Online Store

Geometry.Net - the online learning center
Home  - Mathematical_Logic - Hypotheses And Axioms
Page 1     1-66 of 66    1 

1. JSTOR Die Pythagoreer Religiose Bruderschaft Und Schule Der
According to it, the three sciences geome try, harmonics, and astronomy ought to be based on axioms or Hypotheses. The axioms of geometry were justified by<133:DPRBUS>2.0.CO;2-F

2. An Overview Of Automated Theorem Proving
All of these are tasks that can be performed by an ATP system, given an appropriate formulation of the problem as axioms, Hypotheses, and a conjecture.
Geoff Sutcliffe's Overview of
Automated Theorem Proving
What is Automated Theorem Proving?
Automated Theorem Proving (ATP) deals with the development of computer programs that show that some statement (the conjecture ) is a logical consequence of a set of statements (the axioms and hypotheses ). ATP systems are used in a wide variety of domains. For examples, a mathematician might prove the conjecture that groups of order two are commutative, from the axioms of group theory; a management consultant might formulate axioms that describe how organizations grow and interact, and from those axioms prove that organizational death rates decrease with age; a hardware developer might validate the design of a circuit by proving a conjecture that describes a circuit's performance, given axioms that describe the circuit itself; or a frustrated teenager might formulate the jumbled faces of a Rubik's cube as a conjecture and prove, from axioms that describe legal changes to the cube's configuration, that the cube can be rearranged to the solution state. All of these are tasks that can be performed by an ATP system, given an appropriate formulation of the problem as axioms, hypotheses, and a conjecture. The language in which the conjecture, hypotheses, and axioms (generically known as

3. On The Nature Of Hypothesis
and even socalled positivistic or empirical science is caught in its thrall, for as logic binds these sciences together, mere Hypotheses and axioms
On The Nature of Hypothesis
What is the Empirical basis for the use of Hypothesis in modern science and positivistic philosophy? Is there one? What technique have we to determine the origin of hypothesis which does not make use of hypothesis and therefore rest upon it? What logical mechanism do we possess to create or fabricate the act of hypothesizing within a meaningful context? Does a logical system exist which can create meaningful hypotheses? That is, hypotheses which relate in a meaningful way to their subject matters? Can a logical program ever answer the question why it itself works? Why its own methods can possibly be or must necessarily be true? No. G¶del proved this conclusively. Every logical system ultimately rests upon one final axiom, the principal axiom of all logic, which is the relationship between "if" and "then". Ultimately, all logic boils down to a relationship, but relationships cannot have an Empirical basis. They are infinite in nature. What is the relationship between an atom and the atom next to it? What field equations interject, what fluid forces compel them to dance when close and draws them together when they are separated? Through what medium do they relate, and if it indeed exists, is it truly a medium or merely another substance whose relationship must in turn be analyzed? There is no true vacuum, no such thing as nothing. And so, relationships have themselves no physical basis but only a basis in concept or reason. Number is one expression of such a relationship. Differential another. Hypothesis a third, consequence, origin, all of these combine into a patterned whirlwind of relation. What empirical basis is there for any of it?

4. Uniformitarianism - CreationWiki, The Encyclopedia Of Creation Science
The axioms are universally acclaimed by scientists, and embraced by all geologists — creationary and evolutionary alike. However, the Hypotheses were and
From CreationWiki, the encyclopedia of creation science
Jump to: navigation search Uniformitarianism is defined by the Glossary of Geology as "the fundamental principle or doctrine that geologic processes and natural laws now operating to modify the Earth's crust have acted in the same regular manner and with essentially the same intensity throughout geologic time, and that past geologic events can be explained by phenomena and forces observable today." (Robert Bates and Julia Jackson, Glossary of Geology, 2nd edition, American Geological Institute, 1980, pg. 677) The idea of deep time and an old earth , is based on the philosophy of naturalism . It was promoted in James Hutton's book "Theory of the Earth" and later expanded upon by Charles Lyell in his three-volume series " Principles of Geology " first published 1830-1833. Charles Darwin took Lyell's books on the Beagle where it got him thinking about slow biological change known as gradualism
  • Lyellian Uniformitarianism
    • Unfalsifiable axioms
      Lyellian Uniformitarianism
      Under the term uniformity, Lyell conflated two different types of propositions: a pair of philosophical axioms (required for science to work) and a pair of hypotheses. The axioms are universally acclaimed by scientists, and embraced by all geologists — creationary and evolutionary alike. However, the hypotheses were and still are controversial and accepted by few.

It is the business of this science to test the truth of the Hypotheses and axioms of the lower sciences in terms of Hypotheses of still wider application's Divided Line.htm
PLATO’ S DIVIDED LINE REPUBLIC d ff. B UNDERSTANDING ( no êsis : The Soul’s arrival at first principle) st Division: Intelligible World E THOUGHT (dianoia : The Soul’s conclusion after investigating a hypothesis C pistis st Division Visible World D eikasia A G. M. A. Grube, Plato’s Thought , pp. 24-27 In the physical world then we have the sun from which we derive light, sight and the eye that sees; in the intelligible [world] we have the good from which we derive truth, knowledge and the mind that knows. It is to the good that the sun itself owes its existence. Furthermore, the sun is not only the cause of sight, its light makes existence possible on the physical plane; so the good is not only the cause of knowledge, but causes the very existence of the knowable and, a fortiori of the physical which derives from the knowable. Socrates proceeds at once to make his meaning clear by another image. Starting from the now familiar division of existence into two classes or forms ( ei[dh ), the visible and the intelligible, he bids Glaucon draw a line and divide it into two unequal parts, each part being then subdivided into two further sections, as follows, so that “AD” is to “DC” as “CE” is to “EB” as “AC” is to “CB” (509 d The main division at C is between the world of sense and the world of Ideas.

6. 03Exx
relations, and set algebra); 03E25 Axiom of choice and related propositions 03E60 Determinacy principles; 03E65 Other Hypotheses and axioms
Home MathSciNet Journals Books ...
Contact Us
201 Charles Street
Providence, RI 02904
USA Phone: 401-455-4000
or 800-321-4AMS Or email us at Open Positions
Set theory
  • 03E02 Partition relations 03E04 Ordered sets and their cofinalities; pcf theory 03E05 Other combinatorial set theory 03E10 Ordinal and cardinal numbers 03E15 Descriptive set theory [See also 03E17 Cardinal characteristics of the continuum 03E20 Other classical set theory (including functions, relations, and set algebra) 03E25 Axiom of choice and related propositions 03E30 Axiomatics of classical set theory and its fragments 03E35 Consistency and independence results 03E40 Other aspects of forcing and Boolean-valued models 03E45 Inner models, including constructibility, ordinal definability, and core models 03E47 Other notions of set-theoretic definability 03E50 Continuum hypothesis and Martin's axiom 03E55 Large cardinals 03E60 Determinacy principles 03E65 Other hypotheses and axioms 03E70 Nonclassical and second-order set theories 03E72 Fuzzy set theory 03E75 Applications of set theory 03E99 None of the above, but in this section

7. Truth Evolutionism
From Euclid to Newton, scientists established science system from Hypotheses like axioms and laws, which greatly simplified science foundation.
Cheap Web Hosting Free Web Hosting Credit Card Offers Web Hosting ... Advertise Search the Web
About Evolutionism
Evolutionism means: every change is a link of evolution; connecting all links together will be the whole evolution process from nihility to present existence. It makes the creator a very easy job: just provides the possibility for all kinds of perturbations to emerge. In Darwin's Evolutionism, every life is evolution result from microbe: perturbations from "organic nihility". In "Truth Evolutionism", every existence is evolution result of the corresponding perturbation in nihility. the universe is evolution results from perturbations in "Universe Nihility"; economy is evolution result from perturbations in "economy nihility"; knowledge research is evolution result from perturbations in "knowledge Nihility". Darwinism is just an application of Truth Evolutionism in biology. Evolution of universe is the expanding process: from infinitesimal to infinite large. Origin of the "universe perturbation" is accidental, but the process from perturbation to infinite large is purely evolutionary. On the other hand, in a long time, the origin of such a perturbation is almost certain. About Truth Evolutionism In Truth Evolutionism, every existence origins from perturbation in nihility. So its ultimate goal is to find the evolution process from perturbation to existence and the best methods for expansion.

