I'm away on vacation this week, taking my kids to Disney World. Since I'm not likely to
have time to write while I'm away, I'm taking the opportunity to re-run an old classic series
of posts on numbers, which were first posted in the summer of 2006. These posts are mildly
revised.
Anyway. Todays number is e, aka Euler's constant, aka the natural log base. e is a very odd number, but very fundamental. It shows up constantly, in all sorts of strange places where you wouldn't expect it.
What is e?
e is a transcendental irrational number. It's roughly 2.718281828459045. It's also the base of the natural logarithm. That means that by definition, if ln(x)=y, then ey=x. Given my highly warped sense of humor, and my love of bad puns (especially badgeekpuns) , I like to call e theunnatural natural number. (It's natural in the sense that it's the base of the natural logarithm; but it's not a natural number according to the usual definition of natural numbers. Hey, I warned you that it was a bad geek pun.)
But that's not a sufficient answer. We call it thenaturallogarithm. Why is that bizarre irrational number just a bit smaller than 2 3/4natural?
Take the curve y=1/x. The area under the curve from 1 to n is the natural log of n. e is the point on the x axis where the area under the curve from 1 is equal to one:
It's also what you get if you you add up the reciprocal of the factorials of every natural number: (1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...)
It's also what you get if you take the limit:limn → ∞ (1 + 1/n)n.
It's also what you get if you work out this very strange looking series:
2 + 1/(1+1/(2+2/(3+3/(4+..))))
It's also the base of a very strange equation: the derivative of ex is... ex.
And of course, as I mentioned in my post on i, it's the number that makes the most amazing equation in mathematics work: eiπ=-1.
Why does it come up so often? It's really deeply fundamental. It's tied to the fundamental structure of numbers. It really is a deeplynaturalnumber; it's tied into the shape of a circle, to the basic 1/x curve. There are dozens of different ways of defining it, because it's so deeply embedded in the structure ofeverything.
Wikipedia even points out that if you put $1 into a bank account paying 100% interest compounded continually, at the end of the year, you'll have exactly e dollars. (That's not too surprising; it's just another way of stating the integral definition of e, but it's got a nice intuitiveness to it.)
History
e has less history to it than the other strange numbers we've talked about. It's a comparatively recent discovery.
The first reference to it indirectly by William Oughtred in the 17th century. Oughtred is the guy who invented the slide rule, which works on logarithmic principles; the moment you start looking an logarithms, you'll start seeing e. He didn't actually name it, or even really work out its value; but hedidwrite the first table of the values of the natural logarithm.
Not too much later, it showed up in the work of Leibniz - not too surprising, given that Liebniz was in the process of working out the basics of differential and integral calculus, and e shows up all the time in calculus. But Leibniz didn't call it e, he called it b.
The first person to really try to calculate a value for e was Bernoulli, who was
for some reason obsessed with the limit equation above, and actually calculated it out.
By the time Leibniz's calculus was published, e was well and truly entrenched, and we haven't been able to avoid it since.
Why the letter e? We don't really know. It was first used by Euler, but he didn't say why he chose that. Probably as an abbreviation for "exponential".
Does e have a meaning?
This is a tricky question. Does e mean anything? Or is it just an artifact - a number that's just a result of the way that numbers work?
That's more a question for philosophers than mathematicians. But I'm inclined to say that thenumbere is an artifact; but thenatural logarithmis deeply meaningful. The natural logarithm is full of amazing properties - it's the only logarithm that can be written with a closed form series; it's got that wonderful interval property with the 1/x curve; it really is a deeply natural thing that expresses very important properties of the basic concepts of numbers. As a logarithm, some number had to be the base; it just happens that it works out to the value e. But it's the logarithm that's really meaningful; and you can calculate the logarithmwithoutknowing the value of e.
- Log in to post comments
I'm puzzled by the suggestion that "the number e is an artifact; but the natural logarithmis deeply meaningful."
By what metaphysical argument does one have a greater epistemelogical or ontological status?
The extreme version would be to say that e doesn't exist, except as an apparently random sequence of numerals, while natural logarithms are embedded in the physical structure of space. I know you didn't say that, but see how silly the exaggeration becomes?
It's akin to saying "pi is an artefact, but the ratio of circumference of diameter of circles is innate in every Keplerian orbit."
