Uncommon Descent Serving The Intelligent Design Community

Imaginary Numbers, Once Rejected, Now Commonplace

Barry Arrington
September 4, 2007
Intelligent Design
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Once again I direct our readers to First Things.  This time Amanda Shaw discusses how imaginary numbers, once rejected as “Impossible, irrational, delusionary, absurd, untrustworthy, fictitious, imaginary,” are now a staple of everyday math.  See http://www.firstthings.com/

Is there an analogy to ID here?  The fact that imaginary numbers were not part of the math “system” did not mean they were not out there waiting to be used by those who were willing to look beyond the blinders of the existing paradigm.  Now, as has been argued at this site before, ID can be fit within the existing scientific paradigm; but even if this were not the case, the point is should we cling to a limiting paradigm that prevents us from seeing greater truths?  Shaw’s last paragraphs are very good:

“Scientific positivists, pencil and paper in hand, peer through shatterproof, UV-protected glasses at a world of animals, vegetables, and minerals. But genuine scientists—true seekers of knowledge—are not afraid to let the sunlight dazzle them, not afraid to seek and imagine what our myopic reason calls absurd.

“Impossible, irrational, delusionary, absurd, untrustworthy, fictitious, imaginary: It is always easier to approach—or rather, ignore—mysteries of math by dismissing them as false or unintelligible. And how much more for mysteries of faith. So is God like an imaginary number, waiting to be discovered and accepted in a renaissance of faith? The simile is ridiculous, on its face. But, in a curious way, the ramblings of scientific history remind those who strive for reason just how vast reality is. The realization is at once unsettling and exhilarating: Truth is far richer than our minds—always confined by the here and now—can prove or even imagine.

Comments
Conclusions aside, the observations from experiments like this one, Delayed Choice Quantum Eraser, will bring out the "Double-u Tee Eff" in just about anyone.Apollos
September 6, 2007
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Darn it - I need to proofread better. The end of my first paragraph should read: " and he seems to deride modern physics because it makes assertions that are “anti-intuitive, non-common-sensical, at-variance-from-what-our-senses-tell-us conclusions.Jack Krebs
September 6, 2007
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Charles writes,
If you say to me that a particle travels in two places at once, I call you a liar unless you can show it to be true, and unless that proof is forthcoming, I witdraw my money.
and he says that physics seems to deride modern physics because it makes assertions that are "anti-intuitive, non-common-sensical, at-variance-from-what-our-senses-tell-us conclusions. But this exactly what modern physics has shown us: that our commonsense intuitions about the basic structure of the world are wrong, and that world as correctly understood by science is vastly different than what our senses tell us, and is in fact weird in ways that are really beyond our conception. And yet the theories of modern physics, such as quantum dynamics and quantum electrodynamics (the subject of the little book I suggested Charles read) have produced theoretical conclusions that agree with experimental results to a level of accuracy of 10 ^ -8 power: as Feynman points out, a level of accuracy equivalent to measuring the distance from Los Angeles to New York within a hair's width of the exact distance. The ironic thing is that Charles is also arguing the math only gets its validity via its reference to reality. So here we have some math that agrees with reality just about as well as is conceivable, but Charles doesn't accept it because it's counter-intuituve and because he doesn't understand the "proof" (which is a combination of mathematical rigor and experimental validation.) It is unreasonable, it seems to me, for Charles to reject modern physics because of personal incredulity and because of a lack of confidence in the whole body of scientists who have developed and tested its results when modern physics has met, in spades, Charles' criteria that math be correlated with results in the real world.Jack Krebs
September 6, 2007
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Anyone who is not shocked by quantum theory has not understood it. -Neils BohrApollos
September 6, 2007
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Charles: Pick up a copy of Feynman' little book QED and read it until you understand at least the first few chapters. Then get back to us for discussion. Feynman does about as good a job as anyone of presenting details in an understandable format. Give him a try.Jack Krebs
September 6, 2007
