We were recently having a discussion on consciousness, free will and metaphysics. I have a few friends who work in science and technology who are religious. Just today the discussion about miracles and science came up with one of them.

I asked him if he could explain how miracles fit in a framework of a god that works by rules that are the same yesterday, today and forever.

He admitted that the idea of miracles (which he believes the documented evidence for is high) contradicts the idea that God is consistent. To perform a true miracle (real magic) god would have to violate physics as we know it under all other (non-miraculous) circumstances. Otherwise, it isn't a miracle. It's just an (as yet) unexplained physical phenomenon.

The idea of metaphysics is "beyond physics" which implies that a miracle or any metaphysical "thing" would not be comprehended ever by the formalism of study of matter and energy.

It was suggested that the hypothesized multiverse is in the metaphysics category because by its definition, it has an intractable boundary which can't be crossed. However, it is useful in some formulations of QM and other theories of physics. In this sense, the multiverse is a tool, much like the imaginary 'i' of complex numbers and of phase. QM uses phase, as does just about every branch of physics dealing with wave theory.

As such, the imaginary is a useful tool. This seems to be the same condition one can say about miracles and metaphysics. Not real, but to some, very useful. My friend said to me that religion just works to make him a better person. He acknowledged that many people don't need religion (the imaginary tool) in order to behave themselves, but some do.

These concepts are helping me understand the role religion has for others and to be tolerant.

Oh, and before I forget: God=√(-1)



Edited 2 time(s). Last edit at 03/18/2013 05:51PM by Jesus Smith.

A few questions that might be relevant in the case of multiverses.

1. Do they exist sequentially, concurrently, or both? Or, is
time itself irrelevant, when making such comparisons?

2. What exterior realm or domain or dimension "contains" the
postulated multiverses? Just because they are intangible one
to another, does that fact necessitate their all being
imperceptible to some entity exterior to the entire set?

3. Do these multiverses, all taken together, form some
coherent pattern? If we could "stand back far enough" to
perceive them all at once, would they form the number 42?

4. Toleration of various religions is really only practical
in a situation where two or more of them co-exist, sharing
the same geography. If we could locate one Jew residing in
the tribal areas of western Pakistan, he would not be
demanding toleration (he'd be hiding). Likewise, unless
the multiverses share some extra-cosmic "geography." there
is not much reason for our tolerating their pagan (by our
narrow standards) creator gods and heathen religions.

Just saying...

UD

This is a really interesting idea, JS, and I like your notion that "the imaginary is a useful tool."

Your comment reminded me of Benedict Anderson's book, Imagined Communities, and also of Edward Said's Orientalism. Both scholars make similar arguments about the role the imaginary plays in the ways that humans organize and make meaning. From what I recall, Anderson was more focused on the ways people develop a sense of belonging to a nation, and Said more on the idea of imagined geographies we carry around with us to make sense of the "other."

I am going to dig out those books tonight. Thanks!

[JS] "The idea of metaphysics is "beyond physics" which implies that a miracle or any metaphysical "thing" would not be comprehended ever by the formalism of study of matter and energy."

[COMMENT] This sounds too much like "supernatural" which suggests that there is no naturalistic explanation even in principle. I reject such a suggestion. I prefer a view of miracles, or better "paranormal phenomena," that is based upon ignorance rather than the supernatural--assuming such events occur. Now maybe such things can ultimately be explained by matter and energy on a scale currently too removed from human perception, or maybe matter and energy are not the only aspects of reality that have causal, or measureable, properties.

[JS] "It was suggested that the hypothesized multiverse is in the metaphysics category because by its definition, it has an intractable boundary which can't be crossed. However, it is useful in some formulations of QM and other theories of physics."

[COMMENT] I think that the postulation of multiple universes is indeed metaphysical, considering "metaphysical" as encompassing a large degree scientific speculation. However, not all metaphysical postulations have equal credibility, or equal lack of credibility. A theory being "metaphysical" does not suggest (at least to me) that there are no reasons to postulate such a theory or the existence of a proposed entity. Regarding multiple universes, inflation theory, anthropic considerations, and consistent mathematical models, bring some degree of respectability to such metaphysical entities, as well as some underlying experimental facts that support some aspects of the theory. I would not say the same thing about the Biblical miracles. For example, when particle physicists postuated the Higgs bosen, they were suggesting the existence of an entity for which they had no evidence. BUt they still had good reasons to believe that it existed. Why, because it was required to render the Standard Model consistent, and the Standard Model already had much to recommend it.

