> They “imagined that the next bit of progress will come from some new pieces being dropped onto the table, [rather than] from thinking harder about the pieces we already have,”
When the best minds we have divide into two camps, one saying that (a) "with X and Y we don't have enough information to solve Z" and the other saying (b) we do but we need to think harder, the first camp builds a particle collider, the other creates what, string theory? Aren't we doing "fuzzy science" here?
It seems that the experts in one of the camps should go back and retrace their steps because somewhere along the line they made an assumption based on some data that they (I assume) forgot to encode into their equations and now they have trouble taking it to the next level. Why is it not clear to us that eiher a or b is true?
If we managed to model exactly how a rat thinks, behaves and we create a little fury rat robot that runs around being all rat-like, doing all things real rats do to the degree that the robot is accepted by the local rat community, is that not a system that would be considered equal in complexity to a rat?
Do we have enough information to create a rat? I think so. Is that an A.I.? To be rat-like you have to be able to adopt to new environments, learn new things.
Could we also model in the same way a human baby? Ok, that's perhaps a bit too hard. What if we try to create a model of a mongoloid baby. It would still be considered intelligent, right? Is it an A.I.?
Why does a system need conciousness in order to be able to dominate its environment? To me, clonable robotic rats would be as much of an inconvenience as real rats are.
Common man is brought up within the bubble of their parent's beliefs. Intelligent man breaks out of that bubble, once he find it too constraining. Unintelligent man does not. Unintelligent man also have trouble identifying bubbles.
Earth provide opportunities for success for both men. In the long run, one of them will dominate the other.
Has anyone proven logically that I, an entity, can create a new entity that is more intelligent than me? My intuition might be wrong, but how would that ever be a possibility, other than me aiding in the creation of a new human being?
Relatively weak human arms have enough strength to assemble tools that are much stronger than the original arms. A small, well placed flame can initiate a reaction that burns even hotter and brighter than itself. Our abstract concepts of strength, heat, brightness, and so on, are latent to varying degrees in the environment. Certain arrangements of raw materials release that potential.
If those analogies hold, then intelligence is still another concept expressed by some configurations of matter. Who knows how much "latent intelligence" is available to be released, but I guess the assumption is that it's much greater than what's already manifesting in our brains.
The most well-known rebuttal is that Einstein's mother created someone more intelligent than her. There was no intelligence when the universe began (at least as far as we know), then intelligence was created as an emergent property of biological systems.
Ok, so they DID make it to the top, great! Now, how does this help me get better at graph theory?
Edit: That being said, I mean no disrespect to this... 20 year old article. Hope I didn't hurt your feelings, Article. Just trying to find an entry point into this domain.
Assuming the question is serious I'll attempt an answer. It shows an example of how using graph theory one can solve a problem that seemingly at first has nothing to do with graphs. The proof relies on graph theoretic concepts and understanding the proof helps one get a firmer grasp on these concepts. This particular example and this particular experience may not seem helpful but done often enough you'll end up being good at graph theory.
Thx! And I was being serious. For me, one problem of gaining understanding of the domain is, how do I identify that a problem is indeed a graph problem?
Practice, and read about similar puzzles/problems. Track down reprints/scans of Martin Gardner's Mathematical Games and similar publications.
In general, for identifying if something is a graph problem:
1) Through some mapping, (almost?) everything can be transformed into a graph problem of some sort. Now, that's a bit too big a set and ignores the real question.
2) How do I identify that a problem is practically solved with graphs?
Some heuristics (note, these are heuristics, particularly useful for getting started, but not at all absolutes):
Is it discrete? In the given problem from the link we have a finite number of extreme points (ends of each line segment), but an infinite number of points between. Fortunately, thanks to the problem constraint, there are only a few non-endpoints that we are concerned with. So we can safely ignore the infinity of possibilities by only examining this finite set.
Are there transitions between these states that are easily modeled as edges? In the given puzzle, absolutely. And its symmetric so an undirected graph is suitable (sometimes directed graphs would be more suitable or the only applicable solution).
