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  1. The Lord's Prayer: The Axioms of the Math Model That Solves Our Questions on Salvation: From What, How and for How Long Are We Saved? Are Good Deeds Enough? ... It Predestined? Is Salvation Quantifiable? by Felix Shimata B. Tshinanga PhD, 2009-12-25
  2. Maths (Mentor Learning) by Greg Wilson, 2007-10
  3. Axiom of Choice (Stud. in Logic & Maths.) by T J Jech, 1973-07
  4. The Axiom of Constructibility: A Guide for the Mathematician (Lecture Notes in Mathematics) (Volume 0) by K. J. Devlin, 1977-12-07
  5. Independent Axioms for Minkowski Space-Time (Research Notes in Mathematics Series) by John W Schutz, 1997-10-08
  6. Axiom: Webster's Timeline History, 1316 - 2007 by Icon Group International, 2010-05-17
  7. Head First 2D Geometry by Lindsey Fallow, Dawn Griffiths, 2009-11-24
  8. Probability, Random Variables and Stochastic Processes with Errata Sheet by Athanasios Papoulis, S. Unnikrishna Pillai, 2001-12-14
  9. Number Problem Solving (Mentor Learning) by Greg Wilson, 2007-10

1. Carol's Page 4 - Axioms
Cool math axioms. Math is not a spectator sport. PCTM button. We are continuallyfaced by great opportunities brilliantly disguised as insolvable problems.
http://faculty.nwlehighsd.org/bearc/carols_page_4__axioms.htm
Cool Math Axioms Math is not a spectator sport. PCTM button We are continually faced by great opportunities brilliantly disguised as insolvable problems. unknown Computers are incredibly fast, accurate and stupid; humans are incredibly slow, inaccurate and brilliant; together they are powerful beyond imagination. Albert Einstein Make the mistakes of yesterday your lessons for today. unknown Unless you try to do something beyond what you have already mastered, you will never grow. Ralph Waldo Emerson Do not let what you cannot do interfere with what you can do. John Wooden Genius is the ability to reduce the complicated to the simple. C.W.Ceran back to homepage

2. American Revolution - Template
250Host offers the best value in hosting. Title Math Activity 5 Ben FranklinMathematical Axioms. Subject math axioms. Week Of Week’s Lesson Span 1 day
http://henske.250free.com/pontiac/math/math_5.htm
Title: Math Activity 5 Ben Franklin Mathematical Axioms
Background: the students will brainstorm ways in which some of the saying could be related to math principles. As a writing activity, the students should choose one of the axioms to use to illustrate some property of math or formulate some story problem relating to the axiom. Objectives: The student will use their knowledge of math properties and principles to write story problems or math sayings based on a given situation. State Standards: 6.C.3a, 6.C.2a, 8.C.2, 8.C.3 Materials: copy of sayings from Time needed: ½ class period Vocabulary: axiom Procedure: Provide a list of sayings from Ben Franklin for the students to read in groups or as a class. Brainstorm some ways the sayings could be related to a mathematical principle or situation. Provide an example if needed. Students choose one saying to use as a basis for their own story problem, or other option from Handout 11. Evaluation: assessed in final math rubric
Math Handout 5.11

3. Ifeminists.com Interaction Feminism0201b Draft And Breasts
Most math axioms are accepted by most people. Morality also has many axioms. Mostmath axioms are accepted by most people. Morality also has many axioms.
http://www.ifeminists.com/interaction/forum/feminism0201b/messages/062800893.htm