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Behe is able to present complicated biological concepts and structures in a comprehensible form. Biology has all the complexity of physics. What it lacks is assertions of anti-intuitive, non-common-sensical, at-variance-from-what-our-senses-tell-us conclusions. If you say to me that a particle travels in two places at once, I call you a liar unless you can show it to be true, and unless that proof is forthcoming, I witdraw my money. Or I would if I could. To those that must make decisions about funding, even in private institutions, most will not understand even as well as someone like me with a degree in biology would. I could go in and establish a program and make ridiculous claims and if there are only three people in my field, I have the support of at least a third, and if we surrounded ourselves with enough jargon and Expertese, none from outside could say us nay. If any here would hold, as I do, that the burden of proof lies with the Darwinists who defy what the common conclusion of our senses would be by saying that machines of great intricacy came about without design, than I think you will be hard-pressed, looking at this issue honestly, to argue that that same burden does not lie on those who propound such ideas as multiple states in quantum physics or curved space (whatever the Hell that means).Charles Foljambe
September 6, 2007
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Nonsense, Scooby. When you put out something publicly as fact, you're obligated to have evidence available. My tax money goes to public schools and universities that presents these conclusions as fact. Like with Darwinism, therefore, before my tax dollar goes to fund the teaching of an idea, you need to prove that idea true. Government grants are given to these scientists to continue this work. They ARE obligated to provide in a comprehensible form their results to those who pay the bills.Charles Foljambe
September 6, 2007
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I think that Jack is dead right here. It is completely unrealistic to expect scientists, when writing for professional journals, to write for a popular audience. They are writing for their peers and use language their peers can understand. As Jack goes on to suggest, there are plenty of popular accounts available to anyone who is serious about understanding these issues. Charles wrote (47):
...I’m under no obligation to believe something that sounds like BS until you can demonstrate that it’s true in terms I can understand.
Again, your argument from incredulity carries no weight. Truth is not dependant on any individual's understanding. Furthermore, you are under no obligation to believe anything at all. Similarly, neither I nor anyone else are under any obligation to ensure that you can understand anything. The task of understanding is yours and yours alone. The ScubaredneckThe Scubaredneck
September 6, 2007
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You know damn well that doesn't address my point, or at least I hope you do. Clearly, you are incapable of the metacognition necessary to examine your own prejudices. Negative money is a mental convenience to keep track of positive debt. I can imagine gnomes swarming and skeletonizing unicorns, too, but that doesn't mean that what I imagine actually exists. When you have negative dollars in your pocket, I'll be impressed. As for where these details are, please, someone, provide a link! Prove me wrong! Just one piece that explains it on the web. Show me where they provide the reasoning that justifies their conclusions! Because some of their assertions are quite outlandish, and require justification. Mockery here and, when I ask for an explanation I don't need to get a specific graduate degree to understand, an accusation of unreasonableness there, it's good to see that the conversation here is so strong and informative. Looks like, with some things, you are as unwilling to examine your preconceptions as the Darwinists many (including myself) deride.Charles Foljambe
September 6, 2007
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Joseph, LOLBarryA
September 6, 2007
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I had a negative balance in my checking account once and it wasn't imaginary... D'oh!Joseph
September 6, 2007
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Charles, I have read a number of accounts of the double slit experiments, and I believe they answer the questions you ask. I think you are being unrealistic to expect them to simultaneously give the details you want and also make it an easy layperson's account. If you want to know the details, and not just take their word for it concerning the conclusions, you're going dig in and do the work to understand the papers that describe the details.Jack Krebs
September 6, 2007