After this, proving various things (like that the generalized statement that all properly constructed puzzles have a solution) will require learning some of the basic properties of graphs, and how to restate the premises (such as constraints on altitude and movement) in a graph-theoretic form. It's easy to intuit the proof now that you have a simplified, but complete, model of the problem, but harder to state it with mathematical certitude. That part really will just require more practice and exposure.
At some point you'll develop sufficient vocabulary in the field that you may not be able to prove it straight off, but you'll know what and where to look for the elements you need to construct a proof like they have.
> how do I identify that a problem is indeed a graph problem?
Mathematical intuition. To answer the next obvious question of how to buld mathematical intuition: Solve lots of math problems (I've also heard that reading "Pólya - How to Solve it" is supposed to be helpful for this; I can't say anything about it).
If you are the type of person that absolutely loved each second of each lecture that your math professor held in high school, where he/she tried to prove an equation on the black board, and you acctually managed to pay attention for long enough to acctually understand what he was talking about, and you got a real kick out of that newly gained intuition, and you now long for that type of "profound" enlightenment, how would yo go about gaining in mathematical intuition when you are in your 40ies?
EDIT: Distractions abound so I hit submit before forgetting.
Why: Martin Gardner wrote on a variety of mathematical topics in fields such as geometry, graph theory, number theory, combinatorics, topology, and beyond. His writing is very approachable, and well sourced. This will help to develop a base vocabulary across the mathematical fields that you can use for further research and investigation of your own once you find the areas that interest you, along with being delightful reads just for their own sake for the mentally curious and engaged.
> how would yo go about gaining in mathematical intuition when you are in your 40ies?
Exactly the same way that you would go if you were in your 20ies: Get the relevant textbooks that are typically recommended by professors and read them (sorry, the textbooks that I can recommend for basic studies in mathematics are all in German (my native language); only for main studies in mathematics I can tell English textbooks).
You might think that, because no country has implemented BI, then considered it a failed project, then scraped that system for something better, yet. But one of the countries on this planet might be the first to try that. It's one of those "nobody done it yet thus it can't be done by anyone" ideas that reality so often find is false.
I love the indentation. Much easier to read than a bunch of nested ifs and whiles and good to see a lack of outdated comments inside the method body. So a line-by-line read of that code seemed quite enjoyable to me.
I have ~20 years of experience in programming. The hardest part of my work is knowing if I have everything I need in order to classify a particular problem as one of a particular class so that I then can apply the tools that I know of would fit that problem-space and then use those tools to finish my task. For example, is the problem I'm facing a graph problem? If so, then I have a large set of tools (algorithms) to help me solve that problem. But is it a graph problem? It would suck for my employer or client, if I tried to solve a problem within a particular problem-space with tools suted for a different domain. It's kind of like that part of the Swedish standardized test where they test your math and logical abilities by presenting some information and then asking you, do you have what you need to solve this? If so, what is the answer?
Coding on a white board is useful I have come to find if there is a conversation going on and the interviewer use this opportunity to find out more about how this particular person thinks or works.
Interesting learning streak. I took the same route, started on Vic 20 at the age of 10, but skipped on the amiga assembly, something I suffer from on a daily basis, so good choice.
Thanks! :) The Vic 20 at about the same age here too; assembly was great, but staying for well over a decade on ColdFusion just because there was a lot of work in it (and, comfortable) was not.
I forgot to mention some PHP thrown in there, but I'm glad to have been making the transition to C# in the past year. Working with a strongly typed language is quite a revelation, and some fantastic things are happening with the language and framework these days too (dotNetCore).
When the best minds we have divide into two camps, one saying that (a) "with X and Y we don't have enough information to solve Z" and the other saying (b) we do but we need to think harder, the first camp builds a particle collider, the other creates what, string theory? Aren't we doing "fuzzy science" here?
It seems that the experts in one of the camps should go back and retrace their steps because somewhere along the line they made an assumption based on some data that they (I assume) forgot to encode into their equations and now they have trouble taking it to the next level. Why is it not clear to us that eiher a or b is true?