[JS] "In this sense, the multiverse is a tool, much like the imaginary 'i' of complex numbers and of phase. QM uses phase, as does just about every branch of physics dealing with wave theory."

[COMMENT] I think both the multiverse and complex numbers are more than tools. In the first instance, as indicated above, we have a legitimate scientific speculation that is not empty of all evidentiary support; it is just very weak, and has verification problems. Is it useful, yes. But it also may be true. And we might discover more evidence to support it, for example evidence related to black holes, space-time "bubbles," or other dimensions.

As for complex numbers, the "imaginary" i (square root of minus 1) does in fact find its way into physical reality, and is fundamental to QM, among other things. Its certainly counter-intuitive, but we can solve equations successfully with it about the physical world, which we could not do without it. So, again, I think it deserves more status than just a useful tool to support a pet belief system.

[JS] "My friend said to me that religion just works to make him a better person. He acknowledged that many people don't need religion (the imaginary tool) in order to behave themselves, but some do."

[COMMENT] Honestly, I have a hard time with this. Multiverses and imaginary numbers are not pure falsehoods that are only useful to make us feel better. They are useful in considering the nature of reality, but they also have a great deal of substance behind them, mathematical and otherwise. (Physicist David Deutsh believes that the two-slit experiment proves multiple universes. (See Deutsch, The Fabric of Reality)

[JS] "These concepts are helping me understand the role religion has for others and to be tolerant."

[COMMENT] I am all for tolerance, but I am unimpressed by someone who invokes religious faith for solely pragmatic reasons without a firm belief and conviction that at least on some level the belief system makes sense on a scientific and philosophical level. Equating the ideas of a multiverse or imaginary numbers with the bare imagination required for religious miracles makes be cringe.

Henry Bemis Wrote:
-------------------------------------------------------
> I prefer a view of miracles, or better "paranormal
> phenomena," that is based upon ignorance rather
> than the supernatural

Paranormal seems to indicate something that is (para) besides normal. However, consciousness is not besides normal. It is the most normal. In fact, in philosophy, isn’t there a school of thought that it’s all there really is, ala solipsism? That seems hardly paranormal.

I think what you’re getting at is that if it is unexplained then there might be something beyond our current science. I would agree with that, so far.

The difference we seem to have is you seem (if I am not mistaken) to think that there are aspects about consciousness that cannot and will not ever be explained by science.

Is that correct?

(snip the multiverse discussion. I pretty much agree.)


> I think both the multiverse and complex numbers
> are more than tools.

But you seem to agree with me that they cannot be measured directly. They’re implied in the math models. Measurements are made and mathematics are used to summarize in short hand the ensemble of observations. This is a tool. I know that some here (and I believe HB is one) believe that math has a real qualia to it. That nature is or has inherent mathematics. I disagree. Math is, to me, just a short hand language to describe sets and samples of observations, or using previous results of short-hand ensembles, ways of expressing a prediction of what other observables will be made under different circumstances. At the most basic level, the universe doesn’t know math. The combinations of states that exist among all the particles and energy form ensembles that are quite expressible by the language of math (probability being a very good way).

> Honestly, I have a hard time with this.
> Multiverses and imaginary numbers are not pure
> falsehoods that are only useful to make us feel
> better.

Yeah, I know. The mathematical constructs of complex numbers and the multiverse are not falsehoods to make us feel better. I was drawing an analogy to the usefulness of the math tools to the usefulness of myths. That doesn’t mean I am saying ‘i’ or the multiverse are strictly myth. I suppose the analogy was lost due to over analysis.

[JS] "Paranormal seems to indicate something that is (para) besides normal. However, consciousness is not besides normal. It is the most normal. In fact, in philosophy, isn’t there a school of thought that it’s all there really is, ala solipsism? That seems hardly paranormal."

[RESPONSE] For me "paranormal" refers to any phenomena that cannot be fully or satisfactoraly explained through materialist explanations. This is a broad definition, which includes such things as reports of past lives of children, and consciousness. (Although we disagree on the latter, maybe both)

Regarding philosophy, there is a school of thought called Idealism that basically says that everything is mental. It is based upon "empiricism," which says that all we can know is our own experience. The implication is that what lies beyond our mental experience is unknown and unknowable. Idealism is the extreme empiricist view that there is nothing but our mental experience.

