- 1 Godelian nonsense
- 2 Quotes
- 3 Godel's incompleteness theorem
- 4 Notes
- 5 notes 2
- 6 Notes
- 7 Links
- 8 Other
- 9 Godelian Wall
- 10 Implications
No precise definition of mathematics exists. Math is a dissimilar term for deduction and as such is a subset of language and induction. Axioms are based on induction, deduction uses axioms and is thus based on induction. Language is the metaphysical framework that grounds deduction. Math cannot say anything about that which we designate as meaningless. "Cretan says all Cretans are liers ", is a meaningless sentence and applying deduction such as Godel's incompleteness theorem results a Non sequitur conclusion. Deduction means that if the premises are true, the conclusion must be true by necessity. Which suggests that non tautological deduction doesn't exist because axiomatic inductive inferences aren't necessarily true, deduction is used as a means of escaping Hume's problem of induction. The famous Socrates is a man syllogism as an example of deductive reasoning is obviously inductive: men are mortal not because it is a necessary truth, but because of experience. The term "formal" in formal axiomatic system is used as a dissimilar term for truth. What is truth?
- the writeup below is left as is, they are previous edits because it was not realized that deduction is based on induction
- ....The real reason for Incompleteness in arithmetic is inability to define truth in a language... philosophyideas.com#Truth can't be defined
Godel's incompleteness theorem
"Science is a formal axiomatic system. Therefore, if science aptly describes the universe, the universe itself acts like an axiomatic system. Therefore, it needs something analogous to an axiom outside of itself to be consistent and complete. So God exists." (See http://www.perrymarshall.com/10043/godels-incompleteness-theorem-the-universe-mathematics-and-god/)
- Note that Ironchariots.org have removed their Godel page - why?.
Godels theorem states(Perry Marshall):"...Anything you can draw a circle around cannot explain itself without referring to something outside the circle – something you have to assume but cannot prove..". The syllogism is that Gödel’s incompleteness theorem applies to all logical systems and the universe is logical. Therefore the universe is incomplete(something must be assumed). It’s a progression of inductions that ultimately rest on axioms that are known to be true but are not provable nor refutable. With the term "incomplete" it is meant that something about a description is known to be true but will forever remain unprovable.
Godel's Incompleteness theorem shows that the integers can't be axiomatized - for any finite axiom set there will always be a true statement not provable from the axioms. The only reason we would know it is true, is by faith, the evidence for things not seen.
Gödel noted drily that this represents a problem for philosophy and epistemology rather than for mathematics, which can continue its investigations without ever exhausting the subject. Gödel’s result shows that not even in terms of numbers, the simplest objects we can specify, can natura naturans explain the individuality that we observe. The parallel between Gödel’s attack on the continuum hypothesis and Leibniz’ critique of Spinoza is very strong, and it is remarkable that both hinged on foundational insights into number theory.
Whether or not we believe, as did Gödel, in Leibniz’ God, we cannot construct an ontology that makes God dispensable. Secularists can dismiss this as a mere exercise within predefined rules of the game of mathematical logic, but that is sour grapes, for it was the secular side that hoped to substitute logic for God in the first place. Gödel’s critique of the continuum hypothesis has the same implication as his incompleteness theorems: Mathematics never will create the sort of closed system that sorts reality into neat boxes.
Gödel, of course, did not actually believe in time travel, but he understood his paper to undermine the Einsteinian worldview from within. Yourgrau observes, “The very fact that this inconceivably fast spaceship would return its passengers to the past demonstrated, by Gödel’s lights, that time itself—hence speed and motion—is but an illusion.” Stephen Hawking so abhorred the implications of Gödel’s demonstration that he proposed an ad hoc bylaw for general relativity, the “chronology protection conjecture,” simply to exclude it. Like Einstein, Hawking then believed that a grand theory of the universe would allow humankind to see into the “mind of God.” In recent years, though, Hawking has come closer to Gödel’s point of view, going so far as to conjecture that a sort of Gödelian “incompleteness principle” might exist in physics as well as in mathematics.