4. Www.cis.ksu.edu/~howell/492s02/lec1.txt
The notion of 'agreement' does not have to exist in the math domain; werely on it in the human domain to reach agreement on the math axioms.
http://www.cis.ksu.edu/~howell/492s02/lec1.txt
Introduction for cis492, spring 2002 M. van Swaay Build the scaffold, construct the lattice, weave the net. That is the assignment for the first lecture, or maybe the first pair of lectures. At the end, the structure will be full of holes, which have to be filled in the remaining sessions. But the structure has to be sound, and it has to be complete, before the hole-filling can begin. It is quite a challenge; I hope to rise to it. Ethics: covers not obedience to some set of rules, but consistency and 'moral sense' in the formation of choices/decisions. Choice implies freedom to choose. But that 'freedom' is not unbounded: some - implicit and often unstated - set of rules is generally acknowledged. Fletcher Moulton: Obedience to the Unenforceable. Why did you do X? Because I felt like it. Because my boss told me to. Because the law says so. Because I concluded it was the right thing to do. Why did you conclude that? Because I can argue for X from Y. Why then Y? Persistent questioning of 'why' must eventually lead to an inescapable answer of 'because'. At that level further explanation is both recognized as unnecessary, and admitted as unavailable. In mathematics that level is the place where the axioms are found. Axioms, by definition, are beyond challenge. But are they really? In math we know about Euclidian geometry, but also about non-Euclidian geometry. We have no trouble with that, because math is a domain that 'does not include us': we look at it 'from the outside'. Axioms in the domain of life are not so clearcut and concrete. In math the axioms can be seen as starting premises 'agreed on by mathematicians', on which a structure can then be built. The notion of 'agreement' does not have to exist in the math domain; we rely on it in the human domain to reach agreement on the math axioms. But we cannot define the 'human axioms' by negotiation or agreement: that would imply that both negotiation and agreement are available. We would then have to classify those as axioms, and they must have come into existence without negotiation or agreement. So the 'human axioms' have to pre-exist, before anybody started to think or talk about them. No logical argument can return an answer to the question where they came from. If the human axioms cannot result from negotiation, are they then non-negotiable? That is a profoundly disturbing question. If they are truly non-negotiable, i.e. dogmatic, then we are stuck with them, and we have no argument, or even reason for hope, that the dogma will be shared by others. Observation gives us reason to see the world in far less bleak light. We find that the vast majority of humankind is inclined to sympathy and cooperation. That is precisely the reason why deviations make the headlines. We can speculate and fantasize about the origin of that good luck that appears to be built into humanity. Matt Ridley does exactly that, in his book 'the origins of virtue'. James Q. Wilson approaches the same question from a different angle in his book 'the moral sense'. Kant tries to approach the question philosophically, and arrives at his categorical imperative. But Kant bases his argument on a premise of reason, without answering why 'reason' has to be admitted as a premise. Various religions oblige their followers by declaring the answer as 'god-given', and therefore outside the responsibility of the individual. I propose to dispose of the question where the axioms come from by placing that question in the spiritual domain, and then declaring that domain to be outside the domain of this course. Lest you see this as a copout: I do have a personal belief that lets me be at peace with the question, but I neither can nor want to try to make others share that belief. You have to think that through for yourself. In this context a short comment may be in order, about the nature of faith and religion. I propose to make a distinction between the two terms: in my view faith answers what cannot be reasoned. Faith then has to be personal. Religion is an expressed 'common denominator' for the personal faiths of a group of people, possibly a very large group. The advice that you 'think it through for yourself' reveals that 'axiom' is not the best word to use ... but I have no better. To the outside world one's personal faith is normally considered beyond negotiation. But not necessarily to its 'owner'. I doubt that anyone would challenge the observation that each of us 'matures in faith', and, by implication, challenges his own faith as long as he lives. I submit that a faith that cannot be reconciled with 'being human' cannot survive internal challenge. Of course that raises the question what 'being human' means .... In contrast to faith, religion will have to rest on some expressed dogma, even if that dogma may itself declare tolerance and respect for those who do not subscribe to it. But the idea of religion as a belief that does not allow challenge, even though it can admit other beliefs, makes religion a very tempting cloak in which to wrap fanaticism, in the hope that the refusal to admit challenge will protect the fanaticism. Having admitted to the existence, even necessity, of axioms, can we find reason to hope that they may extend beyond a single person, possibly to all of humanity? Again, that question is not open to proof, but we can make some observations. One of those I made earlier: empirically we find that uncooperative behavior is almost universally seen as abnormal. More reassuringly, theologists tell me that the 'golden rule' is common to all major religions, and to at least one of the major philosophies. Without attempt at proof I submit that the golden rule may well be sufficient, and that we can persuasively argue that it is universal. That does not mean that every person in the world will abide by it, but it does mean that the few who do not will be seen as not only abnormal but also without claim to approval. After all, violation of the golden rule implies asymmetry: 'I am allowed to impose on you, but not the other way around." The suggestion that the golden rule may serve as the core of a universal ethical framework rests in part on an observation: 'there appear to be no arguments against it', and of course in part on belief: 'it should serve'. Just as geometry is vastly more than Euclid's axioms, human life is vastly more than the golden rule. So we need to build some structure beyond the bare axiom(s). We can call those structures 'models of ethics'. Some history is in order here, to encourage some modesty, and to avoid much frustration. People have chewed on ethics questions as long as recorded history. Some say, maybe a bit facetiously, that Plato said it all, and everything after that is just footnotes to his work. The mere fact that today we recognize at least three major models of ethics is evidence that we don't really 'know' yet. That, in turn, leads to the recognition that 'the model of ethics' may not be knowable. But that should not stop us from thinking about it. Ethics models: duty-based, rights-based, consequence-based, relativistic. The U.S. attaches much weight to one more anchor, and rightly so: its founding documents, the Declaration of Independence and the Constitution and its Amendments. I believe these documents derive their strength not merely from the skill and wisdom of those who designed and wrote them, but also from the fact that they reflect the underlying bedrock, and are in tune with it. Without claiming standing as a constitutional scholar, or even a legal scholar, I do want to say a few words about some parts of the Constitution, not to challenge or undermine its stature, but to encourage thought about its place and its meaning. Allow me to throw what appears to be a brickbat, and then argue why it neither is nor aspires to be a brickbat. I submit that the name 'Bill Of Rights' - but not its content! - is misleading at best, and possibly wrong. The preamble to it makes it quite clear that the intent is not so much the assignment of rights as the imposition of restraint on the government: THE conventions of a number of the States, having at the time of their adopting the Constitution, expressed a desire, in order to prevent misconstruction or abuse of its powers, that further declaratory and restrictive clauses should be added, and as extending the ground of public confidence in the government