[JS] "I think what you’re getting at is that if it is unexplained then there might be something beyond our current science. I would agree with that, so far."

[RESPONSE] For me there is no such thing as the supernatural. Everything is subject to explanation in principle, whether we know it, or can know it, or not. I suspect that a full explanation of paranormal phenomena, as well as consciousness, requires more understanding of reality than humans currently have access to, or may ever have access to. But that does not make it supernatural, as if there no conceivable account of what happened based upon some unknown universal principles or natural law.

[JS] "The difference we seem to have is you seem (if I am not mistaken) to think that there are aspects about consciousness that cannot and will not ever be explained by science."

[RESPONSE] Well, I would not go that far. Although, as human beings we have inherent limitations, by our biological and evolutionary heritage, I am open to much more understanding about consciousness. But I think it will involve fairly deep physics on a scale we do not currently have access to. I do not think it will come by neuroscience alone.

[JS] "But you seem to agree with me that they cannot be measured directly. They’re implied in the math models. Measurements are made and mathematics are used to summarize in short hand the ensemble of observations. This is a tool."

[RESPONSE] First, I would not put mathematics and multiple universes in the same metaphysical category. The multiverse is a theoretical construct, with some evidentiary support. Moreover, it is postulated as having ontological status, as something that is part of reality.

Mathematics on the other hand is not generally given ontological status. (Your position, for example), but rather is considered only a mental construct, or as you put it a mental tool. So, it seems to me we are talking about two different things here. Mathematics is only "metaphysical" when we assign it ontological status as part of the real world independent from mind.

[JS] "I know that some here (and I believe HB is one) believe that math has a real qualia to it. That nature is or has inherent mathematics. I disagree. Math is, to me, just a short hand language to describe sets and samples of observations, or using previous results of short-hand ensembles, ways of expressing a prediction of what other observables will be made under different circumstances. At the most basic level, the universe doesn’t know math. The combinations of states that exist among all the particles and energy form ensembles that are quite expressible by the language of math (probability being a very good way)."

[RESPONSE] "Qualia" is not the right word here. Although math has qualitative aspects, what you are getting at is not the mental aspect of math, which is uncontroversial, but again, whether mathematics also has a Platonic type "being" independent of mind. That is what is controversial. I have not made up my mind on this.

However, your bringing complex numbers is particularly interesting. I wish we had time to discuss this in more detail, maybe another post. Put simply, the imaginary number (i) is wholly counter-intuitive. It makes no sense. Nonetheless, when we allow it in our equations, we get much more knowledge about the material world than we could ever get without it. Complex numbers are the foundation for essentially all of modern physics. Doesn't this tell us something about (1) the limits of human understanding; and (2) the deep structure of mathematics as exhibited in the physical world. Both of these ideas suggest to me that reality is very much removed, in fundamental ways, from what we can currently comprehend. The "magic" of complex numbers makes me lean on the side of mathematical realism.

[JS] Yeah, I know. The mathematical constructs of complex numbers and the multiverse are not falsehoods to make us feel better. I was drawing an analogy to the usefulness of the math tools to the usefulness of myths. That doesn’t mean I am saying ‘i’ or the multiverse are strictly myth. I suppose the analogy was lost due to over analysis.

[RESPONSE] The analogy works on some level, but imaginary numbers are definitely NOT merely numbers within the imagination of mathematicians and physicists in the same sense that religious myths are imaginary. I was worried that your suggestion might give too much encouragement to those that might think otherwise.

Henry Bemis Wrote:
-------------------------------------------------------
> I suspect that a full explanation of
> paranormal phenomena, as well as consciousness,
> requires more understanding of reality than humans
> currently have access to, or may ever have access
> to. …
> I think it will involve fairly deep physics on a
> scale we do not currently have access to. I do
> not think it will come by neuroscience alone.

Ok. Refer to statements I made before about the gaps argument. Paranormal (metaphysics or consciousness or free will) is an ever receding pocket of scientific ignorance that is getting smaller and smaller as time goes by. The trend is clearly in favor of physics (or materialism in the broad definition).


> Put simply, the imaginary number (i) is wholly
> counter-intuitive. It makes no sense.
> Nonetheless, when we allow it in our equations, we
> get much more knowledge about the material world
> than we could ever get without it.

