Keine verboten Bilder!

Es gibt jemand (ich weiß, wer er ist), wer schlaget um Internet verboten Bilder meine Freundins an. Ich kann das nicht aczeptieren! Das ist niedrig und unehrlich von ihm! Er hat diese Bilder in meine Freundes LJ und andere Webseite angeschlagen. Schämt euch, wer steht hinter ihm! Schämt euch, wer lächelt nur! Doch meistens über alles, schäm dich, wer macht diese alle!
Denn was immer auch hat meine Freundin dir gemacht? Ihre einzige Schuld war dass sie zu nett war, und sie geht in deine Lügen. Ich zugebe, dass ich nicht alles mit diesem Verhältnis zu tun weiß, doch keine Tat ist schlecht genug für das. Nein, ich kenne Leute, wer sind unehrlich, oder schlecht in der Persönlichkeit, und als sie tun zu tun, macht nichts. Du darfst glauben, dass meine Freundin zu dir unehrlich war, aber ihre verboten Bilder zu anschlagen wird nicht ihr, warum du glaubst das so zeigen. Das meint dass du voll mit Hass bist, und eine Leerheit dass du mit deiner Freundin, wer liebst du nicht, vollenden musst in dem Herz hast. Und wenn du kannst nicht deine Freundin einhalten, du hasst sie, und tust schreckliche Dinge zu ihr. Du bist ein traurig, jämmerlich und boshaft schwach Mann. Ja, du bist der traurigste, jämmerlichste und boshaftest schwachste Mann, wer habe ich immer gewusst.
Normalerweise, ich lasse Leute in Ruhe, doch wenn sie sich mit andere Leute, besonders meinen Freunden einmischen, dann werde ich etwas an dem tun.
Ich weiß, dass du mich fürchtest, so gib fein Acht: Schlag niemal meine Freundins verboten Bilder in mein LJ an! Wenn du das tun, ich werde sie löschen. Du kannst das wie oft als du willst tun, doch ich werde nur gleich tun; du verschwendest nur deine Zeit. Du darfst nicht sagen, dass du dieses nicht lesen kannst, weil ich deises auf Englisch auch geschreiben habe. Wenn du beweisen willst, dass du nicht ein schwach Feigling bist, dann steh wie ein Mann und rede zu mir.
Deine jetzig Freundin tut mir leid, nicht weil sie sich mit dir abfinden muss, sondern dass sie sinnlos ist, und jemand wie dich will. Wahrscheinlich, du denke sie ist nicht so, aber es ist mir egal, wer sie ist. Naja, denn wenn ihr beide glücklich zusammen seid, dann bin ich auch glücklich für euch. Doch bis du hörst andere Leute zu stören auf, werde ich nicht diese Dinge zu schreiben aufhören. Ich bin sicher, du kannst nichts gegen mich machen.
Ich emphfehle andere Leute diese Nachtricht in ihren LJ zu anschlagen; er wird wahrscheinlich weniger Orte, worin er will die Bilder anschlagen haben.

There is someone (I know who he is), who is posting forbidden pictures of my friend. I can not accept that! That is low and dishonest of him! He has posted these pictures in my friends' LJ and other websites. Shame on you, who support him! Shame on you, who only laugh! But most of all, shame on you, who does all this!
For whatever has my friend done to you? Her only guilt was that she was too nice, and fell for your lies. I admit that I do not know everything that has to do with this relationship, but no act is bad enough for this. No, I know people who are dishonest or terrible in personality, and to do as they do does nothing. You may believe that my friend was dishonest to you, but posting her forbidden pictures will not show her why you believe that is so. That means that you are a full of spite, and have an emptiness in your heart that you must fill with girlfriends whom you do not love. And if you cannot keep your girlfriend, you spite her, and do terrible things to her. You are a sad, lame and spiteful weak man. Indeed, you are the saddest, most lame, and most spiteful weakest man whom I have ever known.
Normally, I leave people alone, but if they interfere with other people, especially my friends, then I will do something about it.
I know that you fear me, so pay close attention: Never post my friend's forbidden pictures in my LJ! If you do that, I will delete them. You can do this as often as you want, but I will only do the same; you're only wasting your time. You cannot say that you cannot read this because I have also written this in English. If you want to prove that you are not a weak coward, then stand like a man and talk to me.
I feel sorry for your current girlfriend, not because she has to put up with you, but because she is insane, and wants someone like you. You probably think that she is not so, but it does not matter who she is. Oh well, for if you are both happy together, then I'm happy for you also. But until you stop disturbing other people, I will not stop writing these things. I am sure you can do nothing against me.
I suggest other people to post this message in their LJ; he will probably have fewer places in which he wants to post the pictures.
  • Current Mood
    amused amused

The Elements vs. The Bible

I have often said this, and I will say it again: The bible is not the only book that consists of moral teachings that can help people through life. While the bible may contain certain "teachings", if one wishes to call them such that people follow, it is by no means, the only book which consists of teachings, and most of the ones that exist are infinitely better than the bible, either in expression, in content or in truth. One book, in particular, that I would like to compare against the bible is The Elements, a work written by Euclid thousands of years ago.

There are actually several reasons why I wish to compare the two works. First of all, both of them were written since antiquity, and at this day and age, we may call the accuracy of the truth or revelation of anything written that long ago into question. Secondly, I find quite peculiar contrasts in the expression of both works. Which leads to the third--that the content of both works differ quite drastically. In fact, it is almost as if they are polar opposites of each other. I will address these points in more detail further on.

It is probably not very necessary to discuss the bible at length; the majority of people who may read this already know what it is; an authoritarian collection of scriptures written in 66 books that are divided into what are called the Old and New Testaments that forms the basis of the Christian religion. The bible has generally been rewritten into many different versions to suit particular sects in Christianity, and have had verses altered by priests and other figures in history, either for their own benefit or arbitrary reasons unknown to myself. However, I find it necessary to discuss the authors of the bible, and the manner in which it has been edited.
It is generally agreed upon by most impressionable people that the books were written by the people to which their names are ascribed to--however, is this really the truth? Upon closer inspection, evidence within the bible itself lends itself to questioning the identity of the authors, as the books indicate information of events or things that could not be known by the writers, who supposedly witness said events, or are otherwise blatant contradictions if the name of the book is also the name of the writer and witness of what he writes. Thus, we do not know who the writers are, nor are their accounts necessarily credible since they are not witnesses.
Secondly, it has often been attested to that the bible is considered to be the inspired word of God. But if this is true, why is it that there is no singular version of the bible that says the exact same things in them? How is it that arbitrary people or priests allow themselves to change certain verses for their own benefit, or for no apparent reason at all? If they had a reason to change it, why then are there other versions of bibles, not less "authentic" than the others that people still follow? If the inspired word of God were to have the impact as we supposed it would, there would be no reason to change it. It can be concluded that man has changed these words to suit their own purposes, and is not necessarily divinely inspired.

However, there are few people who are familiar with what The Elements is, what it implies to our modern world today, or who wrote it. Quite simply, The Elements is a 13 volume work which is a compilation of mathematical propositions in geometry--now not all of these propositions were founded by Euclid himself, but he was the one who compiled and organized these propositions from other Greek mathematicians into these 13 books. What makes The Elements so special is the logic which is applied into the propositions--in fact, only 23 definitions, 5 postulates and 5 axioms were all that Euclid needed to support the 465 propositions in The Elements. The Elements is quite simply a milestone in mathematics. It is, by no means, the most perfectly written work on mathematics, but its implications on geometry and math, and its methods are immense and well thought-out.
As for the author itself, there is no historical context, either from secondary sources or from the works itself that suggests Euclid did not write The Elements. However, even if he had not written The Elements, it is of no consequence for dishonesty to be admitted; for the content of The Elements stands on its own, regardless of whether or not he had written the propositions, the proofs of which demonstrate its truths. This is unlike where an author retells an account in writing, for which you have no reason to doubt because we assume the author has no reason to lie about what they saw--but this is not the case if the author did not actually see what he wrote about.
In other words, credibility aside, it does not matter who writes the proofs or propositions in a mathematical document; its validity is ascertained through the validity of its proofs.
It is quite clear that these propositions are not divinely inspired. They are, as stated previously, simply a collection of propositions known from other Greek mathematicians, the proofs of which may be provided by Euclid himself or others. If there was ever a need to edit The Elements, the truth of its propositions is an objective fact that can be stated differently and still mean the exact same thing. If Euclid had made a mistake, it is possible to include footnotes in later editions to state what he had missed. Even if one had neglected to note that Euclid made a mistake and simply changed it without telling the reader, there is no dishonesty to admit that a "divinely inspired" message is being altered for humanity's sake because there was no divine inspiration to begin with, and any changes made are for the sake of mathematical accuracy and revelation.
Regarding the honesty of both works, it is perfectly fine and well for Euclid to make a mistake in The Elements, not that he would want to, but because to admit to writing The Elements means he accepts it was not divinely inspired, and is entirely prone to human error, as it is with any other work--we expect it and admit these errors when they are found by other readers. This is only natural. But for the bible to admit divine inspiration would mean there is no error at all; and certainly, its authors want us to believe it is completely unalterable. That is not the case though; for the bible to have errors and be divinely inspired at the same time makes no sense--a divinely inspired message written with human error defeats itself.

