most complex scientific formula you can think of

[gib65]

I need a really complex scientific forumla. I'm writing a paper and I'm using math as an analogy for physical determinism. I need something really complex to demonstrate how complexity has no baring on making things more "random" (i.e. the math is just as determined whether it's simple or complicated). So what's the most complicated formula from some field of science you can think of?

[woelen]

What do you mean with complex? A complex formula can be complex in the sense of a lot of symbols. It can also be complex in the sense of the underlying concepts.

 

These are equations from electromagnetism, telling something about electromagnetic waves propagate through space and how the two phenomena are related to each other. This may seem very complex to you, but why are they complex? Probably because of the unfamiliarity with the math involved.

 

http://en.wikipedia.org/wiki/Maxwell's_equations

 

The same equations can be summarized in a simpler formula (thanks to Atheist ):

 

[math]\square^2A_a = -\mu_0J_a[/math]

 

This equation seems much simpler, but it describes the same equations as those in the Wiki-page, but using a more advanced mathematical language, which allows one to express things with fewer symbols.

 

It is not important for you to know the meaning of these formulas, but these sets of formulas nicely demonstrate that the concept of complexity is not easily defined. The more advanced the language, in which things can be expressed, the more compact things can be expressed. Things which are overly complex, when described in one language, can look much simpler (and easier to memorize) in a more advanced language. Of course, this is at a cost. One needs to learn (and understand) the more advanced language. Complexity is shifted from a practical level to a meta-level.

[Klaynos]

Well I did a fourier series of a 3 slit repeating pattern the other day, where the coefficients where several lines long for solving fields in a grating. But none of it was complicated, just long...

[swansont]

I think your thesis is flawed. As woelen points out, complexity hasn't been properly defined, and some things are not deterministic (as you appear to have used the term)

[the tree]

The Navier-Stokes Equations are long and have lots of bits to them, they could easily be mistaken for complex.

[gib65]

Yes, by "complex" I mean lots of symbols.

 

...and some things are not deterministic (as you appear to have used the term)

 

I am aware of the non-determinicy of quantum mechanics' date=' but I'm talking about classical mechanics. The formuli used, if they are indeed scientific formuli, are used to predict outcomes of natural phenomena. If they are successful, the phenomena in question must be just as deterministic as the formuli that describe them.

 

Well I did a fourier series of a 3 slit repeating pattern the other day, where the coefficients where several lines long for solving fields in a grating. But none of it was complicated, just long...

 

Great! I'd very much like to see it.

[swansont]

I am aware of the non-determinicy of quantum mechanics, but I'm talking about classical mechanics. The formuli used, if they are indeed scientific formuli, are used to predict outcomes of natural phenomena. If they are successful, the phenomena in question must be just as deterministic as the formuli that describe them.

 

But they aren't, necessarily, for highly nonlinear systems, because of limitations of how well you can know the initial conditions. Chaos theory.

[YT2095]

IMO, although I don`t know How to express it mathematicaly, the most complex is: A=NOT A

 

a bit like a NOT Gate (inverter) where the output is fed back to the input.

what State is A at?

[timo]

IMO, although I don`t know How to express it mathematicaly, the most complex is: A=NOT A

 

a bit like a NOT Gate (inverter) where the output is fed back to the input.

what State is A at?

 

1/sqrt(2) |true> + 1/sqrt(2) |false>

You were talking about quantum computers, were you?

 

EDIT: I should note that I was kinda joking, there. It´s a nice feature that in quantum computers, you can find a complete set of eigenstates for every operator, in this very case even one with an eigenvalue of +1. For completeness: The other eigenstate would be 1/sqrt(2) |true> - 1/sqrt(2) |false> which would be the eigenstate to the eigenvalue -1.

[raivo]

Formulas that are used to calculate exact positions of planets and moon are very long.

 

You could also write equations for the 10 simple particles that can collide and change directions and speeds by simple laws . Suppose they are in the box and each of them has different initial speed and direction. If you want to know what are exact positions of particles after some time when, say, 1000 collisions have occured then you will get very long formula.

[YT2095]

1/sqrt(2) |true> + 1/sqrt(2) |false>

You were talking about quantum computers, were you?

 

not specificly no, but if they share this attribute also, then yes, them too

 

it`s a bit like the old A says B Always tells lies without fail, B says A is correct.

