Tracking the World's Great Unsolved Math Mysteries

coondoggie writes "Some math problems are as old as the wind, experts say, and many remain truly unsolved. But a new open source-based site from the American Institute of Mathematics looks to help track work done and solve long-standing and difficult math problems. The Institute, along with the National Science Foundation, has opened the AIM Problem Lists site to offer an organized and annotated collection of unsolved problems, and previously unsolved problems, in a specialized area of mathematics research. The problem list provides a snapshot of the current state of research in a particular research area, letting experts track new developments, and newcomers gain a perspective on the subject."

Comments

Math cannot exist before wind.

[blackraven14250]

How can math problems exist before people start using mathematics? Last I checked, math was nothing more than a representation of the hypothetical that as closely models our universe as possible.

Re:Math cannot exist before wind.

[Anonymous Coward]

Ever heard of a "figure of speech"?

Re:Math cannot exist before wind.

[John Hasler]

Some say math is discovered. Others say it is invented. You are one of the latter.

Re:Math cannot exist before wind.

[invisiblerhino]

The requirement to model our universe as closely as possible is a requirement of physics, not mathematics. The fact that mathematics, even very abstract mathematics, accurately models the natural world is a deep mystery.

Re:Math cannot exist before wind.

[Anonymous Coward]

Yeah, those are as old as the wind, too.

Re:Math cannot exist before wind.

[JustOK]

they're a dime a dozen, too.

Re:Math cannot exist before wind.

[Anonymous Coward]

You beg the question "does wind necessarily precede intelligence," which it does not.

And yes, I only posted this for the sake of using "beg the question" in a sentence. :(

Re:Math cannot exist before wind.

[CannonballHead]

The fact that mathematics, even very abstract mathematics, accurately models the natural world is a deep mystery.

That depends on your worldview.

Re:Math cannot exist before wind.

[Tynin]

Care to elaborate?

Re:Math cannot exist before wind.

[blackraven14250]

If it is discovered, the solution already exists and the problem was solved before wind existed, because the problem never existed in a state where it didn't have a solution.
If it is invented, the problem didn't exist before the wind.
In either case, the problem isn't older than the wind.

Re:Math cannot exist before wind.

[geekoid]

It shouldn't be a mystery because we created it to model the world around us.

Re:Math cannot exist before wind.

[mario_grgic]

It's not a deep mystery at all. It's by design. We chose our axioms to resemble what we observe. Just take Euclid axioms of the plane geometry, or natural number axioms (like Peano's), etc.

It's no wonder then that what we construct out of them resembles what is out there (at least somewhat). Other modeling problems try to be even stricter and model the observed phenomena even closer than axioms could ever hope to describe.

Re:Math cannot exist before wind.

[jd]

I would claim that the ratio of a circle to its diameter is independent of being observed, or indeed there being an observer. I would also claim that the laws of geometry, probability and topology are universal and also do not depend on the existence of observers, let alone their ability to perform maths.

Radioactive decay follows an exponential decay curve. It will have done so long before anyone could add, let alone handle irrational numbers like e.

This puts me firmly in the category of maths being discovered, not invented. Mathematical tools, however, are invented and not discovered. I consider these to be quite different. If you were to imagine an alien lifeform on some distant world, they'll have an identical math but their experience of it, the way they treat it, the systems they use, those will all be unique to them because those are inventions and not anything fundamental to maths itself.

In a simpler example of the same concept, we can use ancient Greek maths today even though they didn't have a concept of zero and had (to modern eyes) very alien views on the way maths worked. We can use ancient Greek maths because the results don't depend on any of that.

We can use Roman results, too, despite the fact that their numbering system doesn't really follow a number base in any way we'd understand. It doesn't matter, though, because the important stuff all takes place below such superficial details. Even more remarkable, we can read many of the numbers written in Linear A, even though we can't read the language itself and know very little about the culture or people.

None of this would be possible if what lay under maths was invented. It's very hard to rediscover lost inventions, as there's many ways of producing similar results. But when you can rediscover lost number systems with comparative ease - well, doesn't that tell you there has to be something a bit more universal to it?

(I won't get into parrots being able to discover the notion of zero, but it's again pertinent as it's an example of a universality that transcends the invented language it's described in.)

Re:Math cannot exist before wind.

[socceroos]

Mum's the word. We don't want /. starting another one of those threads where everyone tries to continue the joke.

Re:Math cannot exist before wind.

[Obfuscant]

I would claim that the ratio of a circle to its diameter is independent of being observed, or indeed there being an observer. I would also claim that the laws of geometry, probability and topology are universal and also do not depend on the existence of observers, let alone their ability to perform maths.

Since the existance of a perfect circle depends on the thoughts of an observer, the ratio of the diameter to the circumference of such an object must depend on there being an observer. Nature can produce approximate circles, but not perfect ones.

Radioactive decay follows an exponential decay curve. It will have done so long before anyone could add, let alone handle irrational numbers like e.

"Exponential decay curve" and "irrational numbers" are two different concepts. (1/2)^N is an exponential decay curve -- which defines the half-life of a radioactive substance. For no integer value of N is the result "irrational".

This puts me firmly in the category of maths being discovered, not invented.

Right destination, wrong reason.

Re:Math cannot exist before wind.

[Anonymous Coward]

I'm not the parent, but here is my take. Mathematics is the study of well-defined objects. Since natural world seems to be relatively well-behaved, it might be expected that objects which can be well-defined accurately model the natural world.

If you are wondering what I mean about well-defined, consider two examples. The integers can be well-defined. The integers that I think about are the same as the integers that you think about. It doesn't seem that right and wrong can be well-defined. I say this because after centuries of trying, we can't get people to agree on exactly what these terms mean.

Re:Math cannot exist before wind.

[interkin3tic]

Some say math is discovered. Others say it is invented.

And still others (especially those in grade school and high school) say that math should neither have been invented nor discovered.

Re:Math cannot exist before wind.

[PaladinAlpha]

If you discover the answer to a riddle, haven't you solved the riddle?

Re:Math cannot exist before wind.

[PaladinAlpha]

I would claim that the ratio of a circle to its diameter is independent of being observed, or indeed there being an observer. I would also claim that the laws of geometry, probability and topology are universal and also do not depend on the existence of observers, let alone their ability to perform maths.

Since the existance of a perfect circle depends on the thoughts of an observer, the ratio of the diameter to the circumference of such an object must depend on there being an observer. Nature can produce approximate circles, but not perfect ones.

Radioactive decay follows an exponential decay curve. It will have done so long before anyone could add, let alone handle irrational numbers like e.

"Exponential decay curve" and "irrational numbers" are two different concepts. (1/2)^N is an exponential decay curve -- which defines the half-life of a radioactive substance. For no integer value of N is the result "irrational".

This puts me firmly in the category of maths being discovered, not invented.

Right destination, wrong reason.

The circles Nature creates can be approximate only because the "perfect" circle is constant. No perfect circle = no circle = no approximate circles (what is a circle?). The immutables have existed for all time.

The remark about irrational numbers is irrelevant to the point about exponential decay, but for this reason pi has always been a universal constant. All manner of physical relations involve second, third, and fourth roots, which of course easily gives rise to irrational numbers.

Re:Math cannot exist before wind.

[TheVelvetFlamebait]

Actually, mathematics is considerably more than a model for our universe. We can (and do) study mathematical objects which have no immediate instances in the universe.

Re:Math cannot exist before wind.

[Evil Pete]

Hypothetical?

Mathematics is a collection of logically consistent statements about abstractions such as structure and number. "Hypothetical" implies it needs to be tested, a mathematical proof does not need a 'test'.

Re:Math cannot exist before wind.

[Anonymous Coward]

Are you saying that a problem which has a solution isn't a problem?

Re:Math cannot exist before wind.

[Evil Pete]

YES! This has long been acknowledged by people who we usually assume know a little bit about the physical world. It seems reasonable to me, but demonstrating why it is reasonable is another thing.

Re:Math cannot exist before wind.

[blackraven14250]

Yes, because it never existed without a known solution. I might solve the problem of global warming by using renewable energy, but that isn't a 'known' solution, it's just one that is (heavily) hypothesized to work. If it were guaranteed that it would be a solution to global warming, and it were something that had been known at least since the creation of the universe, it would have never been a true problem in the universe to begin with, but rather an intricacy of the universe's workings.

Re:Math cannot exist before wind.

[wwfarch]

Nope, I just googled it

Re:Math cannot exist before wind.

[blackraven14250]

Do you have any proof these constructs don't exist?

Re:Math cannot exist before wind.

[Anonymous Coward]

Long before man had "discovered" how to count, the universe "knew" that the ratio of a circle's diameter to its circumference was a constant number that today we call "pi". We invent the names and notations for the mathematics that we discover.

Re:Math cannot exist before wind.

[Evil Pete]

Ideal circles do not exist. Before human beings they didn't exist, and they still do not exist. We define them. Can you think of a perfect circle? If you can you must have perfect visual processing in your brain. This is a hard problem I admit, and I'm not going to pretend my answer is absolutely correct. However, mathematics proceeds from axioms, which are fundamental assumptions ... sometimes based on physical intuitions, but sometimes not.

I think mathematics is so effective because in the realm of physics our discoveries have few degrees of freedom and can therefore be represented by simple rules. Since the rules must be consistent we have the basis for physics and a tie in to mathematics.

How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? — Albert Einstein

There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology. — Alexandre Borovik

Quotes shamelessly stolen from here.

Re:Math cannot exist before wind.

[Anonymous Coward]

They're children ... give them a break.

Though, to be fair, if someone still thinks that way by the time they enter college, he or she definitely requires some counseling ...

Re:Math cannot exist before wind.

[Anonymous Coward]

Some say math is discovered. Others say it is invented. You are one of the latter.

Math is a language of symbols used to represent patterns observed in nature. Physics is the discipline of actually discovering such rules, and Physics uses the language of Math to describe those rules.

So "math" is invented because it is a language, but the things that math describes are discovered.

Re:Math cannot exist before wind.

[Haxamanish]

I would argue the opposite: a problem is something which has a solution, something without a solution is not a problem but a circumstance.

Re:Math cannot exist before wind.

[Anonymous Coward]

The problem must be invented (i.e. a human construct) before the solution can be discovered (i.e. a natural phenomenon)...

Re:Math cannot exist before wind.

[Obfuscant]

No perfect circle = no circle = no approximate circles (what is a circle?).

Au contraire, mon frere. Nature can produce approximate circles and does so quite happily. The cross section through a bubble is a very good approximation to a perfect circle, and most people not into arguing philosophy would simply CALL it a circle. This is all without knowing the definition of circle that mathematicians have come up with. Nature, however, does not care what pi is.

The remark about irrational numbers is irrelevant to the point about exponential decay,

The OP decided there must be come relevance, else he would not have included it in a paragraph talking about radioactive decay. I was the one pointing out the irrelevance. One need not know anything about 'e' to know exponential decay. The fact that 'e' came along relatively recently doesn't say anything about the origin of math.

...but for this reason pi has always been a universal constant.

I am unclear what antecedant to "this" you are using. What reason? And IS pi a "universal constant? It is only in flat space (like ours appears to be) that the ratio of circumference to diameter in a circle involves pi. In curved space, (the surface of a sphere, e.g.), the ratio will be different. Does the universe consist only of flat space? We are far enough from our sun that the effects are minimal but non-zero. Would living in a region of space in the region of an object more massive than the earth yeild beings who think the circumference of a circle is 3*diameter, and that this "3" is a universal constant? Or would they say "3 if facing east, 4 if facing north?"

Re:Math cannot exist before wind.

[Anonymous Coward]

It's not a deep mystery at all. It's by design. We chose our axioms to resemble what we observe. Just take Euclid axioms of the plane geometry, or natural number axioms (like Peano's), etc.