I will now comment on the expression of both books. How much of the bible was originally intended as truth or as metaphor is not known to me. If it were to be admitted that none of its earliest readers had the slightest glimpse of what sat directly above them or directly below them, then it is easy to see how many of them might believe that everything above the clouds is the same heaven that their God dwells in, and everything sleeping deep beneath the Earth's crust is the same hell that their Satan dwells in. In other words, none of them had the scientific knowledge to know metaphor from truth. Some say the bible is written as a riddle; others say it's a metaphorical piece of literature; and still others read it literally, and can't read it any other way. Any given metaphorical reading may give rise to many different interpretations, the likes of which theologians and others can argue over and never agree. What exactly is meant to be expressed in the bible? Is there some message that is to be made clear from this sea of riddles, jumbled up metaphors and other bizarre lines? The expression of the bible is to be so diluted that the apparently divinely inspired word of God itself is just as diluted.
But for the moment being, let us suppose that we should read the bible as if it were expressed in metaphors. The problem with the metaphor, however, is that it is never exactly clear as to what exactly is to be implied. Does 6 days of creation imply 6000 days, 6000 years or 6 million? Who can say, external contexts notwithstanding? Is the flood as described of in the bible considered global, a physical impossibility which could never have, and never will happen on Earth, or is it a metaphor for a local flood, the meaning of which has been altered because none who read "global flood" could have known without the aid of science that a global flood is impossible? Who can say? Is the Earth really just a flat circle, or is it more spherical, and what are its "four pillars"? Without the knowledge of science, who can say?
The expression of the bible is so diluted and unclear as to make any kind of revelation from it impossible. It is not revelation to derive knowledge from science and claim that the bible predicts it, for a metaphor only tells the reader what he wants to hear. And man cannot want what does not exist; namely, what he does not know.

The Elements, on the other hand, makes clear exactly what its propositions are and how it is proven. For there is only one way to read it--literally. There is to be no metaphor implied in any of its books at all; to do so is highly unmathematical (and there are none, or they are meaningless, and irrelevant to the actual content of the work itself). Being that the content of The Elements is highly mathematical, it is unnecessary to require the reader external knowledge of more advanced or modern mathematics in order to make sense of this; in fact, no knowledge, save for logic itself is necessary (note that in general, save for definitions, this is true for mathematics in general). One can interpret for himself whether or not any of these propositions or their proofs make any sense, free of any other kind of knowledge; its validity can simply be verified by the reader's own thoughts themselves.
The expression in The Elements was clear enough that all the people who have read and analyzed it knew, unambiguously where Euclid's proofs might have holes in them, or when he admitted, albeit indirectly, the uncertainty of his fifth postulate.

It is clear, then, in addition to the consistency and clarity of the expression in The Elements over that of the bible, that the bible offers no protection against the dishonesty of the readers themselves. If the readers refuse to believe that the bible says the Earth was created in 6 days, they can hand wave it away by saying it's a metaphor; they can hand wave away any scientific inaccuracy in the bible by saying it's a metaphor. The Elements, on the other hand, is completely safe against this problem because it can only be read literally. To do otherwise is to defeat the dishonesty of its readers. To deny the proof of Pythagoras' theorem as given by Euclid and say it is not so is absurdity; there can be no other truth, as no counter example destroys Pythagoras' theorem, nor can a proof be given to negate the theorem, as one cannot prove a negative. The reader's dishonesty in The Elements destroys his own credibility, but his dishonesty in the bible destroys its credibility.

Secondly, I would like to call into light an interpretation of the content of both works. The bible generally consists of "historical" knowledge, or in other words, the events that each writer supposedly witnesses, and moral teachings. But what the modern reader would like to know is what exactly does the bible have to offer--does its "rich" history tell him anything that might be useful or interesting, and does its moral teachings make him better as a person? We shall delay the interpretation of the biblical history until the discussion of the truths or revelations of the two works and focus on the teachings.
As I have described the bible previously, it is considered authoritarian, and therefore, is not meant to be questioned. That is, the reader is not meant to question whether or not any of the bible's teachings are "right" or "wrong", and cannot be changed. While the bible does not change, however, times change and people change--society changes. It has changed, so much so, that what might be considered right in the past is no longer considered so in the present, and what is considered wrong in the past may no longer be the case in the present. The bible gives rise to many strange and bizarre morals--no doubt that some of them are ones we still adhere to today, (though I would like to say that not all people adhere to them because they were in the bible) but not all of these morals are preserved through time--slavery is not allowed, homosexuals are not condemned and euphemism is acceptable, even though it involves killing someone. If the bible is authoritarian and we are to take every single moral teaching in it at face value, and then compare it with the morals that society in general accepts, one can begin to see what a difference there is between the two--if the bible were actually an excellent, nay, not even that, but a suitable authority on morality today, then how is it that we have to ignore certain teachings in the bible in order for society to benefit? If we suppose that society today is generally much better than it was in centuries past (and it certainly is in quite many ways; there is less discrimination against certain races and genders or people of different sexual orientations than before), how could this have been achieved by ignoring teachings in the bible? There are two things that can be concluded from this--the bible is not a suitable authority on modern morality, and that its authority cannot be universal.
There is something to be said of the moral teachings in the bible that are still being adhered to, however. And that is that a great number of them are not original. Civilizations before the bible have long had these philosophies and ideals before the bible did, and it is quite possible that the bible had simply borrowed them. There is nothing special about the content of the bible; considering its expression, there are definitely better moral alternatives that are both clearer and suitable for modern morality.
That being said, the content in the bible fails to pass the test of time.

Having explained in general what The Elements is, it is quite clear that its content is highly mathematical. There is no doubt that the propositions in The Elements might already have been considered and not particularly new at all; however, it is the organization of The Elements that makes it most sought after by ancient scribes and mathematicians looking to analyze it alike. Furthermore, it is the ingenuity that Euclid's proofs exhibit that make The Elements so remarkable; it inspires mathematicians, expands readers' perspectives in thinking and solving problems, and is a fundamental work of logic. It is, as previously stated, by no means perfect; it is, however, an excellent start for mathematical thinking, either in pointing out some of its flaws, or in broadening the readers' scope in the usage of logic. Despite some of its flaws, the propositions in The Elements are all entirely accurate, and work. That is, all of them could and are still being used today.
Truth be told, anything in mathematics that is considered to be proven accurately is indeed true, and works universally. Regardless of time, a mathematical theorem or concept, so long as its proof is valid, will be false at no given time, as an abstract concept exists independently of time, and is irrelevant to who discovered it first or whether or not anyone even knows it exists. Mathematical truths are therefore universal, and will always stand the test of time--so long as they really are truths. We will return to this again as we discuss the truths of both works.
But for now, it is sufficient enough to concede that The Elements has passed the test of time.

It may seem unfair that I am comparing the content of a work that is subjective with one that isn't--but one may make subjective implications from The Elements, regardless of whether or not the reader is conscious of them or not. For what is there to be gained from reading either of these? If one were to read the bible, there is a world filled with terror, violence, bloodshed, famines and the wrath of a "benevolent" divine being, and on the other hand, morals that refuse to change with the times, the betterment of humanity and society in general. If the same were to read The Elements, there is logic to be had; the brain at work, rationality to be appreciated, and a widening appreciation of mathematics, logic, problem solving and thought in general--as it is with mathematics in general, this serves not only to make the reader better at mathematics, but problem solving applied to any other problem of reality; the playground need not be an abstract field of mathematical space, and the elements, not numbers or variables--they could apply equally well to problems that occur in everyday life, or even the way in which the reader thinks. If one were to think of both works as compilations, the bible as a compilation of morals and The Elements as a compilation of geometric propositions, then one will see that the bible, as a compilation, is confusing and full of fluff; there is so much sifting required just to remove only a few moral teachings that are still upheld by today's society. The Elements, as a compilation, is full and complete; its meaning or importance, regardless of time, needs no sifting, and none of its propositions may be considered fluff.
I would much much rather have my thinking improved than to follow a set of morals that do not fit the times of my own society.