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Right. I am curious as to how they detect it, and how they rule out other causes for the same data. I understand that the science is complicated, but they are trying to tell us the structure of the universe. They need to find some way to translate it into language at least someone with some scientific literacy, but not their level of Expertese, can understand. If they cannot make analogies, if they cannot simplify it to its essence, then I propose that they themselves understand it little better than we do. For instance, in the double-slit experiment so famous in quantum physics, what exactly is going on? They fire an electron, but in what direction? Right between the slits? Is the whole environment, except the receiver, resistant to electrons? How do they detect it on the other side? How do they know it's the same one? See, I'm of the belief, and I know it's not a valid argument, that I'm under no obligation to believe something that sounds like BS until you can demonstrate that it's true in terms I can understand. If there are no terms, no analogies even, to it, than I propose that if ANY simpler explanation could exist, Occam's Razor at the very least demands it. This is rather pertinent here, given that ID basically says, "Stuff looks designed. You say it wasn't. Prove it."Charles Foljambe
September 6, 2007
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Charles, You wrote: "I have some problem with empirical tests in advanced physics. Looking through the web, I have been able to find many statements of test results consistent with mathematical predictions, but never how those tests were conducted, how the data was gathered. Besides this, physics papers tend to be so cluttered with Greek letters and other Expertese that they are incomprehensible to the layman, and we are expected to just take their word for it. Well, screw them. Until I see how the tests were conducted, I cannot know if there is a more plausible solution that nonetheless does not match the experimenter’s prejudice." As one who has some minor acquaintance with this, my experience is that the test methodology and data gathering IS published. That is an essential part of the peer review process. It's not that laymen are expected to just take their word for it, the problem is that a lot of this work is now so wretchedly complicated that even other physicists outside the specific field can't necessarily comprehend it, never mind laymen. This is all rather unfortunate, but it's an inevitable consequence of the fact that "big physics" has generated such an immense amount of data, and our experimental capabilities so much improved, that you really do have to be an expert in the relevant field in order to identify a more plausible solution. And there are usually a number of teams working on the same issues (e.g. elementary particle physics) so they will be scrutinising each others' work closely. For example, finding the Higgs boson - everyone wants to be the first to find it, so you can be sure there will be a lot of criticism from other groups if one team claims to have found it and their data or methodology means the results are ambiguous.Gareth
September 6, 2007
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ex-xian, "Surely you’re not taking the title literally?" " It does not matter to me, just like it does not matter whether God created the universe in six days or not. "It’s a quote from Kronecker who disliked Cantor’s infinite sets." Thanks, I am ignorant. "There are lots of mathematicians and physicists who have a problem with the god hypothesis." That's expected. The difference and important thing to me is EXPELLED does not happen here, and INTELLIGENCE IS ALLOWED!MatthewTan
September 5, 2007
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Well, sure, any ballistic curve, for instance, can only be approximated by mathematics. Beyond simple quantity (2 apples), measurements are always approximate. My point is that we should remember that, that math is a very useful, but imprecise, abstraction from reality, not reality itself. As for you, ex-xian, your arrogance is telling. Rather than argue why mathematics are so vaunted, you merely dismiss me because you don't think I have your facility with math Expertese. Justify your assertion that mathematics have meanings beyond their referential value.Charles Foljambe
September 5, 2007
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Okay, lemme see how far in over my head I can get on this thread... I may have flunked trig as a freshman, but it seems to me that the things which math describes are themselves abstractions. I mean, there's really no such thing as "a line," "a circle," or "a ballistic curve," right? These are abstractions whose mathematical properties have proven to be precise enough to do things like plan a route on a map, bore out the cylinders of a dragster, and lob mortar shells into the Green Zone. Sort of like when I find myself spending all my time and money on a girl, by that abstraction that can only be called "love."jstanley01
September 5, 2007
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I don't know. Looking at the definition of formalism, I'm uncomfortable with it. I'd rather just say that my position is as stated, and avoid labels. I have some problem with empirical tests in advanced physics. Looking through the web, I have been able to find many statements of test results consistent with mathematical predictions, but never how those tests were conducted, how the data was gathered. Besides this, physics papers tend to be so cluttered with Greek letters and other Expertese that they are incomprehensible to the layman, and we are expected to just take their word for it. Well, screw them. Until I see how the tests were conducted, I cannot know if there is a more plausible solution that nonetheless does not match the experimenter's prejudice. You last point is not one I have any argument with.Charles Foljambe