will best insure the beneficent ends of its institution, RESOLVED by the Senate and House of Representatives of the United States of America in Congress assembled, two-thirds of both Houses concurring, that the following Articles be proposed to the Legislatures of the several States as Amendments to the Constitution of the United States, all or any of which Articles, when ratified by three-fourths of the said Legislatures, to be valid to all intents and purposes as part of the said Constitution, viz. ARTICLES in addition to, and amendment of, the Constitution of the United States of America, proposed by Congress, and ratified by the Legislatures of the several States, pursuant to the fifth Article of the original Constitution. In other words, what would become known as the Bill Of Rights was explicitly designed to impose restraints on the government. It casts a revealing light on the claim of 'I stand on my First-Amendment Rights'. The government has no obligation to defend your choice to spout off at a streetcorner or over the internet. The government merely cannot forbid you from making a fool of yourself, or worse. This may be a good place to warn you about one of the more invidious tricks of the debater - and the activist: the misuse of words that have multiple meaning, and the export of terms beyond the domain in which they are defined. Physicists invariably cringe at the casual use of 'energy' by weather forecasters and advocates of alternative medicine. The word 'right' has two very different meanings. The first is the meaning we find in the Declaration of Independence, and in the Declaration of Human Rights. The second we encounter as 'Miranda Rights', 'first right of refusal' and such. These rights are generally constructed by negotiation and legislation. The 'Inalienable Rights' of the Declaration of Independence pre-exist as part of 'being human'. One would think that the Founding Documents leave little room for relativism, but that has not stopped a well-recognized trend toward "I'm OK you're OK." Admittedly, there is sound value in the Indian advice that one should not judge another human being without first having walked a mile in his mocassins. But it does not intend to say that judgment is 'inappropriate': it intends to say that one should judge conscientiously, and with consideration. I find it ironic that the current fashion that frowns on 'being judgmental' is rarely challenged for its internal contradiction: making judgment is on its face judged inappropriate .... So let us review the four major ethics models I referred to earlier. The duty-based model (deontological model) rests on the premise that some fundamental duties exist for all. The model is mute about where those duties come from, and is mute about what happens when they are ignored. Apart from what one might call 'spiritual duties', the duties in this model consist largely of obligations to one's fellow humans. Under that view the rights-based model, at least under one understanding of the term 'right', is merely the mirror image of the duty-based model. But it has become fahionable to export this notion of 'rights' to a far broader domain, which includes such things as the 'right to a living wage', and the 'right to medical care'. Those are very different from the right to human dignity that we know from the original domain. In particular, they lead to expectations that 'society' is obliged to protect individuals from adversity. It is easy to recognize that extrapolation as logically untenable: it would lead to demands for eternal life, and for reward unrelated to performance. Goal-based - teleological - ethics: Consequentialist ethics rests on evaluation of the outcome of actions. Its most common form is known as utilitarianism. This model can serve quite well as a framework for many practical situations, but it can be shown to fail when pushed logically. For example, we may observe that hacking is intolerable. It might be possible to reduce the appeal of hacking by 'sending a strong message', maybe by publicly dismembering 'a hacker'. Would it matter whether or not the victim actually was a malicious hacker? He would never tell .... So we could just as well pick up a victim at random and do him in. All this would be consistent with the utilitarian model, but is clearly outrageous. The utilitarian model may be useful because we tend to apply it as a secondary filter, after we have dismissed the options that would fail under more fundamental arguments we find so obvious that we leave them unsaid. I submit that the notion of 'fairness' - the golden rule - probably serves to eliminate the unpalatable options before we apply utilitarian criteria. The fairness model, for which Wilson argues extensively, directs us to a form of utilitarianism in which social benefit is justified as a form of investment that will eventually reward us with a better environment. I believe the fairness model is also at the core of Ayn Rand's objectivism. Wilson makes a revealing observation: he relates our apparent failure to discover a universal bedrock to the possibility that we may be seeking it in the wrong place. The nature of 'universal bedrock' will have to be such that everybody will take it for granted. Fairness is precisely such a self-evident notion. Wilson illustrates this by describing a testable form of the 'prisoner's dilemma', which requires three actors: An umpire has control over a quantity of money, but is not otherwise a participant. Two players A and B are bound by two simple rules. Player A must propose a division of a pile of money between players A and B. The pile is placed on the table by the umpire. Player B may accept or reject the division proposed by A, but cannot modify it. If player B accepts, the proposed division is executed; if player B rejects the proposed division, all money reverts to the umpire. What about relativism? Richard Feynman, among others, takes the relativists to task for egregious misunderstanding of Einstein. I don't know to what extent Einstein's theories have influenced philosophers, but one can find at least two arguments to dismiss relativism as untenable. The first is logical: 'it is ALL relative' contains the absolute 'all', contradicting the model it tries to define. The second is that the model would contradict all we can see about the society around us, and that it conflicts with the core of all major religions known today. Finally, it would effectively forbid the making of judgment. We should know better: we pick friends, spouses, presidents, employees, consultants, etc. Admittedly, there is sound value in the Indian advice that one should not judge another human being without first having walked a mile in his mocassins. But it does not intend to say that judgment is 'inappropriate': it intends to say that one should judge conscientiously, and with consideration. I find it ironic that the current fashion that frowns on 'being judgmental' is rarely challenged for its internal contradiction: making judgment is on its face judged inappropriate .... Back to a more practical level: to freedom of speech, privacy, reputation, internet governance ... Does 'freedom of speech' imply that you can say or shout whatever you jolly well please? Hardly. But government, precisely because it wields power from which citizens cannot easily escape, is constrained from using that power beyond what it is granted for. Then what 'rule' defines what one can, and cannot, say? Obedience to the Unenforceable. We are free to choose, but have to concede that we cannot choose arbitrarily. That is the domain of ethics. The domain of ethics is not controlled by some 'ethics law' or 'code of conduct'. Such agreed-on rules are useful, because they can spare us much agonizing thought: the known situations can be answered by reference. But leaders are paid for dealing with the new situations. They are expected to 'think sensibly', not to 'obey slavishly'. So ethics is not about 'knowing the answers': ethics is about 'learning how to think'. Note: not 'knowing what to think.' The 'hard ethics' comes in places where the rules fail, or are absent. The absence of rules implies not only that one 'has to decide', but also that there is no place to hide when the decision is challenged later. The book for this course raises the question what would be the best authority to govern the internet, and then develops a discussion comparing 'states', 'industry', 'netizens', 'code', 'self-government'. To place such a question in context we may find it useful to look at the nature of cooperative groups. We can plausibly