8. List KWIC DDC And MSC Lexical Connection
Hypotheses and axioms other 03E65 hypothesis and Martin s axiom continuum 03E50 hypothesis testing 519.56 hypothesis testing 62F03
hypotheses and axioms # other
hypothesis and Martin's axiom # continuum
hypothesis testing
hypothesis testing
hypothesis testing
hypothesis testing
hypothesis testing # Markov processes:
hypothesis testing # non-Markovian processes:
hysteresis # equations with
hysteresis # problems involving i. rings, rings embeddable in matrices over commutative rings # semiprime p. i.e. PDE on finite-dimensional spaces for abstract space valued functions) # partial operator-differential equations ( i.e. small categories in which all morphisms are isomorphisms) # groupoids ( ideal boundary theory ideal rings # principal ideal theory ideal theory # ideals; multiplicative idealism and related systems and doctrines ideals ideals # distribution of prime ideals # extensions of rings by ideals # injective and flat modules and ideals # operator ideals # other classes of modules and ideals # projective and free modules and ideals # theory of modules and ideals # torsion modules and ideals (nuclear, $p$-summing, in the Schatten - von_Neumann classes, etc.) # operators belonging to operator ideals (Rees ring, form ring), analytic spread and related topics # associated graded rings of

9. 03Exx
and set algebra) 03E25 Axiom of choice and related propositions 03E30 Large cardinals 03E60 Determinacy principles 03E65 Other Hypotheses and axioms
Set theory 03E02 Partition relations 03E04 Ordered sets and their cofinalities; pcf theory 03E05 Other combinatorial set theory 03E10 Ordinal and cardinal numbers 03E15 Descriptive set theory [See also ] 03E17 Cardinal characteristics of the continuum 03E20 Other classical set theory (including functions, relations, and set algebra) 03E25 Axiom of choice and related propositions 03E30 Axiomatics of classical set theory and its fragments 03E35 Consistency and independence results 03E40 Other aspects of forcing and Boolean-valued models 03E45 Inner models, including constructibility, ordinal definability, and core models 03E47 Other notions of set-theoretic definability 03E50 Continuum hypothesis and Martin's axiom 03E55 Large cardinals 03E60 Determinacy principles 03E65 Other hypotheses and axioms 03E70 Nonclassical and second-order set theories 03E72 Fuzzy set theory 03E75 Applications of set theory 03E99 None of the above, but in this section
Version of December 15, 1998

10. Why Science (Natural Philosophy) Is Bullshit
On the basis of these (and still more, this list is just the highlights and not exhaustive) axioms, scientists propose Hypotheses leading to coherent,
Next: Why Logical Positivism is Up: Critique of Specific Philosophies Previous: Why Solipsism is Bullshit Contents
Why Science (Natural Philosophy) is Bullshit
Yes, in a section or three we'll get around to religion, and show how religion, especially organized religion, is philosophical Bullshit however useful or useless you might perceive of it to be from a socio-memetic or ethical point of view. However if I did religion right now I'd be accused of being a Godless Scientistic, and since I'm actually a Godful Scientist I figured I might as well smash my own dolly before smashing anybody else's. So just what is this ``Science'' thing of which I'm about to speak? I'm so glad you asked. Science should properly be called by its true name - Natural Philosophy . It is founded on the very simple idea that if you want to know how Nature works, to find the deepest possible answers to all of those Big Questions, the best way to proceed is to ask it . Metaphorically speaking, of course. The asking and answering of questions in science has been reduced to a ritual. Although Real Science is, in fact, not always or even usually done in strict accord with it , the key step, the step of deciding what one is permitted to conclude, is known as the Scientific Method.

11. 03Exx
03E50, Continuum hypothesis and Martin s axiom. 03E55, Large cardinals 03E65, Other Hypotheses and axioms. 03E70, Nonclassical and secondorder set
Set theory Partition relations Ordered sets and their cofinalities; pcf theory Other combinatorial set theory Ordinal and cardinal numbers Descriptive set theory
[See also Cardinal characteristics of the continuum Other classical set theory (including functions, relations, and set algebra) Axiom of choice and related propositions Axiomatics of classical set theory and its fragments Consistency and independence results Other aspects of forcing and Boolean-valued models Inner models, including constructibility, ordinal definability, and core models Other notions of set-theoretic definability Continuum hypothesis and Martin's axiom Large cardinals Determinacy principles Other hypotheses and axioms Nonclassical and second-order set theories Fuzzy set theory Applications of set theory None of the above, but in this section

12. Weekly Intent - Desh Kapoor : IntentBlog
Its a construct operating according to certain defined mathematical and physical principles (which are themselves Hypotheses or axioms).
Featured Posts
Weekly Intent - Desh Kapoor
Intent - December 08, 2007
Fallacies of God, Force, Matter, Space and Time
God is a matter of faith. Faith that because things "work" even though in a manner not always according to our wishes and "equitably" (as we define the word). Yet, it seems to be so. Random world would leave no room for any hope. No reason to do anything or make an effort. Faith, that there is a reason, gives us life. This reason of why things work (given the assumption, based on faith, that they work) is a mystery. And one that has been unsolved. This reason, we all call God. So lets look at it again: We have faith that there is a "reason" in the world, called God. And because we believe the world has a reason, we believe that there is a "God". See the circular logic? Reason is euphemism for cause and effect. That every cause gives off an effect and every effect has a cause. So, if there was no Cause for any Effect, there would be no need of any reason and, therefore, "God". God is a necessity of faith in cause and effect. Nobody has "seen" God. But God is believed in because its "effects" are seen, based on the assumption that He exists!! Lets put this thing again We believe in the "force" called God without seeing Him, because of the effects of His "existence" - based on the assumption that He should exist!