At last, we now come upon the final comparison of both works--their truths. As in Thomas Paine's own words, I shall describe what is meant by a revelation. A revelation is a universal truth or objective fact. If something is subjective, it can have no revelation. If you have learned something through hearsay, it is not revelation. If a statement is false, it is not a revelation. If the validity of a fact is untested or unknown, it is not a revelation. Literature is not revelation. A metaphor is not a revelation. (I shall explain the last one in more detail as we go along.) So long as what you are being told tells you nothing new that you did not know about before (and is true as it pertains to reality), it is not a revelation.
As I have discussed so far, the expression and content of the bible, it is difficult to believe that there is any sort of revelation in it. Since morality is largely subjective, any of the bible's moral teachings cannot be considered a revelation. To take the metaphorical interpretation of the bible again, we see that any and all metaphors contained in the bible are also not revelation. As we have touched on metaphors previously, we have seen that no one can say what a metaphor must mean because it means only that which we want to believe it means; in other words, it is simply a dishonesty of the reader, quite possibly because the author assumes his readers will know what he means, or he is too lazy to clarify what he means. But naturally, a fact which is something that means what one wants it to mean is nothing new; as I have said before, one cannot want what he does not know. Therefore, what he wants is not revelation; a metaphor, what a reader interprets to be what he wants it to be, is not revelation.
The only thing left in the bible is its "historical" content. But as we have already stated at the beginning, there is no reason to trust the accounts for what they are because it is difficult to accept that the writers who supposedly witnessed the events and wrote about them actually are who they say they are. This dishonesty of the writers makes the revelation of what they write even worse. There are two ways to see this: the writing may in fact, actually be true, independent of what actually happened; in other words, the writers were simply lucky, or what they write about is false, and clearly, there is no revelation. In the former case, we do not consider the fact a revelation because the writers wrote about it, but we call them such because of the methods which we used to determine that the writers were not lying. Of course, if the writers had not lied about their identity to begin with, we might be more lax in accepting their accounts as revelation, but that's clearly not the case (this is still difficult to swallow on the account of the writers alone because it must be admitted that we are merely reading hearsay, not truth). But upon closer inspection, however, there is no revelation whatsoever in the historical content of the bible.
There are two sources that vouch for the accuracy in the bible; one being from science and the other being the bible itself. Since we have already taken care of the metaphorical interpretation of the bible, suppose, for a moment, that we were to take the literal one instead. Now, we read that the Earth was created in 6 days, existed for 6000 years, there was a global flood, and it's a circle with four pillars. When compared against the objective fact of scientific discoveries, we see that the Earth formed over millions of years, existed for approximately 4.5 billion years, there was no global flood, and is shaped roughly like a sphere. It has already been accepted that scientific knowledge, since it is objective and pertains to reality, is indeed revelation--but what we have been told from the bible completely contradicts scientific fact--that is no revelation at all.
I guess we already knew that, but it is also even more important to consider the methods in which the writers, witnesses or observers knew or saw what they saw--that is to say that scientists and mathematicians document quite well and meticulously their findings and explain exactly how they came to their conclusions. Is there any kind of discussion that indicates how the writers knew the shape of the Earth or any of its properties? We are only left with short but unsatisfying answers of "God did it" and "divine inspiration from God's words". God's existence is not a revelation though; it is not proven, nor is it necessarily reality--therefore, it is not satisfactory enough to be considered a revelation.
Then the bible itself is the source of its own undoing. For it is entirely consistent, not of itself, but of its contradictions and blatant lies. As has already been conceded, the evidence that the writers are not who they say they are is readily apparent from the bible itself--Moses saying he is meek is either a lie, or the writer is not Moses, and what they write about betrays their identities. Furthermore, the bible consists of accounts that write of the exact same events over and over again--yet, few, if any of these, ever agree with each other. This is what we call a contradiction, and no contradiction holds truth--there is no revelation in any of them.
The intellectual dishonesty of the writers is clear, either in their identities or in what they write. And intellectual dishonesty holds not a shred of revelation.

The revelation and truth of mathematics is in the validity of their proofs. As I have already mentioned, The Elements is not perfect and does contain flaws--despite this, all of its propositions are still true. This is not because we have accepted this at face value, however, but because we have proven the propositions ourselves, and it is explained rather clearly how Euclid arrived at his conclusions. It is not necessary to call the propositions, the proofs of which contained holes revelations, at least on account of The Elements, but the propositions by themselves, if proven successfully by the reader (or if the holes are to be patched) are revelations. There is, furthermore, also revelation in discovering the holes in Euclid's proofs when they do occur, as they lead to further truths and better proofs. As I have said before, mathematics requires no prior knowledge; all of it can be attained strictly from the mind itself. (That is to say that knowledge of mathematics simply makes it easier to do this; however, memory is not necessary for the problem solving or thinking aspect of mathematics.) Naturally, the revelations in The Elements will occur when Euclid's proofs of the propositions are valid.
As far as I am concerned, there is no intellectual dishonesty in The Elements. If something is found to be false (ie, a hole) in it, it is not ignored or impossible to subject to doubt; Euclid himself did not write these to lie about math, but did the best he could to put these propositions together--if he had intended to lie, then the propositions would not be true, and yet, they are. It is much easier to relate to the intellectual honesty of the author when their true identity is known and not lied about. Even so, it is difficult to lie about mathematics--almost all kinds of dishonesty can be quite easily banished from mathematics from the mere and simple fact that dishonesty defeats the liar in any given discussion of mathematics. In other words, the prudent mathematician would not make a mistake or lie about something on purpose. If he ever did, it would be to admit human error or because he did not think about it--he did not want to make a mistake.

It is almost embarrassing to state how much more credibility The Elements has over the bible, despite the fact that some of Euclid's proofs contain holes. If one were to suggest there is revelation in picking apart the contradictions in the bible or its scientific inaccuracies, I must disagree. For there is nothing to be gained from pointing out that the bible is wrong; you must use what you already know in order to know that it's wrong, and giving revelation to the bible taken from scientific knowledge is simply stealing the credit from science itself. But for mathematics, that is entirely different. To point out a hole, you may have to think about something which you have never done before. After finding the hole, you must find a proof that works. To find the first one that works is a revelation. To find a different one not known to others is also a revelation.

A final remark regarding the truth of both works--it should be noted that both the bible and The Elements may have been subjected to edits over time, whether it's due to the scribe's copying or translation of the work, or because they themselves purposely altered the work. I have also already mentioned the dishonesty of the reader which can easily exert itself upon reading the bible; this is certainly true of the authors; they can change as much as they want as no one is to say whether or not its original was any more true than the current one; a lie changed into another is still a lie, and, contradictions aside, no revelation comes from something which requires external knowledge to know. In contrast with The Elements, however, the dishonesty of the author, as it is with its readers also defeats itself; to make more holes in a proof tarnishes The Elements, something no scribe would wish to do; if any alterations were to be made, they would have to be made for the sake of making the proof more clear or for eliminating truths; this only makes the revelations in the books more clear. If it made any more holes, however, revelation is still possible because no external knowledge is required to realize it. So it is interesting to note that no matter how many alterations may be made to either the bible or The Elements, it is only The Elements that retains its original revelation and truth throughout all this time; no matter how many times the bible has been edited or changed, it still retains no truth.

Had the bible been admitted to fiction rather than the ridiculous notion of an unalterable, divinely inspired message, this comparison might be rendered moot as it is to compare science with fiction (because then we know it's not true).

I cannot quite understand how it is that an embarrassingly ridiculous book as the bible gets read more than The Elements does. How it is that people take morality from a book in which society no longer retains the majority of its moral teachings, find revelation in judgments they already have, or make ambiguous interpretations off of it for themselves I will never know. Perhaps it's because people don't want to think for themselves. Perhaps it's because they deny the truth. Perhaps it's because they fear the converse, despite the fact that the converse is much more plausible than what's written. Perhaps it's because they prefer the drama. Perhaps it's because they like to read what they want to hear. Perhaps it's because many people fear math. Perhaps it's because many people are much too intensely irrational and emotional.
I have, at one point, heard a professor say that The Elements is the kind of book, along with other mathematical works, that people read when they grow old. Why is it that people need to wait until they are old and feeble to stop fearing rationality or mathematics in order to learn about it and improve their minds? Why is it, that such a ridiculous and dramatic book as the bible, with its terror and violence, is encouraged to be read by both children and adults, despite the fact that its content might not be suitable for the former, and why have we to learn from a book that is mostly fluff, and when it isn't fluff, makes no sense at all, or has nothing to do with modern society today? Would it not make more sense to read a book that improves one's own thinking when one is able to think and has an entire life ahead of himself, and not read books that don't tell us anything about today's society, or offer any sort of revelation, rather than to read such books when he will have no more use or benefit for improved thinking? I shall never understand why people are as they are.
But given the choice, I'd rather read The Elements than read the bible.
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The Mathematicians

I think I spent almost two months just working on this. Excluding the time not working on this piece, it probably would have taken me about a month if I went at it non-stop. But now it's all finished.

Here are the mathematicians featured in this picture:

Gauss, Newton, Archimedes, Euler, Cauchy, Poincare, Riemann, Cantor, Cayley, Hamilton, Eisenstein, Pascal, Abel, Hilbert, Klein, Leibniz, Descartes, Galois, Mobius, Jacob, Johann and Daniel Bernoulli, Dirichlet, Fermat, Pythagoras, Laplace, Lagrange, Kronecker, Jacobi, Bolyai and Lobatchewsky, Noether, Germain, Euclid, Legendre

I'm not entirely sure of the accuracy of the biographical information below because it's just from what I remember mostly, but if anyone spots any errors, please point them out. Thanks.

Also note that the pictures might not actually represent how the mathematicians in real life looked like, either because I purposefully made tweaks to their hair or because there weren't good reference pictures of them (or authentic portraits were lacking).

Unfortunately, there are more mathematicians than there is space in this picture to give recognition to each and every mathematician in history that is worthy of it; most of the ones in this picture are mathematicians I am most familiar with from my math history class.

Karl Friedrich Gauss 1777 - 1855
Considered to be one of the three greatest mathematicians in history. Known for constructing a regular 17-sided polygon with only a compass and ruler (this feat was never discovered since the Ancient Greeks, who knew only up to 15 sides), concluding that any polygon with the number of sides equal to a Fermat prime can be constructed, works in his Disquisitiones Arithmeticae on number theory, developed the modulus notation, discovered the fundamental theorem of algebra, calculated the orbit of Ceres, various works on electromagnetism and geodesy, invented the heliotrope, and other contributions too numerous to mention. Did not publish his thoughts on non-Euclidean geometry for fear of being rejected. Considered to be the last universalist before Poincare.

Isaac Newton 1642 - 1727
The second of the three greatest mathematicians in history. Known for discovering gravity, various works in physics, co-inventor of calculus and his best works, Principia. He worked alone. He invented his own telescope and discovered the binomial theorem. He hates disagreeing with people because he hates being wrong; that's why people say he's nasty. But he claims that his work is akin to sitting on the edge of the ocean picking up seashells, never knowing what lies at the bottom of the ocean.

Archimedes ~ 287 - 212 BC
The last of the three greatest mathematicians in history. Known for developing the concept of the lever, inventing the screw pump, ratios of volumes between spheres and cylinders. Rumored to have run through the streets screaming "Eureka!" at the discovery of a method of determining gold from fake gold. He was killed by a soldier during war; the cause of this dispute is unknown; either the soldier had stepped on his work on the ground and angered him, or he had refused to go with the soldier in order to finish the solution to his math problem. He also liked doing math everywhere; if there was soot nearby, he'd write in it. He even wrote in the oils on his skin, which was applied after bathing as is the custom in Ancient Greece.

