In Search of Proofs from “The Book”(quantamagazine.org)
quantamagazine.org
In Search of Proofs from “The Book”
https://www.quantamagazine.org/gunter-ziegler-and-martin-aigner-seek-gods-perfect-math-proofs-20180319/
51 comments
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.
Your phrase "or even what it means to understand that thing" strikes a chord with me. As someone studying math and computer science on my own, without anyone around who shares my research interests, I have to figure out from first principles not only how to gain an understanding of things but define what understanding it even means.
For example, I have recently been pursuing the idea of identifying data streams with bijections with the natural numbers, and higher-order operations on these bijections. The bijection is arbitrary, so the order assigned the sequence "doesn't matter" (the subject of study happens to be associative and commutative in its pure form). However I ran into an inconsistency with this when modeling products of streams, and came to an epiphany: yes, the order (the specific bijection) is arbitrary BUT it is fixed: that is, you can pick any order, but it has to be committed to a priori. (An intuitive analogy would be to consider the difference between a card game where the dealer shuffles the deck before the game and deals from the top, versus a card game where the dealer selects the cards he wants to give you as the game proceeds.)
I felt like that was a moment when I understood what it means to understand this thing I'm studying which is so abstract. I saw it as a fork in the road between mathematics and mysticism: prior to that realization I had been studying mysticism, "assume the (nondeterministic) computer 'magically' chooses the best next item in the sequence to pursue" versus "assume the sequence is a priori in the best order to search (but that order is fixed)". The former is mysticism because it asserts something exists without pinning it down. The latter is mathematical because, although it describes something abstract, the properties of the thing it's describing do not shift during the course of working with it.
For example, I have recently been pursuing the idea of identifying data streams with bijections with the natural numbers, and higher-order operations on these bijections. The bijection is arbitrary, so the order assigned the sequence "doesn't matter" (the subject of study happens to be associative and commutative in its pure form). However I ran into an inconsistency with this when modeling products of streams, and came to an epiphany: yes, the order (the specific bijection) is arbitrary BUT it is fixed: that is, you can pick any order, but it has to be committed to a priori. (An intuitive analogy would be to consider the difference between a card game where the dealer shuffles the deck before the game and deals from the top, versus a card game where the dealer selects the cards he wants to give you as the game proceeds.)
I felt like that was a moment when I understood what it means to understand this thing I'm studying which is so abstract. I saw it as a fork in the road between mathematics and mysticism: prior to that realization I had been studying mysticism, "assume the (nondeterministic) computer 'magically' chooses the best next item in the sequence to pursue" versus "assume the sequence is a priori in the best order to search (but that order is fixed)". The former is mysticism because it asserts something exists without pinning it down. The latter is mathematical because, although it describes something abstract, the properties of the thing it's describing do not shift during the course of working with it.
I think Davis and Hersh say this. Ruben Hersh "what is mathematics really" felt like it was opening the door to the sausage factory.
Well,
It is interesting that a lot of great mathematicians have believed in truth beyond the sausage factory. Not that the reality isn't a sausage factory but belief in an ideal (truth, beauty) seems to often be what keeps them going.
Kurt Godel proved that the machinery of mathematics couldn't create a complete truth system accessible through proofs (which are ultimately a mechanical process) but he definitely was on the side of ideal-truth when he said "Either mathematics is too big for the human mind, or the human mind is more than a machine."
Edit: And Grigori Perelman was certainly famous for rejecting what he saw as the sausage factory of mathematics.
It is interesting that a lot of great mathematicians have believed in truth beyond the sausage factory. Not that the reality isn't a sausage factory but belief in an ideal (truth, beauty) seems to often be what keeps them going.
Kurt Godel proved that the machinery of mathematics couldn't create a complete truth system accessible through proofs (which are ultimately a mechanical process) but he definitely was on the side of ideal-truth when he said "Either mathematics is too big for the human mind, or the human mind is more than a machine."
Edit: And Grigori Perelman was certainly famous for rejecting what he saw as the sausage factory of mathematics.