13. The Axioms Of Probability Calculus (part 1) « Philosophy Of Science
The Logic of Falsification and the Structure of Scientific Hypotheses . to be a very nice summation of the most fundamental axioms of probability.
Philosophy of Science
Can there be falsification in Economics? ... The Logic of Falsification and the Structure of Scientific Hypotheses
The Axioms of Probability Calculus (part 1)
April 4, 2006 1 Comment
Basic Rules of Probability Calculus (part 1) Assumptions: These rules function for a finite group of events/propositions. Elementary set theory is presupposed. If A and B are logically equivalent, then P(A) = P(B). This should be read " the probability of A is equal to the probability of B. (1: AXIOM) Normality This axiom states that all probabilities fall between (no chance of some event occurring) and 1 (a certain chance of some event occurring). (2: AXIOM) Certainty P(certain event) = 1 Thus: P(Ω) = 1 (3: AXIOM) Additivity If two events are mutually exclusive, the probability that one or the other occurs is the sum of their individual probabilities. Thus: P(AvB) = P(A) + P(B In this case think of the probabilities involved in tossing a coin: there is a .5 chance for either heads or tails coming up. Thus, the probability for one or the other event occurring is 50/50 or .5 + .5 = 1, because one event must occur. Overlap (Derivation from 1-3) When A and B are not mutually exclusive, it is necessary to subtract the probability of their overlap. Thus:

14. MathNet-Mathematical Subject Classification
03E50, Continuum hypothesis and Martin s axiom See also 04A30, 54A25. 03E55, Large cardinals 03E65, Other Hypotheses and axioms

15. Sachgebiete Der AMS-Klassifikation: 00-09
03D75 Abstract and axiomatic recursion theory 03D80 Applications of recursion which contradict the axiom of choice 03E65 Other Hypotheses and axioms
Sachgebiete der AMS-Klassifikation: 00-09
nach 90-99 Weiter nach 10-19 Suche in allen Klassifikationen
01-XX 03-XX 04-XX 05-XX 06-XX 08-XX
nach 90-99 Weiter nach 10-19 Suche in allen Klassifikationen

16. Gallery :: Part III: The Illustrated "Zen & The Art Of Motorcycle Maintenance" C
His judgment that the scientist selects facts, Hypotheses and axioms on the basis of harmony, also left the rough serrated edge of a puzzle incomplete.

It could either be a matter of inferential blunders or of generating a contradiction by affirming a possibility which is denied by the axioms, Hypotheses,
Home Page Papers by Peirce Peirce-Related Papers TO BOTTOM OF PAGE

Originally published in Peirce Studies It should be noted initially that the thesis I advance in this paper is part of a broader thesis concerning the general nature of inquiry and a host of surrounding epistemological issues. In the broader context I concern myself with questions such as whether indubitability-in-principle plays any role within Peircean epistemic methodology; whether there are, for Peirce, a variety of epistemically admissible kinds of indubitability; what 'the relation is between truth, indubitability and inquiry; and other related questions as well. Here, however, I limit myself to the specific problem concerning the relationship between indubitability, internal reality and the domain of the fictitious; in particular I try to show the importance, for Peirce, of the distinction between the internally real and the fictitious. We may begin by noticing that if one cannot be wrong in the judgments one makes if one's judgments are indubitable-in-principle then the object of such a judgment is precisely what one thinks it to be, no matter what one thinks it to be. Such judgments are, in effect

18. Learning-Org December 1999: When Is Something Real? LO23493
Conjectures, like Hypotheses, postulates, axioms, logical theorems and empirical laws, are a special kind of propositions. A proposition is
When is something real? LO23493
AM de Lange (
Tue, 7 Dec 1999 00:01:25 +0200
Replying to LO23286
Dear Organlearners,
Greetings to you all.
In LO23286 I wrote:
I then told how we here in South Africa since the fall of apartheid are
learning how to deal more inclusive with reality. I concluded with:
Since then we had the following contributions to this topic: I will quote
what I think is important to each contribution. The result will be
lengthy, but it helps to draw a rich picture. It is now interesting to see how much my own contemplations in PART II on reality reflects and fails to reflect these contributions.

19. HeiDok
03E65 Other Hypotheses and axioms ( 0 Dok. ) 03E70 Nonclassical and secondorder set theories ( 0 Dok. ) 03E72 Fuzzy set theory ( 0 Dok.

20. Gene Expression: Am I Chomsky?
While we re on the subject of mea culpas I ve also been re-evaluating some of my Hypotheses and axioms. I get so worked up over the fact that the
Gene Expression Front Page September 25, 2003 Am I Chomsky? I choose to live in what I think is the greatest country in the world... Noam Chomsky Funny, isn't it? Noam Chomsky is just about the last person you'd expect to sing the glories of the United States of America. But his admission above reveals that while he may believe some of the same things as many other Americans, his continual emphasis on what is wrong with America leaves most readers with the idea that Chomsky hates the United States. Unfortunately, I've recently realized that I myself have become - in some respects - the mirror image of Chomsky. Like a child who repeatedly tongues an aching tooth, my numerous posts on human biodiversity and race in the United States have overstated the racial problems faced by America. Is race a genuine fissure in the United States? Certainly, but constantly searching out the race angle on a story becomes tedious.[1] Which prompts the question: why do it, then? Part of the reason I wrote/write on such issues is a kind of intellectual triumphalism. Just like those Upper West Side liberals who condemn racism in order to feel ennobled by their consequence-free moral posturing, I enjoyed feeling more knowledgeable than "antiracists"[2] innocent of training in genetics or statistics. Was I right on the facts? Sure, but - as I'm starting to realize - arguments like this or this are the equivalent of dunking on middle-schoolers.

21. MathSC2000 < Mizar < Mizar TWiki
03E65 Set theory Other Hypotheses and axioms Primary classification Article Secondary classification Article 1 tarski Tarski Grothendieck Set Theory
Skip to topic Skip to bottom Jump: Mizar

Formal (mathematical) logic is a system for deduction that is always correct when the Hypotheses and axioms are correct. Effectively, all mathematics has
Back to main page
Back to home page
Analyzing Arguments and Evidence
Black, white, and shades of gray
Real world situations always have an effectively infinite number of details. No understanding we have of them will ever be entirely complete. We should always be careful when any issue seems to line up as having two sides which are extreme opposites - right and wrong, good and bad, for us or against us. Almost every situation will have details that we don't know or can't account for. (more)
The dangers of hidden assumptions
Often when we believe something false, that belief is an assumption that we don't even realize we're making. From these we might draw other false conclusions without ever considering whether our original assumption was false. (more)
Cultural assumptions that seem natural
A few hundred years ago it was "obvious" that women were unqualified to vote or hold office. It wasn't an issue that was debated or even questioned. It just seemed natural that men should rule. It was also obvious that educated people should know Latin and a dark suntan was an indication of good health. Some of the hardest assumptions to detect are the ones that are so widely held that they seem natural. We assume these "truths" have been proven over the years. This is not a safe assumption. What false cultural assumptions are we making today? (more)
Testability: is there a way to determine whether a statement is true or false?

23. :: God And Purpose In The Light Of Science And Reason
These Hypotheses have more in common with the most basic axioms and Hypotheses of physical science than may be apparent at first glance.
God and Purpose in the Light of Science and Reason
menu A
item 1

item 2
item 3
item 4
item 5
Search this site

"Intelligence makes clear to us the interrelationship of means and ends.
But mere thinking cannot give us a sense of the ultimate and fundamental
ends. To make clear these fundamental ends and valuations and to set them
fast in the emotional life of the individual, seems to me precisely the most important function which religion has to form in the social life of man."...Albert Einstein "So, in brief, we do not belong to this material world that science constructs for us. We are not in it; we are outside. We are only spectators. The reason why we believe that we are in it, that we belong to the picture, is that our bodies are in the picture. Our bodies belong to it. Not only my own body, but those of my friends, also of my dog and cat and horse, and of all the other people and animals. And this is my only means of communicating with them." Erwin Shroedinger It is with great humility that I must address the issue of why I believe in God. I can not aim to produce a "proof" in the deductive, scientific sense of the word "proof". Rather, I must appeal to an inductive process of understanding. What I will attempt to do is to draw attention to a series of evidences that point to a deep reality.

