Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

In Search of Proofs from “The Book”

quantamagazine.org

Post reply on HN
Print view View original

21–30 of 52 posts

Re: In Search of Proofs from “The Book”

#21

Post by jacobolus »

yantrams wrote: post #11
ggm wrote: post #9

Earlier quoted context omitted.

I think Davis and Hersh say this. Ruben Hersh "what is mathematics really" felt like it was opening the door to the sausage factory.

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/cl…

A big portion of Lakatos’s book is online, https://math.berkeley.edu/~kpmann/Lakatos.pdf

Re: In Search of Proofs from “The Book”

#22

Post by phaedrus »

tacomonstrous wrote: post #7

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.

Re: In Search of Proofs from “The Book”

#23

Post by User23 »

wolfkill wrote: post #17

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.

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.

Re: In Search of Proofs from “The Book”

#24

Post by taoistextremist »

wolfkill wrote: post #17

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.

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.

Re: In Search of Proofs from “The Book”

#25

Post by placebo »

User23 wrote: post #23
wolfkill wrote: post #17

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.

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.

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.

Re: In Search of Proofs from “The Book”

#27

Post by yters »

empath75 wrote: post #19
wolfkill wrote: post #17

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…

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.

Re: In Search of Proofs from “The Book”

#28

Post by jonnybgood »

extr wrote: post #13
tacomonstrous wrote: post #7

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.

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…

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.

Re: In Search of Proofs from “The Book”

#29

Post by dang »

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.

Re: In Search of Proofs from “The Book”

#30

Post by montrose »

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."

Post reply on HN