September 5, 2007
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Charles Foljambe, Surely we don’t have to agree on this—but I’m curious. On the age old contention between formalists and realists, would you say that you are a mathematical formalist? I would surely agree with you that physicists (and philosophers) are able to reason up a lot of dumb things. But if the cosmos was designed then it is in some sense contingent, which means that our hypotheses should be put to the empirical test. Anyway I think the point of this post was that abstract constructs of human reason can (not must) represent reality. And what this says about materialist theories of mind is interesting.Rude
September 5, 2007
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Four may have properties, but only mathematically. It has neither hardness, nor temperature, nor breadth, nor depth. It has no color, flavor, scent or taste. Neither does it have any emotional content. I am no materialist. In saying that emotions cannot be quantified, do I not necessarily state that they are non-material? Yet I think them real. But given that numbers have no properties except within their own system, what makes them real? They gain their reality only insofar as they reference real things. I hold reason in the highest regard, but, as with scientists, mathematcians need to realize that their disciplines are merely offshoots and subordinates to philosophy. You extrapolate bizarre ideas from mathematics, based on an assumption which has not been proven, that mathematics has any meaning separate from that which it references. Nor do physicists need to regard math as you do. Newton did not need to hold math in such bloated esteem to observe and quantify the acceleration of an object in Earth's gravity.Charles Foljambe
September 5, 2007
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Quiz: Who was it that said, "things are numbers" ? 1. Einstein 2. Archimedes 3. Pythagoras 4. Euclid A correct answer (without Googling) deserves a shiny little star glued to your forehead. ;-)Borne
September 5, 2007
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[off topic] Stephen Hawking is the editor of the book with the very interesting title, God Created The Integers. Mathematicians and physicists have no problem with God's creation.MatthewTan
September 5, 2007
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Charles Foljambe: “Positive numbers were originally created (yes, CREATED) to describe quantity.” But that’s one take—the formalist take. Probably most mathematicians are formalists but, as you know, one has to be a realist (if he has thought about the matter) to be a physicist. The realist believes we discovered numbers—we didn’t create them. An argument in favor of this Platonist approach is that numbers have properties. If it is true that a number, say, four, only exists from some human based perspective, like “four apples on a tree”, then why does the number four have all its peculiar properties? But you’re right—not everyone is a mathematical realist. For some reason, why would this be? the hard core materialist always seems to resist this realism. Those in ID, I would suspect, at least when they look into the matter they will tend to be mathematical realists. You bring up the old controversy between observation and reason. The Greek philosophers, as I understand it, would abolish empircism in favor of reason--reason alone was thought sufficient in apprehending the truth. Einstein was not too interested in empirical confirmation, they say. He was a Platonist. You're right--reason alone is not sufficient. But then neither is observation. We have to have both.Rude
September 5, 2007
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And, of course, "the vast majority of mathematicians and philosophers of math would disagree with you (that is, me) about negative numbers being “ontologically lesser.” They are mathematicians and philosophers of math, almost by definition those who make too much of a system that counts things.Charles Foljambe
September 5, 2007
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My point is that you have taken a logical system and MADE IT independent of its original referential nature. While you Pythagoreans may think of "real" numbers as any number without an i in it, I do indeed use it differently. To quote, "On the other hand, some models don’t work so well. Writing a questionnaire about human emotions and putting everything on a 1 to 5 scale is not a very good model..." Indeed. It is an extremely poor model, because emotions, to the eternal consternation of some Mathematikoi, cannot be quantified. You prove my point exactly. Mathematics have been used to "prove" that time is illusiory, that space is curved, and indeed, stretched like a balloon over... um. Math has been used to justify the idea that a universe, expanding from a single point, could have neither center nor boundaries. So much hokum and nonsense comes from people who think far too much of math believing that the universe must submit to it, that because it is something we can understand, that it defines, rather than describes, the universe.Charles Foljambe
