What was the historical problem with Imaginary Numbers?

[FallenApple]

I don't see why imaginary numbers were necessarily so difficult among top mathematicians back then.

From pleano's axioms, we can derive the fact that any negative natural number times another negative natural number must be positive. Then this result extends to the reals, using theorems derived from those axioms as well. ( logic is a+(-a)=0 and multiply both sides by -b, then add ab to both sides)

All this means that that no two real negative numbers can multiply to be positive.

But surely it could be imagined that some mathematical structure times itself produce -1. It is not logically contractictory with the theorems of real numbers. So if it didn't pose any contradictions with previous mathematical work, what was the philosophical hurdle?

If I make up a mathematical structure that is self consistent, then it works, no matter what it is.

 

[fresh_42]

So if it didn't pose any contradictions with previous mathematical work, what was the philosophical hurdle?

How did you come to this conclusion? Which hurdles are you speaking of?

 

[FallenApple]

How did you come to this conclusion? Which hurdles are you speaking of?

Well, I was just reading somewhere that there is a trend in the history of math for new number structures to be met with a lot of scepticism. For example, the acient Greeks with irrational numbers, negative numbers for a few hundred years ago etc. Why when we see so clearly now that none of them produces contradictions.

 

[dkotschessaa]

But surely it could be imagined that some mathematical structure times itself produce -1. It is not logically contractictory with the theorems of real numbers. So if it didn't pose any contradictions with previous mathematical work, what was the philosophical hurdle?

The short and possibly wrong answer is that I don't think people were as accustomed to thinking abstractly as we are. Sure, they behave in predictable ways, but what ARE they? Lots of people new to math still have trouble with this. Yes, I know the definition, but what IS it?

I'm not sure about the statement "It is not logically contradictory with the theorems of real numbers." Surely it involves going outside the real numbers to get a real number, which is why you end up with a whole other set of numbers that are "imaginary." (I attribute later hesitation to bad marketing.. Calling them "imaginary" has not helped their case either.

I know that the numbers started just "showing up" in calculations of 4th and 5th degree polynomials. They eventually worked themselves out of the calculation but people just sort of whistled to themselves and pretended not to see anything. It just took awhile.

-Dave K

 

[fresh_42]

In this generality the only answer can be: People always oppose new methods or concepts. I don't think this applies to the mathematicians of the time.

 

[dkotschessaa]

Well, I was just reading somewhere that there is a trend in the history of math for new number structures to be met with a lot of scepticism. For example, the acient Greeks with irrational numbers, negative numbers for a few hundred years ago etc. Why when we see so clearly now that none of them produces contradictions.

With the Greeks it was a deeply held belief that numbers had to be rational, because their entire religious world view was built on this premise. I don't know if this parallels imaginary numbers, but it might.

-Dave K

 

[dkotschessaa]

This is a decent history: http://rossroessler.tripod.com/

 

[FallenApple]

I'm not sure about the statement "It is not logically contradictory with the theorems of real numbers." Surely it involves going outside the real numbers to get a real number, which is why you end up with a whole other set of numbers that are "imaginary." (I attribute later hesitation to bad marketing.. Calling them "imaginary" has not helped their case either.
-Dave K

What I mean is that while imaginary numbers do not come from the properties of real numbers, defining i and then making it work in a self consistent way is logically valid.

Perhaps the numbers didn't make sense at a gut level to them. But we now know that math does need to make sense at a gut level. A proof is a proof and the results are the results.

 

[FallenApple]

In this generality the only answer can be: People always oppose new methods or concepts. I don't think this applies to the mathematicians of the time.

But isn't it true that currently, we have a very strict axiomatic/theorem based proof system based off of formal logic? So we know that if a proof is done correctly, then it must be valid. Maybe the philosophy back then was whether math can be applicaple to the real world. We now mostly treat math as it's own self contained universe.

For example, I can make a completely new type of number, call it a number, and if it works self consistently(even if it doesn't describe the world in any way), then it must be true(in a mathematical sense) and fall under the category of math.