Whoops -- that brings up the question of to what extent e or pi exist, before we compare exitences. That's the oldest known metaphysical debate in Mathematics, or at least on ontological status of mathematical objects. One one side, Platonists (Transcendent Realists) who hold that 8 and pi and e and triangles and aleph-null exist, if anything, MORE than humans do.
"O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible, -- what would they answer?"
--Plato, "The Republic" [Jowell translation], Chapter 7.
On the second side, the Logicists, following Gottlieb Frege, who hold that mathematics can be known a priori, but suggest that our knowledge of mathematics is just part of our knowledge of logic in general, and is thus analytic, not requiring any special faculty of mathematical intuition. In this view, logic is the proper foundation
of mathematics, and all mathematical statements are necessary logical truths. Cf. Rudolf Carnap (1931).
On the third side, the Formalists, such as David Hilbert, Rudolf Carnap, Alfred Tarski and Haskell Curry, who hold that mathematical statements are equivalent to statements about the consequences of certain string manipulation rules. Some some formalists now propose
that all formal mathematical knowledge should be systematically encoded in computer-readable formats (Cf. QED project).
On the forth side, the Intuitionists whose motto is: "there are no non-experienced mathematical truths" [L.E.J. Brouwer].
Cf. Arend Heyting.
"God created the integers; all the rest is the work of Man."
[Leopold Kronecker, tr. from German].
On the fifth side, Constructivists who hold that only mathematical objects which can be finitely and explicitly constructed in a specific sense properly belong to mathematical discourse.
On the sixth side, Fictionalists, such as Hartry Field published "Science Without Numbers" (1980), rejecting or reversing Quine's argument on indispensability.
On the seventh side, Embodied Mind Theorists who hold that mathematical thought is a natural outgrowth of the evolved human cognitive machinery embedded in our physical universe; hence Math is not universal and does not really exist, except in human brains, which construct (not discover) mathematical objects, in efficacious ways that benefit Darwinian fitness.
Cf. "Where Mathematics Comes From" -- George Lakoff and Rafael E. Núñez; Keith Devlin's "Math Instinct."
On the eighth side, Social Constructivists (Social Realists) such as Imre Lakatos, Thomas Tymoczko, Reuben Hersh, Philip J. Davis, and Paul Ernest, who hold that mathematics is a social/cultural construct, akin to English Common Law, or a Black Queen in Chess. Mathematical objects come from an empirical endeavor dictated by the fashions of the social group performing it, and/or by the needs of the society or power elite financing it, and are ultimately a political struggle of mathematicians seeking sex, money, or power.
Or should I classify Lakatos as a Quasi-Empiricist along with Popper and Hilary Putnam?
There are also Linguistic theorists and Aesthetic theorists of the ontology of Mathematics.
And I'm probably missing whole schools of thought.
Bottom line: we oversimplify this ancient, subtle, and
multipolar debate at peril to our scholarship, if not our right to exist and argue about "e" -- which, by the way, is as barely transcedndetal as any transcendental number can be, which hardly sounds artefactual to me.
I'm out of my depth here, but I did have a wild thought based on the last paragraphs of this essay: What would happen if the value of e was something else? Would something consistent result? Is it even possible?
JVP, that's all very entertaining and you're a towering and impenetrable genius, but I think he means that "3.1416..." is the artifact, as any other representation would be, including the word "pi" or the Greek symbol we use, but the fact that the diameter and circumference of a circle have a certain relationship is the deep reality. In just the same way, we can call it "e" or "about 2.718281828459045" or "macaroni and cheese," but the thing itself is what it is. You might have a photograph of your wife in your wallet and a painting of her in your office, but the deeper reality of your relationship with her is what makes her your wife.
Speedwell, but that's just making the point that the representation of a number, the string of digits it's written out as, is somewhat arbitrary depending on base and writing system etc.
The value is a different thing and is not arbitrary. e is greater than two and less than three, and greater than two plus one-half but less than three, and... whatever the base.
I like this number.
Morgan is correct in correcting speedwell.