I disagree. There are formulations of wave theory, lagrangian, QM and more that do not use complex analysis. They’re cumbersome and counter-productive, but they still result in solutions that never use the imaginary number.

If someone does show me a strict requirement using complex analysis, I will reconsider my current thoughts.

[JS] "Ok. Refer to statements I made before about the gaps argument. Paranormal (metaphysics or consciousness or free will) is an ever receding pocket of scientific ignorance that is getting smaller and smaller as time goes by. The trend is clearly in favor of physics (or materialism in the broad definition)."

[RESPONSE] Agree. But just because science has a good track record for filling gaps, does not mean that all gaps will eventually be filled. And the more sophisticated science becomes, as it approaches the scope limits imposed by human perception and technology; and the more attention that is given to a gap without success, or even progress, the more reasonable it is to take the position that the problem is fundamentally intractable. And, in my judgment, gaps like conscious experience, subjectivity, and mind, just seem to be beyond anything we currently understand about the physical world, even taking neuroscience into consideration.

[JS] "There are formulations of wave theory, lagrangian, QM and more that do not use complex analysis. They’re cumbersome and counter-productive, but they still result in solutions that never use the imaginary number."

[RESPONSE] See below. Certainly, not every aspect of QM requires complex numbers. The question is whether QM as a theory can do without them.

Basically everything having to do with wireless wave transmission (radio tv, wifi, cell phone) simply would have been impossible without complex (yet another poor choice of name) numbers.

In fact, much of modern physics and engineering either requires complex arithmetic, or is vastly simplified by using complex arithmetic. The math that makes your cell phone possible was first developed by Joseph Fourier. He published it in 1805, iirc. Thomas Jefferson was US President at the time. JS was a newborn.

Nearly every mathematical extension since integers (integers good, have integrity) has been met with hostility by the mathematical establishment, which had a very strong tendency to give negative names to stuff they didn't much like, like "negative" numbers.

Zero isn't a very complimentary term.
Nor is the already mentioned "negative".
Cipher, another term for zero, isn' much of a compliment either.
Irrational numbers were way unpopular when they were first discovered. Twenty five hundred years later the term is still a major put-down.
Rational numbers, like integers, were liked by mathematicians
Real numbers too.
Imaginary and complex numbers received major flack from math and physics types right up to the end of the 1800s, even though they had been used to great effect for over a century.

Even mathematicians and scientists are often dragged kicking and screaming into expanded world views. Complex numbers were far too useful to ignore, but they could at least be given insulting names.



Edited 2 time(s). Last edit at 03/18/2013 07:49PM by Brother Of Jerry.

Good point, Bro. The confusion I have about the nomenclature is that I rarely see an actual physical system measuring the imaginary 'i'. The most relevant case is phase. Even then, phase is not physically imaginary. It can be modeled using the complex, but it can also me modeled in real numbers. I'm not a mathematician by training, so I don't that 'i' may be measurable in real space, but I haven't seen it.

One thing that is remarkable about complex numbers is that we can postulate them for convenience, and then when we work through our equations they cancel out, leaving us with theoretical constructs relevant to the physical world that are then validated by experiment. And, we cannot get to that point without them.

At least that is my understanding.

"Imaginary numbers" are no more imaginary than numbers like 7 or
3/11. The whole idea of number is a human-made, abstract
concept. There's a pair of socks, my two hands, a married
couple, a brace of partridges etc. Then there's the abstract
concept of "two." We tend to think of the numbers we use every
day as having some sort of independent reality. It takes a lot
of beginning algebra students a while to get comfortable with
the concept of negative numbers. Electrical engineers and
physicists who use complex numbers on a daily basis and in a
variety of circumstances consider them every bit a real as
counting numbers.

If anyone of you can show me a solution, to a real world (observable, physical) problem, in mathematics that strictly requires complex analysis and cannot use Cartesian or real analysis (no matter how cumbersome it may be), I would like to see that.

In speaking with two PhD mathematicians I work with, neither could think of a case where complex math was the only, strict path to a solution.



Edited 1 time(s). Last edit at 03/20/2013 06:30AM by Jesus Smith.

Here is a quote from my Quantum Mechanic's textbook (without the equations which I cannot reproduce here):

"The wave function is complex. That is, it contains the imaginary number i (italics). Recall that this behavior was forced upon us. We first tried to find a way of satisfying our four assumptions concerning the Schroedinger equation by using a purely real free particle wave function, and we found that there was no reasonable way of doing this. Only when we allowed the free particle wave function to have an imaginary part, by using the free particle wave function in which y (italics) turned out to be equal to i (italics), did we succeed. In this process we also ended up with an i (italics) in the Schroedinger equation."