It's no wonder then that what we construct out of them resembles what is out there (at least somewhat). Other modeling problems try to be even stricter and model the observed phenomena even closer than axioms could ever hope to describe.

The fact you are so nonchalant about the effectiveness of mathematics in the natural sciences tells me you have a limited understanding of mathematics and the natural world.

Re:Math cannot exist before wind.

[Beetle B.]

We chose our axioms to resemble what we observe.

You don't hang around mathematicians, do you? Go play with a number theorist and you'll realize how wrong you are.

It's certainly true that some mathematicians are motivated by modeling the world. But many, and perhaps most, aren't. They'll freely construct mathematical objects that have little basis in the physical world.

Re:Math cannot exist before wind.

[Anonymous Coward]

Are you saying that a problem which has a solution isn't a problem?

because it never existed without a known solution

No, you just said it never existed without a solution. You did not say that the solution had to be known.

Re:Math cannot exist before wind.

[Beetle B.]

I would claim that the ratio of a circle to its diameter is independent of being observed, or indeed there being an observer.

What's a circle? What's a diameter? What's a ratio? Who defines these?

And while we're at it, you do realize that the universe is non-Euclidean? So how do we view the results from Euclidean geometry, given that reality is not Euclidean?

Radioactive decay follows an exponential decay curve. It will have done so long before anyone could add, let alone handle irrational numbers like e.

That some mathematics models the real world does not mean most of mathematics is not invented. It need not be a binary scenario. What would you say of mathematical constructs that have no analogs in nature (but that could be depicted if desired)?

Your arguments are along the lines of "The bicycle was not invented, because the laws of the universe would always have allowed a bicycle to work and exist."

Re:Math cannot exist before wind.

[Beetle B.]

Do you have any proof these constructs don't exist?

Can you prove that there aren't unicorn like creatures in one of the craters of the moon?

Your point?

Re:Math cannot exist before wind.

[Sosetta]

That's just the notation. Math is abstract concepts. Abstract concepts existed before the wind. They're abstract.

Re:Math cannot exist before wind.

[Anonymous Coward]

The universe is nothing more than an imperfect approximation to our mathematical models

Re:Math cannot exist before wind.

[Sosetta]

Exponential decay uses e most of the time, because the derivatives and integrals are a lot less messy. e is irrational. What makes you think you're using 2? And what makes you think that the decay only has values for integer time? What's 1/3 of the way between (1/2)^3 and (1/2)^4? An irrational number.

Re:Math cannot exist before wind.

[TheVelvetFlamebait]

Do you have any proof these constructs don't exist?

I have absolutely no idea. Evidence suggests no, but I have no proof of my ability one way or the other.

/being intentionally dense

Notice that I said "no immediate instances", that is, the mathematics was created before any instances were discovered, if they were discovered at all.

Re:Math cannot exist before wind.

[Anonymous Coward]

Axioms are invented, theorems are discovered.

Re:Math cannot exist before wind.

[Alpha830RulZ]

Some say math is discovered. Others say it is invented. You are one of the latter.

And then there's the one about there being two kinds of people: those that divide things into two groups and those that don't. :P

Maybe it's a bit of both. Non-euclidian geometry seems inventive, pi, seems discoverable.

Re:Math cannot exist before wind.

[lastgoodnickname]

could you imagine a beowulf cluster of threads where everyone tries to continue the joke? ...Profit!

Re:Math cannot exist before wind.

[mctk]

If you were to imagine an alien lifeform on some distant world, they'll have an identical math but their experience of it

I'm not sure if I'm on board with you here. That's quite a claim to make. Just because it's hard to imagine math that's not identical to our own, does not mean it does not exist. I can imagine a quantum sized life-form living in a probabilistic world, never coming up with the integers. Or maybe a universe-sized creature who has absolutely no need for the idea of oneness.

Since I'm posting, here's what I think is a fun problem:
2178*4 = 8712
21978*4 = 87912
219978*4 = 879912

There's one other family of numbers (i.e. a four digit number, expanded in a similar pattern) that does this, if we throw out palindromes and numbers with leading zeros. What is it and can you show that these are the only two such families?

Re:Math cannot exist before wind.

[davek]

Ideal circles do not exist. Before human beings they didn't exist, and they still do not exist. We define them. [...]

And that is the beauty of it all! Can't you imagine a separate universe where PI is a rational number? Somehow, in some separate path of thought, they based their knowledge on 1*pi=1, 2*pi=2, etc.. Their mathematical theorems and laws would be equally true to ours, yet totally based in the supposed non-existent "ideal circle" world that you suggest.

Simply because we do not see it, does NOT mean that it doesn't exist. i.e., even if it exists, we still may not see it.

Re:Math cannot exist before wind.

[Anonymous Coward]

He means he's a postmodern existentialist and believes in all sorts of crackpot theories. All of your logic/science/mathematics/whatsoever means little to him.

Re:Math cannot exist before wind.

[wisty]

OK, I just gotta link to xkcd here: http://xkcd.com/55/

Re:Math cannot exist before wind.

[jd]

No, the bicycle is equivalent to a number base or a mathematical system. It is an implementation OF an underlying system (in this case, Newton's Laws), but Newton's Laws would still remain exactly the same whether Newton - or indeed bicycles - had ever existed.

The definition is also immaterial, as that too is an implementation detail. The underlying principle would remain unaltered whether the definitions of circumference, diameter or pi had ever been developed.

You are confusing the overlaid system with what it overlays. I'm saying you don't need to. Your argument is that the overlaid system is artificial, an invented product. I'm saying you're entirely correct on that. But what I am also saying is that what the product overlays, what is beneath the terms, the dynamics and the fancy Greek lettering is not artificial but exists whether it is known to exist or not.

The problem with assuming the two layers are the same is that you run into the Anthropomorphic Principle - the universe is the way it is because it produced people capable of seeing it. Let us, for a moment, assume the Many Worlds theory of Quantum Mechanics is correct. Then there are universes OTHER than the one we see and the theory falls down. The same would be true if the model of a multiverse as a foam (where each universe is a bubble in that foam) is correct.

But if you're on this site, you should be familiar with layering anyway. Maths - the fundamental, overarching thing that is shown in all mathematical systems that exist, will exist or ever have existed - is a Layer 1 concept in the OSI model. Concepts like numbers and other fundamental but artificial building blocks are Layer 2, which makes Group Theory a layer 2 switch. Anything and everything that MUST be true because of something in layer 2 is arguably also layer 2, which would include Goedel's Theorum. Anything that is true only in a specific implementation of mathematics is layer 3 or above.

Does using an OSI representation make it easier to see how not all maths is the same?

Re:Math cannot exist before wind.

[jd]

My argument is that it is quite immaterial as to whether Pi is universal or not. In any given specific space, there will be -a- constant that denotes the ratio between the circumference and the diameter. The fact that there exists a constant for a given space (whether or not there exists the same constant for all spaces) means that the property of the ratio is fundamental.

Iff* the same constant holds for all spaces, then Pi as we know it is -also- fundamental, but I am unsure this has been proven. My statement that we can split off what is artificial from what is fundamental is unaffected.

*Maths notation: If and only if

The only way it can be proven that mathematics is wholly artificial is to prove that the set of all mathematical "things" that are fundamental is equal to the empty set. ie: there is nothing - not a single property, not a single result - that is true everywhere, including Goedel's Theorum. If even something as simple as Goedel's Theorum is universal, then there exists at least one part of mathematics that is not invented but is wholly natural.

Now, here we run into a problem. If Goedel's Theorum is not a universal result, but an artifice, then it is also false because it would have to be possible to create a counter-example and the theory states no counter-example of this kind can exist.

Surely that seals the argument right there and then. Those who argue mathematics is wholly artificial must be arguing Goedel's Theorum is false. All other cases do not prohibit the theorum from being true. Thus, if there is sound reason for believing the theorum true, there is sound reason for excluding the notion that mathematics is an artifice.

Re:Math cannot exist before wind.

[jd]

Ideal circles do not exist, that is true. So what? The idea that you need an ideal form is Platonic (it comes from Plato's cave analogy). Does there need to be some ideal, in order for an approximation to exist? (Well, C++ and Smalltalk programmers can skip that question.)

Let's try a different example. Let's go for the Second Law of Thermodynamics. Statistically speaking, it's universally true. There are no exceptions on the macro scale of space/time. If you were to examine a small patch of quantum foam over a few picoseconds, it would be lousy even as an approximation.

Does this mean the Second Law is wrong? No, not really. Does this mean the Second Law is artificial, as it's only an approximation? No, I think you'll find the early universe obeyed it long before there were any observers.

So what does it mean? It really doesn't mean very much at all. It means you're asking the wrong questions and not getting useful answers.

As for the quotes, Albert Einstein was very bad at maths and I think Benoit Mandelbrot (amongst a few thousand other chaos and fractal specialists) would beg to differ on the ineffectiveness of mathematics in biology. It's hardly the mathematics fault that biologists are lousy mathematicians.

Re:Math cannot exist before wind.

[masterzora]

And then, upon playing with them for long enough, we discover that they actually do model the world in some manner or another (and then we are disappointed). It's certainly all the evidence (though, not proof) I need that maths is discovered, not invented.

--an algebraic number theorist in training

Re:Math cannot exist before wind.

[hclewk]

First of all "as old as the wind" is just an expression means "really fricken old". It's obviously not meant to be taken literally, so get off your high horse.

Secondly,

If it is discovered, the solution already exists and the problem was solved before wind existed

Just because a solution exists, does not mean you have solved the problem. Think of it this way. You are looking through your telescope at night up at the stars and you notice a new star you have never seen before. You look at all the star-charts you can find and realize that no one has ever documented this star. You've just discovered it.

But you are saying that you did not discover the star, since the star already existed. Of course the solutions already exist for these math problems. However, discovery is the act of documenting an observation (ie, someone has to say "this is the answer"), so while they exist, no one has yet discovered them.

Re:Math cannot exist before wind.

[Anonymous Coward]

I wonder how many AFRICANS will be solving these problems...

How about none?

What is your children's country going to be like when 50% of the population are non-white? How about 80% of the population non-white?

Re:Math cannot exist before wind.

[Boronx]

Mathematics is a rigorous way to describe the relationships between things. In so far as physics is describable, there's no great mystery that a branch of mathematics arises to describe it, so that we may more precisely and unambiguously understand what happens in the universe.

But of course, you're wrong about accurate models. What physics now tells us is that we can very accurately model exactly how the universe cannot be modeled mathematically.

Re:Math cannot exist before wind.

[Sique]

Solutions to problems can also be invented in Mathematics.

Take the real numbers. There are several ways to describe their "continuity", or the idea that any infinite decimal fraction always denotes a well-defined real number.
For instance we have Bolzano-Weierstrass, we have Dirichlet, we have Cauchy. All try to tackle the same problem, and from an axiomatic point of view they are equivalent: Given one, we can prove the others.
The fact that all those approaches are equivalent, was surely discovered. But the approaches themselves are invented.

Re:Math cannot exist before wind.

[Anonymous Coward]

Axioms are invented. Theorems are discovered.

Re:Math cannot exist before wind.

[Boronx]

Didn't the concept of numbers and their properties arise out of observed reality? Is it so mysterious that even the airiest of results would be reflect in the reality from which it was derived?

Re:Math cannot exist before wind.

[masterzora]

Quite mysterious, actually. The fact that the quaternions can reflect part of the real world is mind-boggling. Then you start looking at structures that don't even involve numbers at all....

Re:Math cannot exist before wind.

[daveime]

In Soviet Russia, joke gets you !

Re:Math cannot exist before wind.

[Boronx]

That wiki has a whiff of the tiger who said "It's well I'm named such, as I'm so fierce." Arithmetic, Geometry, Analysis, arose from careful observation of the universe. It's not really a mystery that they are well applied to the universe.

Re:Math cannot exist before wind.

[daveime]

1287 seems to work.

