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IMHO, in order to be able to speak about these subjects, we must either be a mathematician, or have read logicomix (http://www.logicomix.com http://www.logicomi

IMHO, in order to be able to speak about these subjects, we must either be a mathematician, or have read logicomix (http://www.logicomix.com http://www.logicomix.com). It is the easiest book to read. Another usefull one is https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach

throwoutway 3y ago
IMHO, the book was too loose with the truth (even though it was purporting to write about mathematical truths). It's a work of fiction (beyond just filling the characters' words to each other), but that wasn't revealed until the end which left me sour. It pretending to be history right up until the end GED isn't really related to this subject although it contains some of the same themes and characters
crabbone 3y ago
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.

throwoutway 3y ago

IMHO, the book was too loose with the truth (even though it was purporting to write about mathematical truths). It's a work of fiction (beyond just filling the characters' words to each other), but that wasn't revealed until the end which left me sour. It pretending to be history right up until the end GED isn't really related to this subject although it contains some of the same themes and characters

crabbone 3y ago

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.