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What a curious post. Pure mathematics is nothing but ideas - often very deep ones that very few people are capable of originating. If you take Adams literally

What a curious post. Pure mathematics is nothing but ideas - often very deep ones that very few people are capable of originating. If you take Adams literally, then mathematics is worthless. Is the idea of the Ricci flow, which Perelman used to prove the Poincare conjecture, really useless? The problem Adams is addressing, so far as I can see, is that many people think trivial, easy-to-have ideas are somehow valuable. Of course, that's false, as Adams says. Value is associated to scarcity. Easy-to-have ideas aren't valuable, simply because they're not scarce. An idea like the Ricci flow is valuable, because very few people are capable of having it. Ditto the ideas behind deep algorithms. PageRank had value in large part because relatively few people had the depth of insight required to come up with it. Come to think of it, the Turing machine was an idea that Turing invented to formalize Hilbert's intuitive notion of an effective computation. And I think that idea turned out to have some value.

pwhelan 16y ago
"Pure" mathematics sometimes has applications as well - number theory and cryptography, boolean algebra and circuits, etc. I do not think that Adams would argue that pure math is worthless, rather that it has applications that haven't shown how valuable it is or is not. Value is associated with utility as well. If Markov chains didn't help us describe things well, despite it seemingly being difficult to come up with, it would have little value. However, anything that stretches the limits of human understanding has value in that it increases our capacity reason and shows paths that do not bear much fruit.
outotrai 16y ago
I'm shooting from the hip here, but I think maybe there's a distinction to be made between ideas for a product and ideas as a product. The end result of the ideas of mathematicians is mathematics itself - furthering mathematics was the goal all along. Obviously ideas count when you're working within the realm of ideas alone. However, when the ideas are a means to an end (e.g., a movie, say), the implementation is what counts. You might have the world's greatest screenplay in your desk, but if you haven't gotten to the execution stage, what kind of movie do you have? (None.) Whereas if you'd started with something, anything, like walking trees fighting hobgoblins, say, or the idea that maybe humans are just batteries for supercomputers in the future, your movie might have a chance, depending on its execution.
- timwiseman 16y ago
  You make fine points and I agree. But if I may, it can be expanded on. In mathematics, a theorem is an idea combined with proof. In short, the idea is combined with the knowledge that it is a true and accurate fact (at least within the domain you are working in. Things are true in C that are not true in R), along with some idea why it is true emboddied in the proof. It is this combination of the idea with knowledge it is right that makes it valuable. To be slightly more concrete, the idea that P = NP is not overly valuable. It is not yet proven either way and it could be true or false. A full proof would be tremendously valuable, and very surprising to many who are very confident P != NP. Creating a proper proof seems much more akin to writing the novel than it is to having the idea for the novel.
jheriko 16y ago
Even though such ideas are rare, its difficult to measure their value meaningfully until something is done with them. You use the proof of the Poincare conjecture to demonstrate that Ricci flow has value for instance, but without such derivative work can you convince others that it has value?
tjr 16y ago
Sure, in a way, but, at what point do we say the idea has value? PageRank didn't seem particularly valuable until the Google folks built something "useful" out of it. I can imagine that someone who could envision its use (or, more esoterically, someone with lots of money who likes collecting things) might have bought the idea itself, and while the idea itself has intrinsic intellectual value, it is of no good [to society as a whole, or to those who do not enjoy understanding the idea for its own sake] nor of any profit just sitting there as an idea, or as a research paper. Companies spend money regularly to license patents... to pay to use ideas. But what company would find value in simply licensing the patent just to enjoy it? The value is found in licensing the patent and using the idea to create something with it. At the very least, there is more than one kind of "value".
duhprey 16y ago
Respectfully (very respectfully, because your book still haunts my shelf like a specter of how little I understood what was happening to my brain when I took Quantum Computation...) I disagree. At least in part. The ideas you point out are not at all easy to have in the first place, they also, I think, required proper execution to bring them to fruition. Now, Adams is speaking much more to, I think, artistic creativity than mathematical creativity, but that first blush of an idea which starts with "What if I..." is the same and requires a carefully executed (but wildly different) follow through. Imagine the scenario where some guy, say Joe, said to Turing, "What if a machine did it?" to which Alan goes through the effort of building the formalization of the machine. But Joe comes back to say "Well, that was my idea." To come back to my partial agreement. In retrospect, people looking back and may tend to think "well I could have thought of that." And yet the necessary background to effectively wade through the ideas that rage through ones head as he thinks about a problem is non-trivial. Perhaps that's true of either process. When a writer takes what appears to be a really lame idea and says "I can make that work" he or she has probably already, automatically, in the back of his or her head taken a few steps imagining how it would execute.
gsk 16y ago
Interesting. An idea must still have some sort of execution, no? In mathematics, I would have thought that proof is the execution of a purely mathematical idea. Obviously, I do agree with you that some ideas are precious (to take a non-mathematical example, the idea of Freedom). Adams seems to have only had utilitarian ideas in mind. His post didn't strike me as curious, however. I did not expect him to discuss pure mathematics.
- timwiseman 16y ago
  You bring up an interesting topic. Is the idea of Freedom valuable with no execution? Freedom itself is almost priceless. I am well aware that the freedoms I enjoy now was bought with the blood of many who went before me. But that was the very costly execution that made freedom a reality. Is the idea, without tht execution to create freedome itself, of great value?
  - infinite8s 16y ago
    I believe it has great value if it inspires people to execute it.
    - timwiseman 16y ago
      Good point. With that said, the main value lies in having it and thus in the execution. The value of the idea itself without execution is only in inspiring people towards execution.
Vivtek 16y ago
I'd say yeah - without the proof of the Poincaré conjecture (execution), the idea of the Ricci flow is an amusing conceit. And worthless beyond that.
jacobolus 16y ago
That’s an absurd comparison. Generating mathematical conjectures and then proving them takes decades of preparation and careful study and analysis of a difficult problem. Generating a movie plot idea at the level Adams is talking about takes, with no preparation, a few minutes and a dinner napkin. The equivalent kind of “idea” in mathematics might be “I’m going to understand and model heat dispersion in a block of metal”, with the “execution” being developing the underlying machinery (partial differential equations) gathering enough data, and building a model (the heat equation) that fits the data and can be used to compute practical approximations. The “idea” part can be done by anyone; the execution is made up of decades of hard work and hard thought by several of the world’s most famous mathematicians, building on centuries of previous effort.
- michael_nielsen 16y ago
  I wouldn't be entirely surprised if the idea for Ricci flow or PageRank was generated with a dinner napkin. Quite literally, I once used a napkin to explain the Ricci flow to a mathematician, and I've seen dozens of papers start that way, including a few of my own. Such ideas gain value from their scarcity, not from some immense complexity.
  - jacobolus 16y ago
    Okay, but you wouldn’t come up with PageRank until you’d thought about searching and sorting big quantities of data for some time. If you were just trying to sort 10 numbers, you wouldn’t come up with PageRank. It took a “search the whole web” scale problem for the ideas about how to accomplish it to start bubbling up to the surface. (Probably a bit of an oversimplification, but you get the idea.) Ricci flow isn’t even meaningful until you’ve developed up differential geometry – not exactly an afternoon stroll for a layman.
    - michael_nielsen 16y ago
      I completely agree. My original point was that valuable ideas are very difficult to have, and thus scarce. That's why Ricci flow and PageRank are valuable, while "let's build a social network to do X", for some choice of X, is not. Difficult-to-have and can-write-on-a-napkin are not mutually exclusive, of course!