One of the most useful things I ever learnt in school was problems involving "casting out 9s", and it struck me immediately the pattern you used ... 1 + 8 = 9, 2 + 7 = 9

Re:Math cannot exist before wind.

[Boronx]

Why so mind boggling? Quaternions are a straight forward construct from simpler ideas that are well formed representations of reality.

Re:Math cannot exist before wind.

[ignavus]

The fact that mathematics, even very abstract mathematics, accurately models the natural world is a deep mystery.

Of course, you have to pick your maths to find one that models the world.

When I add boxes of apples, they don't happen to conform to modulo 2 arithmetic. Why? 3D space doesn't conform to Euclidean geometry, even though for hundreds of years people thought it did. Why?

There are different arithmetics and geometries. Maybe I could design a world which uses a different arithmetic, or a different geometry to our world. Why does our world make the mathematical choices that it does?

What mathematics the real world uses can only be known empirically.

Re:Math cannot exist before wind.

[bytesex]

The former are Socratists, the latter are Aristotelians. The first group figures there must be an original, perfect form of everything; a blueprint, a divine thing. The second group are just nerds.

Re:Math cannot exist before wind.

[Bromskloss]

Mathematical tools, however, are invented and not discovered.

Indeed. Pen and paper come to mind.

Re:Math cannot exist before wind.

[Anonymous Coward]

???
Imagine a world with absolutely no order. Life cannot exist in such a world as far as I know. If it can, it would still be quite uninteresting to "study" the world because the human mind will soon realize there is no order. So there is some order. This order can always be abstracted into mathematics, or maybe it cannot. Imagine none of it can, then we again would not study it anymore, cause it is just random. Imagine some of it can, then we study this part and ignore the rest in western sciences (remember, occams razor and all that, the spiritual ideas that cannot be put into math are neither explained clearly in words nor accepted in western physics). All this is basic logic as far as I can see. We see no non-mathematical things because we choose to ignore them.

Re:Math cannot exist before wind.

[Anonymous Coward]

Did you mean the opposite? Was it not the Socratean method which was based on questioning?

Re:Math cannot exist before wind.

[mario_grgic]

Ok, I only have masters in pure math from one of the world's most respected universities when it comes to math, and solid background in physics and computer science among other things :D.

Re:Math cannot exist before wind.

[mario_grgic]

Like I said in another post, I'm a pure mathie (masters in it), and minor in CS as well.

Re:Math cannot exist before wind.

[mario_grgic]

I don't think you actually understand math very deeply. Quaternions don't represent or reflect any part of the world, they are just a tool, and found their way into some physical applications that have something to do with the real world.

Just like complex numbers can be used to do analytical geometry (they are a pair of real numbers just like plain coordinates).

Same goes for vector spaces, or non commutative von Neuman algebras, or any infinite dimensional vector spaces.

Re:Math cannot exist before wind.

[lxs]

Or a liberal arts major.

Re:Math cannot exist before wind.

[Anonymous Coward]

Many problems have no discovered solution and are therefore classified as an open problem.

Re:Math cannot exist before wind.

[mcgrew]

Even math is discovered, math problems could only be around as long as there are people for whom it's a problem. Math may have been around since the big bang (indeed, perhaps the universe is nothing but a mathematical construct), but math problems haven't.

Re:Math cannot exist before wind.

[bytesex]

Linkie. Socrates questions his victims, but only in a subjective way, that is, only about their opinions and perceptions. His view of the world below the veneer of Hellenic city-state culture was based on the cave of shadows.

Re:Math cannot exist before wind.

[mcgrew]

First of all "as old as the wind" is just an expression means "really fricken old".

This is a technical site. If you mean "relly frickin old" say "really frickin old" or better yet "since mankind discovered numeration".

Re:Math cannot exist before wind.

[Anonymous Coward]

It's not a deep mystery at all. It's by design. We chose our axioms to resemble what we observe. Just take Euclid axioms of the plane geometry, or natural number axioms (like Peano's), etc.

It's no wonder then that what we construct out of them resembles what is out there (at least somewhat). Other modeling problems try to be even stricter and model the observed phenomena even closer than axioms could ever hope to describe.

How about the surprising applications of complex analysis in Quantum Mechanics? Or Minkowski space in SR? Or Differential Geometry in GR? Group Theory in the standard model? etc, etc. All of these mathematical fields were discovered before , not together, with a physical application. And all of these are not naturally or at least trivially derived from the macroscopical world. Obviously there is a deep connection at work.The interesting thing is to discover what it is. This is one of the most fascinating puzzles we face. Some people speculate about the fact that our universe is a mathematical construction ( http://arxiv.org/abs/0704.0646 ) others say that cellular automata are the underpin of all our physical laws. Another more outlandish propose is that we live in a computer simulation. And I grant you a fourth one, that a sufficient intelligent brain constructed following certain physical laws has the power to inevitably create a formal system which may represent such laws.

Re:Math cannot exist before wind.

[patrickthbold]

"Exponential decay curve" and "irrational numbers" are two different concepts. (1/2)^N is an exponential decay curve -- which defines the half-life of a radioactive substance. For no integer value of N is the result "irrational".

You are insisting on integer values of N. Which is silly. There's nothing special about integer amounts of time. (1/2)^t is irrational for almost all values of t. Furthermore if you were to write down the half life of any radioactive substance in almost any units of time, you would get irrational numbers for their half life.

Re:Math cannot exist before wind.

[anarchyboy]

What is the difference between a mathematical concept representing or reflecting something physical and a mathematical concept that is used as a tool in a physical model?

Seems to me if you're using them in your physical theory then they have as much of a physical role (I'm not saying that they have to correspond to some observable quantity though) as say a symmetry group or any other mathematical object you use.

Re:Math cannot exist before wind.

[The Slashdot 8Ball]

and further, (1/2)^t is defined by e:

We can only make sense of a^p in the naive "multiply a by itself p times" for rational p. If p = r/q for integers r and q then a^p can be defined as "the q-th root of a^r".

However, this naive approach doesn't allow us to make sense of irrational powers (which clearly we need if we want to define (1/2)^t for t in some interval) Consequently we define a^p := exp((p*log(a))) where exp is the exponential function (defined as the power series of e^x). In fact, this is why e^x is called the exponential function as it allows us to define exponents (powers) for all numbers.

This definition makes sense for all p (rational, irrational or complex) and provides the same result as the naive method for rational p.

So, as the parent claims, we need a decent understanding of irrational numbers before we can understand exponential decay.

Re:Math cannot exist before wind.

[The Slashdot 8Ball]

(1/2)^3 = 1/8
(1/2)^4 = 1/16
A third of the way between 1/16 and 1/8 is 1/16 plus a third of their difference
1/16 + 1/3*[1/8-1/16] = 1/16 + (1/3)*(1/16)=1/12 which is rational

Re:Math cannot exist before wind.

[cheftw]

Well maybe is you're stuck on SI time. Personally I calculate with Planck time and so it's always an integer. :p

Exponential decay, by definition uses e. (Or vice-versa, the hint is the "e")

You are absolutely right (IMHO) in your point though.

Re:Math cannot exist before wind.

[Anonymous Coward]

Nah, the boundary between discovery and invention is a human skull.

Re:Math cannot exist before wind.

[Rasperin]

I support our new joke overlords in ruining this thread.

Re:Math cannot exist before wind.

[dkf]

The only way it can be proven that mathematics is wholly artificial is to prove that the set of all mathematical "things" that are fundamental is equal to the empty set. ie: there is nothing - not a single property, not a single result - that is true everywhere, including Goedel's Theorum. If even something as simple as Goedel's Theorum is universal, then there exists at least one part of mathematics that is not invented but is wholly natural.

Since the only real constraint that Goedel's Theorem imposes is that there is not a finite set of axioms that can characterize all "sufficiently interesting" mathematics (i.e., that any finite axiomatization is necessarily incomplete) I don't see where you're going with that. It's a construct all the same and no amount of philosophical bullshit will change that.

Now, here we run into a problem. If Goedel's Theorum is not a universal result, but an artifice, then it is also false because it would have to be possible to create a counter-example and the theory states no counter-example of this kind can exist.

The problem is that you're into the space of self-referential mathematics when you're using Goedel's results (they're a necessary part of it, which is where the "sufficiently interesting" really comes from) so your argument just doesn't work too well. In particular, there most certainly is mathematics possible in systems that do not support the expression of Goedel's Theorem, but it's pretty dull stuff (no integers, for example). You can build on that base in various ways, but once you've got self-referentiality then you can get something equivalent to Goedel, and you're stuck. Or you can plunge on, by adding more rules and axioms which will either let you say more (but not everything) or render the whole edifice bollocks. But not one bit of this says anything about whether it is natural or not.

Surely that seals the argument right there and then. Those who argue mathematics is wholly artificial must be arguing Goedel's Theorum is false. All other cases do not prohibit the theorum from being true. Thus, if there is sound reason for believing the theorum true, there is sound reason for excluding the notion that mathematics is an artifice.

Your argument is full of bollocks. Goedel's Theorem is a consequence of a particular level of complexity of a formal system, and once it holds, you've got to a stage where you know there must be truths about the system that cannot be derived from a finite set of axioms about the system. But such a system can be pure artifice (e.g. by moving symbols around according to clearly stated rules, which is obviously non-real) and if one such system can be non-real, you've not proved that your favorite one ("mathematics") is real on that basis either.

Note that I'm not trying to demonstrate that mathematics is artificial. I'm just pointing out that you've not shown that it is not. (I actually suspect this is a matter for philosophers and not mathematicians since the observable effect is the same in either case.)

Re:Math cannot exist before wind.

[northstarlarry]

I think Benoit Mandelbrot (amongst a few thousand other chaos and fractal specialists) would beg to differ on the ineffectiveness of mathematics in biology

That's because they're mathematicians, not biologists. Stuart Kaufmann thought he was replacing natural selection with self-organization back in the 80's, but he's calmed down now, after arguing with actual biologists for a while. Tom Ray is a guy with a foot in both camps, or at least he used to (I guess now he's doing cog. sci. work?), and even though his Artificial Life projects pretty much inaugurated the field, he doesn't seem to really think of his research as revealing anything fundamentally new.

It's hardly the mathematics fault that biologists are lousy mathematicians.

Well, that's true, but neither is it biology's fault that most of what mathematicians produce has no application. It's not like biologists are stupid; they don't learn knot theory because it doesn't improve their work, not because they are unable.

Re:Math cannot exist before wind.

[sjames]

The problems are intrinsic to a universe where the mathematics are what they are here. We just didn't know about them until we discovered mathematics.

Re:Math cannot exist before wind.

[sjames]

Mathematics can model anything at all that is a logical possibility. Physics is necessarily a subset of that.

Re:Math cannot exist before wind.

[xouumalperxe]

The real question is quite the opposite: How can mathematics exist as an area of study before people have a maths problem they want to solve?

Re:Math cannot exist before wind.

[PaladinAlpha]

Re: bubbles, that's rather what I was saying. Circles very close to true exist in nature. The spherical formation of bubbles itself is the result of a couple of very basic laws. Of course nature doesn't 'care' what pi is, but that doesn't change the value. What do you suppose the ratios of diameter to circumference were before there were observers? Putting a number to it didn't change anything; that's almost the point of mathematics, to describe things in a way that doesn't rely upon them.

Re: flat space, it's true that the locally measured value changes with the curvature of space, but even in the absence of flat space, with access to only a few points of curvature, it would become obvious what the value in an undisturbed medium would be.

Math is simply the observation of what is already there, and what has always been there. No one 'invented' pi -- it's just and simply the measuring of a value that's been there all along. All we did was put a name to it.

Re:Math cannot exist before wind.

[PaladinAlpha]

So you can't 'solve' a riddle if anyone else has ever solved it? Can you not 'solve' a puzzle if anyone else has ever solved it?

Re:Math cannot exist before wind.

[illeism]

That's the way the cookie crumbles...

