or if quarks and gluons behave in a way that can be described using SU(3), does that really imply the existence of the mathematical structure used to describe those things? Where do you get multiplication from? Or tetration? Are they invented while the basic structure of the integers is discovered? What exactly does existence even imply or require in this case? And another thing just came to my mind. In which way do the apples and horses tell us anything about three? I can see that you can probably demonstrate that you can pair your apples and your horses, but that is quite far from telling us anything about three as you can do the same with any number of objects. You also say the common thing is three, but that does not really help to pick out what three is, therevare many common things. Also note that you introduced the idea of a set, where does that come from? Three is not something that applies to one of the apples or horses, it applies to the sets of them. Does the set of three horses exist or is that not something you made up, this, this and that, this are my three horses? And now that we are talking about sets, because that is where three comes into play, why did we talk about apples and horses to begin with, it totally doesn't matter what you put into your sets, just how many things. So did we just lose the relationship to the real world or at least conclude that it never mattered? I would at least say that it is certainly not as easy as you want it to be, there is much more nuance to this than three apples, three horses, therefore natural numbers. It actually sound circular. Let me pick three apples and three horse, now look, here are two sets of three things, so three exists. Maybe it would be a nice challenge to tell me how to find three in nature. You obviously can not say take three horses. It seems tempting to say start without a horse, then add one, then another and yet another. But did you then not just invent that, did you not just come up with the axioms of the natural numbers, just expressed with horses instead of symbols?

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
  [deleted]
- danbruc 3y ago
  What then about mathematical structures that do not exist in the real world? Do you discover some and invent others? Or do you think all mathematical structures are realized? You say that no sane person will deny the existence of three, I am actually willing to go even further, I am willing to deny the natual existence of your apples and horses. The idea of cutting the universe into pieces is arguably an invention. It is of course an extremely useful idea for comprehending the universe, but if you think about it, you are somewhat arbitrarily drawing borders around collections of atoms. There is one forest. And a hundred trees. And millions of cells. Is the water in a tree part of the tree? What about the water inside of the cells? When does it become part of the tree? What about a water molecule just evaporating from a leave, when does it become part of the atmosphere? But even if there are things that can be counted, or if quarks and gluons behave in a way that can be described using SU(3), does that really imply the existence of the mathematical structure used to describe those things? Where do you get multiplication from? Or tetration? Are they invented while the basic structure of the integers is discovered? What exactly does existence even imply or require in this case? And another thing just came to my mind. In which way do the apples and horses tell us anything about three? I can see that you can probably demonstrate that you can pair your apples and your horses, but that is quite far from telling us anything about three as you can do the same with any number of objects. You also say the common thing is three, but that does not really help to pick out what three is, therevare many common things. Also note that you introduced the idea of a set, where does that come from? Three is not something that applies to one of the apples or horses, it applies to the sets of them. Does the set of three horses exist or is that not something you made up, this, this and that, this are my three horses? And now that we are talking about sets, because that is where three comes into play, why did we talk about apples and horses to begin with, it totally doesn't matter what you put into your sets, just how many things. So did we just lose the relationship to the real world or at least conclude that it never mattered? I would at least say that it is certainly not as easy as you want it to be, there is much more nuance to this than three apples, three horses, therefore natural numbers. It actually sound circular. Let me pick three apples and three horse, now look, here are two sets of three things, so three exists. Maybe it would be a nice challenge to tell me how to find three in nature. You obviously can not say take three horses. It seems tempting to say start without a horse, then add one, then another and yet another. But did you then not just invent that, did you not just come up with the axioms of the natural numbers, just expressed with horses instead of symbols?