 

in actual Practice, with an RC and a gate, it`s a great way to make an Oscilator

[woelen]

This type of circuits, as described by YT is called "time dalay oscillator". Any physical gate has a certain time delay between input and output, hence the terminology. This kind of oscillators is related to the class of oscillators, called "phase shift oscillators". The latter are common in analogue electronics, where an n-th order linear circuit is fed back. The dynamics of such feedback circuits, btw. also can be described by complicated formulae with lots of terms (but I would not call that complex, I would only call such formulae tedious and error prone ).

[YT2095]

agreed in Practice yes

 

but where A and NOT A are directly connected Without propogation delay or Phase shifting, the problem becomes somewhat less easy to answer (if at all it Can be).

 

Idealy you would assume A=NOT A would be something like A=A/2 + NOT A/2, giving them a 50/50 state each making a whole of 100.

but since Half states cannot exist it`s either 1 or 0 (never .5) what is the state of A?

[gib65]

But they aren't, necessarily, for highly nonlinear systems, because of limitations of how well you can know the initial conditions. Chaos theory.

 

No, not necessarily, but in highly nonlinear systems, our lack of knowledge is just that - a lack of knowledge. One thing that I do touch upon in my paper is the difference between what I call "ontological determinism" and "epistemic determinism". Ontological determinism simply means that a system in a given state S is determined, independently of anyone's knowledge, to acquire a specific future state S'. Epistemic determinism refers to our ability to know the future state S' given full and accurate knowledge of state S.

 

Obviously, if nonlinear systems are not epistemicly determined, we can't ever know whether or not they are ontologically determined. But here, I should say that I'm not trying to prove they are in my paper. I'm simply explaining what a devout determinist would believe - that everything is ontologically determined, linear or not, epistemicly or not. I'm saying that a determinists thinks of mathematics as the perfect description of nature.

[YT2095]

Gib65, do you mean Complex in it`s implcations, or just some really long winded calculation?

 

My example is simplicity itself (at face value) A=NOT A, and yet the state of A is impossible to determine.

 

I`m not entirely sure what you`re after?

[woelen]

This equation A = NOT A remains a simple equation. Whether this has any solution or not depends on the set in which A may be. If A only may be in the set {0, 1}, then this equation has no solution. You could of course extend the set in which A may exist. Simply by adding a special element with the property Λ = NOT Λ, you can make this equation solvable. Now the set of possible values is extended and it contains at least {0, 1, Λ} and you can solve all kinds of logical equations, with NOT's in it. It however, requires you to introduce a new symbol Λ. You could also make rules for what is 1 AND Λ, 1 OR Λ, 0 AND Λ, 1 AND Λ, Λ AND Λ, etc. In this way, a new "boolean" algebra is created and one can solve all older solvable equations, plus a whole bunch of new equations. Just play around with it and you can grasp the idea.

 

 

Now a more interesting, but still very basic, example:

 

Suppose we have a set {0, 1, 2, 3, 4}, with operators ┼, ●. The ┼ is addition, but the result always brought back to {0, 1, 2, 3, 4} by going modulo 5. E.g. 1 ┼ 4 = 0, 4 ┼ 3 = 2. The ● is multiplication, but also modulo 5, e.g. 3●4 = 2, 2●2=4, 2●3=1.

Now suppose we want to solve the equation x●x = 4. This has two solutions, being 2 and 3 (3●3 is 9 modulo 5, which is 4). Now suppose we have the equation x●x = 2 and x must be in the set {0, 1, 2, 3, 4}. This equation has no solution. Again, we can introduce a new symbol , with the abstract property Λ●Λ = 2. Now we extend the solution space to the set {0,1,2,3,4,Λ}. Now we created a new number system and with the operators ┼, ● we can have expressions like 2┼Λ, Λ●4, etc. By introducing this new symbol and keeping it as a purely abstract entity, with the property Λ●Λ=2, we add a lot of new arithmetic to our simple system and we allow the solution of all equations of the form x●x=b, with b any number from {0, 1, 2, 3, 4}. E.g. 2●Λ is a solution for x●x= 3. (2●Λ●2●Λ = 4●Λ●Λ = 4●2 = 3), the other solution for this is 3●Λ. The original solution space has 5 elements, the extended solution space has 25 elements.

 

YT's example of extended boolean algebra has an extended solution space with 4 elements, 0, 1, Λ, 1+Λ, with the appropriate definition of "+".

 

So, what does this short digression say. YT's example is not anything about complexity, it is just a matter of selecting the solution space in which the equation can be solved. The technique of extending a solution space is common in mathematics. Many equations, which cannot be solved in a certain solution space can be solved in an extension of that solution space. There are quite some problems in mathematics, physics and cryptology that can be solved in this way, which otherwise could not be solved.