Re:Math cannot exist before wind.

[wwfarch]

I agree with your point. I was just making a joke that to discover the answer I did no solving, I just searched for the answer.

Re:Math cannot exist before wind.

[DMUTPeregrine]

If nature produces a singularity it will be a perfect sphere (if stationary) or torus (if spinning). The fact that it is infinitely small doesn't detract from it being as close to a perfect circle as is possible.

Re:Math cannot exist before wind.

[CannonballHead]

It seems to me that calling it a mystery implies you're not really sure why math can accurately model the natural world.

So, if you can explain it, it wouldn't be a mystery.

It seems that many scientists and mathematicians were confident that it could accurately model the natural world for one reason: it was "designed."

Basically, what I'm getting at is this: if your worldview includes a Creator/Designer/whatever of some sort (i.e., God) who created/designed and/or sustains the "natural" world, then it is actually not very surprising or mysterious that it is logical and consistent and able to be accurately modeled.

If your worldview tends toward thinking there was no design aside from what "design" evolution (by chance) "gives" it, then it would seem that accurately modeling the natural world would indeed by a mystery, as it ... does not entirely make sense. Laws do not really make "sense" unless there is an authoritative entity that put those laws in place... that entity could be God, a social construct, a human being, etc. But "laws of nature" don't really make sense if there is nothing actually consistent behind nature.

There's the long-winded version. :)

Re:Math cannot exist before wind.

[CannonballHead]

Since natural world seems to be relatively well-behaved, it might be expected that objects which can be well-defined accurately model the natural world.

That doesn't get at the mystery, though. The mystery isn't really that "math can model well-behaved systems." The mystery is that the natural world appears to be well-behaved, as you put it. Nobody argues - that I've heard, anyway - that math can't be applied to a consistent and logical system... but why the natural world is consistent and logical can be quite a mystery, depending on how you think it came about, "who" is behind it, why it still exists, etc.

Lest anyone think I'm simply arguing for "religion," I would like to mention that the Greeks and Romans were quite religious - and very pagan - and had some rather ridiculous theories about why things happened that tied into their religion. I'm certainly not arguing simply "for religion." I am actually trying to argue rather neutrally and simply pose the question, while having very distinct beliefs about these things. :)

Re:Math cannot exist before wind.

[hawkfish]

Let's try a different example. Let's go for the Second Law of Thermodynamics. Statistically speaking, it's universally true. There are no exceptions on the macro scale of space/time.

Except that the Classical Second Law requires definitions that are based on a particular world view. The way I partition phase space into "useful" and "non-useful" areas is really a definition provided by the observer, not a Platonic reality.

Moreover, the Classical Second Law does not really exist - because the Classical universe is deterministic! You actually need Quantum Mechanics and notions of information transfer to make it work. See Von Neumann's "Mathematical Foundations of Quantum Mechanics" or Henry Stapp's "Mindful Universe" (for a summary).

Now, Statistics itself is an interesting example. Empiricism assumes Statistics, so one might be able to argue that it has some Platonic reality...

Re:Math cannot exist before wind.

[Anonymous Coward]

Baloney. We invented the concept of a circle, which does not exist with our perception of it as such. There is no way to divorce reality from our perception of it. (Think of relativity as a good example of this.) Therefore we invented all mathematics in order to allow our brains to comprehend the physical world in a way that is much more useful than our native perception.

Re:Math cannot exist before wind.

[Z1NG]

Math usually is a liberal arts major....

Re:Math cannot exist before wind.

[avagpingham]

yes, but the time in which a radioactive isotope is reduced by half is in terms of a decay constant divided by ln(2). Which I am pretty sure is irrational and involves e \snark. The only time you have that nice integer decay curve is if the decay constant is ln(2)....so I think JD's point stands.

Re:Math cannot exist before wind.

[brantondaveperson]

Pi is a universal constant in many more ways that it being the ratio of diameter to circumference of a perfect circle - which may not exist in the physical world. It may exist if one considers how forces are distributed in space, but I am not a physicist so I am not sure.

For an example of Pi cropping up without requiring perfect circles consider the fact that the probability of two randomly chosen integers being coprime is 6/pi^2. This quite extraordinary result is proven beyond doubt, and I think we may all be able to agree that nature in some sense 'understands' integers and the concept of primality also.

Re:Math cannot exist before wind.

[Carewolf]

What an interesting load of nonsense...

Here is brain quiz for you: Imagine a universe where fact A is always true everywhere, now imagine another universe where fact A is never true anywhere. Could those two universes be the same universe? If not, then all universes not matter how wierd are governed by logic, which means mathematics applies there as it does here, though the laws of physics may not.

Re:Math cannot exist before wind.

[Obfuscant]

What makes you think you're using 2?

Does the term "radioactive half life" mean anything to you?

Re:Math cannot exist before wind.

[Carewolf]

I think mathematics is so effective because in the realm of physics our discoveries have few degrees of freedom and can therefore be represented by simple rules. Since the rules must be consistent we have the basis for physics and a tie in to mathematics.

No, mathematics is so effective it is true. It would be true in any universe. No universe can exist which disproves its own existence, thus any universe needs to follow a few basic rules of logic, like A equals A implies A equals A. From those very basic rules you can reconstruct any path of logic, and most of what we call mathematics. Axioms are definitions of a common mathematical language not rules or laws. They just have to apply to be able to agree on what we do with the mathematical symbols. Math would still apply if you change every single axiom (they still can not contradict themselves though, otherwise they can't all be true), the math would look different, but the conclusion it draws from evidence would be the same.

Re:Math cannot exist before wind.

[Obfuscant]

What do you suppose the ratios of diameter to circumference were before there were observers?

Before the existance of an observer who could theorize a perfect circle, the "ratios of diameter to circumference" were approximately pi, because there was no "perfect circle" for the ratio to be "exactly pi always." ("Approximately pi" includes the possibility of an imperfect circle actually having pi as the ratio, but the class known as "circles" didn't all have that ratio.)

Putting a number to it didn't change anything;

The point is you couldn't put A number to it until you had IT. You could put various numbers to the members of the class "circle" as exist in nature, but not A number.

Re: flat space, it's true that the locally measured value changes with the curvature of space, but even in the absence of flat space, with access to only a few points of curvature, it would become obvious what the value in an undisturbed medium would be.

Really? The "perfect circle" and "pi" came about a very long time before the concept of curved space-time, so how obvious would it have been to those working on circles back then that they were dependent upon our being in a relatively flat section of space? They would have been as certain that "2" was the ratio and somehow magic had they been in a highly curved space as we are that pi is the ratio and is somehow magic today.

Math is simply the observation of what is already there, and what has always been there.

Oh, so there really are universes that you have observed (or someone has observed) where all lines are parallel and other universes where no lines are parallel? Sorry, Reiman is one of those two geometries but I forget which one and the name of the other one. They are parts of "maths" where one of Euclid's basic assumptions is replaced with a different one.

And someone actually has observed the other 7 dimensions of our theoretical 11-dimensional universe? And strings?

Ok, how about something as simple as a mathematical line? Someone has actually observed this theoretical construct?

I guess that's the long way of saying that no, math is not simply the observation of what is there. It is a description of things that may or may not have yet been seen.

No one 'invented' pi -- it's just and simply the measuring of a value that's been there all along...

Where has it been, since nature and even man cannot create a physical representation of a perfect circle? It has been in the mind of man, since "perfect circle", and even "prime", is a concept born there.

Re:Math cannot exist before wind.

[severoon]

Math is simply a system of reasoning...it, in and of itself, doesn't model or universe or anything in particular, though it can be a useful tool for expressing and creating such models.

Re:Math cannot exist before wind.

[Anonymous Coward]

Nobody argues - that I've heard, anyway - that math can't be applied to a consistent and logical system...

My point about math is that a "consistent and logical system" is necessarily a mathematical object. I don't think that we should say that math can be applied to it. It is math.

but why the natural world is consistent and logical can be quite a mystery

That's a different and very deep question. Trying to tackle that monster is beyond quixotic (IMHO).

Re:Math cannot exist before wind.

[Anonymous Coward]

> Imagine a universe where fact A is always true everywhere, now imagine another universe where fact A is never true anywhere. Could those two universes be the same universe?

Sure, and you could implement it. Just design a cellular automata world where cells can be either on, of, or a third state 'both' which yields true is queried for both on and off state. A is defined as "is this cell on?"

Ternary logic is crisp logic. http://en.wikipedia.org/wiki/Ternary_logic Nothing prevents you to implement a universe with fuzzy logic. Or no logic. In that universe our math would be as useful as astrology is useful to calculate missile trajectories in our.

Re:Math cannot exist before wind.

[lastgoodnickname]

hot cookie crumbles over Natalie Portman.

Re:Math cannot exist before wind.

[minstrelmike]

A problem cannot be solved before it actually exists. It's one of them there time travel paradoxes. That's why we continue to have problems in society and even problems in math and philosophy merely talking about how problems exist.

Re:Math cannot exist before wind.

[Carewolf]

My example is in binary logic. In ternary logic the same problem is just is A "on" or "not on", where "on" covers both on and both. The same basic logical problems exists no matter what logical language you use. That's the point! You can redefine the language but it still describes the same logical space, and all possible universes are describable in logic-space, because otherwise they would contradict themselves.

Re:Math cannot exist before wind.

[mestar]

What are you talking about? There are no circles in real world.

Re:Math cannot exist before wind.

[mestar]

How in the world did you get to this conclusion? This has nothing to do with nature. It only shows that in your idealized mathematical world, many results are pi, or use pi.

Nature understands integers? Thats a pretty big jump from your example.

Re:Math cannot exist before wind.

[Anonymous Coward]

Can't be bothered to login, but:

I think it may be fair that nature understands integers. Or do you not agree that there being 'one' or 'two' of something is a universal truth?

So if this is a truth, then so must primality be since it is a simple logical step from the existence of the integers.

Unless we are to claim that 1 and 2 and counting do not exist except in our own minds? Is this what you claim? There is one planet, there is one photon etc.

It is an interesting question and I'm obviously in no position to argue my position with sufficient erudition. But these things really do seem true to me:

1) Integers exist
2) Primality exists
3) Randomness exists
4) From these three things, the value of pi appears.

Therefore pi exists.

Re:Math cannot exist before wind.

[Evil Pete]

Yes, I understand what you are saying. But why then are there so many linear relationships in the physical world. Why does F=ma and not F=ma^1.3 or some weird relation between velocity and acceleration. Or a universe where the kinetic energy is proportional to the square of the velocity instead of linear ... oh wait. Never mind.

Why are these laws so simple? That is the central question to me. Perhaps they need to be for a life sustaining universe. And the Anthropic Principle raises its head ... is there a "Godwin Rule" for this?

Solve this

[Anonymous Coward]

Why is it that after taking some, the day after I always get a splitting headache?

Oh wait, math mysteries.

Re:Solve this

[dazjorz]

Don't worry, that works for math too, in a metaphorical way of "taking some" and for "the day after" you read "until the day after" ;-)

Re:Solve this

[Anonymous Coward]

Easy. That was methanol, not methamphetamine. Be glad you still can see. Next!

Check out the Collatz Conjecture...

[Oxford_Comma_Lover]

http://en.wikipedia.org/wiki/Collatz_conjecture Speaking of unsolved math mysteries, the 3n+1 problem is a fabulous way to spend days and days of your life. It's particularly fun if you think about it in binary. Whatever the answer is, it's either simple and elegant or complex beyond imagination.

Re:Check out the Collatz Conjecture...

[ae1294]

WHOA... Gotta love that little meme..

If the starting value n = 27 is chosen, the sequence, listed and graphed below, takes 111 steps, climbing to over 9,000 before descending to 1.

        { 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1 }

Re:Check out the Collatz Conjecture...

[Tynin]

Wish I hadn't posted in this discussion, I'd love to toss you an interesting mod. This will no doubt steal hours of my life, thanks :-)

Re:Check out the Collatz Conjecture...