24. MSC 2000 : CC = Other
Other notions of settheoretic definability; 03E65 Other Hypotheses and axioms Riemann and other Hypotheses; 11M41 Other Dirichlet series and zeta

25. Define 'Axiom Postulate Theorem Corollary Hypothesis' - #region Coad's Code
If this statement does not follow from your present system of axioms, you may wish to include an additional axiom or make the hypothesis an axiom itself and
Noah Coad, Microsoft Program Manager
Visual Studio Platform, Former C# MVP
Define 'Axiom Postulate Theorem Corollary Hypothesis'
coad When explaining to a friend of mine some of the fundamentals of mathematics, I found myself giving definitions of these base terms. A quick search with Google returned this great excerpt that efficiently and eloquently defines the words and relations, and merits posting here. Thus, axioms and postulates form the roots of a particular deductive system; theorems and corollaries are the logical consequences that fill out the deductive system; hypotheses drive theoretical development forward. From: The American Heritage Book of English Usage Filed under: General
TrackBack said on June 14, 2005 12:16 AM:
Define 'Axiom Postulate Theorem Corollary Hypothesis'ooeess
TrackBack said on July 16, 2005 02:33 AM:
Define 'Axiom Postulate Theorem Corollary Hypothesis'ooeess
TrackBack said on July 31, 2005 10:06 PM:
Define 'Axiom Postulate Theorem Corollary Hypothesis'ooeess
Leave a Comment
Title (required) Name (required) Your URL (optional Comments (required) Remember Me?

26. Theory
For instance, starting with some basic Hypotheses (or axioms), such as gases are made of tiny, independent particles , one can derive,
Chapter 1 Objectives BHS Mr. Stanbrough Physics ... About Science "Theories are schemes of thought with assumptions chosen to fit experimental knowledge, containing the speculative ideas and general treatment that make them grand conceptual schemes." E. M. Rogers, Physics for the Inquiring Mind (Princeton University Press, Princeton, NJ, 1966).
A Scientific Theory
According to the text , a scientific theory is: "... a synthesis of a large body of information that encompasses well-tested and verified hypotheses about certain aspects of the natural world." (p. 3) Huh? Suppose that, in thinking about the behavior of gases, it occurs to you that all of the facts and laws about the pressure, volume, and temperature of a gas could be explained if gases were made of very tiny, independent particles moving about at high speeds. This is an interesting hypothesis which produces several experimental tests. Suppose that the predictions of this hypothesis all seem to indicate that the hypothesis is correct. Scientists begin to call your hypothesis a theory
Characteristics of Scientific Theories
"Theory" is probably the most misused and ambiguous word in science - misused and ambiguous in its use by scientists as well as nonscientists! Some characteristics of a theory are:

27. Albert Einstein
The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of Hypotheses or axioms.
Albert Einstein

Timeless Aphorisms albert einstein friedrich nietzsche evert jan ouweneel alfred north whitehead ... other subjects
EINSTEIN’S APHORISMS personal discovery and creativity god knowledge, truth and certainty ...

Only one who devotes himself to a cause with his whole strength and soul can be a true master. For this reason mastery demands all of a person.
If A equal success, then the formula is A equals X plus Y and Z, with X being work, Y play, and Z keeping your mouth shut.
Only two things are infinite: the universe and human stupidity; and I’m not sure about the former.
The release of atomic energy has not created a new problem. It has merely made more urgent the necessity of solving an existing one.
The hardest thing in the world to understand is the income tax. He who joyfully marches to music in rank and file has already earned my contempt. He has been given a large brain by mistake, since for him the spinal cord would fully suffice. This disgrace to civilization should be done away with at once. Heroism at command, senseless brutality, and all the loathsome nonsense that goes by the name of patriotism, how violently I hate all this, how despicable and ignoble war is; I would rather be torn to shreds than be part of so base an action! It is my conviction that killing under the cloak of war is nothing but an act of murder. True art is characterized by an irresistible urge in the creative artist.

28. Bacon
In doing induction we come up with what Bacon calls axioms, which we would now call lawlike Hypotheses. These axioms must meet certain standards,
Baconian Induction and Latent Schematisms
In the Novum Organum , Francis Bacon lays out a method of induction, a method which is the correct path to knowledge of Nature. After briefly sketching this method, I shall argue that the making of hypotheses about the invisible internal structure of things plays a crucial role in Bacon's method of induction, and that our knowledge of the laws of nature is inseparable from our knowledge of this invisible inner structure. Bacon states at the beginning of book II of the Novum Organum that "It is the task and purpose of human knowledge to discover the form of a given nature, or its true specific difference, or nature-engendering nature, or source of emanation" (II, 1). Bacon means by each of these phrases the same thing, viz. the law governing the given nature we are interested in: "For when I speak of forms I mean nothing but those laws and definitions of pure actuality, which govern and constitute any simple nature, such as heat, light, weight, in every kind of material and subject that is capable of receiving them. Therefore the form of heat or the form of light are the same things as the law of heat or the law of light" (II, 17). We go about discovering these laws by the process of induction, or "Interpretation of Nature" (I, 115), according to Bacon. In doing induction we come up with what Bacon calls "axioms," which we would now call "lawlike hypotheses." These axioms must meet certain standards, stated in the following "precept":

29. New Axioms For Set Theory
In addition, there are interesting Hypotheses that appear to be true but not yet sufficiently analyzed to be proposed axioms. To be effective, set theory
Dmytro Taranovsky
November 5, 2003 New Axioms for Set Theory Note: This paper is a condensed draft. The base theory is ZFC. Axiom schema: Justification: Justification: only ) not very expressive, but such behavior is a restriction, as the development of the theory will clearly illustrate. Finally, no definable set of real numbers negates the Continuum Hypothesis, so the negation of the CH postulates entire cardinalities of sets of real numbers, none of which are definable. Minimization of unexplainable requires their nonexistence. with ordinal definable payoff set and with the set of allowed moves being an ordinal is determined.
(See my paper "Determinacy Maximum" for analysis.) Justification: The principle captures our intuition that there is no natural way to
pick which elements go first in a well-ordering of a power set, so one has to use the axiom of choice to choose the elements. For example, given two arbitrary real numbers, it appears that there is no way to decide which of them should be first in a well-ordering. It is a maximization principle as it suggests that arbitrary subsets exist rather than only those that have enough regularity to fit in a definable well-ordering. More Principles does not change the theory of H(
Note: Absoluteness under Forcing One way to capture the maximality of the parts of the universe is through absoluteness under forcing: Extending an image of the parts with certain generic sets should not alter the theory (the generic extension should be an elementary extension of the theory) because the parts are already the largest possible, so it should not be possible to make them larger. The statement that L(R) is absolute under forcing provides true canonical theory of L(R), which maximizes which sets are constructible above the reals.

30. Phys. Rev. (Series I) 9 (1899): - New Books
As fundamental concepts are those in terms of which fundamental, Hypotheses or axioms are expressed, the reformulation of concepts should involve a similar
Physical Review Online Archive Physical Review Online Archive AMERICAN PHYSICAL SOCIETY
Browse Search Members ... Help
Abstract/title Author: Full Record: Full Text: Title: Abstract: Cited Author: Collaboration: Affiliation: PACS: Phys. Rev. Lett. Phys. Rev. A Phys. Rev. B Phys. Rev. C Phys. Rev. D Phys. Rev. E Phys. Rev. ST AB Phys. Rev. ST PER Rev. Mod. Phys. Phys. Rev. (Series I) Phys. Rev. Volume: Page/Article:
Phys. Rev. (Series I) 9, 59 - 64 (1899)
Previous article
Next article Issue 1 View Page Images PDF (627 kB), or Buy this Article Use Article Pack Export Citation: BibTeX EndNote (RIS) New Books
DOI: 10.1103/PhysRevSeriesI.9.59
View Page Images PDF (627 kB), or Buy this Article Use Article Pack
Previous article
Next article ... Issue 1 No references available yet for this article
No citing articles available.
APS Journals PROLA Homepage Browse ... Search E-mail:

