16 ms·

In Search of Proofs from “The Book”

94 points by digital55 9y ago

mgamache 9y ago
I have a feeling they really mean Spinoza's God (like Einstein), but this use of 'God' causes confusion for a lot of readers. Some readers might think they are referring to the God of the Bible or Koran.
- gbacon 9y ago
  FTA: Never mind that Erdős doubted God’s very existence. “You don’t have to believe in God, but you should believe in The Book,” Erdős explained to other mathematicians.
- beojan 9y ago
  Spinoza's God couldn't have a book of anything. Then again, the Abrahamic God (being omniscient) wouldn't need a book either, but would simply know the perfect proof (and every imperfect proof) for every theorem.
  - Koshkin 9y ago
    Well, God is the word (i.e. the proof).
tromp 9y ago
> The sixth edition will be the final one. And so will the seventh.
yantrams 9y ago
This book is a must have for anyone who loves Mathematics or problem solving in general. You'll find many celebrated problems attacked using concepts from seemingly disparate domains.
tacomonstrous 9y ago
As a working mathematician, I'm uncomfortable with this apotheosis of 'the proof'. While there's always room for aesthetics, this might support the rather misleading premise that mathematics is about digging up chiseled beauties, when it is really about hours, days and months of flailing around, trying to understand one thing, or even what it means to understand that thing. And it's rarely pretty.
- ggm 9y ago
  I think Davis and Hersh say this. Ruben Hersh "what is mathematics really" felt like it was opening the door to the sausage factory.
  - yantrams 9y ago
    Brilliant book that. The way he explains how a polished rigorous proof hides the process(compared with frontend and kitchen views of a restaurant) by which a mathematician has arrived at it changed my perception of how mathematics is done. I notice the same phenomena in programming too. The final piece of code that works doesn't reflect all the failed attempts at solving a problem. This is why I tend to not delete/clear cells at all from my jupyter notebooks. :) It is always informative/fun to refer to them at a later point in time to relive that problem solving experience. PS: In case you haven't, you should try Proofs and Refutations by Lakatos. Edit: Fixed name of the book mentioned and added some text.
    - ggm 9y ago
      Do you mean "proof and refutations" ?
      - yantrams 9y ago
        Ah yes, thanks for correcting me :)
    - jacobolus 9y ago
      A big portion of Lakatos’s book is online, https://math.berkeley.edu/~kpmann/Lakatos.pdf https://math.berkeley.edu/~kpmann/Lakatos.pdf
extr 9y ago
> It’s a powerful feeling. I remember these moments of beauty and excitement. And there’s a very powerful type of happiness that comes from it. I always try to explain to people this is the reason I love math so much. There is no feeling like getting to the bottom of something and it makes sense in such a perfect way, you can hardly believe the universe works like that. I am no believer but I think it's the closest I can feel to God.
- ythn 9y ago
  > I am no believer but I think it's the closest I can feel to God. It's a shared sentiment: "Mathematics is the language with which God wrote the universe." -- Galileo
wolfkill 9y ago
As a mathematician and a believer in God, I get uncomfortable when people start attributing math to God. I see math as being very much a construct of the human psyche and as astounding as it is, there are enormous gaps between mathematical models and physical observations. So in my mind math is to reality as human is to Divine.
- empath75 9y ago
  The roots of mathematics in the west are fairly inextricably tied up with the development of religion via Pythagoras and especially Plato. Plato (or at least Platonists) taught, for example that God literally was One -- as in the number 1. And that all of reality is the result of successive emanations from 1 -- to the dyad (2), then the rest of the numbers, on through geometry and so on. Those beliefs influenced the philosophical development of all the major monotheistic religions.
  - yters 9y ago
    Math was one of Plato's main methods of showing there was a reality beyond the physical world that we could access with our minds.
- User23 9y ago
  It logically follows that any faith that believes God to be a unitary creator is going to attribute math to Him. Your absolutely correct observation about gaps notwithstanding, the driving question for me here is this: why does math work so well? I personally exclude any explanation that doesn't involve the Logos in some form as incoherent, but I don't have a specific answer and probably never will.
  - placebo 9y ago
    I'm also very interested in this question. My current understanding is that mathematics only works well because the mind is only capable of conceiving a mathematical reality, i.e math is basically the end result of an evolved mind, a tool that divides experience to concepts (including objects, properties, relationships) which at first approximation don't change with time (a lion will remain a lion, a lion will remain dangerous). Later, higher level observations that are constant are also observed (e.g location that linearly changes with time is a constant speed, every triangle on a plain has 180 degrees etc.) - my point is that the mind is geared to find relationships that are constant (in every level of abstraction), and once we find such a constant we call it "truth" (or if we're more honest, we at least admit it's a good model). So I think the bigger question is not why mathematics works (it works because reality, including thought itself, can be seen through the filter of concepts that appear to be self consistent), but rather why reality lends itself to being categorized in the first place into "things" that have internal integrity and consistency.
combatentropy 9y ago
Paul Graham explored why different fields --- like math, art, and engineering --- all use the word "beautiful," http://www.paulgraham.com/taste.html http://www.paulgraham.com/taste.html
dang 9y ago
This fine article was flagged; I'm betting because of the word "God" in the title. That word has triggering effects independently of what it is used to refer to. To assuage the sensitive, we've degodded the title and replaced it with the usual Erdős reference.
montrose 9y ago
This is an interesting point: "To do these short and surprising proofs, you need a lot of confidence. And one way to get the confidence is if you know the thing is true. If you know that something is true because so-and-so proved it, then you might also dare to say, "What would be the really nice and short and elegant way to establish this?" So, I think, in that sense, the ugly proofs have their role."
johnsonjo 9y ago
The book talked about in the article is on Springer Link. Luckily my University is subscribed to Springer Link, so I just had to login to my University's VPN and I was able to download this book for free; score!