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.
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.
A big portion of Lakatos’s book is online, https://math.berkeley.edu/~kpmann/Lakatos.pdf
To those like me, not familiar with the term "sausage factory":
"An unpleasant process, especially one that is hidden from public view, that is used to produce a widely consumed product" - Urban Dictionary
"An unpleasant process, especially one that is hidden from public view, that is used to produce a widely consumed product" - Urban Dictionary
I feel like the article does a good job of addressing this when it talks about the role of "ugly" proofs. It might deserve more emphasis on this; that all pioneering mathematics starts with this, rather than assigning it a space as a "stepping stone" to the "book proof". But it is true that "cleaning up" after the pioneers is a difficult and interesting part of mathematics, that serves to make it easier for new pioneers to get to the end of the paved trail.
> Q: There’s a famous quote from the mathematician G. H. Hardy that says, “There is no permanent place in the world for ugly mathematics.” But ugly mathematics still has a role, right?
> A: You know, the first step is to establish the theorem, so that you can say, “I worked hard. I got the proof. It’s 20 pages. It’s ugly. It’s lots of calculations, but it’s correct and it’s complete and I’m proud of it.”
> ...
> 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.
> Q: There’s a famous quote from the mathematician G. H. Hardy that says, “There is no permanent place in the world for ugly mathematics.” But ugly mathematics still has a role, right?
> A: You know, the first step is to establish the theorem, so that you can say, “I worked hard. I got the proof. It’s 20 pages. It’s ugly. It’s lots of calculations, but it’s correct and it’s complete and I’m proud of it.”
> ...
> 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.
I am also a professional. Sometimes your job deals with the pretty marigold and sometimes with the nasty slug.
Reality has its thing.
And sometimes you need a jeweler’s tool and other times just your fists.
Reality has its thing.
And sometimes you need a jeweler’s tool and other times just your fists.
You sound like a jaded professional, but keep in mind most people are not working mathematicans and never will be, so anything they hear from the pros they're going to take at face value. When it comes to the general layperson, I think popularizing the idea that math is all "months of flailing around, trying to understand one thing" is more unhelpful than portraying it as the pursuit of beauty and simplicity, even if it's more true on a practical level. If you bring up the word beauty in reference to math, most people are a little surprised you even would link the two concepts, let alone get the impression that pursuing clever little proofs is all there is to it. Better to start by giving people get the idea they might enjoy it, and let them find out it's not all about elegant proofs in undergrad or whatever.
I highly disagree. They’re not a jaded professional. It's exactly what a professional goes through. Part of the beauty is in the “months of flailing around, trying to understand one thing”. It’s that “Aha!” and “Eureka!”. The history mathematics is riddled with this. And what if they don’t see this beauty? They’ll give up and not realize they have to persevere.
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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.
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.
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.
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.
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.
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.
Math works so well because we consistently abandon any math that doesn't work well.
And where the math of reality seems ugly or awkward or contrived, we invent notation to make it look neat and simple.
The reason why math education takes so many years is to learn all the complexity, conventions and abuse of notation. As in speech and image interpretation, the adept cannot see the complexity.
You can make anything simple by inventing a language to state it in. Use custom entities instead of multiplying them.
The reason why math education takes so many years is to learn all the complexity, conventions and abuse of notation. As in speech and image interpretation, the adept cannot see the complexity.
You can make anything simple by inventing a language to state it in. Use custom entities instead of multiplying them.
I'm not convinced this does logically follow. There is a view point that God is the source of math only in the sense that God created humanity with the limited capacity for understanding the world around them and one way this capacity has manifested is through the development of Mathematics.
Why does math work so well? Well math is pretty great but I don't think it is magical. There have been thousands of years of slow mathematical advances to get us to where we are now.
Why does math work so well? Well math is pretty great but I don't think it is magical. There have been thousands of years of slow mathematical advances to get us to where we are now.
>why does math work so well?