I am, in fact, a rather arrogant self-centered absent-minded professor. Fortunately, I am kept in my place by my family. As to "you're a towering and impenetrable genius" -- my son is more towering, at above 6' 3" in height at age nineteen, and halfway through law school, having started university at age thirteen. My wife, a Physics professor, since you mention her, is also smarter than I, and that we have an even smarter son is proof at least that she is not impenetrable.
The point is that I remain puzzled by Dr. Mark Chu-Carroll's feeling or opinion (not a claim to fact) that: "number e is an artifact; but the natural logarithm is deeply meaningful."
Now, one might not make sense if one asked a question analogous to speedwell's: "What would happen if the value of 2 was something else? Would something consistent result? Is it even possible?"
So the analogy hinges on whether e is intrinsic, and not contingent on any specific physical universe, as Platonists assert 2 must be intrinsic. Or, on the other hand, that the exact value (not representation in any given base or nomenclature) of e is somehow an accident."
The ratio of circumference to diameter of a physical circle is not exactly pi, but a number contingent on the local curvature of space-time.
Philosophers of Mathematics differ as to whether there are or are not any "accidents" in Mathematics.
Many think not. Gregory J. Chaitin, however, claims that God plays dice not only in quantum mechanics, but even in the foundations of mathematics, where Chaitin discovered mathematical facts that are "true for no reason, that are true by accident." [G. J. Chaitin, Conversations with a Mathematician: Math, Art, Science and the Limits of Reason, Springer-Verlag London, 2002, viii + 158 pages, hardcover, ISBN 1-85233-549-1].
I suspect that Chaitin's right, but that doesn't automatically mean that Mark is right that e is an accident. It can't be an accident if pi is not an accident, because of the supremely beautiful Euler's identity:
e^(i pi) + 1 = 0
Ok, Morgan, I understand what you're saying. Thanks for reducing my question to a simpler one that pointed the way to why it wasn't a legitimate one.
JVP, you may not have appreciated my teasing (meant mostly in fun), but you often give me the impression of being somewhat bombastic. Your dismissive attitude (you're a teacher? Really?) doesn't do much to dispel this impression. The fact is that if I know enough to ask such a question, I know enough to understand the answer that you think you're above having to answer. I suppose I'm to be quelled and abashed by your idle observation that the question doesn't make sense?
Re: various issues but mainly speedwell's question -
In analogy to pi, and as jvp pointed out, the numerical value of the 'constant' is only in euclidean (flat) space. Now, given we know this to be a special case, we resort to basic definitions, independent of local circumstances. For pi this is "the ratio of circumference to radius", while for e it would be "where the curve is equal to it's gradient" which also changes with local curvature.
I'm asking questions such as - "at what values of curvature are pi, e rational?" and "do they still satisfo euler's identity?"
Bit of a cock-up by you maths guys, why didn't you make pi and e both 3 when you were designing the number systems?
Tsk, tsk, is it too late for a re-design?
And no jvp, because you're clearly a bit lacking in the brain department, I am not being serious, it's a far more subtle point that you won't understand, so please don't bother with your "I'm the worst teacher of your nightmares" bleat!
I for one are thankful to Jonathan Vos Post for writing Euler's Identity in its real beauty, including 1 and 0:
What makes it profound in my view isn't the numerical value of Ï or e, nor the question if there ist any deep epistemic meaning hidden in any one of those constants; it's the elegance in the relationships expressed in this very identity.
"The existence of a coincidence is strong evidence for the existence of a covering theory"
"Are There Coincidences in Mathematics?"
Philip Davis
The American Mathematical Monthly, vol. 88, 1981, pp. 311-320.
Mark, just so you know, some of your emphasized words are coming through without spaces. For example, "We call it thenaturallogarithm" in the fourth paragraph.
Jonathan Vos Post, that was a very interesting run-down of schools of thought. I guess I'm a Formalist/Logicist who recognizes a "filtering" role for the other schools.
I thought thatmakingspaces optional was the latestfashionin grammar andhavebeen practising allday.
worth noting that e shows up in analyzing how things change in a proportional manner (i.e. percentage changes); also that the Euler formula is the formula
e^(i theta) = cos(theta) + i sin(theta) when theta = pi. Maybe he hit this formula in another entry.
My favorite equation where e seems to appear mysteriously (deeper consideration makes it less obvious) is x^x. It has an absolute minima at 1/e. I actually figured out first that x^-x has an absolute maxima at 1/e back in HS calculus (too much fiddling with a graphical calc can be enlightening).