argue that within such groups no scheme is possible that maintains a uniform (flat) distribution of power. So, like it or not, there will arise some hierarchy of power, i.e. some form of government. The question then is not what entity can best fill the role of government, but what people can serve best in government. Jane Jacobs has made some interesting observations about that issue. She recognizes two types of social syndromes: guardian and commercial. The guardians work at maintaining a workable environment, and the commercialists work at being productive within that environment. But if people, or institutions, develop ambition to participate in both activities, that will lead to severe social instability. It makes sense: the mixing of the two positions leads to blatant conflict of interest. It may still be useful to ask from which domain one is most likely to recruit suitable people to accept the task of governance. But the gorilla in the room is called 'democracy'. Personal freedom is not free, nor can it be guaranteed. It has to be earned. I missed the first conference on Computers, Freedom and Privacy (CFP), but attended a string of them after that. The conferences were born from a major attempt to 'catch hackers'. As luck would have it, that appeared to be easy: catch the people who attend the 'hacker conference'. What struck me at the CFP conferences was the delicate dance of members of two camps, neither of which was willing to entertain the thought that members of its side might not all be perfect, or that 'the other side' might not be unequivocally evil. So the 'acronym spooks' tried to present themselves as decent folk who had only the most laudable intent for the country, and the 'free spirits' tried to present themselves as innocent babes who only want to 'be left alone to dance'. As the years progressed, the two camps did appear to get to know each other better .... There is a widely quoted statement, I believe attributed to Jefferson (but I have not been able to confirm that attribution) that 'That government is best which governs least'. Few people are aware that that is only half of the sentence, and that the important part comes afterward: 'for its citizens discipline themselves.' I expect that you will spend much time this semester discussing 'scenarios': defined situations that pose dilemmas. It may be wise to alert you to the fact that discussion of a scenario cannot be compared with reaching a decision in real life. When you get tired of the discussion you can declare the matter closed, sling on your backpack and walk out. After you have made a decision you have to live with the consequences, forever after. It may sound corny, but those who make major ethical decisions have to be prepared to stand behind those decisions with their life: they have no rule to hide behind. Discussion of scenarios is useful, not because the discussion can lead to an 'answer', but because it offers an opportunity to develop your skills at reasoning - and judging - toward an answer. Please be well aware: 'doing ethics' implies making judgment. Those who would tell you not to be judgmental should think about the contradiction in that dictate. Making judgment is not 'luxuriating in the power of it': it is the agonizing process of deciding what is right, and sometimes what is wrong. The need for judgment arises precisely where the rules end, and where they fail or conflict with each other. So how does all this fit into your ambitions for a career as computer scientist? Is not competence at writing Java script vastly more important? Ask yourself what your software is intended to do. Any nontrivial software will affect the lives of people, and much of it may have big effects on many people. Ask yourself whether you can be at peace with what your work does to all those people. Your ambitions may initially be no higher than a reasonable income and a comfortable cubicle. But those who aspire to 'be somebody' will wind up making decisions, on behalf of others. In that capacity, they act as 'professionals' in the original sense: they are 'called forth' to act on behalf of their clients. I know that the term is used loosely in other contexts. But it is useful to look at what it meant originally, and to deduce from that what it implies. Among the key characteristics of professionalism is that the professional is expected to choose on behalf of his clients, and that his choices are not bound by rules that would reduce those choices to mere obedience. In other words, the professional is expected to 'make his own law' about his actions: his work requires autonomy. But autonomy is not 'anomy': the absence of law/rule. On the contrary, it is the presence of, and obedience to, a 'higher law' that resides in the integrity of the professional. One more topic, to illustrate conflicting imperatives: We have come to value 'privacy' very highly, almost reverently. But an attempt to define what this 'privacy' is that we wish to protect soon reveals that it is a very slippery thing. Very crudely, privacy deals with control over what others are allowed to know about us, and how we might want to control what they can do with that knowledge. But it is not so simple to identify who 'owns' this sort of information, and even less simple to define who can control it, and to what extent. Moreover, the elevation of privacy must eventually clash with a very sensible and defensible ambition to 'build a reputation'. Computer scientists should have little trouble recognizing that 'reputation' shares its root with 'computer'. The Latin verb 'putare' may be translated as 'to reckon'. Computers combine pieces of information and then reckon with them. People tend to do the same, repeatedly, with the stuff they learn about those around them. A rigid enforcement of privacy would then interfere with the activity we know as reputation-building. To build a reputation you will have to make visible who you are. Of course that implies that you must first think through who you want to be. It makes little sense to 'put on an act' if your aim is to let people find out who you really are. Beyond that, you presumably want to build a good reputation. You will want people to like and respect you for who you are, and for how you think and act. But you cannot 'just tell them' what to think of you: you have to persuade them to think well of you .... If I were given the opportunity/authority to define the purpose of this course I would hope that it will serve to extend mental discipline beyond 'cogent thought' to 'conscientious thought'. Cogent thought is what you will need to do good science. Conscientious thought is what prevents a good scientist from becoming a 'mad scientist'. ========================= A handful of little books from and about Richard Feynman: Six Easy Pieces ISBN 0-201-32842-9 Six Not-So-Easy Pieces ISBN 0-201-40825-2 QED ISBN 0-691-02417-0 The Meaning of It All ISBN 0-201-36080-2 Surely You're Joking, Mr. Feynman ISBN The first three are extracted from the freshman physics course Feynman tought at CalTech in an attempt to make the course more exciting. Feynman himself wonders about its success: he notes that as the course progressed his audience contained fewer and fewer undergraduates, and more and more graduates and fellow faculty. It is quite obvious that the text is transcribed from the spoken word (which was recorded: much of it can be purchased on audio tape and maybe CD). The 'not-so-easy pieces' contains the comments by Feynman about misunderstood interpretation of Einstein's theories by some philosophers. QED is about the work that earned Feynman the Nobel prize. The fourth book contains a set of three lectures Feynman was invited to give at the university of Washington at Seattle. The book carries the subtitle "Thoughts of a Citizen-Scientist"; Feynman used the lectures to reflect on religious, political and social issues of the day (1963). The last book is autobiographical, contains little science, and lots of humor and playfulness. ======================== Fletcher Moulton: Law and Manners Atlantic Monthly, july 1924 (Hale) Richard Mitchell: The Gift Of Fire ISBN (Manhattan Public Library) James Q. Wilson: The Moral Sense ISBN 0-684-83332-8 1993 Matt Ridley: The Origins of Virtue ISBN 0-670-87449-3 1996 Charles Sykes: A Nation of Victims ISBN Ayn Rand: Atlas Shrugged Ayn Rand: The Virtue of Selfishness: anthology Ayn Rand: Essays in Objectivist Thought Jane Jacobs: Systems of Survival: A dialogue on the Moral Foundations of Commerce and Politics M. van Swaay: The Value and Protection of Privacy (reprint)