 

[Logical Dog]

I don't see why imaginary numbers were necessarily so difficult among top mathematicians back then.

From pleano's axioms, we can derive the fact that any negative natural number times another negative natural number must be positive.
.

How so? i thought peano only defined the natural number..and there are no negative natural numbers.
1. Zero is a number
2. The successor of any number is a number.
3. Zero is not te successor of any number.
4. Any property common to zero and its successor and its successors successors is a property of all numbers. (basically any property common to zero, 1 and 2 is a property of all numbers)
5. No two numbers have the same successor.

and then define addition between two numbers as:

a = s(s(s(s(0))) + b = s(s(s(s(b))) where b = s(s(s(s(0)))

I am confused by the post a little, I know the issue here is that the even root of a negative REAL number, cannot be negative REAL as the multiplication of a negative even times into itself gives a positve number. It also cannot be positive REAL, because the multiplication of a positve number even or odd times itself give positive.

I disagree, just because it is logically valid does not make it true in any sense of the word..look at economics. LOT OF FANCEY math but not gronded in reality!

 

[FallenApple]

How so? i thought peano only defined the natural number..and there are no negative natural numbers.
1. Zero is a number
2. The successor of any number is a number.
3. Zero is not te successor of any number.
4. Any property common to zero and its successor and its successors successors is a property of all numbers. (basically any property common to zero, 1 and 2 is a property of all numbers)
5. No two numbers have the same successor.

and then define addition between two numbers as:

a = s(s(s(s(0))) + b = s(s(s(s(b))) where b = s(s(s(s(0)))

I am confused by the post a little, I know the issue here is that the even root of a negative number, cannot be negative as the multiplication of a negative even times into itself gives a positve number. It also cannot be positive, because the multiplication of a positve number even or odd times itself give positive.

Oh sorry about that. I haven't studied analysis in a while, so I get somethings mixed up. Basically, my point is that you can eventually build up to the fact that negative time negative=positive. I must have confused peanos axioms with something else.

 

[TeethWhitener]

But isn't it true that currently, we have a very strict axiomatic/theorem based proof system based off of formal logic? So we know that if a proof is done correctly, then it must be valid. Maybe the philosophy back then was whether math can be applicaple to the real world. We now mostly treat math as it's own self contained universe.

For example, I can make a completely new type of number, call it a number, and if it works self consistently(even if it doesn't describe the world in any way), then it must be true(in a mathematical sense) and fall under the category of math.

First off, the notion that provability implies validity is known as soundness, and it's not a property that holds for all formal systems in general (although we tend to discard unsound systems for the reason that they're not particularly useful). Second, the notion that logic is kind of a "game" that we play, where the results are the results, regardless of any deeper meaning, is a philosophical stance, and a very recent one at that (originating in about the 1950's or so). But there's always some contentiousness of this type in math to be found. Just look at the initial resistance to category theory as "abstract nonsense" for a recent example.

 

[Aufbauwerk 2045]

Mathematics was developed for most of its history as a tool to help us calculate about the real world. The idea of developing math as a logical game which may not apply to the real world is fairly new. For example, Newton would not have developed calculus unless he found it useful for his physics. But complex numbers did turn out to be very useful indeed in physics and of course we all learn about Euler's formula which Feynman described as our mathematical gem.

 

[fresh_42]

Mathematics was developed for most of its history as a tool to help us calculate about the real world.

What about Euclid's geometry, Pythagoras' addiction to numbers? Some of their results may have helped to solve engineering problems, but they were certainly not driven by them. What about Diophantine equations? In the 17th and 18th century mathematicians wrote letters to others, in which they posted questions which they had the answer to, only to see whether the other one will find it, too. Also has mathematics been regarded as a philosophical branch. The connection to natural sciences is actually the new perspective. Large parts of number theory had been developed to solve Fermat's last theorem.

So what do you mean by

The idea of developing math as a logical game which does not necessarily correspond to the real world is fairly new.