Leonhard Euler 1707 - 1783
"Analysis Incarnate", as some call him. Known for various works in number theory, the sum of 1/n^2, development of the concept of functions with D'Alembert, and capable of calculating large calculations entirely in his head. Liked children and had many of them. Slowly went blind, and was completely blind by the time he was 70. His blindness did not hinder his mathematical insight, but rather, it increased after he became blind.

Jules Henri Poincare 1854 - 1912
Considered to be the last universalist in mathematics. Known for conjecture on three body problem and concepts related to the development of relativity theory--some say that he should deserve all the credit for it instead of Einstein. The circle slightly above him is a Poincare disk model, used to visualize lines in a sphere in hyperbolic geometry.

Augustin Louis Cauchy 1789 - 1857
A contemporary of Gauss. Known for developments in calculus including certain concepts of limits and continuity, some algebra and complex analysis. The formula next to him is known as Cauchy's Theorem, used in complex analysis, and below it, the well known Cauchy's inequality.

Bernard Riemann 1826 - 1866
A German mathematician whose originality in thought impressed Gauss. Known for ideas in non-Euclidean geometry and integrals. He died young from illnesses. The sphere next to him is a stereographic projection of a Riemann sphere.

Georg Cantor 1845 - 1918
A German mathematician whose methods were consistently criticized by Kronecker. However, he is known for developing the concept of set theory. Though his ideas were accepted by Hilbert and other great mathematicians, he could not get over Kronecker's criticism and admitted himself to a mental institution. The fractal next to him is a Cantor set.

Arthur Cayley 1821 - 1895
A British mathematician. Found the theory of invariants with his friend Sylvester, and succeeded in having women admitted to Cambridge. Also known for the concept of n-dimensional geometry.

William Rowan Hamilton 1805 - 1865
Considered to be the greatest Irish mathematician. By the time he was 14, he knew as many languages as he was old. Known for discovery of complex variables in the fourth dimension and the algebra of quaternions, the former of which he discovered when he could not find a way to represent complex variables in the third dimension. Had drinking problems in his later life.

Ferdinand Gotthold Max Eisenstein
A brilliant mathematician and pupil of Gauss. His mentor considered him to be one of his greatest students and one of the greatest mathematicians. Unfortunately, he died young.

Blaise Pascal 1623 - 1662
Originated the mathematical theory of probability. Was a French mathematician who posed cycloid problems to other mathematicians and also known for his converse of Descargues' theorem in projective geometry. The triangular array of numbers in front of him is Pascal's triangle, and are also the coefficients of the terms in a binomial expansion.

Niels Henrik Abel 1802 - 1829
A Swedish mathematician who lived in poverty. He taught math and did some work on algebra. He died young before his contemporaries could give his work recognition.

David Hilbert 1862 - 1943
One of the successors to Gauss' former position as the director of the observatory at Gottingen. Made some contributions to algebra. Supported Cantor's set theory. Tried unsuccessfully to get Emmy Noether a faculty appointment at Gottingen. He was also known to be slow at grasping new concepts in an attempt to understand it completely.

Felix Klein 1849 - 1925
Another of Gauss' successors at the observatory of Gottingen. Made contributions to algebra, and also known for the concept of a Klein bottle (pictured).

Gottfried Wilhelm Leibniz 1646 - 1716
One of the founders of calculus with Newton. However, the competition between himself and Newton was bitter. He was also skilled in other areas besides mathematics, including philosophy, politics, law and history.

Rene Descartes 1596 - 1650
Well known for his phrase, "Cogito ergo sum" and the Cartesian coordinate system, thereby creating an entire system of geometry. The phrase is often misinterpreted to mean one exists because he thinks, but it means that the act of thinking is the only truth that exists.

Evariste Galois 1811 - 1832
A brilliant mathematician whose genius was not well recognized. His examiners had difficulty understanding his explanations, and he often proclaimed most of them were so easy as to not require an explanation. He wrote very little in his career and accurately predicted he would die in a duel. Known for work in group theory, Galois theory and algebra.

August Ferdinand Mobius 1790 - 1868
A German mathematician from whom the Mobius strip is named after. The Mobius strip is an object which has only one side. Also made contributions to algebra.

The Bernoullis (Jacob 1654 - 1705 (pictured left), Johann 1667 - 1748 (pictured right) and Daniel 1700 - 1782 pictured (below))
The Bernoullis are a family of brilliant people, some of which are mathematicians. Daniel Bernoulli was the son of Johann Bernoulli, and made many contributions to applied mathematics. His father and Jacob Bernoulli were in competition with each other, and fought often. One of their disputes involves the question of what shape a string should be in order for a bead to travel from one end to the other most quickly (the correct answer is a cycloid). Daniel Bernoulli was often excluded from disputes between Euler and D'Alembert.

Peter Gustav Lejuene Dirichlet 1805 - 1859
One of Gauss' pupils, whose works in number theory were inspired by his mentor. Apparently, on his jubilee lecture, Gauss wanted to burn the original of his Disquisitiones Arithmeticae, and was about to light his pipe with it, when Dirichlet saw him doing that and saved the original in time (I don't actually know if this is true though; I read it somewhere).

Pierre de Fermat 1601 - 1665
Considered to be the greatest mathematician of the seventeenth century. Known for his work in number theory, and his last theorem (which he claimed to have proven, but no evidence of this has been found), which has caught the attention of many mathematicians and other challengers. He also created the Fermat primes, which have later been shown not to be primes. Gauss was not interested in proving his last theorem.

Pythagoras 572 - 492BC
His well known theorem regarding right angle triangles is actually a proof of a Babylonian theorem. However, he is credited for his abstraction of numbers, including the property of even or odd numbers. He suggests that all things are considered to be numbers.

Pierre-Simon de Laplace 1749 - 1827
A French mathematician who made many contributions to mathematical astronomy and physics. Known for his Laplace equation in calculus and Laplace transforms. Some consider him to be as great a scientist as Newton, and call him a French Newton.

Joseph-Louis Lagrange 1736 - 1813
A mathematician with bad eating habits. He first proposed the mean value theorem in calculus, and did a little bit of work on number theory. However, his Mecanique is consider his best work.

Leopold Kronecker 1823 - 1891
A mathematician who did work in algebra and number theory. He mastered Galois' theory of fields before others, but was critical about using mathematicians using irrational numbers, and said mathematics should be based on relationships between integers; he said to Lindemann that irrational numbers don't exist. He was also critical towards Cantor, and did not agree with his concepts. this eventually caused Cantor to admit himself to a mental asylum.

Carl Gustav Jacob Jacobi 1804 - 1851
A mathematician whose reputation is often mistaken with his brother's. Known for his work in number theory, algebra and Abelian functions.

Janos Bolyai 1802 - 1860 and Nikolas Ivanovitch Lobatchewsky 1793 - 1856
Both mathematicians were the first to introduce the concept of non-Euclidean geometry to the public (remember that Gauss did not do this). Their ideas were challenged due to the popularity of Kant's Critique of Pure Reason, in which the idea of non-Euclidean geometry would be made absurd. While Gauss commended both mathematicians for their work, only Lobatchewsky received support from Gauss in being admitted to Gottingen, but in his letter to Bolyai, Gauss claimed that giving credit to him would be like giving credit to himself. Lobatchewsky also challenged Euclid's fifth postulate, using non-Euclidean geometry for counter examples.

Emmy Noether 1882 - 1935
A mathematician who was one of two female students out of a thousand students in the university of Erlangen. She was influenced by Hilbert and Klein, and although Hilbert tried to help her get an appointment in Gottingen, he did not succeed. She is known for her original work in noncommutative algebra.

Sophie Germain 1776 - 1831
A mathematician whose parents discouraged her from pursuing the sciences. She was influenced by Gauss' work in number theory, and when she made some discoveries on quadratic reciprocity, she wrote to Gauss about them under the disguise of a man (because she feared he would not accept her if he knew her gender). However, when she did reveal her identity, Gauss was impressed with her work and admired her even more because it would have been harder for a woman to succeed in sciences, due to society's prejudices.

Euclid ~325 - 265BC
A Greek mathematician known for his works in geometry in The Elements. His works, however, are restricted primarily to plane geometry, and some of his postulates, including the last one do not work on non-planar surfaces. However, his ideas in geometry have been well accepted for centuries.

Adrien Marie Legendre
A mathematician with some works in number theory. His theory of quadratic reciprocity was never successfully proven by himself, but by a younger Gauss, whom Legendre was mostly jealous of.

Anniversarius Dedicatio Duocentesimus Tricesimus Primus Carolo Frederico Gauss

Haec dedicatio est causa celebrantis Caroli Frederici Gauss, Principis Mathematicorum, hac die pridie kalenda Maia, MMVIII.