"Since a wave function of quantum mechanics is complex, it specifies simultaneously two real functions, its real part and its imaginary part. This is in contrast to the wave function in classical mechanics."

(Robert Eisberg and Robert Resnick, Quantum Physics (1985) page 134)

Note also the following quote from Roger Penrose, The Road to Reality, page 73. Penrose is one of the worlds foremost mathematicians and theoretical physicists:

"Over the four centuries that complex numbers have been known, a great many magical qualities have been gradually revealed. Yet this is a magic that had been perceived to lie within mathematics, and it indeed provided a utility and a depth of mathematical insight that could not be achieved by use of the reals alone. . . . It would, no doubt, have come as a great surprise to all those who had voiced their suspecion of complex numbers to find that, according to the physics of the latter three-quarters of the 20th century, the laws governing the behavior of the world, at its tiniest scales, is fundamentally governed by the complex number system."

Note: In mathematics proper, or pure mathematics, your associates may indeed be correct. But here we are talking about complex numbers as specifically applied to modern quantum theory. The proper question to ask is whether in QM as a theory all solutions to the equations can be found without complex numbers, or whether the equations can even be formulated without complex numbers. The answer to that question appears to be no.

"Vector Theory of Noncomplex Quantum Mechanics"
A. S. Krausz and K. Krausz (2004) Vector Theory of Noncomplex Quantum Mechanics. Physics Essays: March 2004, Vol. 17, No. 1, pp. 14-23.

http://dx.doi.org/10.4006/1.3025626

abstract:
Real vector state functions are proposed for a quantum mechanics analysis of probability density, dispersion, probability current density, conservation of mass, orthonormality, interference, and the Hamiltonian. The conclusions are equivalent to those of the usual representations and hold the promise that quantum mechanics may be described with real vector state functions instead of complex functions and the imaginary number.



Edited 1 time(s). Last edit at 03/19/2013 06:27PM by Jesus Smith.

O.K. But I don't have access to this essay, and probably would not understand its details even if I did.

But, given its date (2004), and the tentative tone of the abstract, I would think that if anything became of this proposal, it would have made it into the popular scientific journals. The Penrose book quoted above was published in 2007, and makes no reference to this line of inquiry. I also did a google search to see if there was anything more on this, but found nothing. It certainly was not the mainstream view in 2004, and for what I know, it is not now.

But who knows. On this issue, I certainly have to defer to experts.

I do not know Penrose's full response to this topic. However, the idea of noncomplex Hilbert space and its use in QM has been around for a long time.

http://jmp.aip.org/resource/1/jmapaq/v22/i11/p2404_s1?isAuthorized=no

The uncertainty principle expressed in only real space:

http://adsabs.harvard.edu/abs/2006PhyEs..19..499K

And many complex coordinate to cartesian transformations will give you a real-space version to formulate the problem.

miracle, n. An act or event out of the order of nature and unaccountable, as beating a normal hand of four kings and an ace with four aces and a king. (Ambrose Bierce)

Also Prayer: A petition by the admittedly unworthy for the temporary suspension of the laws of the universe for immediate personal advantage (or something like that)

So OK then, I underestimated the crowd. :)

I used to know a passable amount about complex function theory, but that has largely evaporated over the years. I can limp passable well through signal processing math, but that can be "faked" using 2D real vectors.

I would argue that a 2D vector is just another way to write a complex number, and that is precisely what Hamilton (of Hamiltonian fame) did. I'm not sure if there is anything that "must" be done using complex numbers. That's a question for a specialist in complex analysis. I have seen one result that I would not have a clue how to do using just real numbers.

In "An Imaginary Tale: the Story of √-1" by Paul Nahin, p 105-7 outlines a result from a 1914 paper by Edward Kasner, where, for a complex valued complex function (input and output are complex numbers), it is possible to have an arc length between two points where the length of the arc is ***less than*** the length of the straight line connecting the two points.

IOW, in complex space, a straight line is not necessarily the shortest distance between two points. Neither I nor Nahin have a clue how to visualize this result, nor would I have a clue about how to do it using just real analysis, not that that says anything about whether it is possible. Just not possible for me!