31. Philosophical Foundations Of Occam's Razor
For the first, creative step, the scientist is trying to find a set of Hypotheses or axioms which imply the observed phenomena.
Philosophical Foundations of Occam's Razor As you try to make sense of Occam's Razor you find yourself bumping up against a number of related issues in the foundations of science and philosophy. Here we address some of these issues. The questions involved are big enough that our answers can hardly be considered conclusive, but they should at least help illuminate the areas of dispute. Models vs Classifiers As discussed in the interpretations section, one interpretation which we call "Occam's Razor proper" states that if two models make equivalent predictions, the simpler is to be preferred. This principle tells us how to choose between different models which correspond to the same classifier. It may seem counterintuitive that such models can exist. What do we mean by these words model and classifier In our usage, a model is a set of assumptions (aka axioms, premises) we make to explain our observations. A classifier (or decision rule ) can be defined as a function which takes an input point from the feature space and assigns it to one of a number of classes (or categories). It is important to see that the same classifier can result from different models. Consider an example. Suppose our domain is the integers from 1 to 10, and we are told that 1, 2, 3, 5, and 7 belong to class A and 4, 6, 8, 9, and 10 belong to class B. How could we explain these observations? One model might take as its axiom that numbers whose digits contain fully closed areas (as 4 has a triangle, 8 has two circles, etc) belong to class A, and numbers with no closed areas belong to class B. Another model could assume that prime numbers (including 1) belong to class A and composite numbers belong to class B. These two models make quite different assumptions, but the classifiers they produce (over the domain 1-10) are identical. A more practical example of different models producing (essentially) the same predictions is the case of Kepler's laws of planetary motion and Newton's universal law of gravitation, discussed on the

32. Axiom Quotes & Quotations
“The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of Hypotheses or axioms.”
Advanced Search My Account Help Add the "Dynamic Daily Quotation" to Your Site or Blog - it's Easy! ... AX 1-6 Quotations of
Axiom quotes
It is an old psychological axiom that constant exposure to the object of fear immunizes against the fear. Maxwell Maltz quotes (US plastic surgeon, motivational author, and creator of the Psycho-Cybernetics, 1927-2003) Similar Quotes Add to Chapter... The idea that to make a man work you've got to hold gold in front of his eyes is a growth, not an axiom . We've done that for so long that we've forgotten there's any other way. Add to Chapter... It has long been an axiom of mine that the little things are infinitely the most important Arthur Conan Doyle, Sr. quotes ... Scottish writer, creator of the detective Sherlock Holmes, Similar Quotes Add to Chapter... It is an axiom , enforced by all the experience of the ages, that they who rule industrially will rule politically. Aneurin Bevan quotes ... Politician Add to Chapter... Perhaps the truest axiom in baseball is that the toughest thing to do is repeat. Walt Alston quotes ... Similar Quotes . About: Baseball quotes Add to Chapter...

33. About Proofs
Start with a collection of Hypotheses, which you assume to be true, for the sake of argument. Aristotle divided these Hypotheses into axioms or
Learning Mathematics in College
(Mathematical) Proofs
These pages are composed and maintained by Greg McColm , Department of Mathematics, University of South Florida. One of the less popular kinds of exercises is a proof : proofs tend to be very difficult to do right, and since one already knows the answer (unless the sadistic professor has assigned a trick question) there seems to be little point to it. Why do professors assign proofs?
  • A legitimate proof will not prove something that is not true . That is the point of a proof: to verify that something is true.
  • It takes skill and experience to be able to determine if a proof is legitimate . And the lazy person who can't be bothered, or the unthinking one who just goes by what feels right, is the lawful prey of politicians, advertizers, and opportunistic office mates.
When Pythagoras introduced proofs to Europe, he introduced them as a way to settle what he regarded as religious issues: the stakes were high. And in applied mathematics, when the issue is whether the plane will fly or whether the electrical grid will run, the stakes remain high. Secondly, proofs are a mathematical version of a kind of exercise we've all seen:

34. Radical Therapy - Book From Origin Press
The hypothesis of the Unified Field of love validates the reports of facts by logical deduction from the smallest number of Hypotheses or axioms.
Back to Radical Therapy Contents Chapter 1 Chapter 2 Chapter 3
Chapter 2
The Search for the Unified Field
I have proposed in the first chapter that the fundamental underlying and uniting force of the universe is an infinite The hypothesis of the Unified Field of love validates the reports of mystics who directly (and subjectively ) experienced the same reality that scientists only approach through an objective and outward process of observation, experimentation, and deduction. In other words, the same phenomena that science describes as "energy" or "light" can be directly experienced as love— if experienced from within . In other words, light is a constant of the physical universe, as Einstein showed, but love is the absolute constant of the universe, as the heart reveals. Coming to such a conclusion is, of course, relative to your state of consciousness and is entirely observer-dependent. In radical therapy, I help my clients open their hearts and experience directly (meaning subjectively ) their deepest inner connection to this universe of love. And it is my hope that scientists will realize just how this truth is observer-dependent, and will be able as a result to link this subjective experience of the Unified Field with their own more limited theories based on their objective observation of nature.

35. Metamath Solitaire
Each axiom has one assertion and zero or more Hypotheses. The first 3 have no Hypotheses. The axiom axmp has 2 Hypotheses and is usually called an
Home Metamath Solitaire (Java Applet) Run the applet in a popup window Run the applet in main browser window Welcome to Metamath Solitaire, a Java applet that lets you build simple proofs in logic and set theory. You can play around with it for curiosity or fun, or if you're serious it will be the hardest "game" in the world! This is because built into it are the axioms of logic and ZFC (Zermelo-Fraenkel with Choice) set theory, which are generally held to encompass essentially all of mathematics . It's up to you to unlock its secrets!
  • What is the goal of Metamath Solitaire? Can I make a mistake? I'm impatient and want to see something happen right away! I only see the axioms for propositional calculus. How do I get set theory? ... (For logicians) Where can I find references for the other logics shown on the 'Select Logic Family' screen?
  • Most philosophers agree that contemplation of Reality is the highest form of happiness. So, if you want happiness, play Metamath Solitaire all by yourself.
    New 2-Dec-2006 Croatian [external].

    36. Mark “Halcy0n” Loeser » Albert Einstein
    The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of Hypotheses of axioms.
    @import url( );
    Albert Einstein
    Albert Einstein quotes
    The fairest thing we can experience is the mysterious. It is the fundamental emotion which stands at the cradle of true art and true science. Ethical axioms are found and tested not very differently from the axioms of science. Truth is what stands the test of experience. Laws alone cannot secure freedom of expression; in order that every man present his views without penalty there must be a spirit of tolerance in the entire population. The State is made for man, not man for the State. Science is the attempt to make the chaotic diversity of our sense-experience correspond to a logically uniform system of thought. We should take care not to make the intellect our god; it has, of course, powerful muscles, but no personality. Man like every other animal is by nature indolent. If nothing spurs him on, then he will hardly think, and will behave from habit like an automaton. In a healthy nation there is a kind of dramatic balance between the will of the people and the government, which prevents its degeneration into tyranny. Everything that is really great and inspiring is created by the individual who can labor in freedom.