pwhelan 16y ago

"Pure" mathematics sometimes has applications as well - number theory and cryptography, boolean algebra and circuits, etc. I do not think that Adams would argue that pure math is worthless, rather that it has applications that haven't shown how valuable it is or is not. Value is associated with utility as well. If Markov chains didn't help us describe things well, despite it seemingly being difficult to come up with, it would have little value. However, anything that stretches the limits of human understanding has value in that it increases our capacity reason and shows paths that do not bear much fruit.

outotrai 16y ago

I'm shooting from the hip here, but I think maybe there's a distinction to be made between ideas for a product and ideas as a product. The end result of the ideas of mathematicians is mathematics itself - furthering mathematics was the goal all along. Obviously ideas count when you're working within the realm of ideas alone. However, when the ideas are a means to an end (e.g., a movie, say), the implementation is what counts. You might have the world's greatest screenplay in your desk, but if you haven't gotten to the execution stage, what kind of movie do you have? (None.) Whereas if you'd started with something, anything, like walking trees fighting hobgoblins, say, or the idea that maybe humans are just batteries for supercomputers in the future, your movie might have a chance, depending on its execution.

timwiseman 16y ago
You make fine points and I agree. But if I may, it can be expanded on. In mathematics, a theorem is an idea combined with proof. In short, the idea is combined with the knowledge that it is a true and accurate fact (at least within the domain you are working in. Things are true in C that are not true in R), along with some idea why it is true emboddied in the proof. It is this combination of the idea with knowledge it is right that makes it valuable. To be slightly more concrete, the idea that P = NP is not overly valuable. It is not yet proven either way and it could be true or false. A full proof would be tremendously valuable, and very surprising to many who are very confident P != NP. Creating a proper proof seems much more akin to writing the novel than it is to having the idea for the novel.