Gauss fuit unus trium mathematicorum maximorum qui mirantem prudentiam mathematica habuit. Natus MDCCLXXVII, Brunswick, Germania, ingenium atque tantum tres anni aetate demonstravit. Dum suus pater inceptum mathematica studente recusavit ingeniumque mathematicae ingoravit, ipse studio mathematica cum auxilio matri et Carolo Guilielmo Ferdinando praevaluit. In mathematica, omne area Gauss fuit dexter. Etiamsi hoc, arithmeticam reginam mathematicae nominavit; mathematicam, reginam scientiam, et fortasse maximam scriptarum operarum Disquisitionem Arithmeticae scripsit. Non modo ipse fuit mirabilis mathematicus, sed etiam mirabilis astronomus, ac rectorem observatorii Gottingene factus est. Fortasse in astronomia maximum factum fuit computatio ambitus Cereris. Fuit nihil mathematica in illo die quod Gauss non facere potuit. Quicquid aequales comperiverunt, ipse iam comperiverant--et plus. Multa haec in mathematica ephemere quod etiam iam difficile lectu tentus sunt. Tantum fuit studium mathematicae ut recognoscens idem factum ceteris ipsi pauca interfuit famaque fortuna non coegerunt. Gauss fuit dexter, non modo in mathematica, sed etiam in lingua. Enim non modo Germanam linguamque Latinam linguam sed etiam multam linguam Europa didicit, et cum sexaginta duo anni aetate fuerit, Russiam linguam conatus est, ac paucis annis, omnino hanc linguam perdidicit, ipseque bene legit ac scripsit. Talis fuerunt facta tanti viri ut Gauss. Et tamen, alicui ut dextro ut ipsi, cuius scivit paucum edidit. Sua sententia fuit "pauca sed matura", et arbitratus est quid esse levis aequalesque quid potuerunt uti non edidit. Etiam cum scivit ceteros suas sententias non intellexerit non edidit. Gauss mortuus est ante diem septem kalenda Martia, MDCCCLV, Gottingene, Germania. Postea Principem Mathematicorum nominatus est.

Tamen, etiamsi magnum ingenium Caroli Frederici Gauss in mathematica, ac honorque reverentia quas ipse meruit, vita asperior aut tristior ipsi non esse potuit. Nam sua uxor, quam quam aliquas in orbe terrarum amavit modo quinque annis post conubium vixit, tum mense post tertium infantem natum mortuus est, fotusque Carolus Guilielmus Ferdinandus saeve Napoleone occisus est, paucosque amicos quod magnum ingenium, tantum super ceterorum solitudinemque hostesque quod facta ceterorum non recognovit habuit, melancholiamque ac hypochondriam ac alios morbos senectute passus est. Mathematica fuit tantum suum solaciumque medicamen. Et tamen, nihil horum facientem magnum virum impedivit. Nam ipse fuit magnus, non modo in mathematica ac ingenio, sed etiam in more. Humanusque lenis, placidusque mollis fuit, certaminaque pugnas non amavit, religiosamque intolerantiam non praedicavit, tamen, potens in animo ac ingenio.

Id est magna cum honore quo anniversaria dedicatio duocentesimus tricesimus primus Carolo Frederico Gauss celebramus, viro cuius moresque ingenium sunt dignus celebrantis. Ita huic occasioni, hanc effigiem Caroli Frederici Gauss cum uno eius noto dicto, "Si ceteri veri mathematica meditentur ut penitus ac perpetuo meditatus sim, mea inventa faciant." ac "Regina scientiae est mathematica, ac regina mathematicae est arithmetica." traxi.

This dedication is to commemorate Carl Friedrich Gauss, Prince of Mathematicians, on this day of April 30, 2008.

Gauss was one of the three greatest mathematicians, who had an amazing insight in mathematics. Born in 1777, Brunswick Germany, he displayed his talent as soon as he was only three. While his father resented his attempts in studying mathematics and ignored his genius in mathematics, he prevailed in his study for mathematics with the help of his mother and Duke Carl Wilhelm Ferdinand. Gauss was well versed in every field in mathematics. Despite this, he dubbed arithmetics queen of mathematics; mathematics was queen of science, and he wrote the Disquisitiones Arithmeticae, perhaps the greatest of his written works. Not only was he an extraordinary mathematician, but also an extraordinary astronomer, and became the director of the observatory in Gšttingen. Perhaps his greatest achievement in astronomy was his calculation of the orbit of Ceres. There was nothing in mathematics in his day which Gauss could not do. Whatever his contemporaries discovered, he had already discovered--and more. Many of these were kept in his mathematics journal which even today are difficult to read. So great was his devotion to mathematics that he cared little to recognize the same achievements of others, and fame and fortune did not drive him. Gauss was skilled, not only in mathematics, but also in languages. For he learned not only German and Latin, but also many European languages, and when he was 62, he tried Russian, and in a few years, had mastered the language completely, and read and wrote it well. Such were the achievements of so great a man as Gauss. And yet, for someone as skilled as himself, he published little of what he wrote. His motto was "few but ripe", and did not publish what he thought to be trivial and what his contemporaries could use. When he knew that others would not understand his ideas, he also did not publish. Gauss died on February 23, 1855 in Gšttingen, Germany. Afterwards, he was dubbed the Prince of Mathematicians.

And yet, despite the great genius of Carl Friedrich Gauss in mathematics, and the honor and respect he deserved, life could not have been more harsh or more sad for himself. For his wife, whom he loved more than anyone else in the world lived only five years after marriage, then died a month after his third child was born, and the cherished Carl Wilhelm Ferdinand was brutally killed by Napoleon, and he had few friends because of his great genius, so much above that of others and isolation, and made enemies because he did not recognize the achievements of others, and suffered melancholia, hypochondria and other illnesses in his later life. Mathematics was his only solace and drug. And yet, none of these things prevented him from becoming a great man. For he was great, not only in mathematics and genius, but also in character. He was kind and gentle, calm and sensitive, and did not like conflicts, and did not preach religious intolerance, yet, he was strong in spirit and intellect.

It is with great honor that we commemorate the 231st anniversary dedication to Carl Friedrich Gauss, a man whose character and genius are worthy of commemorating. So, for this occasion, I have drawn these pictures of Carl Friedrich Gauss with one of his well known quotes, "If others would reflect on mathematical truths as deeply and continuously as I have, they would make my discoveries." and "Mathematics is the Queen of Sciences, and arithmetic, the Queen of Mathematics."

Here also is a Gauss Widget for those using a mac (OS X Tiger or higher).

And also commemorating this event, are some cool math problems I've encountered on exams, from other people, and on my own, which I've successfully solved on my own. You'll find some of these easy, and some of them hard, although I included the easier ones because they were still interesting. I may choose to post the solutions later. I've divided them into three sections, the Pidgeonhole Principle, Logic and Proof Related (you'll find the logic stuff the easiest).

For those not familiar, the pidgeonhole principle is the idea that if you are given n sets and m number of elements, where m>n, then there is at least one set which contains at least m/n elements rounded to the nearest integer (up).

Pigeonhole Principle

1) Socks in a Drawer
A drawer contains 6 black socks, 8 white socks, and 10 blue socks. One night you need to get some socks but the lights go out.

Without being able to look inside the drawer, what is the smallest amount of socks that you need to take from the drawer to guarantee that you have picked up a pair of identically colored socks?

What is the smallest number of socks you need to take from the drawer to guarantee that you have picked up 5 socks of one color?

What is the smallest number of socks you need to take from the drawer to guarantee that you have picked up 5 blue socks?


1) Five people, Five Occupations, and Two Genders
Five people with family names Dow, Elliot, Finley, Grant and Hanley have exactly one of the following occupations: appraiser, broker, cook, painter and singer.
Use the following information to determine each person's sex and occupation.
-Three of these five people are men.
-The broker and the appraiser are father and son.
-The singer, the appraiser and Grant like Italian food.
-Dow and Hanley are playing in a ladies golf tournament.
-The singer told Finley that he likes hockey.
-The cook owes Hanley $50.

2) The Prisoners in The Tower
An elderly queen, her daughter, and little son, weighing 195 pounds, 105 pounds and 90 pounds respectively, were kept prisoners at the top of a high tower. The only communication with the ground below was a rope passing over a pulley with a basket at each end, and so arranged that when one basket rested on the ground the other was opposite the window. Naturally, if the one were more heavily loaded than the other, the heavier would descend; but if the excess on either side was more than 15 pounds, the descent became so rapid as to be dangerous, and from the position of the rope the captives could not check it with their hands. The only thing in the tower to help them was a cannon ball, weighting 75 pounds. They, not withstanding, contrived to escape.

How did they manage it?

3) A Traditional River Crossing Problem
Three men, traveling with their wives, came to a river that they wished to cross. There was but one boat, and only two could cross at one time. Since the husbands were jealous, no woman could be with a man unless her own husband was present. In what manner did they get across the river?

Bonus: Try this problem with four or more couples.

4) Handshakes at Dinner
Four couples attended a dinner. At the beginning of the dinner, they shook hands (a couple does not shake hands with each other, and one does not shake hands with oneself). When the host asked how many people shook hands, there were seven different answers. If you know how many people shook hands with the host, how many people shook hands with the host's wife?

Proof Related

1) Laplace Equation
Verify that if u(x,y) and v(x,y) are two functions that satisfy the Laplace equation, ∇²f = f_xx + f_yy = 0 and ∇u∙∇v = 0, then the function uv(x,y) = u(x, y)v(x, y) satisfies the Laplace equation.

2) Ceva's Theorem + Area of a Triangle in a Triangle
Consider a triangle ABC. Take points A₁, B₁, C₁ such that A₁ ∈ BC and BA₁: A₁C = 2:1, B₁ ∈ CA and CB₁: B₁A = 2:1, C₁ ∈ AB and AC₁: C₁B = 2:1. Consider the triangle which sides belong to the lines AA₁, BB₁, CC₁. Prove that its area is equal to 1/7 of the area of the triangle ABC.

3) Number Theory
Suppose, for some positive integer, n, that 2n + 1 is prime. Prove that 2n + 1 | M_n or 2n + 1 | M_n + 2, where M_n = 2ⁿ - 1.
(Hint: use M_n(M_n +2).)