    37. Global Spiral :: Article
    Secondly, the Hypotheses underlying a theory are unconfirmed statements on what by logical deduction from the smallest number of Hypotheses or axioms.
    The Global Spiral is an e-publication of Metanexus Institute . Through articles, essays, book reviews, and news, the Global Spiral explores humanity's most profound questions and challenges. Home Subject Authors Columns ... About Free Updates! Email
    If you enjoy this article, consider making an online donation to support the Global Spiral.
    Email This Article

    The Criterion of Simplicity
    By Varadaraja V Raman
    In any event, Ockham's principle became a matter of much philosophical and theological controversy. But it has always appealed to the scientist's mind. Newton, in his Principia , noted that "more is vain when less will serve." In our own times a number of philosophers of science have analyzed this concept in detail, and tried to provide a logical justification for it. The criterion of simplicity may be justified on at least two grounds. First, it reflects the efficiency of a scientific theory or explanation because it accomplishes more with less. Secondly, the hypotheses underlying a theory are unconfirmed statements on what is not directly observed or observable. Hence, the simpler and the less the number of such claims, the more acceptable they should be. There is also a third reason, partly theological, for accepting the principle. It is that God would not make a universe governed by complicated laws. But this notion was challenged by Pierre Duhem who regarded such an attitude as obsolete. "There was a time," he wrote, "when physicists supposed the intelligence of a Creator to be tainted with the same debility, when the simplicity of these laws was imposed as an indisputable dogma in the name of which any experimental law expressing too complicated an algebraic equation was rejected, when simplicity seemed to confer on a law a certainty and scope transcending those of the experimental method which supplied it.... We are no longer dupes of the charm which simple formulas exert on us; we no longer take the charm as evidence of a greater certainty." But the criterion has certainly not been given up since Duhem wrote these lines in 1906.

    38. AmosWEB Is Economics: Encyclonomic WEB*pedia
    While axioms cannot be verified directly with real world data, as is the case for Hypotheses, they can be checked indirectly.

    39. ORO: Oxford Reference Online: Premium Collection Sample Entries
    aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest possible number of Hypotheses or axioms.
    Premium Collection sample entries
    About Oxford Reference Online About the Premium Collection Key features Sample entries ... Public area site map

    from The Oxford Dictionary of Quotations
    German-born theoretical physicist; originator of the theory of relativity on Einstein: see Anonymous (43), Squire (2)
    Science without religion is lame, religion without science is blind.
    Science, Philosophy and Religion: a Symposium (1941) ch. 13
    E = mc2.
    (the usual form of Einstein's original statement: 'If a body releases the energy L in the form of radiation, its mass is decreased by L/V2')
    in Annalen der Physik
    God is subtle but he is not malicious.
    remark made during a week at Princeton beginning 9 May 1921, later carved above the fireplace of the Common Room of Fine Hall (the Mathematical Institute), Princeton University; R. W. Clark Einstein (1973) ch. 14
    I am convinced that He [God] does not play dice.
    (often quoted as: 'God does not play dice') letter to Max Born, 4 December 1926; in Einstein und Born Briefwechsel If my theory of relativity is proven correct, Germany will claim me as a German and France will declare that I am a citizen of the world. Should my theory prove untrue, France will say that I am a German and Germany will declare that I am a Jew.

    40. Axioms Cannot Prove Falsity? Text - Physics Forums Library
    A rule is finite if it has finitely many Hypotheses. Then define theorems as follows i) The axioms of S are theorems of S; ii) If all Hypotheses of a rule
    Physics Help and Math Help - Physics Forums PF Lounge General Discussion Philosophy ... PDA View Full Version : Axioms cannot prove falsity? roygabv Hello.
    I am working on a paper detailing Godel's Incompleteness Theorm and I came across this statement.
    "An axiomatic base where all the axioms are true cannot prove anything to be false."
    Is this correct? honestrosewater What's an axiomatic base? Dooga Blackrazor From my interpretation of the sentence, I'd say an "axiomatic base" is the foundation in which a axiom is made up. Basically, an idea founded entirely from axioms cannot prove falsity.
    I think the sentence leaves one open to broad interpretations, but I would agree with the statement. "An axiomatic base where all the axioms are true cannot prove anything to be false."
    Is this correct?
    I'm assuming by axiomatic base they mean those axioms which a system needs and no more :confused:. The irreducibles, I suppose. But, in ANY system, any that I'm aware of any ways, the axioms are taken to be self-evident truths. We speak of axiomatic set-theory, and we hold that its axioms are both self-evidently true and not reducible to any combination of the other axioms. And yet, we prove statements to be either true or false using the rules of logic and these suppositions. Or, I suppose, they are found to be either consistent or inconsistent within our system. But we generally take inconsistency to be falsity, at least within the system. It seems to me that this statement should read "An axiomatic base where the axioms are false cannot prove the truth of a statement." I'm a bit confused, though, and may be entirely wrong. Still...

    41. Science Musings By Chet Raymo
    the greatest possible number of empirical facts by logical deductions from the smallest possible number of Hypotheses of axioms. Simplicity. Parsimony.
    @import "";
    Science Musings by Chet Raymo
    Site Navigation
    Recent Musings
    Musing Archives
    Sunday, July 23, 2006
    Occam's razor
    I n the introduction to my book Skeptics and True Believers, I defined two frames of mind: Skeptics are children of the Scientific Revolution and the Enlightenment. They are always a little lost in the vastness of the cosmos, but they trust the ability of the human mind to make sense of the world. They accept the evolving nature of truth, and are willing to live with a measure of uncertainty. Their world is colored in shades of gray. They tend to be socially optimistic, creative and confident of progress. Since they hold their truths tentatively, Skeptics are tolerant of cultural and religious diversity. They are more interested in refining their own views than in proselytizing others. If they are theists, they wrestle with their God in a continuing struggle of faith. They are often plagued by personal doubts and prone to depression. True Believers are less confident that humans can sort things out for themselves. They look for help from outside from God, spirits or extraterrestrials. Their world is black and white. They seek simple and certain truths, provided by a source that is more reliable than the human mind. True Believers prefer a universe proportioned to the human scale. They are repulsed by diversity, comforted by dogma and respectful of authority. True Believers go out of their way to offer (sometimes forcibly administer) their truths to others, convinced of the righteousness of their cause. They are likely to be "born again," redeemed by faith, apocalyptic. Although generally pessimistic about the state of this world, they are confident that something better lies beyond the grave.

    42. FormalSystemsLanguage - ESW Wiki
    So, the rules of inference tell us how, with the benefit of axioms, and possibly some hypothesis, (wait for it,) we can figure out new things.
    ESW Wiki Search:
    Formal Systems Language
  • Formal Systems Language
  • Formal Systems Theory Symbols ... Discussion
  • Formal Systems
    I don't fully understand what "Formal Systems" mathematics is all about, but I have learned some of it's terminology, which comes up in SemanticWeb technology. My limited understanding is that this is a formalism for how we make logical arguments. It's a lot like Geometry, where you say, "because this is an axiom, and this is something we proved with other axioms, we can reason this, this, and this, and then we can prove this, which was what we wanted." It's also is like Computer Science formalisms of computer languages. That is, where you do things like: Break a language down into the itty bitty words you can express. And then you talk about "is it grammatically correct to arrange the words in this way." And there you have the "Syntax, Semantics, and Pragmatics" split, talking about the issues in the language. So, this is the frame of mind to be in as you read the rest of the terminology.