[mister_playboy]

I noticed that too... LOL.

That conjecture is interesting and simple enough to be understandable without being a math person. I looked at the wiki article for the above mentioned Riemann Hypothesis, and that's a bit too complex for me.

Re:Check out the Collatz Conjecture...

[Shikaku]

This is something Ruby is DESIGNED for.

http://pastebin.com/m25fc9de4

I popped this out in a few minutes, but if it can be modified to save every valid Collatz number it finds and not recalculate anything at all it can go pretty fast for very little code and eat all your RAM in the process :)

Re:Check out the Collatz Conjecture...

[Shikaku]

Oh, I should mention that I mash very large random numbers into this ruby script and it doesn't overflow. Instead it gets a stack error...

So small update, nonrecursive edition:

http://pastebin.com/m29c38ed3

It worked fine for a 40+ digit whole number pasted about 20 times...

Re:Check out the Collatz Conjecture...

[ae1294]

That conjecture is interesting and simple enough to be understandable without being a math person. I looked at the wiki article for the above mentioned Riemann Hypothesis, and that's a bit too complex for me.

Yeah same here, not really a math person but you gotta love how simple it would be to write a program to play around with it. Who knows, if you picked the right starting number you might even prove it wrong! but I'm not really sure how one would ever be able to prove it right!

Re:Check out the Collatz Conjecture...

[ACS Solver]

I have fond memories of that one. On the subject of teaching and education...

One of my math teachers once showed me the problem. The teacher knew I'm decent at math and would occasionally show me interesting or unusual problems. The interesting part is, the teacher told me to have a try at proving the proposition of this problem, without telling me that it's an unsolved problem. So I had a good amount of fun trying to prove this. Of course, it's not like I could make a proof with my high school knowledge, but it challenged my mind and was a fun thing to do. And had the teacher told me right away that it's an unsolved problem, I wouldn't have had the motivation to think about it, knowing beforehand that I wouldn't be able to find a proof.

That was one of my educational highlights, though. Way to provide a mental challenge!

I'm still amazed by how part of the problem's beauty is that it's easy to understand the actual proposition. That isn't true for most unsolved problems, after all. Take the recently proven Poincaré conjecture, just understanding what it states takes some math knowledge, though it has a nice approximation in layman's terms. As for the example of the Hodge conjecture, I probably don't know half the mathematical concepts required to understand the problem.

Re:Check out the Collatz Conjecture...

[Undead NDR]

"Very little code"? Bah! Kids these days...

This will run on any system where `dc` is installed.

Re:Check out the Collatz Conjecture...

[TheVelvetFlamebait]

Whatever the answer is, it's either simple and elegant or complex beyond imagination.

Actually, if you believe this guy, it's not only complex beyond imagination, it's complex beyond any possible finite representation, that is, it's unprovable.

Re:Check out the Collatz Conjecture...

[SanguineV]

A python version that returns the cycle length is available here. Of course it can be optimised by storing known concluding cycles and terminating immediately if you hit one. But the code works for stupidly large numbers without any issues (can't paste example as the filter complains)

Re:Check out the Collatz Conjecture...

[Anonymous Coward]

CAF is a notorious usenet troll. You can safely ignore anything he writes. (Also note: arxiv is not peer reviewed.)

Re:Check out the Collatz Conjecture...

[Anonymous Coward]

Anyone can say a number that dont work for 3n+3? (ending with 3, not 1)

Re:Check out the Collatz Conjecture...

[hazem]

If you like life-stealers like that, you might want to check out Project Euler: http://projecteuler.net/

But don't say I didn't warn you!

Re:Check out the Collatz Conjecture...

[Shikaku]

http://pastebin.com/m67281bd6 :D I have nothing better to do. Optimized to save known values and not recalculate them.

Re:Check out the Collatz Conjecture...

[Anonymous Coward]

So, what you're effectively saying is that ruby is _not_ designed for this kind of stuff ?

the recursive version should run just fine with no stack overflow. That's called tail recursion: http://en.wikipedia.org/wiki/Tail_recursion

Re:Check out the Collatz Conjecture...

[Anonymous Coward]

What's the difference in 3n+1 and 5n+1? Why not just say any odd number * n + 1????
All we're doing is turning an odd number into an even. Even n + 1 would work right? 3->4->2->1 5->6->3 7->8->4 9->10->5 etc. etc.
What am I missing here?

Re:Check out the Collatz Conjecture...

[Tiggan]

Anyone can say a number that dont work for 3n+3? (ending with 3, not 1)

3 -> 12 -> 6 -> 3
5 -> 18 -> 9 -> 30 -> 15 -> 48 -> 24 -> 12
7 -> 24 -> 12
9 -> 30 -> 15 -> 48 -> 24 -> 12
11 -> 36 -> 18 -> 9
13 -> 42 -> 21 -> 66 -> 33 -> 102 -> 51 -> 156 -> 78 -> 39 -> 120 -> 60 -> 30
15 -> 48
17 -> 54 -> 27 -> 84 -> 42
19 -> 60
21
Can you find me one that does work? I haven't done even numbers because they are either a power of 2 or they eventually halve down to an odd number. Powers of two always work.

I have this proof.

[140Mandak262Jamuna]

I have this wonderful proof for this conjecture, but unfortunately the 80 char limit for sig in slashdot is too small for it.

Re:I have this proof.

[JustOK]

try twitter, it goes up to and includes 140 chars

Re:I have this proof.

[Anonymous Coward]

HA! I was just thinking that.

Hey wait... HOW DARE YOU STEAL MY COMMENT! I had a scathing reply to your post all thought up, but unfortunately the 80 char limit for sig in slashdot is too small for it.

Re:I have this proof.

[interkin3tic]

I have this wonderful proof for this conjecture, but unfortunately the 80 char limit for sig in slashdot is too small for it.

And thus was born the famous "140Mandak262Jamuna's Last Theorem" which was not fully proven until 2367 AD.

Re:I have this proof.

[selven]

Use a real website, char limits are stu

Re:I have this proof.

[Anonymous Coward]

I proved that there are infinite primes in a twitter comment for a friend doing her homework:

Assume some set of primes is all of them. Multiply them all. Add 1. None of primes go into new #; its factors are additional primes

Re:I have this proof.

[Evil Pete]

But the attention span doesn't change.

Re:I have this proof.

[Anonymous Coward]

god dammit fermat, just give us a hint!

Strange point

[2.7182]

Julius Shaneson and Sylvain Cappell claimed to have solve a famous problem about counting the lattice points in a circle. It's been out for years, even earlier than this arxiv paper:

http://arxiv.org/abs/math/0702613

Thing is, even though it is a famous problem, no one cares enough to check. So this notion of "famous" is shaky.

Re:Strange point

[Feminist-Mom]

It's even worse than that. The problem of counting lattice points is closely related to the Riemann Hypothesis, the "most" important unsolved math problem. Clearly that is what Shaneson and Cappell are after. I've looked at the paper, and it is only 40 pages (compare with the 200+ of Wiles work), and these guys are respected mathematicians. No one has said it is wrong. I don't know the area, but it shouldn't be as hard to check as the Wiles paper. Maybe people are waiting to see if they announce a proof of the Riemann-Hypothesis.

Re:Strange point

[Anonymous Coward]

Nobody bothers to check it because it's published on Arxiv. In the math community, Arxiv is basically a very detailed blog. Say anything you want, and some people will read it and maybe be interested. But nobody will really take you seriously. After all, there's tons of flat-out wrong papers on Arxiv and no form of quality control whatsoever.

There are many, many peer-reviewed journals. If this paper is good, it should be published in one of those. The fact that it's not raises doubts about its quality.

Re:Strange point

[Anonymous Coward]

Indeed. Arxiv is a pre-print archive. I was about to post about how it's not fair to judge it just because it's in Arxiv when the peer review and publication process can take some time, but then I thought I'd better check the date of the paper, which meant I had to rewrite this post. If it had merit it would have been published by now.

Re:Strange point

[Anonymous Coward]

The preprint first appeared less than three years ago, and as ridiculous as it may seem, some journals do take that long or more to publish papers. The Annals of Mathematics, for example, can take several years between the decision to accept and the final publication, and since many journals can take a year or more to referee a paper (especially one with as much detailed computation as this one) before that decision is even made it's not impossible to believe that this paper is silently working its way toward publication as we speak.

As for the comment above this, the math community most certainly does *not* view the arXiv as a blog. Most papers are put there before they're submitted to journals, so that they can be freely and quickly accessed, and from respected mathematicians like Cappell and Shaneson it's expected that the papers are worth reading and correct. People do read papers on the arXiv regularly and take them very seriously -- it's the only way to stay up to date in certain fast-moving areas of math -- and if a mistake is found and the authors aren't cranks, they'll either post a new version correcting it or retract the paper completely. Since nothing of the sort has happened with this paper, and nobody has pointed out any mistakes, it's more likely that the paper is correct and just stuck in the middle of a slow editorial process.

Re:Strange point

[Anonymous Coward]

But this work was circulating around in the late 90's. They only put it up in 2007. No one has wanted to read it or listen to them for 12 years or so.

Re:Strange point

[Anonymous Coward]

That is a little unusual, but it happens sometimes. For example, Thurston's paper "A norm on the homology of 3-manifolds" was circulated for most of a decade before anyone bothered to get it published, but it was both interesting and correct and in the meantime Gabai used it extensively in his seminal thesis on sutured manifolds.

I'm not an analytic number theorist so I don't know what the experts think of the Cappell-Shaneson paper, and maybe nobody does want to read it, but they claim in the arXiv version that it's a significant rewrite of an older paper and they name half a dozen mathematicians who have read the current version carefully, so without talking to experts I can't just dismiss their work.

Nice Move

[Anonymous Coward]

So they finally discovered the Merits Of The Bug Tracker.

But i hope they don't switch to math 2.0 anytime soon, that'll just introduce a bunch of regressions and won't do anyone any good. They've already spent some thousands of years just to get their project management straight, it's about time they delivered.

The Millennium Prize Problems

[lazy_nihilist]

They can probably add the remaining unsolved millennium prize problems to the list.

1 + 1 = 3

[Anonymous Coward]

For very large values of 1

Re:1 + 1 = 3

[JustOK]

or small values of 3

Re:1 + 1 = 3

[lastgoodnickname]

or normal uses of "+" and "="

Re:1 + 1 = 3

[laejoh]

And with a margin that's large enough to contain these words!

Sadly...

[cosm]

their servers will explode when they take a stab at Navier-Stokes. I asked Wolfram-Alpha, but it simply returned the exact solution of a degenerate case, the solution being 'Fuck you.'

Re:Sadly...

[TapeCutter]

You need one of these to get a good aproximation.

Re:Sadly...

[DoninIN]

Can you imagine a Beowulf cluster of these?

Re:Sadly...

[Anonymous Coward]

But does it run Linux?

If you really want to run into trouble

[JoshuaZ]

See http://rjlipton.wordpress.com/2009/11/12/more-on-mathematical-diseases/ for unsolved problems which are all really simple and also really addicting to think about. For many of these, the best way to stop thinking about one of them is to start thinking about another.

I'm actually a bit puzzled as to why this is a Slashdot article. If I wanted to point to something new in the way people are doing math I'd point to Math Overflow http://mathoverflow.net/ where many professional mathematicians, grad students and others are active. It is essentially a centralized system for people to post math questions and get math answers from people who know. It is very cool. It also is highly addictive to read.

Re:If you really want to run into trouble

[tonycheese]

The site you gave seems to be like a WikiAnswers for math problems. The site this article is talking about is more of a collaboration tool on unsolved problems that will allow any and all known progress be open to be expanded upon.