? Sometime between the Sumerians, Babylonians and the Greek?

 

[TeethWhitener]

What about Euclid's geometry, Pythagoras' addiction to numbers? Some of their results may have helped to solve engineering problems, but they were certainly not driven by them. What about Diophantine equations? In the 17th and 18th century mathematicians wrote letters to others, in which they posted questions which they had the answer to, only to see whether the other one will find it, too. Also has mathematics been regarded as a philosophical branch. The connection to natural sciences is actually the new perspective. Large parts of number theory had been developed to solve Fermat's last theorem.

So what do you mean by
? Sometime between the Sumerians, Babylonians and the Greek?

The word "geometry" literally means "to measure the earth," reflecting the practical considerations and connection to the real world that have driven math throughout its history. Were mathematicians driven exclusively by practicality? Of course not, but I think it's disingenuous to deny that a great deal of math was inspired by the real world. I wonder how long the development of e.g., variational calculus would have taken without the brachistochrone or celestial mechanics. Would Fourier have come up with his series if he hadn't been interested in heat transport? How about Gibbs (an engineer) and vector calculus? Do you really think that group and representation theory would be where it is today if Wigner hadn't noticed how useful it was in quantum theory?

But this is all really a digression from the fact that the vast majority of mathematicians until quite recently viewed their proofs as unassailable truths about the world, at least in some respect. Whereas the view nowadays seems to be that the axioms we pick are largely chosen as a matter of convenience and pragmatism, and they and whatever theorems follow from them don't have any special ontological status outside that formal system. I think that's what David Reeves and I meant.

 

[Aufbauwerk 2045]

What about Euclid's geometry, Pythagoras' addiction to numbers? Some of their results may have helped to solve engineering problems, but they were certainly not driven by them. What about Diophantine equations? In the 17th and 18th century mathematicians wrote letters to others, in which they posted questions which they had the answer to, only to see whether the other one will find it, too. Also has mathematics been regarded as a philosophical branch. The connection to natural sciences is actually the new perspective. Large parts of number theory had been developed to solve Fermat's last theorem.

So what do you mean by
? Sometime between the Sumerians, Babylonians and the Greek?

I think von Neumann's famous article The Mathematician is relevant. Here is part of what he says about ancient geometry.

"Geometry was the major part of ancient mathematics. It is, with several of its ramifications, still one of the main divisions of modem mathematics. There can be no doubt that its origin in antiquity was empirical and that it began as a discipline not unlike theoretical physics today. Apart from all other evidence, the very name "geometry" indicates this."

Of course "geometry" means "earth measurement." The earliest civilizations developed geometry so they could figure out how to divide plots of land, make buildings, and so on.

Von Neumann goes on at great length to talk about how when math becomes too far removed from its empirical source it becomes more like art for art's sake. Which I suppose is fine if that's what you like. I'm just looking at it from a physics standpoint.

 

[fresh_42]

The earliest civilizations developed geometry so they could figure out how to divide plots of land, make buildings, and so on.

That's why I mentioned the Sumerians and Babylonians: they measured soil and did accounting. But since the Greeks, mathematics has been a theoretical subject in its own right. The Greek society had slaves for their daily work and thus the time to deal with geometrical objects in Plato's sense of ideals. And this didn't really change ever since. Did Euler had applications in mind? I seriously doubt this. Why did Hamilton spend years to find a field extension of complex numbers? The problem with application driven science is, that it narrows your perspective. Of course have achievements been used whenever they were available, but the logical game as you call it is by no way a modern phenomena.

 

[fresh_42]

Do you really think that group and representation theory would be where it is today if Wigner hadn't noticed how useful it was in quantum theory?

Oh yes, very much so. Its usage in QM came definitely after its main parts had already been developed. If at all, it had been differential geometry, i.e. differential equations and surely not QM.