    43. Utah Skies Astronomy Resource | The Amateur Astronomer's Resource
    The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of Hypotheses or axioms.
    Utah Skies
    " Bringing the Joys of Astronomy to the Public Through Awareness, Advocacy, and Education ."
    Light Pollution Weekly Report Contact Tips ... Search HMO TCO SLC Newsfeed Eventsfeed Podcast eMail Newsletter HOME Solar System Sun Mercury Venus Earth Moon Mars Jupiter Saturn Uranus Neptune Pluto Deep Sky Constellations Messier Galaxies Clusters Nebula Stars Hubble NGC IC ARP AOTM Favorites APOD Spaceweather Astro Photos
    Constellation Report Our Constellation report is an easy way for people to become familiar with the nighttime sky. We’ll discuss myths associated with the various constellations as well as describing the numerous deep sky objects residing in the area. No equipment is required to view the constellations, though a star chart can be quite helpful. This ease of observing makes constellations a natural place to begin your journey to the stars. This weekend we move on to the constellation Pisces the Fishes . Pisces can be found to the south and east of The Great Square of Pegasus . As the name indicates, Pisces is the fishes...plural. One fish runs north/south to the east of Pegasus. The other runs east/west just to the south of Pegasus. These fish are tied together using some fishing line with the knot being Alpha Piscium or Al-Rischa, "The Cord". The constellation is not excessively bright, but it is not invisible either. Under reasonably dark skies you should have no problem finding it.

    44. E-G Quotes
    The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of Hypotheses or axioms.
    Home About Me Pictures Links ... Contact Albert Einstein
    (1879-1955) physicist/creator of the theories of relativity/major contributor to quantum theory
    "What really interests me is whether God had any choice in the creation of the world."
    "The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of hypotheses or axioms."
    "Common sense is the collection of prejudices acquired by age 18."
    "Too many of us look upon Americans as dollar chasers. This is a cruel libel, even if it is reiterated thoughtlessly by the Americans themselves."
    "Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute. THAT'S relativity."
    "I cannot imagine a God who rewards and punishes the objects of his creation, whose purposes are modeled after our own a God, in short, who is but a reflection of human frailty. Neither can I believe that the individual survives the death of his body, although feeble souls harbor such thoughts through fear or ridiculous egotisms." obituary in New York Times, 19 April 1955

    45. Albert Einstein - Biography
    The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of Hypotheses or axioms.
    Now Playing Movie/TV News My Movies DVD New Releases ... search All Titles TV Episodes My Movies Names Companies Keywords Characters Quotes Bios Plots more tips SHOP ALBERT... DVD VHS CD IMDb ... Albert Einstein Biography Quicklinks categorized by type by year by ratings by votes by genre power search credited with biography publicity news articles message board miscellaneous photographs Top Links biography by votes awards news articles ... message board Filmographies categorized by type by year by ratings ... tv schedule Biographical biography other works publicity photo gallery ... message board External Links official sites miscellaneous photographs sound clips ... video clips
    Biography for
    Albert Einstein
    advertisement photos board add contact details Date of Birth 14 March Ulm, Germany Date of Death 18 April , Princeton, New Jersey, USA. (heart failure) Height Mini Biography Son of Hermann and Pauline Einstein. His father was a featherbed salesman. Albert began reading and studying science at a young age, and he graduated from a Swiss high school when he was 17. He then attended a Swiss Polytechnic, where he met his first wife. He graduated in 1900, and became a Swiss citizen in 1901. He began working at the Swiss Patent Office, and continued his scientific studies. He taught at universities in Prague, Zurich, and Berlin, and continued his research in physics. The onset of World War II led him to move to the United States, and he was granted a post at the Institute for Advanced Study in New Jersey. Einstein was heavily involved in attempting to bring about world peace in his later life, and he continued his scientific research until his death in 1955.

    46. Continuum Hypothesis - Wikipedia, The Free Encyclopedia
    Later work by Paul Cohen established that the continuum hypothesis is neither provable nor disprovable from the axioms of ZermeloFraenkel set theory with
    var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia";
    Continuum hypothesis
    From Wikipedia, the free encyclopedia
    Jump to: navigation search This article is about a hypothesis in set theory. For the assumption in fluid mechanics, see fluid mechanics In mathematics , the continuum hypothesis (abbreviated CH ) is a hypothesis , advanced by Georg Cantor , about the possible sizes of infinite sets . Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers . His proofs, however, give no indication of the extent to which the cardinality of the natural numbers is less than that of the real numbers. Cantor proposed the continuum hypothesis as a possible solution to this question. It states:
    There is no set whose size is strictly between that of the integers and that of the real numbers.
    In light of Cantor's theorem that the sizes of these sets cannot be equal, this hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. The name of the hypothesis comes from the term the continuum for the real numbers.

    47. DC MetaData For: The Proper Forcing Axiom And The Singular Cardinal Hypothesis
    Abstract We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof uses ideas of Moore from \cite{mooSCH} and the notion of
    Matteo Viale
    The Proper Forcing Axiom and the Singular Cardinal Hypothesis

    Preprint series:
    ESI preprints
    03E65 Other hypotheses and axioms
    03E75 Applications
    Abstract We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis.
    function on pairs of ordinals.
    Keywords: MRP, PFA, SCH, minimal walks

    48. 03E: Set Theory
    Natural axioms which imply the negation of Continuum Hypothesis; How does (Set Theory axiom) V=L prove the Continuum Hypothesis?
    Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields
    POINTERS: Texts Software Web links Selected topics here
    03E: Set theory
    Naive set theory considers elementary properties of the union and intersection operators Venn diagrams, the DeMorgan laws, elementary counting techniques such as the inclusion-exclusion principle, partially ordered sets, and so on. This is perhaps as much of set theory as the typical mathematician uses. Indeed, one may "construct" the natural numbers, real numbers, and so on in this framework. However, situations such as Russell's paradox show that some care must be taken to define what, precisely, is a set. However, results in mathematical logic imply it is impossible to determine whether or not these axioms are consistent using only proofs expressed in this language. Assuming they are indeed consistent, there are also statements whose truth or falsity cannot be determined from them. These statements (or their negations!) can be taken as axioms for set theory as well. For example, Cohen's technique of forcing showed that the Axiom of Choice is independent of the other axioms of ZF. (That axiom states that for every collection of nonempty sets, there is a set containing one element from each set in the collection.) This axiom is equivalent to a number of other statements (e.g. Zorn's Lemma) whose assumption allows the proof of surprising even paradoxical results such as the Banach-Tarski sphere decomposition. Thus, some authors are careful to distinguish results which depend on this or other non-ZF axioms; most assume it (that is, they work in ZFC Set Theory).

    49. PlanetMath: Axiom
    However, at this date we have no way of demonstrating the consistency of modern set theory (ZermeloFrankel axioms). The axiom of choice, a key hypothesis
    (more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... RSS Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia



    meta Requests



    talkback Polls
    Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About axiom (Definition) In a nutshell, the logico-deductive method is a system of inference where conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (syllogisms, rules of inference Tautologies excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions ( theorems , if we are talking about mathematics) must be proven with the aid of the basic assumptions. The logico-deductive method was developed by the ancient Greeks, and has become the core principle of modern mathematics. However, the interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the

    He would have also recognised that the use of the terms hypothesis or axiom , including when qualified as fundamental and when expressed with the help
    Michel Bitbol Home First Published (In French) in: Philosophia Scientiae, 3 (cahier 2), 203-213, 1999 (with the title: " L'alter-ego et les sciences de la nature, Autour d'un débat entre Schrödinger et Carnap"), and also as chapter 2-13 of: Michel Bitbol, Physique et philosophie de l'esprit, Flammarion, 2000. Recently revised English version: Phenomenology and the Cognitive Science, "All the premises on which science depends, when they are not of a purely conventional nature, rest on experience". It is with this programmatic sentence that Carnap concludes an article published in French, in 1936, in the journal Scientia. This article was entitled "Existe-t-il des prémisses de la science qui soient incontrôlables?" (Are there premises of science which are beyond control?).
    Schrödinger's article, modestly titled "Quelques remarques au sujet des bases de la connaissance scientifique" (Some remarks on the bases of scientific knowledge), contains a thesis which is outrageous for Carnap, explaining his prompt and vigorous response. That thesis, all the more provocative to Carnap because it had been formulated by one of the greatest physicists of the time, is as follows: "Science is not self-sufficient; it needs a fundamental axiom, a basic axiom from without". An "axiom" which is radically outside the system of science because it is neither empirically testable nor assimilable to a convention. A basic axiom which Carnap thereby identifies as a trans-empirical premise which should be called metaphysical.