On a more general note, the concept of subtraction has been around since before even rudimentary writing, at least 4,000 years. I'm not sure when negative numbers became a widely used concept, but it was met with considerable resistance. I have even seen a poster here on RFM state that negative numbers don't really exist, because nobody has ever seen "minus two" cows.

True enough, but whenever someone sees "Combo Discount -$0.47" on a fast food receipt, they certainly think that negative number is real enough. Besides which, even 1+1=2, while it is true of coins in a cup, is not true of clouds. Context is everything.

Back to negative numbers. Ever notice how most corporate reports put negative numbers in parentheses, or red ink (or both), rather than numbers with a minus sign in front of them? Turns out this is essentially tradition. Corporate reports have been around longer than the tradition of indicating negative numbers by sticking a minus in front of them. That only came into vogue around 1750. George Washington was alive. We don't bat an eye at seeing a number with a minus sign in front of it. THat was a long fight to get there, and still today, some people, even on RFM, don't "believe" in negative numbers.

A lot of people don't "believe" in imaginary numbers, mostly because of the unfortunate name. Most people are OK with irrational numbers, because they don't understand them. If they did, they probably wouldn't believe in them either. :)

I'm of the opinion that 1+1=2, negative numbers, zero, irrationals, imaginary, and complex numbers are all of precisely the same level of actual existence. There are places where they don't fit (1 cloud plus 1 cloud equals one cloud) and places where they make perfect sense (a negative number indicating a discount, or multiplying two complex numbers is equivalent to adding two phase angles).

BTW, the Greeks hated zero. How could a good Platonist consider the non-existence of anything? Everything always existed as a platonic ideal. The Greeks damn near created calculus back around the beginning of CE, but without a zero, taking the limit as a variable approaches zero becomes an extraordinarily difficult concept to express. Good thing. Imagine if the Industrial Revolution, a more or less direct outgrowth of the invention of calculus, had happened a thousand years earlier, and nuclear weapons had been created back before the Magna Carta.

There's a scary thought.

Find a solution to the following algebraic equation

x^2 + 1 =0 (or in English: X times x + 1 = 0. )

Fact: There exists no real number which will satisfy this equation. Every solution to the problem (there are two solutions) is a complex number that is not a real number.


*********


Or put another way: The Fundamental Theorem of Algebra is only true if you allow the use of complex numbers to find solutions. If you restrict your set of numbers to real numbers, the Fundamental Theorem of Algebra is false.

The Fundamental Theorem of Algebra can be stated as follows: Every (non-constant) polynomial of degree n has exactly n roots (counting multiplicities). This just means that you can factor every polynomial of degree n into n linear factors-- But this is absolutely not true if you do not allow the use of complex numbers.

Also, there are certain definite integrals that can only be solved in closed form using complex arithmetic.

There are lots and lots of examples that I can think of where complex numbers are required to solve a mathematical construct.

--This retired math teacher didn't know she was going to have this much fun reading RFM tonight. . .

longtallsally Wrote:
-------------------------------------------------------
> Find a solution to the following algebraic
> equation
>
> x^2 + 1 =0 (or in English: X times x + 1 =
> 0. )

...
> There are lots and lots of examples that I can
> think of where complex numbers are required to
> solve a mathematical construct.


Hi longtallsally. From one tall person to another, thanks for replying. I agree completely, as far as mathematical constructs are concerned, there are conditions where complex solutions are required.

However, point out to me where in physical reality the imaginary root of your problem above can be measured. In physics, that root is tossed out and the real number kept, because only one root represents an actual solution in real (observable) space.

To paraphrase my son, who's loosely followed this discussion: all numbers are anthropogenic constructs, and imaginary numbers are only as good as any other numerical construct when speaking strictly in mathematical construct terms.



Edited 1 time(s). Last edit at 03/20/2013 07:30AM by Jesus Smith.

"I agree completely, as far as mathematical constructs are concerned, there are conditions where complex solutions are required. However, point out to me where in physical reality the imaginary root of your problem above can be measured. In physics, that root is tossed out and the real number kept, because only one root represents an actual solution in real (observable) space."

It doesn't make sense to demand that the imaginary root "be measured" in some physical sense. What we are talking about are complex mathematical constructs that have application to the physical world within the theories in which they are found. Thus, the imaginary numbers of QM are not what is measurable, or confirmed by experiment. It is the solutions to the equations. When a measurement is made in quantum theory, colapsing the wave function, the result is a real number measurement. However, the probabilities established by the wave function, i.e. by the Schoedinger equation, involve complex number theory.