Why don't you mathematical towering intellects admit it, you're all in that driven-by-power-sex-money group, right?
We mere lurkers, of course, are pure of heart.
Which one? There are something like twelve high flying mathematical Bernoullis in the first four generations. Hermann Hesse's wife was a mathematical Bernoulli and there are still Bernoullis occupying chairs for mathematics in Switzerland.
JVP, lovely rant!
Some blog I'm reading (and I'm hoping it's not this one or I'll sound like a total idiot) has spent a fair amount of time defining power series and their derivatives, all to be able to solve a few simple differential equations.
x' - x = 0
x" + x = 0
The first gives you exp(x), the second gives you sin(x) and cos(x). You can also get Euler's Identity from them if you let x be complex.
The author's point is that e comes from the first equation. There's nothing odd about it, it's just an identity value in some deep way like 0 and 1 are for addition and multiplication.
'pi' comes from the second equations. Not in relation to circles in Euclidean space, but solely as the smallest positive value where the sin(x) value crosses zero.
These values pop up in a lot of other places, of course, but you can define them entirely in terms of two simple differential equations. That's pretty staggering when you think about it. It makes me wonder about the solutions to other simple differential equations and where those values might pop up. x" - x = 0 is still exp(x), as are all higher derivatives.
Aside: another blog was pointing out how a few simple invariants under spatial translation give you special relativity. Not just that, those invariants require that any massless particle travel at a 'c' defined by that transformation. We just happen to call 'c' 'the speed of light' for historical reasons, but you can argue that it really goes the other way.
There can be no meaning to either the number e, or the natural logarithm. They are both artifacts of our minds that were created from a history of random chances. To look for meaning is to completely mis-understand the advances in science that has been made since Darwin's brilliant theory of natural selection. Random chance has been responsible for the creation of every living creature, and our minds are no more relevant than that of the tiny cockroach. The great intellectuals like Denton have discovered that the universe has created itself out of nothing. Everything afterward is the product of random chance, so any mathematical system can not represent a deeper meaning. To do so would imply a deeper purpose, and we don't want to invoke a higher intelligence like those IDiots.
I had a month ago been pondering the nature of e myself. Rather the nature of the curve of e^x to be more precise. Even though I had been exposed to the concept in calculus it had never really struck me how weird it was. I am rather tickled to see that it will not become blase' as I learn more.
I must also confess that I posted in part because it bothered me to think that the last response of the year to this post would be from a creationist troll.
Max said:
I take offense to your name calling. Could you please end the new year with an argument. I believe everything I said was easily found in many evolutionary texts.
Sorry, Peter. I agree with Max that you're a Creationist troll. To me, you are pathetic. My impovershed ghetto teenaged students understood Evolution by Natural Selection better than you do.
You are stomping on a straw man, and think that you are revealing profound errors in the neodarwinian synthesis. You are merely exposing the depths of you ignorance and stupidity.
"Random chance has been responsible for the creation of every living creature"
No, you poor fool. The VARIATION of individual organisms from their parents has a random component. Actually, several random components, as there are different probability distributions for different types of gene and chromosome mutations.
But NATURAL SELECTION is not random at all.
You miss the point. You show us all that you miss the point. Are you smarter than an 8th grader who has been systematically deprived of a decent education?
No. You are far, far worse off than my students. Most of my students got it. You don't.
Try making one New Year's resolution:
"I, Peter, promise to try reading an actual good Evolution textbook, with an open mind, and ask questions of experts instead of spitting in their faces, in the hope that maybe the brainwashing that I have received from liars and frauds may be somewhat mitigated."
I may be wasting my time. You can't be cured until you admit that you are sick.
My students, at least, were willing to learn. And a third to a half of them had Creationist members of their families. Can you improve your state? Can you light a candle, or will you trudge through your weary life cursing the darkness, and blogging: "behold! I've lit a candle that only I can see!"
Sad.
Hi Mark,
Eli Maor [Loyola, Chicago], '"e": The Story of a Number' now available in paperback, includes a discussion of Napier developing logarithms and almost, but just missed devloping e.
Maor also wrote 'To Infinity and Beyond' [and other books on mathematics] before Buzz Lightyear appeared in Toy Story.
