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This subject actually doesn't belong in mathematics: it's philosophy of mathematics. I.e. no matter how much you study mathematics you will not be able to answ

This subject actually doesn't belong in mathematics: it's philosophy of mathematics. I.e. no matter how much you study mathematics you will not be able to answer (or even attempt to answer) these questions because no mathematical tools or disciplines are designed for that. I also believe that the emphasis on "rigor" here is misplaced. The argument isn't about whether mathematical rules are rigorous or not. The argument is about whether mathematics exists independent of mathematicians (and they discover it in a way how an astronomer peers into telescope and discovers new stars) vs mathematics being created by mathematicians' minds (similar to how an architect designs a building: there weren't one before, and now there's a concept of a new building with so many walls, floors, windows etc.) I believe that mathematics is art, not science. I.e. mathematicians create new rules, they don't discover them. The whole argument to support this point would be too long to write it in a single post, but the general idea is that mathematics is a system that can easily describe counterfactual worlds. We use it to also describe our physical world because, of course, it can do that. But then asking the question about the "surprising effectiveness" is moot: we deliberately made it to be as effective as possible, so how is it so surprising that it is?

AnimalMuppet 3y ago
> The whole argument to support this point would be too long to write it in a single post, but the general idea is that mathematics is a system that can easily describe counterfactual worlds. We use it to also describe our physical world because, of course, it can do that. But then asking the question about the "surprising effectiveness" is moot: we deliberately made it to be as effective as possible, so how is it so surprising that it is? The "surprising effectiveness" is, I think, one of three things. First, the surprise is that we can create mathematics that describes our physical world. The second surprise is that, when we find out something new, it often takes the form of existing mathematics that we didn't design to describe the physical world. (Though, from your point of view, I suppose you could say that we created mathematics to describe everything that could be described by mathematics, and so it's not surprising that something was there to describe reality.) The third surprise (maybe this is just a restatement of the first one) is that mathematics really describes the physical world. It's not that we find some math that describes it, and then we change the situation a little bit and we need to find some new math. The surprise is that the math describes what is going on so well that it applies to situations that we didn't know about when we devised the math that applied. That is, it's predictive, not just descriptive.
- crabbone 3y ago
  Well, the last one is a kind of surprise like... if you believe in real numbers, then there are some nonrepeating real numbers (pi is believed to be one of them) that embed all other real numbers. I mean, to someone unfamiliar with the concept it sound amazing and improbable that some number embeds all other numbers (all of which are infinite). But this is because we aren't used to dealing with infinite things. Similarly, if mathematics is so powerful as to be able to describe any physical reality, it shouldn't be surprising that it can describe ours, no matter how complex and detailed.
snarfy 3y ago
All ideas are discoveries, not creations. The fact that anybody else can come up with the same idea lends credence to this. That's how ideas work - they are transcendental.
- crabbone 3y ago
  Your argument is similar in spirit to the claim that all, even infinite sets can have single element selected from to form a new set (so called "axiom of choice"). I.e. you equate the mere possibility of existence with a proof of existence (it's not a slur, there are a lot of mathematicians who do just that). I don't see the reason to think that coincidentally arrived at ideas mean that ideas are discovered. We use the same framework, with the same rules. It's not unlikely that we'll create same ideas, because we use all the same rules, but the framework is so vast... it has so many rule combinations it's mind-boggling how would you think all ideas already exist in this framework. More practically, we call an action a "discovery" when we (unbeknownst to us) faced the consequences of the phenomenon being discovered, but didn't know why we were facing them. An astronomer who finds a "new" star was ever so slightly influenced by that star's light, gravity etc. A marine biologist who discovers a new deep-water fish was ever so slightly affected by that fish through a complicated chain linking many different species through biosphere. An artist drawing a new painting isn't discovering it in the same sense. She isn't interested in how an existing painting was connected to the consequences of her life or the lives of the whole human species. By selecting of all possible ways the paint can be laid down on canvas her particular way of doing that she creates something genuinely new, or as new as it can possibly be. It's counterproductive to label this activity "discovery" because then we lose an important distinction between the nature of the work of an astronomer and that of a painter (or a mathematician). There are sculptors who'd jokingly say that they "discover" the statue in a stone slab. But they do so in a sarcastic kind of way, really (well, artists are weird and will make a lot of nonsense claims just to trigger non-artistic audience). But, deep down, nobody believes that they are searching and finding good images, melodies or novels. I cannot really imagine a mechanism through which I'd discover the answer writing to you. It's a lot easier to explain what i wrote by saying that I meant to write it.
  - 3y ago