5. Math 366, Axioms For Plane Geometry
Next The axioms and related. math 366, axioms for plane geometry
http://www.cs.umsl.edu/~benson/m366/m366axioms/m366axioms.html
Next: The axioms and related
Math 366, Axioms for plane geometry
This handout summarizes the axiomatic foundations for geometry employed in Math/366. We have followed the metric approach , due in its present form to G. D. Birkhoff (1884-1944).

Chal Benson

6. Math 302: The Axioms
math 302 The axioms for Straight Lines. Given any space of points(probably a surface in threespace, but perhaps something more
http://torus.math.uiuc.edu/jms/m302/handouts/axioms.html
Math 302: The Axioms for Straight Lines
Given any space of points (probably a surface in three-space, but perhaps something more abstract) and given a notion of which curves in that space are to be considered lines (which always means straight lines), we can consider whether or not each of the following statements is true. In other contexts, straight lines could be defined by means such axioms: we would take lines to mean those paths in a surface which satisfy the specified axioms. In this course we instead take the approach or using various other criteria to determine which paths should be considered lines in the spaces we study. (For a smooth surface in three-space, the lines will be the geodesics.) Then, for each space that we study, we will check which of the following axioms are true. For most spaces, some of these axioms are true but others are false. (We break each axiom down into several parts so that if it fails we can tell exactly which part fails.)
  • The "incidence axiom": There is at least one line through any two given points. There is at most one line through any two given points.
  • 7. Math 302: The Axioms
    math 302 The axioms for Straight Lines. The incidence axiom Thereis at least one line between any two given points. There is
    http://torus.math.uiuc.edu/jms/m302/01Sp/handouts/axioms.html
    Math 302: The Axioms for Straight Lines
  • The "incidence axiom": There is at least one line between any two given points. There is at most one line between any two given points. The "ruler axiom": Along any given line, you can travel an infinite distance forwards or backwards. As you travel forwards along a line, you never pass over the same point twice. The "protractor axiom": There is at least one line through any given point in any given direction. There is at most one line through any given point in any given direction. The "halfplane" axiom: If you cut the surface along a line, you get exactly two pieces. If H is one such piece, and x and y are two points in H , then:
  • There is a line segment from x to y which is contained in H
  • Every line segment in the surface from x to y is contained in H
  • The "mirror axiom": There is a local reflection across every line. There is a global reflection across every line.