From PUP:
"His thesis was on an unusual subject: using mathematical methods to investigate problems in musical acoustics. This reflected his long interest in the relations between science and the arts, and in particular, music. His article, "What is There so Mathematical About Music?" received first award by the National Council of Teachers of Mathematics as the best article on teaching the applications of mathematics."
JVP:
Again with the insults. Don't you know that resorting to insults is the weakest form of argument. You would know that if you were half as smart as you think you are. Insults are not persuasive. In fact, they only persuade that the insulter's position is indefensible.
Well at least this is some form of argument, but this is mostly double-talk. It's random, but it's not really random. Is that so? It is really convenient to have it both ways isn't it? Evolution is continuous until the evidence shows that it is not and then it is punctuated evolution. It is really convenient to be all things and change the definition when it suits you.
You do mean ask, but not question. That is your form of education. I am here to learn. Give me your best argument. I can follow the evidence no matter were it leads. Can you? I saw "Expelled, no intelligence allowed." I had to see for myself if its claims were true. Well JVP you have shown the the caricature of the doctrinaire teacher is true. No questioning is allowed in your classes.
You wouldn't be if you made a strong argument, but you resort to name calling. I however have not been wasting my time. I have learned that you have nothing to offer to support the theory of evolution.
This has been very informative JVP, but only of the vacuity of your opinions.
Peter:
While JvP can be a bit of a bombastic ass at times, I'm in complete agreement with him that you're nothing but a troll. Your comments:
(A) Have nothing to do with the actual content of the post. They're a non-sequitur.
(B) They're pointless regurgitations of misrepresentations
of what scientists actually say about various topics.
And I challenge to back up your statement that what you said can easily be found in evolution textbooks. Name a single evolution textbook that says "The universe created itself out of nothing". Name a single evolution textbook that in any way even suggests that evolution says anything about whether or not ideas have meaning.
If you want to claim that you're not just a troll, then go ahead and cite one single evolution textbook that does either of the above. Just one.
Mark:
Good to hear from you. I come to this sight because, as far as I know, it has the highest caliber arguments, with the most educated people. I hope I am not detracting you from your holidays. I am on holidays too but I am starting a class in evolution next week and am doing some research here.
I couldn't find a good definition of a troll in your meaning. If it means someone who is only interested in causing a commotion that is not me. I am here to learn. I am obviously a skeptic though.
I was relying on my faulty memory when I referred to Denton above. I should have said Daniel C. Dennett, "Breaking the Spell." He is very well known and undeniably an evolutionist. While this book is not a textbook, it is nevertheless a book about evolution. In this popular book Dennett says:
I trust this satisfies your requirement. You limited the requirement to textbooks, but that is not the only source of all scholarly work on evolution. Also, while Dennett earned a PhD in philosophy he was given an Honorary Doctor of Science in 2007.
As to the issue of meaning and evolution I will have to get back to you later on that with a quote. My interest lies mainly towards the philosophical area. So when I talk about meaning I am referring to an overarching framework of meaning. So I am not saying that numbers have no meaning at all. They obviously do. What I am saying is that if the universe created itself out of nothing as Dennett says, and life is the produciton of random variation, then whatever meaning we ascribe e or the natural logarithm will eventually be meaningless. While an abstract argument, I believe it is on topic
Peter the Troll:
Mark is right that I "can be a bit of a bombastic ass at times." Yet I don't think you grasp why he and I agree about your illness.
As a teacher, I don't really care what you believe, as your attempt to show it is utterly riddled with misunderstanding, fallacy, and brainwashing. As a teacher, I don't teach my students WHAT to think. I teach them HOW to think. And that's where you fall on your face.
You may be well intentioned, but you wildly misunderstand even what you quote.
"It [the concrete Universe] we have seen, does perform a version of the the ultimate bootstrapping trick; it creates itself ex nihilo, or at any rate out of something that is well-nigh indistinguishable from nothing at all."
This is NOT about Biological Evolution. Nor is it, strictly, about Cosmic Evolution (same word, different meaning). It is about particular theories of the Big Bang, with which you Creationists should agree because it is consistent with one of the 3 creation myths in Genesis.