 

[TeethWhitener]

This is going to become too philosophical for this forum pretty quickly (and I have a feeling we agree on this more than we disagree), but the point I think David and I are trying to make can be illustrated thus: We can ask the question "is it true that 1+1=2?" Until very recently, the answer was simply "yes." A more modern answer is "Yes, assuming certain axioms." So the statement "1+1=2" only says something about a system of axioms, and not necessarily something true about the world. It's related, somewhat tangentially, to the debate about whether math is discovered or invented.

Essentially, it wasn't until Godel (and Tarski with his undefinability theorem) crushed Hilbert's program that people started to accept that there were many different logical systems and that no one system was more intrinsically "true" than any other. There may have been hints at this fundamental shift earlier (my mind jumps to Gauss and Lobachevsky rejecting the parallel postulate) but my guess is that the full shift didn't come along until at least Russell, and probably not until after Godel and Tarski.

 

[micromass]

What about Euclid's geometry, Pythagoras' addiction to numbers? Some of their results may have helped to solve engineering problems, but they were certainly not driven by them.

They were directly driven by the desire to describe various aspects of nature. That was the main goal of the Greeks.

 

[Logical Dog]

though simple addition holds for real objects, its laws of commutatitity and associativity don't hold. Eg.. chemicals. if we use addition in the colloquial sense of trying to combine them. Though i m sure there is some area of mathematics that can easily model this.

 

[dkotschessaa]

They were directly driven by the desire to describe various aspects of nature. That was the main goal of the Greeks.

Sort of, except they imagined nature to be much more orderly than it was. The Pythagoreans in particular had an elaborate and fantastical mythology surrounding numbers. Even by the standards of the day it was more religious than practical.

Have you ever heard Euclid praised for its practical value? It is almost always praised for its structure and consistency.

-Dave K

 

[lavinia]

Imaginary numbers came up in the 16'th century in attempts to find formulas for solving special cubic equations. They were seen as a calculation tool. They were not viewed as corresponding to anything real. Even as late as the 17'th century imaginary numbers were viewed as not corresponding to reality. For instance , Descartes referred to them as "impossible numbers" . Euler used imaginary numbers in complex functions but I am not sure that he considered them to be anything other than a computational convenience. Gauss apparently made imaginary numbers more acceptable by showing that they could be viewed as the complex plane and that multiplication could be seen as angle addition and length multiplication. But this was not until the 19'th century.

Gauss questioned the "metaphysical" nature of complex numbers so it seems that he also struggled with what they were.

It seems historically that numbers were not viewed as mere abstractions or symbols. They had to somehow be real. The search for their reality required going beyond sensory intuition and this was not easy.

- It is interesting that negative numbers also took time for acceptance.

 

[fresh_42]

- It is interesting that negative numbers also took time for acceptance.

I think that even zero first was also only a balancing tool. (India some 5,000 years ago, if I remember correctly)

 

[Logical Dog]

a = s(s(s(s(0))) + b = s(s(s(s(b))) where b = s(s(s(s(0)))

this definition is wrong. Just clarifying, I was doing some arithmetic and it failed spectacularly. I don't have time to post the correct version right now.

 

[FactChecker]

What would you call a number that you can not use to count or measure anything? The imaginary numbers are very different from any number system before it. Zero and negative numbers were bad enough and Pythagoras killed a man who proved that √2 was irrational. There would be great opposition to complex numbers and an effort to replace its use some how with real numbers.

 

[willem2]

this definition is wrong. Just clarifying, I was doing some arithmetic and it failed spectacularly. I don't have time to post the correct version right now.

You have to define s(a+b) = a + s(b) and a+0 = 0. (it's a nice exercise proving that addition is commutative from this)

 

[Svein]

You have to define s(a+b) = a + s(b) and a+0 = 0. (it's a nice exercise proving that addition is commutative from this)

Er... a+0 = a.

 

[Svein]

Complex numbers are habit-forming. Things that are tricky in the real domain often become routine in the complex domain. I have gotten into the habit, so have a tendency to answer problems by taking them to the complex domain and solving them there.