  • Straight lines could be defined by means of axioms like these: we would take "lines" to mean those paths in a surface which satisfy the specified axioms. In this course we will take a different approach. We will use various criteria to determine which paths should be called "lines" on the surfaces we study, eventually arriving at a definition of "line". Then, for each surface (or "space") that we study, we will check which of the axioms above are true. For most spaces, some of these axioms are true but others are false.

    8. [math/0302003] Quantum Torsors With Fewer Axioms
    math.QA/0302003. From Peter Schauenburg schauen@mathematik.unimuenchen.de Date Fri, 31 Jan 2003 211737 GMT (6kb) Quantum torsors with fewer axioms.
    http://arxiv.org/abs/math.QA/0302003
    Mathematics, abstract
    math.QA/0302003
    Quantum torsors with fewer axioms
    Authors: Peter Schauenburg
    Comments: 7 pages
    Subj-class: Quantum Algebra
    MSC-class:
    We give a definition of a noncommutative torsor by a subset of the axioms previously given by Grunspan. We show that noncommutative torsors are an equivalent description of Hopf-Galois objects (without specifying the Hopf algebra). In particular, this shows that the endomorphism of a torsor featuring in Grunspan's definition is redundant.
    Full-text: PostScript PDF , or Other formats
    References and citations for this submission:
    CiteBase
    (autonomous citation navigation and analysis)
    Links to: arXiv math find abs

    9. Penrose On Godel - An Astronomy Net God & Science Forum Message
    to Penrose)is that a given system of math, axioms plus derived theorems, will contain true solutions that cannot
    http://www.astronomy.net/forums/god/messages/14689.shtml
    Forums: Atm Astrophotography Blackholes CCD ... God and Science Post Penrose On Godel
    Forum List
    Follow Ups Post Message Back to Thread Topics ... In Response To
    Posted by Richard David Yannopoulos-Ruquist, on January 22, 2002 15:59:15 UTC What Godel says (according to Penrose)is that a given system of math, axioms plus derived theorems, will contain true solutions that cannot be derived from the axioms.
    It is true that we know the solutions to be true (proven) based on larger systems of axioms that include the smaller system as a subset.
    Does that make math imperfect. From all that I have read it does mean that math cannot be made rigorous. But that is because we cannot derive the emmergent solutions of a given system of math.
    For me that is better than perfection. We are getting something for nothing. It means to me that nature, based on a fundamental set of postulates or laws, can evolve properties that go beyond what can be calculated from the laws. For me that is like magic, and I find that magical nature of nature appealing.
    It still all comes from math. And it is math that we can ultimately know. Natural laws are finite- a finite number of axioms. But we math physicists can invent a larger system of math that includes nature's math as a subset, and which is able to prove the emergent properties of nature as well.

    10. [math/9805104] Axioms For Weak Bialgebras
    mathematics, abstract math.QA/9805104. From Dr. Florian Nill florian.nill@physik.unimuenchen.de Date Fri, 22 May 1998 173747 GMT (49kb) axioms for Weak
    http://arxiv.org/abs/math/9805104
    Mathematics, abstract
    math.QA/9805104
    Axioms for Weak Bialgebras
    Authors: Florian Nill
    Comments: 48 pages, Latex
    Report-no: Preprint SFB 288/11/5/98
    Subj-class: Quantum Algebra
    Full-text: PostScript PDF , or Other formats
    References and citations for this submission:
    CiteBase
    (autonomous citation navigation and analysis)
    Links to: arXiv math find abs

    11. Formal Systems - An Astronomy Net Blackholes Forum Message
    1) The formal system that comprises math (axioms, theorems, definitions, etc) isn't just a methodology practiced by
    http://www.astronomy.net/forums/blackholes/messages/4388.shtml
    Blackholes Forum Message Forums: Atm Astrophotography Blackholes CCD ... Blackholes I Post Formal Systems
    Forum List
    Follow Ups Post Message Back to Thread Topics ... In Response To
    Posted by Harvey on August 22, 2001 01:01:20 UTC Mark,
    You say that the rules of math exist 'out there'. Okay, let's figure what that means:
    1) The formal system that comprises math (axioms, theorems, definitions, etc) isn't just a methodology practiced by mathematicians, but it also describes the workings of reality.
    2) This formal system is built upon axioms (using rules of inference, definitions, etc) to construct theorems. Hence, reality itself is 'built' upon the same principle. That is, statements of reality depend on axioms, which is why statements of reality are to be considered true. If the axioms of reality are wrong, then the statements are also wrong.
    You have to give account for the meaning of axioms in your model if you want to use a 'math world' as your metaphysical foundation.
    Warm regards, Harv Follow Ups:

    12. Math Forum - Ask Dr. Math
    at 173049 From Doctor Achilles Subject Re Geometry, proofs Hi Allie, Thanksfor writing to Dr. math. To do this, you start with definitions and axioms.
    http://mathforum.org/library/drmath/view/52373.html

    Associated Topics
    Dr. Math Home Search Dr. Math
    Why Proofs? Definitions and Axioms
    Date: 09/16/2001 at 18:49:30 From: Allie Subject: Geometry, proofs Why is a proof needed? Why do you think proofs are important in the development of a mathematical system such as geometry? Thank you, Allie http://mathforum.org/dr.math/ Associated Topics
    High School About Math

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    [ Choose "whole words" when searching for a word like age. all keywords, in any order at least one, that exact phrase parts of words whole words Submit your own question to Dr. Math Math Forum Home Math Library Quick Reference ... Math Forum Search Ask Dr. Math TM http://mathforum.org/dr.math/

    13. Math Forum - Ask Dr. Math
    The points may LOOK as if they determine a plane (and in fact they will, becausethese axioms are true of any geometry you Doctor Peterson, The math Forum http
    http://mathforum.org/library/drmath/view/61011.html

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    14. Separation Axioms
    in Topology by Lynn Arthur Steen and J. Arthur Seebach, Jr. Separation axioms. Let (X O) be a topological space.
    http://www.math.toronto.edu/jjchew/math/topology/separation.html
    Back to John Chew's Home Page Much of the following is based on Counterexamples in Topology by Lynn Arthur Steen and J. Arthur Seebach, Jr.
    Separation Axioms
    Let (X,O) be a topological space.
    T (Kolmogorov) Spaces
    (X,O) is a T space if for every pair of points a and b there exists an open set U in O such that at least one of the following statements is true:
  • a lies in U and b does not lie in U.
  • b lies in U and a does not lie in U.
    T
    (X,0) is a T space if for every pair of points a and b there exists an open set U such that U contains a but not b . To say that a space is T is equivalent to saying that sets consisting of a single point are closed. All T spaces are T . The 'particular point' topology (where the open sets are the sets containing a particular point a ) is T but not T
    T (Hausdorff) Spaces
    (X,0) is a T space if for every pair of points a and b there exist disjoint open sets which separately contain a and b . Some people say in this case that open sets separate points. All T spaces are T . The 'cofinite' topology on an infinite set (where the open sets are those with finite complement) is T but not T
    T (Regular) Spaces
    (X,0) is a T
  • 15. Math 302 The Axioms
    math 302 The axioms. An axiom means A proposition that commends itself to generalacceptance; a well established or universallyconceded principle (OED2).
    http://www.math.uiuc.edu/~stolman/m302/handouts/axioms.html
    Math 302 The Axioms
    An axiom means "A proposition that commends itself to general acceptance; a well established or universally-conceded principle..." (OED2). Often, one assumes the following statements are true. However, they are not true on every space. Therefore, we will check if each statement is true on each space.
  • The "incidence axiom" There is at least one straight line between any two points. There is at most one straight line between two points. The "ruler axiom" You can travel an infinite distance along a straight line in either direction. As you travel along a straight line, you never pass over the same point twice. The "protractor axiom" There is at least one straight line through any point in any direction. There is at most one straight line through any point in any direction. The "half-plane" axiom If you cut the surface along a straight line, you get two pieces. Every straight line segment that connects two points on one of the pieces is contained entirely in that piece. The "mirror axiom" There is a global reflection through every straight line.
  • 16. Sci.math Topic
    All Discussions sci.math Topic. Date. Topic. Author. 12 Jun 02. Physics from empirical axioms and deductive reason.
    http://mathforum.com/discuss/sci.math/t/417693
    The Math Forum
    sci.math
    Search

    All Discussions
    sci.math Topic
    Date Topic Author 12 Jun 02 Physics from empirical axioms and deductive reason. Charles Francis 12 Jun 02 Re: Physics from empirical axioms and deductive reason. David C. Ullrich 13 Jun 02 Re: Physics from empirical axioms and deductive reason. Charles Francis 13 Jun 02 Re: Physics from empirical axioms and deductive reason. David C. Ullrich 15 Jun 02 Re: Physics from empirical axioms and deductive reason. Mike 15 Jun 02 Re: Physics from empirical axioms and deductive reason. David C. Ullrich 16 Jun 02 Re: Physics from empirical axioms and deductive rea... Mike 16 Jun 02 Re: Physics from empirical axioms and deductive re... David C. Ullrich 16 Jun 02 Re: Physics from empirical axioms and deductive r... tadchem 16 Jun 02 Re: Physics from empirical axioms and deductive ... Mike 16 Jun 02 Re: Physics from empirical axioms and deductive ... David C. Ullrich 16 Jun 02 Re: Physics from empirical axioms and deductive... tadchem 16 Jun 02 Re: Physics from empirical axioms and deductiv...