How the Cosmos began is not related to Darwin. Nor, for that matter, is Biogenesis (how life began) related to Evolution by Natural Selection.
You say Philosophy, but your rant about Evolution versus Meaning is repackaged (bad) Theology. Instead of addressing the point about whether "e" has meaning, or what numbers are, you are caught up in the era of early Medieval philosophy where Aristotle influenced Jewish and Arabic thought, which trickled down to Christian thought, and then was dumbed down recently for ignorant Americans to Intelligent Design.
"In one stunning passage of the Guide, he proclaims that he [Moses ben Maimon (1138-1204)] would be willing to discard the core religious tradition of an ex-nihilo creation if he felt that the Aristotelian argument for the eternity of the world was incontrovertible."
The Great Eagle
Maimonides:
The Life and World of One of Civilization's Greatest Minds
by Joel L. Kraemer
Doubleday. 640 pp. $35.00
Reviewed by David C. Flatto
January 2009
Now, what do you think that 18th-century mathematician Leopold Kronecker meant by (as usually translated):
"God created the integers, all the rest is the work of man."
Do you think that God created "e", or that Man created "e"?
Show your work. Give proper citations to actual textbooks and refereed papers.
Even closer to the original topic of this thread:
xkcd forum: "e" For the discussion of math. Duh. Moderator: gmalivuk
JVP said:
I am afraid I am going to have to disagree with you. Dennett included a discussion on the origin of the universe in his book on the evolution of religious thought. Not biological evolution, but evolution nevertheless. So I agree with Dennett that the ORIGIN of the cosmos is related to the ORIGIN of the diverse life forms, as well as the ORIGIN of the first life.
Now that is an interesting question and why I post here. Novel ideas force me to consider and learn material I would never have considered otherwise. I think this may come closer to what Mark was talking about when he talked about philosophical meaning. I know scientist have what they call the beauty principle which says that the most beautiful solution to a problem is usually the correct approach. Some ponder the meaning of the beauty principle. Why would the most elegant solution be more often then not the correct ones. Some have conjectured that the elegance underlying the universe is the product of the mind of the creator.
BTW, this view was around a long time before ID, so don't attribute it to the 'creationists.' There are religious scientists, and some very intelligent ones at that. I won't give you the list because it is extremely long, would be incomplete, and you must be aware of that anyway.
Getting back to your question, I would have to say, being a Theist, that the (mathematical) order underlying the universe is a result of the necessary condition for its creation. So, the underlying order was created by God and the expression e was created my man to describe this order.
Regarding post #28, the word meaning was never used once in that forum, while it is central to this.
Peter is typical of Intelligent Design trolls, in how he drags anything he wants into discussion, and falsely claims that it relates to Evolution by Natural Selection:
The Biologic Institute, Bill Dembski, and ID Research in 2008
Category: Intelligent Design
Posted on: January 2, 2009 4:30 PM, by John Lynch
So there you have it. Four very different papers with no apparent connection to the desiderata I [John Lynch] mentioned in my original post: "(a) evidence for design, (b) a method to unambiguously detect design, or (c) a theory of how the Designer did the designing".
Even though I warned about the irrelevant Dennett quote: "This is NOT about Biological Evolution. Nor is it, strictly, about Cosmic Evolution (same word, different meaning)." we still get Peter trolling away undeterred:
"Dennett included a discussion on the origin of the universe in his book on the evolution of religious thought. Not biological evolution, but evolution nevertheless. So I agree with Dennett that the ORIGIN of the cosmos is related to the ORIGIN of the diverse life forms, as well as the ORIGIN of the first life."
This time he is fixated on two fallacies:
(1) Any use of the word "origin" is a chance to debunk Darwin;
(2) If both subject A and subject B are discussed in the same book C, then subject A and subject B are so closely linked as to be synonymous.
As to (1) does Peter think that the (0,0) point in the Cartesian Plane has something to do with Biogenesis, or with Evolution by Natural Selection?
This is typical of ignorant people who have been sent out as the Army of God by fraudulent demogogues.
As to Cosmic Evolution, go look at the Tufts University web ages: "Cosmic Evolution: An Interdisciplinary Approach"
Link to Intro Movies: Arrow of Time and Cosmic Origins
When consciousness dawned among the ancestors of our civilization, men and women perceived two things. They noted themselves, and they noted their environment. They wondered who they were and whence they came. They longed for an understanding of the starry points of light in the nighttime sky, of the surrounding plants and animals, of the air, land, and sea. They contemplated their origin and their destiny.