September 5, 2007
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Charles, I assume you know you are using "real number" here in a different way than mathematicians use that phrase, as negative numbers are a subset of the real numbers. I also assume that you know that the vast majority of mathematicians and philosophers of math would disagree with you about negative numbers being "ontologically lesser." Math as a logical system is independent of any physical referents. When we make a mathematical model to refer to the real world, we have to empirically check to see if that model works. Using the positive integers to count things is a model that works very well as long as the real things are clearly distinct objects. On the other end of the spectrum, modeling the behavior of photons using complex numbers (with i in them) as part of the theory of quantum electro-dynamics also works very well. On the other hand, some models don't work so well. Writing a questionnaire about human emotions and putting everything on a 1 to 5 scale is not a very good model, for instance, nor is using a normal distribution as a model for the outcomes of throwing a loaded die. So the applicability of math to the real world is a significantly different subject than the math itself. e^i*pi + 1 = 0 is true whether it applies to anything in the real world or not. P.S. Technically e^i*pi + 1 = 0 is an identity, not a formula. It contains no variables - it's just a fact about numbers. P.P.S. I have no idea why the word "serendipitous" is being used to describe this identity. The discover of it might have been serendipitous (although I don't think that is the case), but I don't think it even makes grammatical sense to say that the identity itself has "serendipitous elegance."Jack Krebs
September 5, 2007
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As for its tremendous serendipity... what, you still don't get the joke?Charles Foljambe
September 5, 2007
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Okay, so, if you give me $10 dollars, I'll give you $20i dollars. Here, I'll even give them to you up front. There you go. What, are you saying I haven't given you the money? Prove it. Describe $20i, so that we can see you don't have it. Just one example, but the point is that i doesn't reference anything. Positive numbers were originally created (yes, CREATED) to describe quantity. Unlike any other sort of concept called number, they are capable of referencing real, objective objects. In that sense, they are independent of us. There are trees with 2 apples on them. Hell, moving away from positive numbers, there are many trees with 0 apples on them. There is no tree, though, with -2 apples on it. In some situations, negative numbers are convenient, as in temperature, particularly with Celsius degrees. Knowing how the current temperature relates to the freezing point of water can be very useful. If it's -4 degrees Celsius out, I'd best be using my tire chains. Still, despite the utility of such a measure, it is an artificial measure of convenience. There is an absolute 0, 0 Kelvin, which is the true 0 for temperature. Since we'll never encounter that, though, we use a different 0 point, one that relates better to our actual experience, and doesn't render the freezing point of water at a number so cumbersome as 273. I am not joking. I would argue that symbols referencing real quanities DO differ ontologically from those that reference artificial quantities (negative numbers), or no quantities at all (complex numbers). Now, Josephus, your proof assumes what is in question, that putting something to the power of i is valid. Further, I maintain that the proofs of the identity sited, insofar as they incorporate ontologically lesser negative numbers, are deceptions derived from useful ideas which themselves are the purely conceptual additive inverses of real (that is, referential) numbers.Charles Foljambe
September 5, 2007
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Your post reminds me of this from mathematician David Berlinski: “If proofs are stripped to their syntactic shell, they have no interest. The human language that Mr. Ruelle is eager to dismiss reasserts itself the moment the mathematician asks about the meaning of the proof.” I’m also reminded of Noam Chomsky’s theory of syntax. To be admitted to the theory a grammatical form had to be freed of any universal function, with the consequence that syntax shrunk to include very little if anything. And so the functionalists arose to the rescue—true universals were meaningful, functional. The relationship between form and function must not be forgotten. But then that leads to an uncomfortable situation similar to that posed by mathematical realism. Can the materialist permit such a link between mind and ultimate reality? The formalist wants a world of logic stripped of all meaning, just as the postmodernist wants a world of meaning stripped of all logic. Intelligent Design is invested in both worlds—the world of reason and the world of reality.Rude