This question really really bothers me. Why can I completely describe gravitational attraction (at certain distances) by just solving for f=g(m1*m2)/r^2? It's truly disturbing when you start appreciating this for the first time.
This question really really bothers me. Why can I completely describe gravitational attraction (at certain distances) by just solving for f=g(m1*m2)/r^2? It's truly disturbing when you start appreciating this for the first time.
It must have been just as disturbing for the ancients to see how one could count the sheep using a sack full of pebbles. (We, modern people, are too used to the wonders given to us by the - quite abstract, in fact - notion of a number and never question our faith in the applicability of arithmetical operations to the real world.)
You can hear a band play music from a radio, even though there is no band inside your radio. How? Because electrons in a wire will slosh back and forth in near synchronous response to electrons in a different wire far away. We live in an interactive world where the fact that correlations can occur in matter interactions means that communication is possible.
And the ability to count sheep by counting pebbles is just a generalization of the same principles. One antenna can move in sync with another, without literally being the 'same' antenna. This is indirection or abstraction, depending on how you want to slice it.
The point being, there is no question of why arithmetical operations apply to the real world. The real world permits an infinite variety of valid and useful abstractions. I'd wager it is impossible to imagine a reality where this weren't the case.
And the ability to count sheep by counting pebbles is just a generalization of the same principles. One antenna can move in sync with another, without literally being the 'same' antenna. This is indirection or abstraction, depending on how you want to slice it.
The point being, there is no question of why arithmetical operations apply to the real world. The real world permits an infinite variety of valid and useful abstractions. I'd wager it is impossible to imagine a reality where this weren't the case.
What kind of gaps do you mean? As someone who studies mathematics, I feel like this is a vague criticism I've seen some make without any concrete justification.
I work in applied mathematics and specifically in CFD. This may make me not a "true mathematician" to some of the mathematical community. In fluid dynamics nearly everything is an approximate solution to an imperfect model. In fact, there are whole branches of mathematics dealing with trying to figure out exactly how imperfect models are. "Perfect math" has limited applicability, and only perfectly reflects reality in contrived situations. For example, linear algebra works amazingly well in the contrived environment of a computer, but if you want to perfectly model the electron flow that is involved in that computation one must necessarily use an imperfect approximation.
That's a problem with the model, though, not math itself. People who attribute divinity to mathematics aren't considering these types of things as its the application of math into another field, not an intrinsic truth within the field of mathematics.
I'm not convinced that such an intrinsic truth exists apart from application.
That's a different take on mathematics than many people I've met who study it. For me, mathematics is an extension of philosophy, geared towards finding intrinsic truth in a quantitative domain, and it just happens to have applications.
Not the OP, and he/she mentioned "physical observations" so maybe he/she was only referencing hard sciences like Chemistry or, indeed, Physics, but I am of the same opinion to him/her when it comes to the gap between mathematics and social (for a lack of better word) sciences, which social sciences (and the underlying social component behind them, i.e., us, humans) play a very important role in, well, how the Universe runs, or is seen as running.
In other words, mathematics is very bad at modeling and explaining human behavior, be it in economics, history or even political science (even though one of the best political scientists that ever was, Hobbes, wrote his most famous book by trying to imitate Euclid's "Elements"). This is starting to become particularly important now because we try to build some "AI" functionalities that should imitate humans (and even surpass them) based mostly on mathematics (and some underlying data), but it is my opinion that because of this "gap" between how humans are and what mathematics can tell us about how humans are and behave, it is my opinion I say that those "AI" functionalities will never "become" human enough. Stanislaw Lem's "The Cyberiad" does a much better job compared to me at showing this gap between humans and "machines built on mathematics".
In other words, mathematics is very bad at modeling and explaining human behavior, be it in economics, history or even political science (even though one of the best political scientists that ever was, Hobbes, wrote his most famous book by trying to imitate Euclid's "Elements"). This is starting to become particularly important now because we try to build some "AI" functionalities that should imitate humans (and even surpass them) based mostly on mathematics (and some underlying data), but it is my opinion that because of this "gap" between how humans are and what mathematics can tell us about how humans are and behave, it is my opinion I say that those "AI" functionalities will never "become" human enough. Stanislaw Lem's "The Cyberiad" does a much better job compared to me at showing this gap between humans and "machines built on mathematics".