The issue I see with this is how do you dish out credit for something like this? If 20 different people solve varying amounts of a problem and then one last person pieces everything together, who solved the problem? Or will we move away from "Smith's Proof" into "Proof of May 2009" by a list of 90 people? I always thought that part of the fun of solving impossible math problems was to take the glory at the end of the day... but, then again, I'm not a mathematician.

Massively collaborative "Polymath" efforts

[Anonymous Coward]

As of about a year ago, a new kind of collaborative math project known as "polymath" is emerging. These research projects are completely open for any interested scholar to drop in and make contributions to the problem at hand. The technical infrastructure is based on well-known tools such as wikis and forum discussions

The very first such project successfully explored a new approach to the density Hales-Jewett thorem--a significant problem in combinatorics--in about six weeks of effort, with a fully preserved record of about a thousand contributions from dozens of participants.

See Polymath Wiki for the details. This new contribution from the AIM will provide a focus point for such efforts and encourage similar massively collaborative projects.

And of course, the emerging field of computer-verified mathematics is also dependent on massive collaboration, in order to translate existing results into a fully-formalized form that computes will understand and verify as correct. A wiki-based project could be a great help there as well.

why not hide them in video games so we can

[Joe The Dragon]

why not hide them in video games so we can get more people to look at them.

Re:why not hide them in video games so we can

[robinesque]

Higher math crowd-sourcing. I like this idea.

Re:why not hide them in video games so we can

[Anonymous Coward]

Because this isn't Stargate: University

Re:why not hide them in video games so we can

[Inda]

Excellent idea!

---

You enter a dark room. Inside the room is a large door with the words "Entrance to the second level" scratched in the paintwork. Below the door handle is a riddle.

"Extend the Kronecker-Weber theorem on abelian extensions of the rational numbers to any base number field."

Re:why not hide them in video games so we can

[Anonymous Coward]

STARGATE IS NOT REAL! While its true many SciFi shows, movies, and books can have real scientific insight and value, I'm sorry but that show does not. Particularly the SGU ones that you apparently have watched. Its more Science Fantasy than Science Fiction.

Encyclopedia of Integer Sequences

[Rockoon]

Only (very) loosely related but deserving mention is the Encyclopedia of Integer Sequences.

This encyclopedia has proven very useful for me in that I have avoided 'solving' many problems with it.

These problems are easily solved...

[jkiller]

maybe (AIM) should ask Smarterchild for the answers.

Meh.

[jd]

Mathematically modelling the brain would seem to be a very trivial problem. The problem is that there's a lot of brain to model. I've posted (admittedly non-rigorous) mathematical models of the brain on Slashdot before, but narry a grant check from it. Bah.

Computational fluid dynamics for foams, liquid crystals, et al, isn't any harder than for anything else. The equations are chaotic by nature, but chaotic systems can be well-behaved on aggregate under certain conditions. CFD as generally done relies on some specifically hand-picked special case or cases being universally true. They never are, which is why most CFD differs from how systems actually behave in practice.

If you were to treat CFD as a problem in chaos theory, rather than as isolated collections of imperfect examples of special cases, there would be no problem. It is always when engineers try to take shortcuts and oversimplify the maths to make it easy on themselves that they run into problems. They should be locked up for their own safety. If you want to really annoy them, lock them up with some airgel foam.

The problem with chaotic systems is that the system is sensitive to initial conditions, which means the only way to get "correct" results is to use infinite precision and zero step sizes. This isn't useful, but is a good way to annoy experts in CFD.

This leaves two options - use very very big, very very fast computers (the option used by F1 teams), or find an equivalent problem you CAN solve (the idea behind transforms).

Ok, does chaos look like a good place to use transforms? If you could identify and classify the Strange Attractors in the system, can you do anything useful? Probably not, at least not in the "solving the problem" sense. Chaos is fully deterministic, but it is utterly unpredictable. The only solution is the whole solution.

What knowing the Strange Attractors might tell you is how to vary the precision and step-size to get the best results for a given level of compute power. But it's going to be all raw horsepower from thereon out.

The best way to invest money on such work is to design a co-processor that performs a handful of fairly high-level maths functions directly (optimized purely for speed, not physical or logical space) so that you can do Navier-Stokes almost at the level of raw hardware rather than through clunky software. Then cluster the living daylights out of the co-processor.

It's necessary to optimize commodity hardware for space, because chip real-estate is expensive. However, if you're building what is basically a SOP (single-operation processor) for a dedicated market that can afford things like Earth Simulator, the only time you care about space is when it impacts speed.

Ideally, if the speed of light wasn't an issue, you'd want each bit in the output to be produced by wholly independent logic, duplicating the input bits as necessary to accomplish this. In practice, you'd probably want to start with that conceptually but in reality have something that was somewhere between that and a highly compressed form. Too parallel and the delays in communication exceed the benefits from the parallelization.

But this is all obvious. Anyone here who has done multi-threading or any other form of parallelization knows about synchronization issues and communication overheads. It's even one of the biggest chunks of any course on the subject of parallel design. There's nothing new there, certainly nothing "unsolved".

But, yeah, a well-designed Navier-Stokes co-processor would likely give you greater accuracy and greater performance than the modern pure software solutions. Especially those using ugly protocols to do the communications.

If Intel can conceptulalize 80 Pentium 4 cores on a wafer, it would seem reasonable enough to imagine modern fabrication methods being able to put at least a couple of hundred dedicated Navier-Stokes processors into the same space. Since the input for an iteration would be based on output from that and other processors, there's no

Re:Meh.

[NewbieProgrammerMan]

<asshole>

Mathematically modelling the brain would seem to be a very trivial problem. The problem is that there's a lot of brain to model. I've posted (admittedly non-rigorous) mathematical models of the brain on Slashdot before, but narry a grant check from it. Bah.

You must be either one of the greatest geniuses of all time, or uninformed on the topic of neurological modeling. People doing real research generally tend not to waste their time trolling Slashdot to find insightful theories, so you might want to try to get it published in a journal instead.

If you were to treat CFD as a problem in chaos theory, rather than as isolated collections of imperfect examples of special cases, there would be no problem. It is always when engineers try to take shortcuts and oversimplify the maths to make it easy on themselves that they run into problems.

What specific treatment, pray tell, would suffice as a "one algorithm fits all" approach to solving Navier-Stokes (let alone when mixed with extra behavior like crystal growth, chemical/thermal diffusion, etc.)? If it works, you should code it up, sell it, and make yourself rich.

The reason people "oversimplify" Navier-Stokes--by using reasonable assumptions in the context of the problem at hand--is because it often lets them determine analytically how the system will behave. If they *do* have to resort to numerics, making those assumptions greatly reduces the computing power required, so you don't need some high-powered (or highly customized) machine.

</asshole>

Re:Meh.

[Tomfrh]

Mathematically modelling the brain would seem to be a very trivial problem.

Yours perhaps...

Re:Meh.

[jd]

People doing useful and interesting research frequently post on Slashdot, so I don't see what your problem is. It doesn't take a genius to mathematically model a brain and that isn't something people have bothered much with doing.

Some things people have tried to do are build models of compartments of the brain (bad idea), simulations of some poorly-specified upper-level functions of the brain (even worse idea) and discrete/binary simulations of individual neurons assuming them to be stateless and/or with a rigid topology (talk about dense).

The first is like trying to build a model of one part of a Mandelbrot set. A complete waste of time, since the maths doesn't work that way. The second is stupid because without a good specification, there's nothing meaningful to simulate. And since neurons are neither discrete, stateless nor in a fixed network (even adult brains have a surprisingly dynamic topology), all you get is a simulation of something that never existed instead of a simulation of the thing you want.

Why do people do these things? Because they're very doable. Neural networks are a doddle to code up, logic chains and decision trees are trivial on a computer, and since more people are interested in medical applications than AI, understanding compartments is far more practical than understanding the brain itself.

In short, people want to be paid far more than they want to discover, especially since discovering the mathematics of the brain won't do you any good as it'll be well outside the capacity of any machine out there (including the 100 million core one) to do anything sensible with such a model. Nobody likes inventing things that can't be used for another 50-100 years.

However, the fact that nobody WANTS a real mathematical model of the brain doesn't change the fact that the brain is an extremely simple device (mathematically-speaking). The unwritten part of the challenge is that they want a mathematical representation they can use and it is that which does not exist and will not exist for at least the next 50 years, simply because of the technology. The maths is a non-issue.

As far as Navier-Stokes is concerned, there are no reasonable assumptions. Particles do not move with a uniform speed, speed follows a bell curve. Well, almost, as a bell curve has infinite tails in both directions but in physics you're bounded. Particles are strictly between 0 and C and cannot take on either value or anything outside of the range.

In practice, since you don't see too many Bose-Einstein Condensates or even hypersonic particles when boiling water for an egg. However, even in a pan of cold water, there'll be water molecules moving fast enough to leave the liquid, and even when the water is boiling, there'll be water molecules that have the kinetic energy of a slug. Not many, but there will be some.

That's your first problem, because the first simplification is to decide what sort of range of speeds particles are likely to move at. The reality is "all of them, at some point or another".

The second problem is this differentiation between compressible and non-compressible fluids. In the same way that speed is non-uniform, density is also non-uniform. That means all fluids will have a mix of the two characteristics.

The third problem, as I've already pointed out, is that the system is chaotic. This means you need an infinitely fine grid and an infinitesimal time interval between iterations. Neither of these is possible. However, chaotic systems don't necessarily improve as you improve resolution, which is why CFD is often far more coarse-grained than you might expect. It has nothing to do with context, or even compute power, it has to do with experimentally finding a resolution where the results are similar (through the property of self-similarity) to what you might get if you could work at infinite resolution.

Self-similarity is NOT the same as identical, though, which is why most competent hardware engineers treat CFD as being a first approximation at

Re:Meh.

[dkf]

Mathematically modelling the brain would seem to be a very trivial problem. The problem is that there's a lot of brain to model. I've posted (admittedly non-rigorous) mathematical models of the brain on Slashdot before, but narry a grant check from it. Bah.

You sound like a pure mathematician. (You know what I mean: "a solution has been shown to exist, so it is trivial".)

The problem is that the brain is a non-linear system on many scales, and it's not clear that the nature of the non-linearity is the same at all scales. This makes even approximate modeling rather difficult. And there's a lot of detail, and a lot of different scales. Right now, it's easier to let poets and psychologists write the higher-level models than to derive them either numerically or analytically...

Re:Meh.

[jd]

Oh, it's practically a given that the non-linearity won't be the same at all scales. But complex behaviour can be produced by very simple non-linear systems - the Mandelbrot Set being the best-known example, so the presence of non-linearity merely creates a problem of practical computability rather than a problem of mathematical computability.

(Remember, to be computable in the mathematical sense, the algorithm has to complete in finite time. Which can mean anywhere from a few picoseconds to an hour after the heat-death of the Universe. To be practical, though, the model must produce results within the time the results are useful. To be commercially practical, it also has to produce results faster than other methods of getting those results.)

So we're looking at nested non-linear systems, no matter what starting point we're using.

Let's start with a bottom-up approach.

In the biological world, each cell has multiple mechanisms running in parallel where each mechanism is non-linear. The cell itself is a non-linear construct of these. There are different types of interconnect and these are also non-linear, so any network of cells is a non-linear construction of non-linear components. The brain has topological constraints, but unless there's grounds for believing those constraints to fundamentally alter the maths, the maths should be independent of implementation details.

This says we're looking at a nesting 3 deep. So we're looking at a chaotic system in which potentially all of the parameters are themselves chaotic systems in which potentially all of the parameters of that are also chaotic systems.

What else do we know? We know that the lowest-level systems are fundamentally unchanged from how they were 3.5 billion years ago when cellular life first arose. They may be chaotic but the building-blocks are all very simple. The only real internal changes have been in the organization of the building-blocks. All other changes within cells deal with interactions and mathematically interactions are on a different level.