September 5, 2007
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Here is a little more detail on Euler's identity, That will help those who, like me, have a hard time relating to what is being discussed here. Connecting the constants The final number comes from theoretical mathematics. It is Euler's (pronounced "Oiler's") number: eΠi. This number is equal to -1, so when the formula is written eΠi+1 = 0, it connects the five most important constants in mathematics (e, Π, i, 0, and 1) along with three of the most important mathematical operations (addition, multiplication, and exponentiation). These five constants symbolize the four major branches of classical mathematics: arithmetic, represented by 1 and 0; algebra, by i; geometry, by Π; and analysis, by e, the base of the natural log. eΠi+1 = 0 has been called "the most famous of all formulas," because, as one textbook says, "It appeals equally to the mystic, the scientist, the philosopher, and the mathematician." The reason for this wide-ranging appeal is its utter serendipity. First, there is the ubiquitous number e, which pops up in the most unexpected places. It was first discovered in an attempt to make multiplication easier. In 1614, John Napier figured that adding exponents was easier than multiplying multi-digit numbers, so he (and others) calculated the logarithms of all integers from 1 to 100,000, expressing these numbers as powers of 10. Later mathematicians found it more convenient to express logarithms as powers of the natural log e, a number close to 2.71828. This number also appears in banking, because it is the limit for growth of compound interest. Let's say one invested $1,000 in a very generous bank that paid an annual interest of 100%. If interest were compounded annually, at the end of the year, the money would have grown to $2,000. If, however, the bank compounded interest four times a year, the money would grow to $2,441.41. If the bank compounded interest continually, the deposit could grow to $2,718.28, which just happens to be the value of e times the original investment. Finally, e turns up at the origin of calculus, where it is the function equal to its own derivative (if y = ex then dy/dx = ex), and it equals the limit of (1+ 1/n)n as n approaches infinity. e is irrational, so it can never be written exactly in decimal form, but it is a very useful and fascinating number in its own right. When we combine e with Π, we are introducing the oldest irrational number. Two thousand years before Christ, the Greeks knew that Π was the ratio of the circumference of a circle to its diameter and that it could not be expressed as the ratio of any two integers. It is essential in geometry, but it also turns up in waves of air, water, electricity, and light, and it even helps actuaries calculate how many 50-year-old men will die this year. The number i is a relative latecomer, proposed in the 1600s as an imaginary number and defined as the square root of -1. It was proposed to help solve equations like x2+ 1 = 0, but today it is useful in science and engineering. George Gamow, in his book One, Two, Three … Infinity, even uses i to locate buried treasure with an outdated map. The idea that these two irrational numbers should combine with an imaginary one to yield so utilitarian a result is breathtaking. It is like deconstructing a chemical necessary for life (salt) and finding that it consists of two ly poisons (sodium and chlorine). That these three strange numbers with such diverse origins should work together to produce a result so basic to mathematics argues that there is a profound elegance or beauty built into the system. The discovery of this number gave mathematicians the same sense of delight and wonder that would come from the discovery that three broken pieces of pottery, each made in different countries, could be fitted together to make a perfect sphere. It seemed to argue that there was a plan where no plan should be. Because of the serendipitous elegance of this formula, a mathematics professor at MIT, an atheist, once wrote this formula on the blackboard, saying, "There is no God, but if there were, this formula would be proof of his existence." Today, numbers from astronomy, biology, and theoretical mathematics point to a rational mind behind the universe. To be sure, they do not point to the personal God of the Bible as such. Yet they are not inimical to the biblical God, either. The apostle John prepared the way for this conclusion when he used the word for logic, reason, and rationality—logos—to describe Christ at the beginning of his Gospel: "In the beginning was the logos, and the logos was with God, and the logos was God." When we think logically, which is the goal of mathematics, we are led to think of God. Charles Edward White is professor of Christian thought and history at Spring Arbor University in Michigan. http://www.christianitytoday.com/ct/2006/march/26.44.htmlbornagain77
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