4) The Last Digit of Every Fermat Number
Prove that the last digit of every Fermat number, 2^2^k + 1 is 7, where k ≥ 2.

5) Divisibility Tests
Prove the following:
A number is divisible by 2 if its last digit is divisible by 2 (is even).
A number is divisible by 3 if the sum of its digits are divisible by 3.
A number is divisible by 4 if its last two digits are divisible by 4.
A number is divisible by 5 if its last digit is 0 or 5.
A number is divisible by 7 if the difference of the sum of its digits except the last one and the last digit doubled is divisible by 7.
A number is divisible by 8 if its last three digits are divisible by 8.
A number is divisible by 9 if the sum of its digits are divisible by 9.
A number is divisible by 11 if the difference of the sum of its alternating digits is divisible by 11.

Bonus: Find a divisibility test for 2^n and 5^n and prove it.

6) Right Triangle within a Circle
Construct a triangle inside of a circle with one side the diameter of the circle, and one vertex along the circle itself. Prove that any such triangle you construct must be a right triangle.

7) 1 = 0.999...
Prove that 1 = 0.999...

8) The nth Residues of a Prime
Prove that the nth residues for a prime, p occur the same number of times, for 1 ≤ b ≤ p-1 in a ≡ b^n mod p.

9) The nth Residue, m - 1 of m
Prove that for the nth residue of m, such that a^n ≡ b mod m, (m-1)^n ≡ (-1)^n mod m.

10) Geometry in a Square
Construct a square, ABCD. Now construct a bisector of ∠BAC that meets on BC at a point F. Now construct a line from a point E on AC to F, where AE = AB. Prove that FE is perpendicular to AC and that BF = FE = EC.

11) Menelaus Problem
Prove that in any non-isosceles triangle ABC the three points of intersection of the bisectors of its external angles with the opposite sides belong to one line. Hint: If P is the point of intersection of the bisector of the external angle A with the extension of the side BC, then PC:PB = AC:AB. Prove it (using similar arguments to our proof of a similar statement for the interior bisector).

12) 6p Congruence
If a is an integer and p is prime, and p > 3, show that a^p ≡ a mod 6p.

13) Angles in a Circle
Suppose there is a circle with two line segments, each intersecting the circle at two different locations, (denote these points, A and B) and which intersect each other at another point on a circle, C. Prove that if there are two other line segments passing through the same two points as the first two segments and which meet at a different point on the circle, D, the angles ACB and ADB are the same.

Bonus: For a regular n-gon inscribed in a circle, draw line segments from one vertex of the n-gon to the other n-1 vertices. Prove that the angles between these lines and the first vertex are all the same.

14) Ratios in a Triangle
Suppose there is a triangle with vertices, A, B and C. Suppose D is the median of AB, and E is the median of DC. Prove the ratio of the sides along AC when it is cut by a line through BE is 2:1.

15) Relation between Pythagorean Triples and their Product
Prove that for any Pythagorean Triple (a^2 + b^2 = c^2; a, b, c are integers), their product abc is divisible by 60.

And also, I will now introduce a mathematical object in R^3: a polyhedron, with regular faces, the octahedron. You can take some paper and make one for real! Here's how:
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Something you can do with a Sheet of Paper

Hey look, it's math even normal people can do!

I actually thought about this while doing an assignment for geometry class, which involved drawing a figure on graphing paper, of which I had none at the moment. I'm not poor, and it's probably unusual for a math geek such as myself to not have graphing paper, although I guess I don't have any because I don't believe in measuring stuff in geometry. It's the theory that counts. So when you draw a function, for example, it's not exactly important to know exactly where every point is, only that you know basically what it looks like and its properties. I prefer working with general cases that work on lots of things rather than only a few specific things.

But I digress. Anyways, not having any graphing paper, and not wanting to lose any marks for not drawing the picture, I decided to still draw it; at least I'll get something for knowing what the picture looks like. Even more unusual to some people is that I don't use rulers, compasses or other aids for geometry. Well, the reason I don't is for the same reason as above--the general picture is sufficient enough; not the exact picture.

Anyways (again) here is what we were asked to do:

"Take a sheet of graph paper, draw a horizontal segment of length 11 (which occupies roughly the middle third of the sheet), draw a line through the left endpoint A of the segment with the slope of -45 degrees, draw another line through the right endpoint B of the segment with the slope of 45 degrees. Draw a series of lines through the following pairs of points: pick an integer x between -9 and 9, move the point A along the first line by x squares and get a new point A(x), move the point B along the second line by x squares and get a new point B(x); (for positive x A(x) and B(x) lie to the right of A and B respectively); and then draw a line through A(x) and B(x). The resulting picture will clearly show a conic section (parabola) that is tangent to all lines that you have drawn."

So I didn't have any rulers, no other geometrical aids and no graphing paper--how did I accomplish this?

I actually had a lot of room left on the side of the page I was writing the solution on, so it was easy to do this, but for those who want to try this, you can take any blank sheet of paper you want--you don't need any rulers, compasses, geometrical aids or graphing paper to do this--YOU DON'T EVEN NEED TO WRITE ANYTHING AT ALL! The only reason I did was that I was afraid the person marking my assignment wouldn't notice it. Now, onto the procedure itself:

1) Fold your sheet of paper diagonally both ways so that you now have an X across one side of your sheet. Don't worry if your sheet of paper isn't square; just fold the diagonal over so that the corner touches the opposite edge of the sheet and the other corner is folded exactly in half. Now make sure your sheet is completely unfolded.

2) I did this on an 8.5 x 11, presumably, so a third of the page is approximately 2.8 in., but the "X" on my page only left 2.5 in. of extra space above it, but it was pretty close, so I used that as a guide. I will be referring to "horizontal" as the width of the page that's the shortest and "vertical" as the length of the page that's the longest--so your page should be facing you from a "portrait" perspective (see the page setup command under your browser to see what I mean) If your page is the same dimension as well, simply fold the sheet horizontally so that there's a crease across where the edge of the page meets both ends of the "X". Then fold the sheet in half again (in the opposite direction of the first fold) so that there's another horizontal crease somewhere before the middle of the "X". On my sheet of paper, the second horizontal crease would be approximately 5 in. down from the side of the page farther from the "X". Once again, unfold your sheet completely.

3) To avoid confusion, it would be best to mark the second horizontal crease now with a pencil or pen. Marking down creases with pencil is just about as easy as using a ruler to draw straight lines. Of course, there are other ways of drawing straight lines without rulers, including using the edge of the page as a straightedge. I used to do this a lot in high school. Now make any horizontal crease (fold it in the direction opposite of the second horizontal crease) you want between the two horizontal creases from the second step. Now fold the page over the second horizontal crease (but keep the new crease you made folded). Now fold a new horizontal crease on the "X" where your new fold meets the rest of the page. Now unfold the page.

4) Look for two points in which two of your creases meet each other:
-one point is between the second to last horizontal crease you made and one of the diagonals you made at the beginning of the procedure.
-the other point is between the last horizontal crease you made and the other diagonal.
Now fold a crease between those two points. You may repeat this step with two other points by switching the diagonals.

Repeat steps 3 and 4 until you can start to see the conic.

Unfortunately, I haven't gotten my assignment back yet, so I can't scan it and show it to people. However, I did take a picture of this one so that it will be easier to follow the instructions. This is actually the back side of the page, which I took the picture of because the creases show up better, and you can actually see the conic better (it's a parabola above the deep crease in the middle). The horizontal segment from the problem itself is the one in the middle which curves the page differently from the other horizontal creases.

How this works (there are some math proofs ahead, so you may want to skip them if you don't get it)

1) The diagonals:
I have never seen too many sheets of paper that didn't have 90 degree corners--most conventional sheets of paper are rectangular or square, and therefore, have 90 degree corners. Any bisector of such a corner must be 45 degrees, which are the 45 degree lines on the endpoints A and B in the assignment. I admit I made the diagonals before I made the horizontal line segment, although it doesn't really change the fact that you'll still end up with a conic.

2) The two horizontal creases:
The first one was made to mark out where 2.5in. of the page would be, and since the diagonals are 45 degrees with the edge of the page, the second horizontal crease and the first one form two sides of a square (a square has diagonals at 45 degrees). This would mean that the diagonal crosses the second horizontal crease 2.5 in. into the page on both sides--the remaining section is roughly a third of 8.5 in, although it would actually be a bit bigger, since I didn't have a ruler to measure out where 2.8 in. should be. In either case, it doesn't really matter, but I wanted the line to be as close as possible to what the assignment asked, but for your own purposes, it won't be necessary. You can place the line above the intersection of the diagonals wherever you want. Your resulting conic will just look slightly compressed or dilated.

3) The other horizontal creases:
I just used these as guidelines for the tangents. Since the diagonals are both at 45 degrees to the bottom of the page, making two creases both parallel with the bottom of the page and the same vertical distance away from the second horizontal crease makes the diagonals between them the same distance. They are, afterall, forming the same squares if you draw vertical lines between the intersections of the diagonals with one crease to the second horizontal crease, and one from the intersection of the diagonals with the second horizontal crease to the other crease--you'll end up with four boxes, all the same size. This is the same as choosing an x and the points A(x) and B(x) the same distance away from the second horizontal line.

4) The two points and the crease between them:
This is the supposed tangent of the conic--we've already calculated where the points should be from the guidelines above, so all we need to do is just make them.