94 points by digital55 9y ago

I have a feeling they really mean Spinoza's God (like Einstein), but this use of 'God' causes confusion for a lot of readers. Some readers might think they are referring to the God of the Bible or Koran.

gbacon 9y ago
FTA: Never mind that Erdős doubted God’s very existence. “You don’t have to believe in God, but you should believe in The Book,” Erdős explained to other mathematicians.
beojan 9y ago
Spinoza's God couldn't have a book of anything. Then again, the Abrahamic God (being omniscient) wouldn't need a book either, but would simply know the perfect proof (and every imperfect proof) for every theorem.
- Koshkin 9y ago
  Well, God is the word (i.e. the proof).

tromp 9y ago

> The sixth edition will be the final one. And so will the seventh.

yantrams 9y ago

This book is a must have for anyone who loves Mathematics or problem solving in general. You'll find many celebrated problems attacked using concepts from seemingly disparate domains.

tacomonstrous 9y ago

As a working mathematician, I'm uncomfortable with this apotheosis of 'the proof'. While there's always room for aesthetics, this might support the rather misleading premise that mathematics is about digging up chiseled beauties, when it is really about hours, days and months of flailing around, trying to understand one thing, or even what it means to understand that thing. And it's rarely pretty.

ggm 9y ago
I think Davis and Hersh say this. Ruben Hersh "what is mathematics really" felt like it was opening the door to the sausage factory.
- yantrams 9y ago
  Brilliant book that. The way he explains how a polished rigorous proof hides the process(compared with frontend and kitchen views of a restaurant) by which a mathematician has arrived at it changed my perception of how mathematics is done. I notice the same phenomena in programming too. The final piece of code that works doesn't reflect all the failed attempts at solving a problem. This is why I tend to not delete/clear cells at all from my jupyter notebooks. :) It is always informative/fun to refer to them at a later point in time to relive that problem solving experience. PS: In case you haven't, you should try Proofs and Refutations by Lakatos. Edit: Fixed name of the book mentioned and added some text.
  - ggm 9y ago
    Do you mean "proof and refutations" ?
    - yantrams 9y ago
      Ah yes, thanks for correcting me :)
  - jacobolus 9y ago
    A big portion of Lakatos’s book is online, https://math.berkeley.edu/~kpmann/Lakatos.pdf https://math.berkeley.edu/~kpmann/Lakatos.pdf