First let me try to state in clear terms exactly what he proved, since some of us may have sort of a fuzzy idea of his proof, or have heard it from someone with a fuzzy idea of the proof.. The proof begins with Godel defining a simple symbolic system. He has the concept of a variables, the concept of a statement, and the format of a proof as a series of statements, reducing the formula that is being proven back to a postulate by legal manipulations. Godel only need define a system complex enough to do arithmetic for his proof to hold.
Godel then points out that the following statement is a part of the system: a statement P which states "there is no proof of P". If P is true, there is no proof of it. If P is false, there is a proof that P is true, which is a contradiction. Therefore it cannot be determined within the system whether P is true.
As I see it, this is essentially the "Liar's Paradox" generalized for all symbolic systems. For those of you unfamiliar with that phrase, I mean the standard "riddle" of a man walking up to you and saying "I am lying". The same paradox emerges. This is exactly what we should expect, since language itself is a symbolic system.
Godel's proof is designed to emphasize that the statement P is *necessarily* a part of the system, not something arbitrary that someone dreamed up. Godel actually numbers all possible proofs and statements in the system by listing them lexigraphically. After showing the existence of that first "Godel" statement, Godel goes on to prove that there are an infinite number of Godel statements in the system, and that even if these were enumerated very carefully and added to the postulates of the system, more Godel statements would arise. This goes on infinitely, showing that there is no way to get around Godel-format statements: all symbolic systems will contain them.
Your typical frustrated mathematician will now try to say something about Godel statements being irrelevant and not really a part of mathematics, since they don't directly have to do with numbers... justification that might as well turn the mathematician into an engineer. If we are pushing for some kind of "purity of knowledge", Godel's proof is absolutely pertinent.
In addition, some known mathematical phenoma already exhibit the Godel incompleteness property. For instance, in set theory mathematicians define different degrees of infinity based on the number of members of the set of all integers, rational numbers or reals. The first degree of infinity, called (aleph-nought), is the number of integers or the number of rational numbers (these numbers are the same "degree of infinity"). The second degree of infinity is aleph-nought raised to the power aleph-nought. For a long time people were trying to decide whether 'C', the number of real numbers, was the same as the second degree of infinity. Finally it was proven that whether C and 2nd infinity were equivalent came down to the truth or falsehood of a statement that could not be proven from the existing axioms of mathmatics. This statement was absorbed as a new axiom, just as Godel statements would have to be. So there is the first of many Godel-style statements that we'll probably see popping up in mathematics.
Of course, a more familiar example is the parallel-postulate axiom, since it cannot be proven from any other axioms of Euclidean geometry, and in this case the way you define it leads to at least three different self-consistent systems.
In any case, what does it mean that a symbolic system based on deriving truth from axioms is incomplete? Could we make a complete system? The only way I can see to do that would be to include an infinite number of axioms, which deterministicly describe all happenings in the past, present and future. This would only work in a deterministic universe, and it would be difficult to draw a distinction between the data of this 'complete' system and reality itself.
Thinking of the data required is perhaps the right direction to move in: it is the reason the symbolic system is incomplete. The symbolic systems we use to describe the universe are not separate from the universe: they are a part of the universe just as we are a part of the universe. Since we are within the system, our small understandings are 'the system modelling itself' (system meaning reality in this case). Completion of the model can never happen because of the basic self-referential paradox: the model is within the universe, so in effect the universe would have to be larger than itself. Or you can view it iteratively: the model models the universe. The universe includes the model. The model must model itself. The model must model the model of itself.. ad absurdum.
So Godel's incompleteness is something to expect. It is even something that can be intuitively understood without a mathematical approach and proof: the incompleteness concept appears in clearly recognizable form in Zen Buddhism. So it brings to mind how to solve the paradox. There is the idea that consciousness might be a kind of superset of the universe, and thus through consciousness we might understand the universe. Yet we must realize that consciousness and the universe represent a yet larger system or universe to "understand" ( if that word still applies ). This continues iteratively as well. We can perhaps move beyond the self-referential part of the paradox by moving beyond the self: becoming through some higher dimensionality or level of complexity something with no coherant self, or clear perception- point.