AnimalMuppet 3y ago

> The whole argument to support this point would be too long to write it in a single post, but the general idea is that mathematics is a system that can easily describe counterfactual worlds. We use it to also describe our physical world because, of course, it can do that. But then asking the question about the "surprising effectiveness" is moot: we deliberately made it to be as effective as possible, so how is it so surprising that it is? The "surprising effectiveness" is, I think, one of three things. First, the surprise is that we can create mathematics that describes our physical world. The second surprise is that, when we find out something new, it often takes the form of existing mathematics that we didn't design to describe the physical world. (Though, from your point of view, I suppose you could say that we created mathematics to describe everything that could be described by mathematics, and so it's not surprising that something was there to describe reality.) The third surprise (maybe this is just a restatement of the first one) is that mathematics really describes the physical world. It's not that we find some math that describes it, and then we change the situation a little bit and we need to find some new math. The surprise is that the math describes what is going on so well that it applies to situations that we didn't know about when we devised the math that applied. That is, it's predictive, not just descriptive.

crabbone 3y ago
Well, the last one is a kind of surprise like... if you believe in real numbers, then there are some nonrepeating real numbers (pi is believed to be one of them) that embed all other real numbers. I mean, to someone unfamiliar with the concept it sound amazing and improbable that some number embeds all other numbers (all of which are infinite). But this is because we aren't used to dealing with infinite things. Similarly, if mathematics is so powerful as to be able to describe any physical reality, it shouldn't be surprising that it can describe ours, no matter how complex and detailed.

snarfy 3y ago

All ideas are discoveries, not creations. The fact that anybody else can come up with the same idea lends credence to this. That's how ideas work - they are transcendental.

crabbone 3y ago
Your argument is similar in spirit to the claim that all, even infinite sets can have single element selected from to form a new set (so called "axiom of choice"). I.e. you equate the mere possibility of existence with a proof of existence (it's not a slur, there are a lot of mathematicians who do just that). I don't see the reason to think that coincidentally arrived at ideas mean that ideas are discovered. We use the same framework, with the same rules. It's not unlikely that we'll create same ideas, because we use all the same rules, but the framework is so vast... it has so many rule combinations it's mind-boggling how would you think all ideas already exist in this framework. More practically, we call an action a "discovery" when we (unbeknownst to us) faced the consequences of the phenomenon being discovered, but didn't know why we were facing them. An astronomer who finds a "new" star was ever so slightly influenced by that star's light, gravity etc. A marine biologist who discovers a new deep-water fish was ever so slightly affected by that fish through a complicated chain linking many different species through biosphere. An artist drawing a new painting isn't discovering it in the same sense. She isn't interested in how an existing painting was connected to the consequences of her life or the lives of the whole human species. By selecting of all possible ways the paint can be laid down on canvas her particular way of doing that she creates something genuinely new, or as new as it can possibly be. It's counterproductive to label this activity "discovery" because then we lose an important distinction between the nature of the work of an astronomer and that of a painter (or a mathematician). There are sculptors who'd jokingly say that they "discover" the statue in a stone slab. But they do so in a sarcastic kind of way, really (well, artists are weird and will make a lot of nonsense claims just to trigger non-artistic audience). But, deep down, nobody believes that they are searching and finding good images, melodies or novels. I cannot really imagine a mechanism through which I'd discover the answer writing to you. It's a lot easier to explain what i wrote by saying that I meant to write it.
- 3y ago