extr 9y ago

> It’s a powerful feeling. I remember these moments of beauty and excitement. And there’s a very powerful type of happiness that comes from it. I always try to explain to people this is the reason I love math so much. There is no feeling like getting to the bottom of something and it makes sense in such a perfect way, you can hardly believe the universe works like that. I am no believer but I think it's the closest I can feel to God.

ythn 9y ago
> I am no believer but I think it's the closest I can feel to God. It's a shared sentiment: "Mathematics is the language with which God wrote the universe." -- Galileo

wolfkill 9y ago

As a mathematician and a believer in God, I get uncomfortable when people start attributing math to God. I see math as being very much a construct of the human psyche and as astounding as it is, there are enormous gaps between mathematical models and physical observations. So in my mind math is to reality as human is to Divine.

empath75 9y ago
The roots of mathematics in the west are fairly inextricably tied up with the development of religion via Pythagoras and especially Plato. Plato (or at least Platonists) taught, for example that God literally was One -- as in the number 1. And that all of reality is the result of successive emanations from 1 -- to the dyad (2), then the rest of the numbers, on through geometry and so on. Those beliefs influenced the philosophical development of all the major monotheistic religions.
- yters 9y ago
  Math was one of Plato's main methods of showing there was a reality beyond the physical world that we could access with our minds.
User23 9y ago
It logically follows that any faith that believes God to be a unitary creator is going to attribute math to Him. Your absolutely correct observation about gaps notwithstanding, the driving question for me here is this: why does math work so well? I personally exclude any explanation that doesn't involve the Logos in some form as incoherent, but I don't have a specific answer and probably never will.
- placebo 9y ago
  I'm also very interested in this question. My current understanding is that mathematics only works well because the mind is only capable of conceiving a mathematical reality, i.e math is basically the end result of an evolved mind, a tool that divides experience to concepts (including objects, properties, relationships) which at first approximation don't change with time (a lion will remain a lion, a lion will remain dangerous). Later, higher level observations that are constant are also observed (e.g location that linearly changes with time is a constant speed, every triangle on a plain has 180 degrees etc.) - my point is that the mind is geared to find relationships that are constant (in every level of abstraction), and once we find such a constant we call it "truth" (or if we're more honest, we at least admit it's a good model). So I think the bigger question is not why mathematics works (it works because reality, including thought itself, can be seen through the filter of concepts that appear to be self consistent), but rather why reality lends itself to being categorized in the first place into "things" that have internal integrity and consistency.

combatentropy 9y ago

Paul Graham explored why different fields --- like math, art, and engineering --- all use the word "beautiful," http://www.paulgraham.com/taste.html http://www.paulgraham.com/taste.html

dang 9y ago

This fine article was flagged; I'm betting because of the word "God" in the title. That word has triggering effects independently of what it is used to refer to. To assuage the sensitive, we've degodded the title and replaced it with the usual Erdős reference.

montrose 9y ago

This is an interesting point: "To do these short and surprising proofs, you need a lot of confidence. And one way to get the confidence is if you know the thing is true. If you know that something is true because so-and-so proved it, then you might also dare to say, "What would be the really nice and short and elegant way to establish this?" So, I think, in that sense, the ugly proofs have their role."

johnsonjo 9y ago

The book talked about in the article is on Springer Link. Luckily my University is subscribed to Springer Link, so I just had to login to my University's VPN and I was able to download this book for free; score!