    17. Homework 1 For Math 302, Fall 2000
    Refer to the axioms for straight lines which were handed out in class (they arealso available at http//www.math.uiuc.edu/~stolman/m302/handouts/axioms.html
    http://www.math.uiuc.edu/~stolman/m302/homework/hn1.html
    Math 302 , Assignment 1, Fall 2000
    Due Wed August 30 (at the beginning of class)
  • Using that the formula for a straight line is ax + by = c , prove that there exists a line through any two points.
    Refer to the axioms for straight lines which were handed out in class (they are also available at http://www.math.uiuc.edu/~stolman/m302/handouts/axioms.html ). Consider the subset of the plane consisting of all points (x,y) so that y is greater than 0. This is called the open upper half plane. Determine whether each of the axioms is true or not on the open upper half plane Explain each of your answers. Diagrams may be helpful. Build an approximate model of the hyperbolic plane, useing the sheet of cutouts given to you and following the directions below. You should build this carefully, with scissors and tape. Please construct your model with care, as you will use it for many explorations of hyperbolic space. Bring it to class in your shoebox.
  • The handout is a sheet with many annular arcs to cut out. This is used to build the "annular hyperbolic plane" as described in Chapter 5 of Henderson's book. To build this model, cut out each piece, and note that it is one-sixth of an annulus, with an inner rim, an outer rim, and two short radial ends. Next, tape these together. The rules are as follows. The inner rim of each piece gets taped to (most of) the outer rim of another one. A short radial end of one piece can be taped to such an end of another piece, oriented so they form a larger part of one annulus (and NOT in an "S" shape). No pieces should be allowed to overlap.

    18. Math 302: The Axioms
    math 302 The axioms for Straight Lines The "incidence axiom" There is at least one line between any two given points. There is at most one line between any two given points. The "ruler axiom"
    http://www.math.uiuc.edu/~jms/m302/01Sp/handouts/axioms.html
    Math 302: The Axioms for Straight Lines
  • The "incidence axiom": There is at least one line between any two given points. There is at most one line between any two given points. The "ruler axiom": Along any given line, you can travel an infinite distance forwards or backwards. As you travel forwards along a line, you never pass over the same point twice. The "protractor axiom": There is at least one line through any given point in any given direction. There is at most one line through any given point in any given direction. The "halfplane" axiom: If you cut the surface along a line, you get exactly two pieces. If H is one such piece, and x and y are two points in H , then:
  • There is a line segment from x to y which is contained in H
  • Every line segment in the surface from x to y is contained in H
  • The "mirror axiom": There is a local reflection across every line. There is a global reflection across every line.

  • Straight lines could be defined by means of axioms like these: we would take "lines" to mean those paths in a surface which satisfy the specified axioms. In this course we will take a different approach. We will use various criteria to determine which paths should be called "lines" on the surfaces we study, eventually arriving at a definition of "line". Then, for each surface (or "space") that we study, we will check which of the axioms above are true. For most spaces, some of these axioms are true but others are false.

    19. Math 4606, Summer 2002 Inductive Sets, N , The Pe Ano Axioms, R E
    math 4606, Summer 2002 Inductive sets, N, the Peano axioms, Recursive Sequences ver. 1 Page 1 of 13 A set S R is an inductive set if 0 S and if, whenever x S then x + 1 S. In logic this is ( In principle, we can use the fourfollowing statements as ourfundamental axioms, then dene the integers, the
    http://www.math.umn.edu/~jodeit/course/InductiveSets.pdf

    20. The Axioms And Related Definitions
    next up previous Next Consistency and models Up math 366, axioms forPrevious math 366, axioms for The axioms and related definitions.
    http://www.cs.umsl.edu/~benson/m366/m366axioms/node1.html
    Next: Consistency and models Up: Math 366, Axioms for Previous: Math 366, Axioms for
    The axioms and related definitions
    The structure that underlies our theory is:
    Here S is a set, whose elements are called points , and is a family of subsets of S , whose elements are called lines . We do not, however, define the terms ``point'' and ``line''. The object d is a map
    which we call a distance function. For points , we usually write AB for the number d A B Using the map d we make the following definitions
    • Betweenness relation A B C for distinct collinear points A B C means AB BC AC
    • Segments for distinct points
    • Rays for distinct points
    • Angles for distinct non-collinear points A B C
    • Triangles for distinct non-collinear points A B C
    • Convexity : A subset of S is said to be convex if for every pair of distinct points A B in
    Note that the concepts segment, ray, angle, triangle and convexity all are derived from the betweeness relation. We let denote the set of all angles. The object m is a map
    which we call an angle measure . Using d and m we define:
    • Congruence for segments means AB CD
    • Congruence for angles means
    • Congruence for triangles means and
    We next impose the following axioms:
    (L-1)
    For every pair of distinct points there is exactly one line with and . (We denote this line by
    (L-2)
    Every line contains at least two distinct points.

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