jheriko 16y ago

Even though such ideas are rare, its difficult to measure their value meaningfully until something is done with them. You use the proof of the Poincare conjecture to demonstrate that Ricci flow has value for instance, but without such derivative work can you convince others that it has value?

tjr 16y ago

Sure, in a way, but, at what point do we say the idea has value? PageRank didn't seem particularly valuable until the Google folks built something "useful" out of it. I can imagine that someone who could envision its use (or, more esoterically, someone with lots of money who likes collecting things) might have bought the idea itself, and while the idea itself has intrinsic intellectual value, it is of no good [to society as a whole, or to those who do not enjoy understanding the idea for its own sake] nor of any profit just sitting there as an idea, or as a research paper. Companies spend money regularly to license patents... to pay to use ideas. But what company would find value in simply licensing the patent just to enjoy it? The value is found in licensing the patent and using the idea to create something with it. At the very least, there is more than one kind of "value".

duhprey 16y ago

Respectfully (very respectfully, because your book still haunts my shelf like a specter of how little I understood what was happening to my brain when I took Quantum Computation...) I disagree. At least in part. The ideas you point out are not at all easy to have in the first place, they also, I think, required proper execution to bring them to fruition. Now, Adams is speaking much more to, I think, artistic creativity than mathematical creativity, but that first blush of an idea which starts with "What if I..." is the same and requires a carefully executed (but wildly different) follow through. Imagine the scenario where some guy, say Joe, said to Turing, "What if a machine did it?" to which Alan goes through the effort of building the formalization of the machine. But Joe comes back to say "Well, that was my idea." To come back to my partial agreement. In retrospect, people looking back and may tend to think "well I could have thought of that." And yet the necessary background to effectively wade through the ideas that rage through ones head as he thinks about a problem is non-trivial. Perhaps that's true of either process. When a writer takes what appears to be a really lame idea and says "I can make that work" he or she has probably already, automatically, in the back of his or her head taken a few steps imagining how it would execute.

gsk 16y ago

Interesting. An idea must still have some sort of execution, no? In mathematics, I would have thought that proof is the execution of a purely mathematical idea. Obviously, I do agree with you that some ideas are precious (to take a non-mathematical example, the idea of Freedom). Adams seems to have only had utilitarian ideas in mind. His post didn't strike me as curious, however. I did not expect him to discuss pure mathematics.

timwiseman 16y ago
You bring up an interesting topic. Is the idea of Freedom valuable with no execution? Freedom itself is almost priceless. I am well aware that the freedoms I enjoy now was bought with the blood of many who went before me. But that was the very costly execution that made freedom a reality. Is the idea, without tht execution to create freedome itself, of great value?
- infinite8s 16y ago
  I believe it has great value if it inspires people to execute it.
  - timwiseman 16y ago
    Good point. With that said, the main value lies in having it and thus in the execution. The value of the idea itself without execution is only in inspiring people towards execution.

Vivtek 16y ago

I'd say yeah - without the proof of the Poincaré conjecture (execution), the idea of the Ricci flow is an amusing conceit. And worthless beyond that.

jacobolus 16y ago

That’s an absurd comparison. Generating mathematical conjectures and then proving them takes decades of preparation and careful study and analysis of a difficult problem. Generating a movie plot idea at the level Adams is talking about takes, with no preparation, a few minutes and a dinner napkin. The equivalent kind of “idea” in mathematics might be “I’m going to understand and model heat dispersion in a block of metal”, with the “execution” being developing the underlying machinery (partial differential equations) gathering enough data, and building a model (the heat equation) that fits the data and can be used to compute practical approximations. The “idea” part can be done by anyone; the execution is made up of decades of hard work and hard thought by several of the world’s most famous mathematicians, building on centuries of previous effort.

michael_nielsen 16y ago
I wouldn't be entirely surprised if the idea for Ricci flow or PageRank was generated with a dinner napkin. Quite literally, I once used a napkin to explain the Ricci flow to a mathematician, and I've seen dozens of papers start that way, including a few of my own. Such ideas gain value from their scarcity, not from some immense complexity.
- jacobolus 16y ago
  Okay, but you wouldn’t come up with PageRank until you’d thought about searching and sorting big quantities of data for some time. If you were just trying to sort 10 numbers, you wouldn’t come up with PageRank. It took a “search the whole web” scale problem for the ideas about how to accomplish it to start bubbling up to the surface. (Probably a bit of an oversimplification, but you get the idea.) Ricci flow isn’t even meaningful until you’ve developed up differential geometry – not exactly an afternoon stroll for a layman.
  - michael_nielsen 16y ago
    I completely agree. My original point was that valuable ideas are very difficult to have, and thus scarce. That's why Ricci flow and PageRank are valuable, while "let's build a social network to do X", for some choice of X, is not. Difficult-to-have and can-write-on-a-napkin are not mutually exclusive, of course!