Most of those lowest-level systems are common to heart cells, skin cells and brain cells. Now, this will include communication mechanisms and those we DO have to consider. Basic housekeeping that is a product only of it being biological can be ignored. Systems specifically activated in neurons and NOT common across all cells also have to be considered, even if housekeeping, as state is persistent in neurons by means of such housekeeping.

Now, the mechanics of these functions aren't what's important. What's important is what they do to the logic of a neuron to make it capable of data processing.

The cell itself is a network of these. In standard computer network terms, you're looking at the equivalent of a multicast-capable routing-capable ad-hoc network of moderate size. This is just for a single neuron, we're not even up to networking these things. Actually, strictly speaking, it's multiple such networks. In biological cells, you've independent chemical and electrical paths. Different latencies and different bandwidths.

Unless there is firm evidence that this is an implementation detail that does not alter the specification, I believe that it is wisest to assume it DOES alter the specification, that signal delays and other signal characteristics are important. Some variables from iteration X of the system are fed into iteration X+1, but others are fed into iteration X+N (where N can't be guaranteed to be a constant). This is what makes it a chaotic system of chaotic systems rather than merely a bigger chaotic system.

Now, the network of cells is basically more multi-path networking where again different types of interconnect have different properties. Further, not only are the nodes in the network effectively mobile and multicast, but the number of nodes is variable.

(We can ignore the number of connections a given neuron has by looking at the superset of functions exhibited by all types of cell in the brain, whether neuron, axion, or wha

sp-called experts steal work

[gr8_phk]

The problem list provides a snapshot of the current state of research in a particular research area, letting experts track new developments

So when a promising idea comes along, the "expert" can follow up and hopefully get credit for the solution. I see this is the workplace and on the net in various places. Technical discussion forums are lurked by "experts" in industry who look for ideas without contributing anything to the discussion. Some people don't mind, others don't realize, and others are bothered by it.

Re:sp-called experts steal work

[Anonymous Coward]

ideas are a dime a dozen

you have ideas. why aren't you rich?

Hilbert problems

[aws4y]

I am pretty sure that some of the problems at least will be Hilbert Problems that do not currently have a solution. http://en.wikipedia.org/wiki/Hilbert_problems

Re:Hilbert problems

[Sique]

There is only one Hilbert Problem which still doesn't have a solution, and that's the Riemann Hypothesis.

Re:Hilbert problems

[Anonymous Coward]

That's not what the Wikipedia page GP linked says. Feel free to update it with references to the solutions of the others.

This is a great idea.

[Phantasmagoria]

This is a great idea. It whould promote more interest in the specific problems and unsolved math problems in general. Besides, more collaboration should result in better research.

Yes.

[gbutler69]

There exist no empty sets in the Universe! We can construct all of Mathematics beginning only with the Empty set. So, all of mathematics can be constructed from something that does not exist in the real world. Hmm? Makes you think.

Re:Yes.

[brantondaveperson]

We can construct all of Mathematics beginning only with the Empty set

I'm afraid that we can't - we have to take some of it on faith and that's just the end of it. Something to do with Goedel.

Christmas gifts,look,nike air max jordan shoes,

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Disambiguation reveals the simple answer

[Brain-Fu]

Those things which we use mathematics to describe (relationships of every variety) are discovered (by observation and experience)

The language with which we describe them (symbols, axioms, and rules of transformation) is invented (and refined over time, as a quick review of the history of mathematics will promptly reveal)

Additional products of this language (logical consequences of the axioms we have invented) are subsequently discovered.

We equivocate the term "Mathematics" to mean all three of these things (that described, the language of description, and logical consequences of the axioms of that language). When the word means all three of these things at once, it seems that we have both discovered and invented it, and lively (though misguided) debate ensues.

When we establish clarity about our topic of discussion (through disambiguation of our terms), then whether it was invented or discovered becomes clear, as I have just demonstrated.

Re:Disambiguation reveals the simple answer

[samwhite_y]

Let us suppose for a moment that mathematics is invented and not discovered. Just how far do you want to push this.

Was the concept of 0 and 1 discovered or invented? You can argue that because of quantum mechanics and the probability that there is a nonzero (though immeasurably small) chance that any particle (or group of particles) could exist at any point in space and time, that the idea of 0 and 1 cannot truly be represented in nature (if you tried to show me 0 blocks and 1 block, quantum mechanics would say that there is a small probability that my 0 block might actually be a small fraction of a block -- of course the odds against this are ridiculously small, but that is not the point). So even for something this simple, you can claim that mathematics is only a model that is not truly represented in nature.

The existence of 0 and 1 is an "axiom" in mathematics (for set theorists, they usually describe this as the existence of Set with nothing -- the nothing is 0 and 1 is the Set that contains nothing). It is not provable, but it does not mean that it cannot assume it to be true and work from there.

I want to find anybody rational that believes that there can be "intelligence" of any reasonable complexity and sophistication that does not intuitively understand the difference between the absence of something and something. I want to go further and say that this idea was discovered and not invented. The symbols and notations that we use to represent this idea were invented but the underlying idea exists and is true even if all of existence were to vanish and nothing ever existed anywhere.

Once you believe in 0 and 1, there is a nicely built up sequence of logic that will lead you to circles and PI. Some of it requires advance graduate mathematics to fully understand, but there is an unavoidable discoverable chain of logic. For example, the existence of numbers following 1. This is one of the Peano axioms. Again you have to assume it is true, but nothing breaks down in logic if you do and you work from there. Again, in Set theory, which is one way to build up the axiomatic foundations of mathematics, if 1 is the Set that contains nothing, then 2 is the Set that contains the Set that contains nothing. Once you believe in positive integers as being discovered and not invented, then the rest of the big construct called mathematics followed and was "discovered" just as much as 0 and 1 were "discovered".

HHGTTG

[Anonymous Coward]

42

can we get the good will hunting guy to work on th

[Joe The Dragon]

can we get the good will hunting guy to work on them or rainman. may even Kazan?

Why 3n?

[snowwrestler]

Why is the alternative to halving, 3n+1? Why 3? I'm curious. If it were just n+1 it seems like it would converge to 1 pretty quickly (since most non-even numbers become even if you add 1).

10 gives you:
10 5 6 3 4 2 1

100 gives you:
100 50 25 26 13 14 7 8 4 2 1

What if it were 4n+1? Then 10 gives you:
10 5 21 85 341 1365 5461 21845 uh oh

What if it were 5n+1? Then 10 gives you:
10 5 26 13 76 38 19 96 48 24 12 6 3 16 8 4 2 1

Re:Why 3n?

[Tejin]

4n+1 doesn't work because 4n gives an even number, then adding 1 makes it odd. I assume this holds for any even integer.

.999...

[Anonymous Coward]

Maybe they can explain why most math "teachers" think 1 = .999...

Retards.

Re:.999...

[kipling]

I think the more interesting question is why people would ever think that 1 is not equal to 0.999...?
I mean, what do they expect to find in between?
(Disclaimer: I am one of them Math "teachers", I suppose)

Re:.999...

[Anonymous Coward]

1 = .555...

Try it out in binary. It's represented as .111...

x=.111...
10x=1.111... (10 = 2 in decimal)
10x-x = x
1.111...-.111... = 1

Hence .555... = 1 Wow, magic!!

You CAN'T represent 1/3 in DECIMAL!! .333... does NOT equal 1/3
Go back to first grade. Divide 1 by 3. You alway have a remainder of 1 with a divisor of 3.. i.e. 1/3.

1/3 + 1/3 +1/3 = 1

.333...+.333...+.333... = .999...
You can't add infinity. Infinity is a concept not a #.
.333... or .999... is a concept. You can't add .333... and .333... and get .666...

Re:.999...

[rayharris]

Let's try this...

Start with:
a = 0.999...

Multiply both sides by 10 to get:
10a = 9.999...

Subtract the first equation from the second:
10a - a = 9.999... - 0.999...
9a = 9
a = 1

Substituting back into the original equation:
a = 0.999...
1 = 0.999...

Who's the retard?

Re:.999...

[quadrox]

My immediate response to that would be that 9.999... != 10 x 0.999... . I'm not sure exactly how to explain it though, other than "you just added 9, you didn't really multiply by 10 because it is "impossible" to do so due to 0.999... having an infinite amount of decimals."

I'm not a math teacher though, so what do I know.

Re:.999...

[Anonymous Coward]

Let x = sum(n=1..inf) 1/10^n = 1/10 + 1/100 + 1/1000 + ... = 0.111...
Thus 9x = 9 * sum(n=1..inf) 1/10^n = sum(n=1..inf) 9/10^n = 9/10 + 9/100 + 9/1000 + ... = 0.999...
and 10x = 10 * sum(n=1..inf) 1/10^n = 10/10 + 10/100 + 10/1000 + 10/10000 + ... = 1 + (1/10 + 1/100 + 1/1000 + ...) = 1.111... = 1 + x.

In other words: what's 10x - x? Answer: (1 + x) - x = 1. Therefore 9x = 1, so x = 1/9, and 9 * (1/9) = 0.999... = 1.
Alternately, 10x = 10x = 9x + x = 0.999... + x, yet above we see that 10x = 1 + x, so 9x = 0.999... = 1, also.

Re:.999...

[Lisandro]

This is a common misconception. 0.9999... is EXACTLY the same as 1, they're two representations for the same number. This can be proved in a lot of ways, the most basic being that if 1/3 = 0.33333..., then 3 * 1/3 = 1 = 0.999... Another way is, if these two numbers are not equal, what's the distance (difference) between them? Can you find it?

More info here

Re:.999...

[rayharris]

The problem is your failure to understand infinity.

You can easily see that 0.123 * 10 is 1.23, right?

Well, just because I multiplied by 10 and brought one of the 9's over to the left side of the decimal, there's still an infinite number of nines on the right side.

Even though I "borrowed" a 9, infinity - 1 is still infinity.

Re:.999...

[2obvious4u]

Semantics != Mathematics

There is a lot between 1 and .999 if your frame of reference is 1 second and the value you are measuring is the speed of light.

If you are measuring the length of a shoelace in inches then 1 inch and .999 inches might as well be the same thing.

Really if you just change the frame of reference and name your units something so that .999 is irrelevant then there for all practical purposes is no difference. However if your scale is wrong then those repeating decimals make a big difference.

Re:.999...

[Anonymous Coward]

The problem is you can't add and sting of infinitely long #'s.

.333 repaint for ever does not = 1/3. You can't ever convert 1/3 into decimal form. We pretned we can and then then we make 1/3 EQUAL to .333 with a line over it. Well, of course if you add 3 of those representations you get 1. 1/3 + 1/3 + 1/3 =equals one.

Where your little .999... equals one falls apart is EVERY OTHER # system.

If you go to the 9s number system 1/3 = .3 and in 9s .3 + .3 + .3 = 1!

Try doing you LITTLE MAGIC show of .999 in HEX and guess what .999.. does not equal 1, but you can do you magic on .fff... which looks like it equals one.

But you said .999.. equals one, which is 9/10th +90/100 = 900/1000th.

But in hex you have .fff... which is 15/16th + ......

Where it really starts to fall apart is in # systems that have less room/precision.

Start doing your "fancy math" in BINARY.

.555... = 1
.666... = 1
.777... = 1
.888... = 1
.999... = 1
.aaa... = 1


Does .555... really equal one because that little trick works in BINARY!?!

Hell, no, it means you plain CAN'T represent .888... (i.e. 8/9) perfectly in binary.

So same with .999... go to a # system with more precision like the 11s, 12,s etc and you can then represent .999...


Other wise you are just doing divison from 3rd grad. 3 goes in to 1.0000 , you get .3 with a remainder of !!!! 1/3!!! You're back were you started.

So 1/3 in decimal is really .333r1/3 . That r1/3 is there wherever you stop.

.333r1/3 + .333r1/3 + .333r1/3 = .999r9/9 which of course is 1..

The fact that you can't realize just putting a line over it means r1/3 is sad. Of course the real representation of 9 repeating after the . litteraly just means keep moving (adding) 9/10th closer than the last power of ten.