 

There is a whole branch of mathematics, devoted to this subject, which I just touched upon. I have tried to explain this in layman's terms and it is a fascinating branch of mathematics. It is the theory of algebras (not to be confused with what high school pupils call 'algebra'). Again, here you see a nice example of simple equations, but with a more "complex" language, allowing us to express things, which cannot be expressed in a simpler language.

[gib65]

Gib65, do you mean Complex in it`s implcations, or just some really long winded calculation?

 

My example is simplicity itself (at face value) A=NOT A, and yet the state of A is impossible to determine.

 

I`m not entirely sure what you`re after?

 

Yes, long windedness is good. It's not so important what the equation means, just that there's a lot of symbols and operations. The reader needs to feel overwelmed by it (although I will take a moment to explain what it means).

 

I like The Tree's link. I like the one for conservation of energy. That one's a monster, especially with the expansion of [math]\phi[/math]. Now, I will need to explain what this formula stands for, at least in one or two sentences. I read part way through the article that [math]\rho[/math] is the density (mass per volume) of the liquid. So if I took the expression in the brackets that's being multiplied by [math]\rho[/math] and divided both sides of the equation by it (thus getting an equation for [math]\rho[/math] by itself), could I call this the formula for liquid density? It would just be better for my purposes to have a formula in this form.

[woelen]

gib65, if you rally go for long windedness, then lookup the solutions for third and fourth degree equations. Just as their is a formula for second degree equations a*x² + b*x + c = 0, there also are formula's for third degree equations a*x³ + b*x² + c*x + d = 0 and fourth degree equations. Especially the latter one is EXTREMELY longwinded, but it is a closed form expression in the coefficients a,b,c,d,e for a fourth degree equation.

 

http://en.wikipedia.org/wiki/Cubic_equation

http://en.wikipedia.org/wiki/Quartic_equation

 

The formula for the solution of the quartic equation is so longwinded, that no textbook or website simply gives the formula, but they only give an outline (recipe) for how to solve the equation by means of a few steps.

[matt grime]

Well done, everyone. You've just convinced me that this is not a site ever worth visiting again in the hope that there was some interesting and informed mathematical discussion about anything. The sheer idiocy on this thread alone is amazing. Goodbye.

[gib65]

Well done, everyone. You've just convinced me that this is not a site ever worth visiting again in the hope that there was some interesting and informed mathematical discussion about anything. The sheer idiocy on this thread alone is amazing. Goodbye.

 

Okay

[the tree]

Oh don't be a mean Matt. We may not be informed mathematitians (I'm working on it), but it'd be nice if you could start some interesting mathematical discussion about anything.

 

I tottally get what Gib is talking about, when I was volunteering at my school's open day we decided to fill all the whiteboards with big scary looking maths, stuff with lots of symbols in it. It only had to fool parents so it didn't have to be all that geniunely complex, in fact it didn't really need to be genuine but we figured it had better be just in case. People often mistake big scary sums for being impossible to understand and having unpredictable results when in reality this just isn't true.

 

Gib, you don't need an overly complicated equation. Simply modeling a falling particle should prove your point just fine.

[YT2095]

http://upload.wikimedia.org/math/3/7/5/3753fdaad566b516775d38df5aa9a877.png

 

Gib65, take a look at that one, it looks pretty complex to me.

 

failing that, the Drake equasion`s pretty long too, although most people nowadays will probably recognise it right away.

[gib65]

Thanks for the support, The Tree (or should I just call you "Tree"?). You're right about the ominous equations coming off looking like real equations to the untrained eye - and so almost anything we come up with, no matter how fallacious, might pass as a genuine scientific formula. But I should mention, just so no one gets the wrong idea, that this digresses a bit from the main point I'm trying to make in my paper. I'm only trying to say that the determinist, whether right or wrong, thinks of physical nature in the same way we think of mathematics - that is, as perfectly determined.

 

Anyway, you say describing a falling particle would do the job just as well. It seems like a much simpler idea - much more parsimonious - but I'm not sure how it would be generalized to the entire universe. Can I ask you to elaborate on this?

[gib65]

http://upload.wikimedia.org/math/3/7/5/3753fdaad566b516775d38df5aa9a877.png

 

Gib65, take a look at that one, it looks pretty complex to me.

 

failing that, the Drake equasion`s pretty long too, although most people nowadays will probably recognise it right away.

 

That is a big one. What does is represent?

[Klaynos]

That is a big one. What does is represent?

 

Looks a bit like the schrodinger equation to me, but I'm quite tired atm...

 

Here's my equation.