DISCOVERY VS INVENTION

[Wel lEnTaoed]

What is the difference?  For example did Benoit Mandelbrot discover or invent the mandelbrot/set?

[teamfresh]

I think he discovered it.

[cbuchner1]

I think he discovered it.

I think his computerized method of plotting the M-set on a plane is more of an invention, really (however not one that fulfills any criteria for patentability - software patents are pretty uncool anyway).

The term "discovery" rather applies to the formula, its mathematical properties and all the chaos theory behind it.

[teamfresh]

I think he discovered it.

I think his computerized method of plotting the M-set on a plane is more of an invention, really (however not one that fulfills any criteria for patentability - software patents are pretty uncool anyway).

The term "discovery" rather applies to the formula, its mathematical properties and all the chaos theory behind it.

different inventions show the same discovery ie:Mset
so now I think Mandelbrot did both

[Tglad]

I think velcro, the lightbulb and the bicycle are all discoveries, but were probably inventions at the time.

[Wel lEnTaoed]

Discovery is something that has always existed.  I see things that are man made such as velcro, bicycles etc. as inventions. 

Now the Mset ..that's a lil trickier.  hmmn could be like in archeology (the tool neccessary) "invention" to discover the mandelbrot that as always been there waiting to be discovered?   

[hobold]

This question is almost as old as mathematics itself. There are many mathematicians on record who, through the ages, voiced their feelings of discovery and exploration when they practiced their abstract work with their abstract tools. That kind of belief is shared by a majority of mathematicians, as far as I can tell. But there is a very significant minority group, the constructivists, who see themselves more as builders rather than explorers, and in fact they do a slightly different kind of mathematics. This difference isn't easy to explain in depth, but it is rooted in the "Axiom of Choice". The constructivists "don't believe" in that axiom, so to say, and cannot rely on it in their proofs.

The Axiom of Choice, very loosely speaking, assumes that certain abstract sets do exist, even if you haven't fully specified them (as long as nothing contradicts their existence, of course). The constructivists find that assumption a bit too strong, and would rather fully specify every set before they reason about it. The constructivists are indeed on more solid ground (as far as that is possible in a purely theoretical environment), but they have to work quite a bit harder, having to construct everything by themselves.


There are other lines of reasoning that the non-constructivists could cite to back themselves up. For example: if extraterrestial aliens developed mathematics, why should their math be fundamentally different from ours? How could it be different at all?

If you like to occupy yourself with that kind of science philosophy, you might want to check out the books "Gödel Escher Bach" by Douglas Hofstadter and "Anathem" by Neal Stephenson.

[Wel lEnTaoed]

This question is almost as old as mathematics itself. There are many mathematicians on record who, through the ages, voiced their feelings of discovery and exploration when they practiced their abstract work with their abstract tools. That kind of belief is shared by a majority of mathematicians, as far as I can tell. But there is a very significant minority group, the constructivists, who see themselves more as builders rather than explorers, and in fact they do a slightly different kind of mathematics. This difference isn't easy to explain in depth, but it is rooted in the "Axiom of Choice". The constructivists "don't believe" in that axiom, so to say, and cannot rely on it in their proofs.

The Axiom of Choice, very loosely speaking, assumes that certain abstract sets do exist, even if you haven't fully specified them (as long as nothing contradicts their existence, of course). The constructivists find that assumption a bit too strong, and would rather fully specify every set before they reason about it. The constructivists are indeed on more solid ground (as far as that is possible in a purely theoretical environment), but they have to work quite a bit harder, having to construct everything by themselves.


There are other lines of reasoning that the non-constructivists could cite to back themselves up. For example: if extraterrestial aliens developed mathematics, why should their math be fundamentally different from ours? How could it be different at all?

If you like to occupy yourself with that kind of science philosophy, you might want to check out the books "Gödel Escher Bach" by Douglas Hofstadter and "Anathem" by Neal Stephenson.

Thanks!  I like the idea of combing the builders with explorers.

[Calcyman]

I think velcro, the lightbulb and the bicycle are all discoveries, but were probably inventions at the time.

The bicycle and incandescent lightbulb are definitely inventions. As for velcro and the lightbulb in general, I think that these might have already existed in nature. Certain forms of bioluminescence could be considered to be examples of natural lightbulbs (e.g. angler-fish), and both plants and insects are endowed with myriad hook-like barbs akin to velcro.

As for whether the Mandelbrot Set was discovered or invented, let's take a simpler example: the Sporadic Groups. Did they exist all along? I would say so, along with all exceptional objects in mathematics -- E8, the Leech lattice, the Platonic solids, etc. As for RSA cryptography, I would probably consider that to be an invention, even though the underlying properties of modular exponentiation were discoveries.

So, is the Mandelbrot Set an application of mathematics, a mathematical concept, or an exceptional object?

Platonists would almost definitely consider all mathematics to be 'discovered' rather than 'invented'.


Would you consider 2^43112609-1 to be a discovery, or an invention?

I think his computerized method of plotting the M-set on a plane is more of an invention, really

That would be Argand's invention, if anything: the idea of mapping the set of complex numbers to a Euclidean plane.

[cbuchner1]

That would be Argand's invention, if anything: the idea of mapping the set of complex numbers to a Euclidean plane.

Hmm, I tend to disagree. Mandelbrot's sampled this set on a regularly spaced grid, performing a computerized iteration until hitting a pre-defined iteration limit (or until the point escapes). Then he created a graphical representation of the (approximate) set by plotting all points reaching the iteration limit into a bitmap. This method resulted in the first known black & white prints of a Mandelbrot set.

So there's more to it than just representing complex points on a euclidean plane.

Mandelbrot's Method = Argand + Zuse + Fatou  

[Bent-Winged Angel]

Inventions as patentable!