Thousands of years ago, all these basic queries were treated as secondary, for the primary concern seemed well in hand: Earth was presumed to be the stable hub of the Universe. After all, the Sun, Moon, and stars all appear to revolve around our planet. It was natural to conclude, not knowing otherwise, that home and selves were special. This centrality led to a feeling of security or at least contentment--a belief that the origin, maintenance, and fate of the Universe were governed by something more than natural, something supernatural.
The ancients thought deeply and well, but not much more. Logic was paramount; empiricism less so. Their efforts nonetheless produced such notable endeavors as myth, religion, and philosophy.
Eventually, yet only a few hundred years ago, the idea of Earth's centrality and the reliance on supernatural beings were shattered. During the Renaissance, humans began to inquire more critically about themselves and the Universe. They realized that thinking about Nature was no longer sufficient. Looking at it was also necessary. Experiments became a central part of the process of inquiry. To be effective, ideas had to be tested experimentally, either to refine them if data favored them or to reject them if they did not. The scientific method was born--probably the most powerful technique ever conceived for the advancement of factual information. Modern science had arrived.
Today, all natural scientists throughout the world employ the scientific method. Normally it works like this: First, gather some data by observing an object or event, then propose an idea to explain the data, and finally test the idea by experimenting with Nature. Those ideas that pass the tests are selected, accumulated, and conveyed, while those that don't are discarded--a little like the evolutionary events described on this Web site. In that way, by means of a selective editing or pruning of ideas, scientists discriminate between sense and nonsense. We gain an ever-better approximation of reality. Not that science claims to reveal the truth--whatever that is--just to gain an increasingly accurate model of Nature.
Peter the Troll writes of his archaic and discredited view: "this view was around a long time before ID, so don't attribute it to the 'creationists.'"
Sorry, Peter's view is a mishmash of fallacies, misunderstandings, and misrepresentations. Saying that these fallacies, misunderstandings, and misrepresentations are old does not make them true, it just makes it more pathetic when they are trotted out, as the Intelligent Desig loons insist on doing.
Peter the clueless Troll writes: "Novel ideas force me to consider and learn material I would never have considered otherwise."
This in his response to my quote from the 1700s: "God created the integers, all the rest is the work of man."
If that is "novel" to him, then he is confessing his ignorance.
The ignorance of Good Math and the ignorance of what the Neodarwinian Synthesis explains go hand in hand.
I'm interested:
Is it possible to define functions sin, cos (or e^z) and Pi-constant, without refering geometry circle, that periodicity of sin/cos will be evident? Direct algebraic represention of curve length via integration sqrt(1-x^2) is supposed to be cheating :) And gluing of function pieces to make sin/cos to be defined over all reals also is seems 'unnatural'.
sin/cos x defined as 1/0! - x^2/2! + x^4/4! - x^6/6! does not give nor evident periodicity property, nor Ï Pi definition.
Are there other ways to define sin/cos/Pi, without circle or tailor series? Or can be Pi/periodicity directly derived from tailor representation?
Sorry for my bad English %)
--
Ivan Dobrokotov.
One more way to define: as diffirential equation (f''(x) + f(x) == 0). But periodicity and Ï are still in question.
--
Ivan Dobrokotov.
Well, I thought about this a little more, and I think I can derive from f''(x) + f(x) == 0 periodicity of f(x).
Thinking about this as spring oscillation process helps to find formal proofs.
Please allow me to tie together the "e" thread with the "continued fraction" thread through this gem, from AMERICAN MATHEMATICAL MONTHLY, October 2006
"Almost Alternating Sums", Kevin O'Bryant, Bruce Reznick, and Monika Serbinowska, pp.673-688:
2/(e-1) = [1; 6, 10, 14, 18, 22, 26, 30, ...]
which the very industrious can derive from the Taylor expansion of e^x at x=1:
e = SUM[from n=0 to infinity] 1/n!
= 1 + (1/2)(1 + (1/3)(1 + (1/4)(1 + ...)))
please i need for natural numbers