Math can be fun and made accessible to people.
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Please Leave me Alone

It seems that there are these interesting topics that people like discussing in debate sections, and I do enjoy talking about them as well. But a few things bother me. In particular, I would like people to leave me alone about the following:

I know I talk lots about religion in debates concerning the said topic, but I would like people to quit pestering me about changing my religion. It doesn't happen on every forum I go to, but there are particular forums with people that can't seem to get it through their heads that I'm perfectly fine and happy as an atheist-agnostic, even if they think I'm going to hell, not as happy as could be, logic is just a tool, there's better things to waste time on than math and logic, etc, etc, etc. Well, I'll say it again. I'm perfectly happy with just logic and math. If it bothers you because I'm not buying into your idea of religion, well too bad. If it makes you any happier, I'm going to hell according to your religion, woe is me, and you can pray for me for all you care about, but it changes absolutely nothing.

Now I realize there are particular experiences that we'll all never get to experience. There are particular pleasures or kinds of happiness that we'll never experience even once in our lives. But why let that bother you because I'm missing one experience of human life? Is it such a difficult thing to understand that missing out on something is completely natural, particularly when life is only finite?

Perhaps I shall never appreciate or experience the love of God that people keep telling me about, but why should I care when I am indeed, extremely happy with the revelations of mathematics, particularly when math requires logic and rationality, and faith, which is required to love a being whom we cannot see or interact with runs completely counter to logic and rationality? How could I enjoy something based off of an ideal in life that I take no pleasure in? It makes absolutely no sense to me whatsoever that I might enjoy something like this. Granted, I could be completely wrong, but the probability of that is quite low. I'll say it again. I need no saving, I need no religion, I absolutely need no faith, and I certainly would appreciate it if people would stop telling me the following:

God owns your soul.
Everyone inherently believes there is a God.
You're going to hell forever if you don't believe.
I will pray for you (however, you're free to do this if you wish; it just means nothing to me that you're saying it to my face; you may as well just save your breath and refrain from saying something pointless)

You can keep your religious proselytizing to yourself and save it for people who actually need it (ie, people who clearly are asking for it--literally). It's really not all that hard.

I also realize that the reason I'm getting hounded at on particular forums is because they're Christian-based. But since when does a Christian forum become a place where people go to be converted? I went there for the interesting features, the religious debates and the broad range of topics. I absolutely didn't go there to get converted, and I doubt that's the sole purpose for that forum existing anyways. Please, please, please learn the difference between "God makes me happy, he'll make you happy too if you believe in him..." and "God makes me happy, so I continue praying for him". Big difference.

Alright, I admit I'm probably the last person on this planet who will actually find someone I might actually love. Someone alive, that is. Not that I'll even find someone, or am even looking for one. And it's because my perspective of love is that it's an emotion, not a rationality, and has little, if anything at all, to do with logic or reasoning. The fact that I am insensitive and quite strongly oriented by logic and reasoning rather than emotions easily explains why I see love the way I do. And yet, people continue to pester me about how great love is, and that I should try it. I don't mind if we're simply going to debate about what's so great about love or what stinks about it, but I draw the line when people start telling me what I should do about it.

I'm not here to seek advice or help about love; save that for someone who's actually had their heart broken or actually cares about having someone to love. I don't particularly care if you see love as a great thing; if you like it, then good for you. I hope you don't think I'm trying to tell you that you shouldn't love, only that I have my reasons for why I don't love. If you want to love someone, go ahead and do it all you want. It has nothing to do with me. But I would appreciate if others would do the same to me. My love life (or lack thereof) is of no concern to anyone else, and I do not need to be told that I need to try it.

As mentioned above, there will always be certain experiences that we miss out in life. I'm fairly happy as I am now, and don't need love to make my life more full. I consider a life filled with logic and rationality already quite full, and what more could I ask except from mathematics?

Love, an emotion which lacks all logic and rationality, and therefore, all control of the mind cannot make me happy. I do not derive happiness from losing myself to irrationality, nor do I wish ever to do something as spontaneous and lacking in control as that which involves love. It is not the feeling from which I derive pleasure (or lack thereof), but the principle of it which bothers me most, and I would appreciate it for those not to call me a coward for not choosing to love. For it is not cowardice to deny irrational feelings, but complete and utter folly to do otherwise in my opinion. I have said it before, and I shall say it again: there is a fine line between stupidity and cowardice.

Although that fallacious view that if you don't do something you're a coward is rather too generalized and too easily abused. It bothers me that people say it, but it must just be because they don't realize the problems in the things they attribute it to. I've seen a few:

"You don't have sex because you're afraid of it."

No actually, there's a good reason why people might not have sex. One being that abstinence is a good way of not getting AIDS or unwanted pregnancies. If you are to suggest that they're too "chicken" to get AIDS, I suggest you seriously rethink what that means. And an unwanted pregnancy could shatter your dreams. Yes, I know some of you might say child support is a solution, but don't forget that not everyone has access to it, nor do people necessarily have the time or energy to raise children. Simply put, some of us don't make for good parents, and suggesting that we're "chicken" for not wanting pregnancy when having no children is a perfectly acceptable style of life is also ridiculous.

"You don't like feeling emotions because you're afraid of them."

I admit one of my friends told this to me, but in expressing what I think about this, I mean her no offense. But it is an example of one of the things that abuses the phrase above. I don't feel emotions because there are consequences to feeling them. Get angry, and you lose control. Feel scared and you can't do anything. Fall in love, and you lose all sense of rationality and forget where you are in life, and become oblivious to the fact that people might be deceiving you. I don't know how it is that people are perfectly fine with loving people that deceive them, and continue doing it even after people have told them they're being deceived and even after they've realized it. Makes little sense to me.

It has been suggested that these things can be controlled, and it is a misconception that they can't, but not feeling such emotions at all is simply a solution to not losing control. It's just not the same as learning to control them, and I can't really see the difference. If someone could prevent the problem of emotions by learning to control their emotions, and someone else could do the same by not feeling them at all, then I fail to see the difference. It's the same in the end.

I could suggest that learning to control emotions means they go away after you've controlled them, so you've effectively prevented yourself from feeling the full blast of it, and if you had felt it, then you didn't really control your emotions--the fact that controlling them means you stop feeling them could mean technically that you're afraid of them, so you feel the need to control them, particularly if you suggest the same of not feeling them at all. But by realizing that it's impossible to do anything right under the influence of emotions, it is only a rational consequence to realize that feeling emotions must be prevented in order to think rationally. It is not cowardice, it is a revelation.

I guess I could also mention that it's not that I'm afraid of emotions, but rather that it is fairly unattractive and unappealing to me to submit to things as irrational as emotions. I would say it's a matter of taste--not bravery.

There can be no intelligence in the presence of emotions. How many times have you ever heard people being considered "brave" for daring to delve deeply into something as abstract and as complicated a subject as mathematics, something the majority of people have fears about and avoid whenever possible? Could these same people, driven by emotions, seriously think that I'm a coward for being more interested in this subject that they fear than emotions, which make them think they're brave for even submitting to them at all?

And if I could not feel, then I should also be oblivious to fear as well. For there is truly very little that I fear; if I cannot feel emotions, then I also cannot fear as well. It is a contradiction that I must fear feeling emotions if I cannot feel as well. I particularly fear very little of what most people fear. Loneliness, the unknown, the future, etc, etc, etc. that many people worry over. It doesn't make much sense that I might be considered a coward for not doing the things they do which have little rational basis.

"You don't love because you're afraid of it."

And finally, we come to the main problem itself. Although I suppose the above example answers this one fairly well because love is also an emotion. I do not know whether to take the word of the people who say love isn't as bad as I say it is, when I have seen enough instances of love gone wrong, and these same people can only tell me about love from their own experiences, and have had their hearts broken very little. They may say that I treat love as if it were as drastic as a gamble, and be wrong about it, and that I am simply avoiding something in which I have little to lose.

But man in the throes of real love would sacrifice everything for that one being whom he loves, whether it's his money, his belongings, and everything else important to him--his life and his heart are for that one person that he loves. And the more he loves, the more he sacrifices--oblivious of the fact that the person he loves may not love him back, or may deceive him. For what worth is there in loving a person who has no honesty or virtue, and in sacrificing everything for someone? Why should I be compelled to believe that it was worth loving someone and losing everything--rationally? The belongings and the material possessions may not matter; the money is not important. But to love someone and give away your life and heart to someone else, never knowing if they'll do the same for you as well, and to lose it and have no heart or life in return is the worst possible investment one could ever make. If you suggest to say that love could not involve the sacrifice of everything, then perhaps it is not as great as people say it is, and is as worthless as I had perceived it to be. For the more weight something has, the more there is to have lost when it happens.

My opponents say that it is cowardly not to give up and sacrifice everything for the sake of one person, regardless of whether or not he or she is a fraud. Perhaps they believe the probability of fraud is low, or perhaps they don't think much of it because they themselves have never been deceived. But it is not the probability that is the problem, or their lack of experience necessarily, but the principle of losing rationality which harms me the most. For I would not give up rationality for anything at all in the world. If I could love someone and know if they were deceiving me, I would stop loving them and find someone else. But it is not possible in the throes of love to know this because the emotions of love compel me to make a sacrifice that I wish not to make; of rationality, the one thing I value more than anything else in this finite life.

Is it cowardly because I wish not to give away the one thing most valuable to me in my life for the sake of some irrational emotion, wasted on one person alone? Is it cowardly because I find irrationalities such as love unattractive, and therefore, a waste of my time? Is it cowardly because I don't choose to pursue certain pleasures that other people find pleasing, particularly when it does not suit my lifestyle?