Henry Bemis Wrote:
-------------------------------------------------------
> It doesn't make sense to demand that the imaginary
> root "be measured" in some physical sense. What
> we are talking about are complex mathematical
> constructs that have application to the physical
> world within the theories in which they are found.
> Thus, the imaginary numbers of QM are not what is
> measurable, or confirmed by experiment. It is the
> solutions to the equations. When a measurement is
> made in quantum theory, colapsing the wave
> function, the result is a real number measurement.
> However, the probabilities established by the wave
> function, i.e. by the Schoedinger equation,
> involve complex number theory.


And yet, using complex formulations are not the only way to get the values of those measurable observables. Hence as I have been arguing, complex math is nothing more than a tool.

We don't deal with reality, we deal with our mental construct of reality. Hopefully our mental construct is only one step removed from objective reality. Sometimes it is not one step removed, as in dreams. If your mental construct varies too far from reality too often, you are deemed mentally ill. Or Mormon (cue rimshot)

Incidentally, human newborns and birds and mammals seem to understand small numbers. If crows see 5 hunters walk into their neighborhood grove of trees, and 4 walk out, they know it is not safe, and if 5 walk out, they know it is safe to return. Try the same thing with ten hunters, and the crows are no longer sure when all the hunters have left.

Clapping to a beat and synchronized group dancing seemed to be uniquely human mathematically oriented abilities as well, but thanks to YouTube videos, researchers have found parrots that can move to a beat. So, math isn't solely anthropogenic. I'm still waiting for a group of parrots that can line dance. :)

And where it fits, which happens often, math does a spectacularly good job of both describing reality, and providing hooks for manipulating that description. We use math to predict the existence of subatomic particles and slingshot orbits for satellites. That's how much we trust the math to accurately describe reality.

Very good point, Bro of Jerry. It's all idealism (solipsism) in the end anyway.

Jesus Smith: You say In physics, "that root is tossed out and the real number kept, because only one root represents an actual solution in real (observable) space". Here is an example where you need both parts of the complex number to represent observable physical reality. Here is my example. The solution to the algebraic equation x^2 +4x + 5 =0 has solution, x=-2+i . This algebraic equation is the characteristic equation of the differential equation y"+4y'+5y =0, which means that a solution of the algebraic equation is an eigenvalue of the differential equation. Thus we use the solution to the algebraic equation in solving the differential equation. This differential equation models vibration--say a mass on a spring. The eigenvalues tell about the behavior of the system. The real part of the solution, the -2, tells about the damping taking place in the mass-spring system, and the imaginary part, the 1 times i tells about the oscillatory behavior of the system. Consider a system with an eigenvalue of -2 instead of -2 + i (where you have "tossed away the imaginary part). The behavior of this system is totally different--the mass on the spring never reaches equilibrium but only slowly approaches it from one side. If your eigenvalue includes the imaginary part then you get the correct behavior--the spring crosses the equilibrium point, going back and forth, over and over again--vibration is taking place. So you can't simply "toss out" the imaginary part and get the correct results. The fact of the existence of the imaginary part tells us the "physically observable" fact that the system is oscillating. If the solution is pure real, then the system is not oscillating. Plain and simple--the imaginary part is measuring the existence of oscillation. In addition,the magnitude of imaginary part of the solution tells us something more--it determines the frequency of the oscillation--the bigger the number the higher the frequency. Since magnitude is a real number, we have indeed "stripped" the solution so that we are using only real numbers, in talking about frequency, but the fact that we have a vibration and thus can talk about frequency is a complete consequence of the fact that the solution to the characteristic equation has imaginary roots. So, to summarize, the imaginary solution means that vibrations in the system exist, and only pure real solutions mean lack of vibrations. We are using imaginary numbers to "measure" in "observable space" something quite tangible here (the existence or lack of existence of vibrations).

Because (as has been pointed out by other writers adding to this thread) the complex system can be represented in terms of a real vector space of dimension 2, (with certain special properties on the operation) all of complex mathematics can be performed without saying or using the letter "i", but by simply manipulating pairs of real numbers. (This is very common largely because it is less cumbersome.) But changing the representation doesn't change the underlying concepts--the properties of "i" are carried over when you impose those special operational properties necessary to make the equivalence.