Early attitudes towards negative and imaginary numbers

Khwarizmi

Apparently, many early mathematicians and philosophers resisted the concepts and considered them absurd.

Given how fundamental they are to modern mathematics, I can't help but wonder how so many intelligent individuals could be so ignorant.

Super Lazuli

Well, think about it from the perspective of people who know everything about the world that they know by looking at it unaided.

The concept of having less than none of something would have rightfully been seen as absurd. Even today, it's kind of a hard concept to wrap your head around.

Khwarizmi

But it's an abstraction, isn't it?  It doesn't have to be directly applicable to the real world, it just has to be useful in terms of calculations.

That said, negative numbers, at least, have obvious real world applications.  Debt, for instance.

Super Lazuli

Well, when you say "early" how early are we talking?

I assumed you meant "absolute, earliest, dawn-of-civilization, before math was really a unified thing" early.

Khwarizmi

I'm talking about ancient Europe through to round about the 18th century AD.  So, it was an attitude that prevailed long after the inventions of algebra, geometry and trigonometry, and some decades after the invention of modern calculus.  I find that astounding.

Super Lazuli

Oh.

That is an entirely different "early" than I was thinking.

Yeah, that's stupid.

Pykrete

People only really jumped on imaginary when Euler started screwing around with Taylor series and found out cos x + i*sin x = e^{^ix} and suddenly OMG IMPLICATIONS!  Geometry!  Cyclic functions!  And impedance for some reason!  Most useful applications of imaginary numbers are because Euler's little discovery works, and even then it's more of an intermediary step than something that actually behaves i-ishly.

Also superscript doesn't work.

Khwarizmi

Yes, I can certainly see that the practical applications of imaginary numbers were less readily apparent, but it still seems unscientific to dismiss the entire concept as nonsense.

thatguythere47

I still don't get -1 squared. It hurts my brain to look at it. 

Gelzo

Look, it's simple. Draw a line that's -1 inch lon-

Wait.

Khwarizmi

It's simply a double negative, isn't it?  The way I had it explained to me was along the lines that if you walk backwards while facing the wrong direction, you move in the right direction, if that makes sense.

Ponicalica

But it's an abstraction, isn't it? It doesn't have to be directly applicable to the real world, it just has to be useful in terms of calculations.

Well, the problem is that it wasn't obviously useful for a while. I think the first time imaginary numbers were seen as «useful» was when it turned out that the method for solving cubic equations depends on imaginary numbers even when the roots are all real.

I assumed you meant "absolute, earliest, dawn-of-civilization, before math was really a unified thing" early.

Math wasn't really «a unified thing» until the early 20th century with Bourbaki and Principia Mathematica and the like.

Yes, I can certainly see that the practical applications of imaginary numbers were less readily apparent, but it still seems unscientific to dismiss the entire concept as nonsense.

Math isn't really a science, and even in science you frequently get people who are dismissive of new ideas that turn out to pretty much be true. The example that comes to mind first is the story of how Boltzmann was driven to suicide because of the academic...bullying...he received due to resistance against his theory of atoms.

I still don't get -1 squared. It hurts my brain to look at it.

If you're talking about i as the square root of -1, consider the complex plane as a plane, and i to be the point (0,1). Complex multiplication by a point on the unit circle is a rotation, and the amount of rotation is equal to the angle between the point, the origin, and the X axis--in the case of i, 90 degrees. So if you rotate (0,1) 90 degrees, you reach (-1,0), which is -1 on the real number line.

thatguythere47

you lost me. I'll stay away from mathy threads from now on. 

Khwarizmi

Good grief.  I had never heard about the Boltzmann thing.  This is not doing wonders for my faith in the scientific community.

Ponicalica

As for the scientific community, my faith in them is conditional and based mostly on the fact that they still suck less than everyone else.

Gelzo

I think I heard something about how mathematicians used to try to kill each other to steal discoveries.

TheyCallMeTomu

Well, "abstraction" might have some theological constraints to. Remember, there were a lot of philosophers who threw around Platonic Ideals and crap like that while trying to avoid being branded heretics. All of these "Real" objects are basically "abstractions." So, the "weight" of an abstraction back then might not be the same "weight" as an abstraction now.

Actually, I'm just BSing, but still.

Myrmidon

Http://tinyurl.com/d4oma ;

Further proof that the world of Particle Physics is made up of incoherent sophistries based on force of personality and internal politics.

trollface

[Deleted User]

"Early"? I wish. I've seen this attitude among modern high schoolers.

Longfellow

I've never considered the absurdity of a number any reason to stop playing with it. Even if i and -1 had no import they'd still make for good mathematical masturbation.

[Deleted User]

But mathematics is already masturbation compared to physics, which has been termed sex. I guess that makes engineering prostitution?

Pykrete

I guess that would make CS cybering until you give yourself a rash.