You can't take an infinitly long # like .333... and do math to it. It's impossible. If you pretend it is .333r1/3 then you can and .333... +.333.. repeating is really .333r1/3 +.333r1/3 = .666r2/3.

Try it. In decimal take 2 and divide by three. You always will have 2/3 left. You can't magically get rid of it by a line.

Try to cut a 1/3 of a pie. Now try to cut .3333333.... of a pie. You can't because they are not the same thing.

Re:.999...

[Anonymous Coward]

10a - a = 9.999... - 0.999...

There's your problem. What makes you think you can take an infinitely long number and ADD or SUBTRACT it to another infinitely long number?

Please work out the whole addition sequence for me till you reach the end.

Thanks,
Reality

Re:.999...

[rayharris]

You claim that 0.555... = 1?

Let's do my "fancy math" and see what it equals...

a = 0.555...

10a = 5.555...

10a - a = 5.555... - 0.555...

9a = 5
a = 5/9

Convert 5/9 to a decimal and you get 0.555...

So your arguement that 0.555... = 1 is wrong.

Once again, you fail to understand the nature of infinity.

Re:.999...

[rayharris]

10a - a = 9.999... - 0.999...

There's your problem. What makes you think you can take an infinitely long number and ADD or SUBTRACT it to another infinitely long number?

http://en.wikipedia.org/wiki/0.999
http://www.straightdope.com/columns/read/2459/an-infinite-question-why-doesnt-999-1
Algebra
Calculus
et alii ad infinitum...

Re:.999...

[Anonymous Coward]

Try thinking for yourself instead of post other people crap.

2nd did you RTFA you posted from Wikipedia!?! It shows all the "Mystical" ways people try to show .999~ = 1 and then GOES ON to show IT DOESN'T!!

Since you didn't read it I'll quote it for you,

Breaking subtraction
Another manner in which the proofs undermined is if 1 - 0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include commutative semigroups, commutative monoids and semirings. Richman considers two such systems, designed so that 0.999... < 1.

And look at the second article, it states the same false crap, 1/3 = .333~ no, 1/3 does not equal .333~

You can "say" it does and then add three together and say 1!!!

That's like saying, I'll let 1/3 = 4. So, 4 + 4 + 4 = 1 1/3 should not be represented by 4 anymore than it should be represented by .333~

Cut a pie. You can cut 1/3 of the pie slice. Now cut a pie in .333~ You never make a cut, because you you move 3/10 or the way around the pie from where you start, then you start to slice, OH WAIT, you need to move 3/100 further, now you get ready to cut again, OOOPS wait, you need to move 3/1000 closer now, get ready to cut, HOLY CRAP you need to move again.

This is like using the 9s number system. 1/3 = .3 In the nines number system I can cut .3 or 1/3 of a pie. But gues what!?! There is no way to represent 1/4 in the 9s number system. 1/4 in 9s is .2222~ You can't represent 1/4 in 9s number system. It would be absurd to try. Just like it is absurd to try to represent 1/3 in Decimal. 3 WILL NEVER divided 10 or any power of 10, ever. Then just putting a line over it and going well, .333~ is "close enough" lets multiply by 3 and get 1.

Re:.999...

[Anonymous Coward]

In said convertered to BINARY ! .5 = 5/10th = 1/2
In Binary that is .1

So in binary .111~

If I do you magic math IN BINARY. .111~ = 1



Binary, Trinary, etc this all happens. You can do it with any # system. Each one you get closer to 1. So if in decimal .999~ = 1 then in binary .111~ equals one.

This happens in every number system. Go to hex and .fff~ will equal one with your flawed math.

Again this is because you can't represent certain numbers in certain # systems. 1/3 != .333333~ There is no way to represent 1/3 in Decimal. If you go to the 9s number system, 1/3 is represented very well, .3

What you're doing by saying .999~ is really tricking yourself. Look again at 1/3. Dived 1.000 by 3. You always have a remainder 1/3. That is one third of the last value you stopped at.

So for example. 3 into 1.0, I get .3 with a remainder of 1/3 of a tenth. Now instead of just saying it's 1/3 of a tenth left over we stick a bar over the 3. Problem is when we add .3 with a bar over it we are in essence adding the 1/3. Hench .3 with the 1/3 of a thenth remainder. So you when you add .333~ plus .333~ what you are doing is getting .666~ r2/3 Now the bar = r2/3

Hence when you do this with: 1/9 you get .1111 r1/9 2/9 you get .2222 r2/9

.3 r1/3 + .3 r1/3 + .3 r1/3 = .9 r3/3 , that's 3/3 of a tenth which is one whole tenth so it's .9 + r.1 so =1

You are just treating the remainder as r3/3 or r9/9

So x = .9 r9/9
10x = 9 r9/9
10x - 9x = 9 r9/9 - .9 r9/9 === all you are doing is taking 10-1.

So you are letting the line over it be a whole. This different then saying .999999 to infinity. You are literaly saying keeping going closer, but NEVER touch one. Then to say it touched one contradicts what you meant it to mean.

.999~ is being molested into rounding your fraction off.

This happens in every # system. Except it the amount of error is worse the small the # system. Like binary, where .1111~ = 1 or in trinary .2222~ = 1

The same thing happens when you use number systems with MORE precision than 10s, like HEX, you get .ddd which approaches one much faster than .999 in decimal. However, in HEX at .fff~ you do the same thing if you are trying to add it you are really adding .fff rf/f .

Basically just sticking a line over something that can't be converted is not logically. I can't write 1/3 in decimal. .333~ is not 1/3.

Re:.999...

[masterzora]

I said I'd never reply to an AC on this topic, but this one is being particularly bad, so I have to do it.

Yes, .111~ in base 2 is 1. And .222~ in base 3 is 1. Incidentally, .111~ in base 2 = .222~ in base 3 = .999~ in base 10. There is no difference in precision, and it's silly to even suggest. It indicates that you are not thinking in infinity, and that you believe an infinitely repeating decimal has an end point.

First, note that .111~ in base 2 and .222~ in base 3 and .999~ in base 10 and all other similar cases share a similar sum: $\sum_{n=1}^\inf (r-1)(\frac{1}{r})^n$. This is a geometric sum starting with n=1, so we can quickly find the sum to be $\frac{(r-1)}{1-\frac{1}{r}} - (r-1) = \frac{(r-1)}{\frac{r-1}{r}} - (r-1) = r - (r-1) = 1$, so, yes, this "trick" does work in every base.

Your problem is that you are thinking of the number .999~ as a process, when it's really a set number with an infinite number of decimal places, just like every other decimal number (the fact that some numbers have most of their fractional digits be 0 doesn't mean they don't exist or that you can ignore those digits).

Of course, the multiply-by-10-and-subtract proof is just a quick and easy-to-understand proof. I much prefer using some concepts from real analysis, but they would clearly go over your head.

Re:.999...

[masterzora]

2nd did you RTFA you posted from Wikipedia!?! It shows all the "Mystical" ways people try to show .999~ = 1 and then GOES ON to show IT DOESN'T!!

Actually, it shows that it is true, and then goes on to show constructions people specifically to make it not true. In fact, your "breaking subtraction" quote specifically states that it was part of a construction intended to make .999~

And look at the second article, it states the same false crap, 1/3 = .333~ no, 1/3 does not equal .333~ You can "say" it does and then add three together and say 1!!! That's like saying, I'll let 1/3 = 4. So, 4 + 4 + 4 = 1 1/3 should not be represented by 4 anymore than it should be represented by .333~

Except that the equivalence 1/3 = .333~ is provable and 1/3 = 4 isn't (in our usual system). Other than that huge fact, yeah, they're exactly the same.

Cut a pie. You can cut 1/3 of the pie slice. Now cut a pie in .333~ You never make a cut, because you you move 3/10 or the way around the pie from where you start, then you start to slice, OH WAIT, you need to move 3/100 further, now you get ready to cut again, OOOPS wait, you need to move 3/1000 closer now, get ready to cut, HOLY CRAP you need to move again.

Except numbers don't work that way. The number .999~ is a specific number, one which we can define in terms of a process or sequence if we wish, but it is not that process or sequence itself. If I want to cut .333~ of a pie, I'll make my first cut and then move .333~ of the way around the pie and slice. Incidentally, if I start from the same 0 point, this slice will be identical to the one I'd make if I went 1/3 of the way around the pie.

Really, the only way I'd agree with you that 1/3 != .333~ in decimal is if you complained that 1/3 is integer division and that 1/3 is really 0, but then we're just being silly.

Strange Attractors *could* be useful

[S3D]

If you could identify and classify the Strange Attractors in the system, can you do anything useful? Probably not, at least not in the "solving the problem" sense. Chaos is fully deterministic, but it is utterly unpredictable. The only solution is the whole solution.

That is not entirely correct.
First strange attractor is usually embedded into lower-dimension manifold. Reducing dimensionality can make problem a lot more tractable, especially if original system was infinitely-dimensional (like Navier-Stocks). Estimation of dimensionality of that manifold or any other information about it can help a lot in numerical simulation.
Second strange attractors have non-trivial statistical properties and specific geometry. Exact solutions is not always the target of calculation, statistical properties or averages of solutions, or its geometrical properties could be target of calculations too.
That is probably more potential that practice for now. While I've read paper (long ago) about statistical (ergodic theory) and algebraic-geometrical approach to Lorentz attractor I don't know if there are any works on dimensionality or statistical properties on attractors of other "real world" systems.

here is one

[nerdyalien]

1^0 = 1
1^1 = 1

hence 1^0 = 1^1

taking log to the base 1 from both sides,

0 = 1

by extending to 1^2, 1^3 etc...

0 = 1 = 2 = 3 .....

Re:here is one

[rayharris]

The base of a log can be neither 0 nor 1. That's like dividing both side of an equation by zero to "prove" that 1 = 2.

Re:here is one

[Anonymous Coward]

You kinda forgot that log(1) = 0, and that log(a^b) = b*log(a), so log(1^k) = k * log(1) = k * 0 = 0.

So you just tired to say for all numbers k, 0 * k = 0; therefore k = 0/0; therefore all k are equal.

Fail.

Here's mine...

[rayharris]

P = NP?

I so want it to be true. Quantum computing is our best hope right now of shedding light on this problem.

And it's not on their list...

Re:Strange Attractors *could* be useful

[jd]

Ok, I would agree with all that. (I can't mod you informative for obvious and non-chaotic reasons.)

galois theory

[Anonymous Coward]

galois theory can be used to show that some constructions do not exist. for example, it is impossible to trisect an angle in general by using a straight edge and compass in a finite number of operations (although 180degrees can be trisected by constructing three 60deg divisions). it can also be used to show that there are fifth order polynomials that have root that cannot be expressed using a finite number of the operations addition, subtraction, division, multiplication and taking powers/radicals. these roots are real (or complex numbers) that cannot be constructed using the conventional mathematical operators a finite number of times.

similarly you cannot construct a 7 sided polygon. its quite interesting to think about the nature of these truths. that you can show that something is impossible, is not constructive. when you have a constructive proof it is easy to see when the answer is correct. but to verify these proofs is more subtle and requires considerable thought and time. you could expect to look at galois theory towards the end of a degree in mathematics, but how many people can dedicate the time and effort required. how can they trust that these assertions are correct. how we enable trust in such non obvious truths.

A math problem I would love to see solved...

[Shawn Way PE]

is why there never is enough money in my account... --- Shawn Way...

Math + Chicks

[toilettext]

I always respected my engineering friends in college who were busy solving complicated problems. I was a humanities major, so I was far more interested in solving this math problem with the hot chicks in my social sciences classes: let's add ourselves together, subtract our clothes, divide our legs and multiply.

Too Easy

[SleazyRidr]

P   = NP
P/P = NP/P
1   = N

Any time now

[PPH]

High school students begin posting math homework problems there.