The act of rationality is not cowardice. It is a sign of intelligence. Something which few people dare to venture into and refuse to think about. I value this more than I value the trivialities of love, and if you cannot see that, then I suggest you give up trying to persuade me into love. You cannot judge or compare what you don't know or understand with love.

If you will concede that it is cowardly not to love, then I will say it is cowardly not to pursue intelligence.

A Final Comment: I didn't write this because I didn't enjoy my debates, either with my religious opponents or with my star-crossed opponents. I wrote it because I believe a good debate can be enjoyed without the need to persuade me into religion or love. Afterall, the main topics of those debates were not for the sake of conversion or pushing others into loving. If that were the case, then people would ask for them. A debate is a debate because it is an exchange of ideas; not a means of counsel.
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The World May Grow Weary...

The weeds may grow long,
The scent of compost, rank and strong,
The walls may be crawling with wild vines, unchecked and free,
And the cobwebs, aggregated here, there, and everywhere the eye can see,

But it does not matter, so long as this abstract beauty entertains me,
The dimensions of which it is a part of, and yet, not a part of,
And the continuum of which exhibits shapes, shapes, and more shapes, well formed and described as vast as a sea,
Which no weeds, regardless of their length,
no odors, regardless of their strength,
no vines, no matter how tall,
and no cobwebs, no matter how they crawl,
May move me away from these abstract shapes to worry over them.

The mouth may beg for bread and water,
The body, for trinkets and fashion hotter,
The attraction of wealth may be distracting,
That of material wealth, exacting,

But it does not matter, so long as I have this abstraction,
Worth more to me than any amount of gold,
Which can buy no mountains and mountains of equations and variables in action,
Problems of the abstract world more worthy of being solved than any of attaining possession may hold,
Which not even a thousand loaves of bread, in my mouth mashing,
nor ten thousands of trinkets, flashing,
nor millions of coins, a mountain high,
nor any number of possessions I could buy,
Or all, may move me to buy them, the value of which is nothing before this one enigmatic beauty.

The world may grow weary,
The mind, depressed and gray,
The people may become uninteresting and dull,
These matters, fruitless and null,

But it does not matter, so long as my mind sees this abstract science,
Which no one can steal, and nothing can compensate for,
The beauty of which is infinitesimally more so than any world, or people, or any other science,
Which any world, if alive,
any mind, if vibrant and full,
any people, if they strive,
any matters, if it is revelation they pull,
Can be no greater than this abstraction, unless they are equivalent, and move me when all other treasures and realities fail to do so.

For this enigmatic beauty, mathematics, has shown my mind the greatest revelations that could ever be comprehended.

[I suppose it's highly unusual that I write poems, but I guess I did write one on math. It is one of my favorite subjects afterall.]

Two Kinds of Faith

Well, it seems that the term "faith" gets thrown around a lot, and even I myself believed the word to be defined unambiguously to have one meaning. Apparently though, this isn't quite the case, as it seems there are at least two different "faiths", the former of which I use more: religious faith, and probabilistic faith.

I formerly thought of faith as the word that describes the concept of a particular belief or trust in something, for which you have absolutely no basis of what you believe or trust--that is, it is entirely possible for you to take something as truth for no reason at all (or in lesser cases, for lesser or bad reasons). It is for this reason that I have always labeled faith as being blind. There is simply no direction, revelation or enlightenment in trusting so irrational a thing as faith, when you simply believe something in the absence of reason. This description of my former view of faith most accurately describes what is known as religious faith, the blind kind of faith of believing in things without either evidence, reason or logic.

Now I admit that "religious faith" and "probabilistic faith" are terms that I have coined for the purpose of writing this piece, but to see how I have come up with the idea of "religious faith", it is because the kind of faith involved in believing in truth without evidence or reason is most often apparent in religion. In fact, religion is the single largest explanation for "religious faith". Upon closer inspection, it is not difficult to see how or why this is the case. For it is often the case that religious texts, of which most religions are based off of are completely devoid of any kind of reason or logic at all. The three most popular holy texts, the Koran, Christian bible and the one in Judaism are all repetitive, contradictory and inconsistent. Therefore, they are considered to be lacking in reason and irrational. To believe that any of these books constitutes to truth is to commit religious faith--you would have to ignore reason and external evidences to even believe and accept any of this as the truth.

It is then not surprising that I have interpreted the meaning of faith to be closer to the definition of religious faith, as it is most commonly used in religious debates. The probabilistic faith is rarely ever talked about, if ever mentioned, and when it is mentioned, it is not really labeled as faith, but as probability, which has nothing to do with religious faith.

Unlike religious faith, probabilistic faith is more reliable, and less blind. For probability, being a portion of the domain of mathematics, is at least reasonable, and to have probabilistic "faith" in something that is rationally based is more reasonable than having religious faith. But to have a better grasp of what probabilistic faith is, let us first consider the concept of probability itself.

Probability, even among particular mathematicians, is not a very comforting subject. It is still comforting in the sense that it is logical, it works, and it constitutes to mathematical beauty, but it is sometimes not simple or easy to grasp or use. The principle behind probability, however, is to consider all possible outcomes, and events, which are subsets of of all the possible outcomes. Probability, therefore, is the number of ways an event can occur compared to all the possible number of outcomes that can occur. The more ways an event can occur, the more "probable", or likely it is to happen. The fewer ways, the less likely it is. Probability might be considered a very open-minded approach to any particular situation. Rather than grounding oneself to one event being able to occur, and one event only, probability influences you to think of every possible outcome imaginable, and rather than restricting oneself to a single event, one can now perceive all the possible things that can happen, and accept that any of them might be possible. The denial that comes from restricting oneself to one event occurring when it does not is less apparent when one relies on probability; at least you didn't expect it wouldn't happen.

Despite the fact that probability states that many things could happen, in reality, only one of these outcomes occurs. Probability is only an analytical tool in interpreting what actually happens; to go one step further from the given analysis, we now consider probabilistic faith. Since we now know the number of ways any given events can occur and all the possible outcomes, we are able to know the probabilities of these events. If one could predict what would happen, and we could pick only one event from probability that were to happen, then probabilistic faith tells us that the event that is most probable is the one that will happen. The only blindness of this kind of faith is that sometimes, the most probable does not happen. It is entirely possible to win the lottery. It is entirely possible to slip and fall in the middle of the street while riding a bicycle and get hit by a car and die, even if you ride meticulously. It is entirely possible to fail a test, despite spending hours and hours of studying. All of these things might be considered improbable, yet they can still happen. Only by accepting that probability does not tell you what will happen, but tells you what could happen can one accept both the nature of probability and what actually happens, regardless of whether or not it is what you wanted to happen.

The other usage of probabilistic faith applies to experience. While religious faith does not rely on reason or evidence, probabilistic faith might rely on experience, though in considering experience, you are instinctively relying on probability, though not conscious of it much--this is the same with some of the lower levels of mathematics being used all the time, yet most people don't realize that they're using it. Because of experience, you are more well aware of what might happen--if something happened in the past, that has repeated itself continuously in the past, your experience tells you that such an event, if the occasion arises, is most likely to occur--you automatically skip the probability calculation processes, and rely on probabilistic faith--it is the event that will most likely occur, and you rely on it. To consider this purely in the sense of probability, however, what you experience might still not happen, even if you have seen it happen a lot before--it might be improbable, but by no means, impossible. In terms of mathematics, the "impossible" can only occur if there is a contradiction or independence of events--it is because there are few cases where contradictions are involved that probability allows us to accept that anything can happen. And in doing so, we are more accepting of the things that can happen. We are not limiting ourselves to one event, or setting ourselves up to devastation because what we want to have happen does not always happen--by allowing ourselves to accept what we do not necessarily desire could happen, we attain a revelation of how these things occurred, and through acceptance, we learn and grow. Man cannot grow if he will only restrict himself to the things he wants to have happen, rather than the reality that our desires are not always reality.

Because probabilistic faith is probability based, and more rationally sound than religious faith, it is much wiser to rely on probabilistic faith than it is to rely on religious faith. Religious faith does not much for the growth or learning of humanity. It only continues to blind man and devoid him of any kind of real revelation that might exist. This dangerously confines man to only his desires, but removes him further from reality. In doing so, he will never come to accept that he cannot always get what he wants.

And people often wonder how or why it is I am able to predict what happens on the Internet; I am, by no means, either a prophet, psychic, palm reader, or someone who otherwise predicts the future. I am only someone who relies on probability. It is because the most probable event has occurred that people are baffled at how I am able to "know" certain things, when I do not constitute this as knowledge, but an educated guess based on the event that is most probable. Once again, probability does not tell you what will happen. It only tells you what could happen. But it is because of probability that I can consider these things that happen. For I do not care for whether or not what I want to happen happens, but what is to come will come through time, regardless of whether or not it is probable. It is only because people consider other options as being impossible that they are shocked into denial.

A person who relies on religious faith is easily drawn into denial. This is often the case because they believe in only one thing, and one thing only happening, which is usually something that they want to happen. They do not consider that it is possible for other things to happen, and for absolutely no reason at all; but to consider only one thing as being possible and all else as being impossible makes for a very small subspace of all possible outcomes. By eliminating every single possibility save for one, it is very easy and probable for the many "impossible" outcomes (which aren't actually impossible) to occur, and very improbable for the one event that the person wants to happen to occur. On the other hand, the man who relies on probabilistic faith is aware of many events that could occur. Any even that occurs does not surprise him because he expects that one of them would occur. It only shocks the man of probabilistic faith if an event that he considered to be a contradiction, and therefore, impossible were to happen. But because such people are inclined to consider as many outcomes as possible, it is rare that they ever come across such circumstances.
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