In my opinion this is less of a gap and more of a misapplication of what mathematics is. Mathematics isn't a tool for describing human behavior, though social sciences may use results of mathematics (and I would argue to much greater accuracy than they'd otherwise have) and the fact that mathematics can't describe human behavior is not due to gaps (results of mathematics are consistent with reality, as far as I'm concerned) but because we aren't answering questions in the domain of mathematics.
The physical observations, though, I'm still waiting on an answer from OP about that. It's fairly weasely to say something like that with no example.
The physical observations, though, I'm still waiting on an answer from OP about that. It's fairly weasely to say something like that with no example.
> but because we aren't answering questions in the domain of mathematics.
And then one can ask “what is the domain of mathematics?” or even “does mathematics have a domain?”, questions which lead us into a “philosophy of science” discussion with no end in sight.
I’ve felt for quite some time that the fact that mathematics can model/answer some aspects related to physical reality is just a happy coincidence at best, which we shouldn’t insist too much upon, for fear of then risking to miss the forest because of some trees that absorb our view, like “isn’t this mathematical equation perfectly describing how galaxies interact billions of light-years away?” might obstruct from us the very dire truth that there is no math to describe what will be my cat’s movings around the room in the next 5 minutes (and it’s not for lack of trying, just look at the billions of dollars invested by hedge-funds into mathematics so that they could “model”/predict the future; I don’t think they’re scientifically anywhere close to that).
And then one can ask “what is the domain of mathematics?” or even “does mathematics have a domain?”, questions which lead us into a “philosophy of science” discussion with no end in sight.
I’ve felt for quite some time that the fact that mathematics can model/answer some aspects related to physical reality is just a happy coincidence at best, which we shouldn’t insist too much upon, for fear of then risking to miss the forest because of some trees that absorb our view, like “isn’t this mathematical equation perfectly describing how galaxies interact billions of light-years away?” might obstruct from us the very dire truth that there is no math to describe what will be my cat’s movings around the room in the next 5 minutes (and it’s not for lack of trying, just look at the billions of dollars invested by hedge-funds into mathematics so that they could “model”/predict the future; I don’t think they’re scientifically anywhere close to that).
> And then one can ask “what is the domain of mathematics?” or even “does mathematics have a domain?”, questions which lead us into a “philosophy of science” discussion with no end in sight.
I think these are useful conversations even if we can't foresee them ending. It's what's helped us move physics beyond stuff like Newtonian mechanics where we expect things to line up with these nice equations, and apply math in a more appropriate fashion to our observations. i.e. We treat math as something we apply to our observations, and reconsider models as we run into problems, as opposed to demanding our observations line up with our initial model.
I think these are useful conversations even if we can't foresee them ending. It's what's helped us move physics beyond stuff like Newtonian mechanics where we expect things to line up with these nice equations, and apply math in a more appropriate fashion to our observations. i.e. We treat math as something we apply to our observations, and reconsider models as we run into problems, as opposed to demanding our observations line up with our initial model.
> Can you elaborate on how Hobbes tried to imitate Euclid's "Elements?"
Please!
Please!
Can you elaborate on how Hobbes tried to imitate Euclid's "Elements?"
So, are the natural numbers human? (Kronecker would famously disagree: Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.)
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.
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.
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.
> 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.
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.
> 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
It's a shared sentiment:
"Mathematics is the language with which God wrote the universe." -- Galileo
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.
To assuage the sensitive, we've degodded the title and replaced it with the usual Erdős reference.
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."
"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."
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.
Paul Graham explored why different fields --- like math, art, and engineering --- all use the word "beautiful," http://www.paulgraham.com/taste.html
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!
> The sixth edition will be the final one.
And so will the seventh.
And so will the seventh.