    51. The Scientific Method/Components Of The Method - Wikibooks, Collection Of Open-c
    1 Principles; 2 World View (axioms, Postulates); 3 Theory. 3.1 Hypothesis; 3.2 Predictions; 3.3 Theorems. 4 Verification. 4.1 Observations; 4.2 Experiments
    var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikibooks";
    The Scientific Method/Components of the Method
    From Wikibooks, the open-content textbooks collection
    The Scientific Method Jump to: navigation search The Scientific Method
    • Principles World View (Axioms, Postulates) Theory
      edit Principles
      The laws of nature as we understand them and the bases for all empirical sciences. They are the result of postulates (specific laws) that have passed experimental verification resulting in principles that are widely accepted and can be re-verified (using observation or experimentation).
      edit World View (Axioms, Postulates)
      The word "axiom" comes from the Greek word αξιωμα (axioma), which means that which is deemed worthy or fit or that which is considered self-evident. To "postulate" is to assume a theory valid due to be based on an a given set of axioms, resulting on the creation of a new axiom, this is so due to be self evident, "axiom," "postulate," and "assumption" are used interchangeably. Generally speaking, axioms are all the laws that are generally considered true but largely accepted on faith, they cannot be derived by principles of deduction nor demonstrable by formal proofs—simply because they are starting assumptions, other examples include personal beliefs, political views, and cultural values. An axiom is the basic precondition underlying a theory.

    52. 20th WCP: The Heuristic Function Of The Axiomatic Method
    This axiomatic method is usually restricted to a nonphilosophical approach to pure e.g., by the creative methods to find best Hypotheses (abduction),
    Philosophy of Science The Heuristic Function of the Axiomatic Method Volker Peckhaus
    Erlangen University, Erlangen, Germany
    ABSTRACT: This lecture will deal with the heuristic power of the deductive method and its contributions to the scientific task of finding new knowledge. I will argue for a new reading of the term 'deductive method.' It will be presented as an architectural scheme for the reconstruction of the processes of gaining reliable scientific knowledge. This scheme combines the activities of doing science ('context of discovery') with the activities of presenting mathesis universalis in the Cartesian-Leibnizian sense according to which mathematics is the syntactical tool for a general philosophy of science, applicable to all scientific disciplines. In this function, mathematics takes its problems from the sciences. Hilbert did not deny that mathematics should play a role in explaining the world. The analysis-synthesis scheme helps to provide a consistent interpretation of these two sides of Hilbert's attitude towards his working field. Educating humanity is a big task for philosophy which should stand as a guiding idea behind all philosophical activities. In epistemology and in the philosophy of science the idea would be served by considering at least the following three interrelated questions:

    53. Popper's Philosophy Of Science
    Suppose we have a simple theory with one axiom, one middle level statement which, for this example, we will call the hypothesis , and only one basic
    Popper's Philosophy of Science
    Anyone who has attended a summer Seminar Laboratory Workshop at the Institute of General Semantics in recent years has heard Stuart Mayper speak of Sir Karl R. Popper's philosophy of science as best illustrating the methodological view espoused by General Semantics as a discipline. Stuart reviews some of the key features of Popper's philosophy and shows how major paradigm shifts in scientific beliefs over the millennia conform to the system described by Popper. For those of you who have never been to an Institute seminar, or are otherwise not familiar with the recent philosophy of science, I will review the salient features of Popper's philosophy of science. Almost everyone is familiar with the classical method of reasoning know as modus ponens . The well known example goes as follows: If Socrates is a man then Socrates is mortal.
    Socrates is a man.
    Therefore, Socrates is mortal. Few know that the progress of science no longer depends primarily upon this method, but on the less familiar form known as modus tolens , which goes like this: If Socrates is a god, then Socrates is immortal.

    54. § 9. Axiom / Postulate / Theorem / Corollary / Hypothesis. 4. Science Terms. Th
    9. axiom / postulate / theorem / corollary / hypothesis. 4. Science Terms. The American Heritage Book of English Usage. 1996.
    Select Search All All Reference Columbia Encyclopedia World History Encyclopedia Cultural Literacy World Factbook Columbia Gazetteer American Heritage Coll. Dictionary Roget's Thesauri Roget's II: Thesaurus Roget's Int'l Thesaurus Quotations Bartlett's Quotations Columbia Quotations Simpson's Quotations Respectfully Quoted English Usage Modern Usage American English Fowler's King's English Strunk's Style Mencken's Language Cambridge History The King James Bible Oxford Shakespeare Gray's Anatomy Farmer's Cookbook Post's Etiquette Bulfinch's Mythology Frazer's Golden Bough All Verse Anthologies Dickinson, E. Eliot, T.S. Frost, R. Hopkins, G.M. Keats, J. Lawrence, D.H. Masters, E.L. Sandburg, C. Sassoon, S. Whitman, W. Wordsworth, W. Yeats, W.B. All Nonfiction Harvard Classics American Essays Einstein's Relativity Grant, U.S. Roosevelt, T. Wells's History Presidential Inaugurals All Fiction Shelf of Fiction Ghost Stories Short Stories Shaw, G.B. Stein, G. Stevenson, R.L. Wells, H.G. Reference Usage American Heritage Book of English Usage ... SUBJECT INDEX A Practical and Authoritative Guide to Contemporary English.
    4. Science Terms: Distinctions, Restrictions, and Confusions

    55. MOT - SYSTEM I
    (4) also called the KKthesis or the axiom of self-awareness. If a method knows a hypothesis, then the method´knows that it knows the hypothesis
    SYSTEM I - The arrows from realism/anti-realism, AFK, epistemology and synchronic/ diachronic principles boxes point to the box of S4 ... We are interested in epistemic strength. There exist axioms designed to determine epistemic strength of a knowledge acquiring inquiry method. The axioms considered here most notably include:
    • (D) also called the axiom of consistency . If a method knows a hypothesis, then it does not also know its negation.
      K h ¬K h
    • (T)) also called the axiom of truth . If a method knows a hypothesis, then the hypothesis is true:
      K h h
    • (K) also called the axiom of deductive cogency . The knowledge of a method is closed under implication:
      K h l) (K h K l).
    • (4) also called the KK thesis or the axiom of self-awareness . If a method knows a hypothesis, then the method´knows that it knows the hypothesis:
      K h K K h
    • (5) also called the axiom of wisdom or negative introspection . If a method does not know a hypothesis, then the method knows that it does not know the hypothesis:

    56. [FOM] On The Continuum Hypothesis, Part 2
    The solution presented to the Continuum Hypothesis (if it is a solution) can be considered as a semiaxiom as it is our best but not certain guess about
    [FOM] On the Continuum Hypothesis, part 2
    Dmytro Taranovsky dmytro at MIT.EDU
    Mon May 15 21:25:00 EDT 2006 More information about the FOM mailing list

    Page 1     1-66 of 66    1