In Search of Proofs from “The Book”
quantamagazine.org
quantamagazine.org
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.
"An unpleasant process, especially one that is hidden from public view, that is used to produce a widely consumed product" - Urban Dictionary
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.
Reality has its thing.
And sometimes you need a jeweler’s tool and other times just your fists.
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.
> 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.
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.
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.
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.
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.
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.
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".
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.
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).
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.
Please!
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.
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.
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.
It's a shared sentiment:
"Mathematics is the language with which God wrote the universe." -- Galileo
To assuage the sensitive, we've degodded the title and replaced it with the usual Erdős reference.
"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."
And so will the seventh.