The Zen answer to what to do next is that real truth is in everyday life. This may well be so: in a universe where knowledge defeats us, what can we do but be what we are? We have to ask why it is that it matters that knowledge of the universe be moved into symbolic representation in our minds. The information we seek is in existence around us at all times, happening in the patterns we seek to understand and quantify. What good is there in this understanding? Clearly we are evolutionarily driven to this attempted understanding, but is there a better reason to be had?
Tarski on truth
Godel and Tarski showed that Truth itself can't be derived from first principles, we must assume something about language(logic) that we know to be true, but will never be able to prove. Dissimilar terms like incomplete,finite are the shorthand to express Godel's incompleteness theorem conclusion that in any logical description there will always be something about the description or system that must be assumed, known to be true but unprovable. Wikipedia science article states that @....Science is an enterprise that... organizes knowledge in the form of testable explanations...@ The sentence itself can't be tested, hemmed in by a Godelian Wall: is Popper falsifiability itself falsifiable? Truth and Life1 are universal principles, Tarski proved mathematically that is impossible to define truth, all attempts leads to a contradiction. Jesus Christ said: ... I am Truth, Life and the Way, no man comes to the Father but by me..... God himself can't be reduced to a mathematical construct. Do atoms exist in reality or is it some semantic placeholder: when uranium decays and turns into thorium - what exists in between these two stages? Paraphrasing Christ: "......I am Alpha and Omega, Reality, Truth and Life, no man comes to the Father but by me, who has given me all authority in Heaven and Earth(Platonic opposite) ....." Atheists state that they only accept 'scientific reality' but not where we can buy a 'reality measurement machine' calibrated for its span and zero.
The Wireless World July 1981 editorial, ‘The decline of the philosophical spirit’, contrasts the nineteenth century, when scientists were interested in and capable of distinguishing between the physical real and the mere mathematical construct, and today, when scientists no longer care about the difference, and have even developed a philosophy of science which confuses them (Popper K., Conjectures and Refutations, R.K.P., 1963, p100.)
Falsifiability is a subset of unfalsifiability. We use unfalsifiable tautological assertions or axioms , they are merely assumed. Godel showed that in any logical system there will always be something you must assume and will never be able to prove.
If God could be reduced to a falsifiable construct, He won't be God, since by definition God is outside of the Godelian Wall that hems in our knowledge. In other words the only way you will have an experience with God himself is if you unreservedly believe what he tells you as He thrusts his hand through the Godelian Wall to rebirth your fallen spirit - John 3.
When the Lord Jesus told Moses .... I am that I am ..... he made an unfalsifiable statement. Falsifiability is a subset of unfalsifiability because something about falsifiability must still be assumed, something that will never be able to be falsified. Hence having unfalsifiable beliefs isn't a fallacy. Atheists state that they have not beliefs, the only way they could know this is by believing it. Thus have a belief that the have no belief a self-refutational position
Physics equations as falsifiable constructs must assume unfalsifiable tautological assertions(axioms). Tautological assertions This is consistent with Godel in that a logical system will assume things it will never be able to prove. Since we consider ourselves as logical beings and our descriptions as logical, there will always be something hidden or assumed about our descriptions.
d/dx is a universal operator in math that is applied to a function. The study of Life (biology) lacks this operator because we can't define life. Epicureans try to emulate d/dx by invoking the term NS, which as some sort of universal mechanism is no more plausible than a single differential equation explaining all of physics. Alchemy was only stopped after Gold and lead were defined in terms of molecular theory. Prof. Cleland from Nasa astrobiology pointed out that before the discovery that water is H20, acids were called 'weak' and 'strong' water. Likewise the very thing that Epicureans claims to study - Life(biology) itself - isn't defined, and thus we don't know whether anything analogous to d/dx is applicable - Life1 . See also calls Darwin's theories 'weak'.
- https://math.stackexchange.com/questions/771836/how-is-the-g%C3%B6dels-completeness-theorem-not-a-tautology , http://www.logicmatters.net/resources/pdfs/gwt/GWT2f.pdf , https://math.stackexchange.com/questions/884959/godel-incompleteness-theorem
Penrose points out, correctly, that Godel's Theorem proves computers, no matter how powerful, are subject to fundamental limitations. He then claims, Tipler thinks incorrectly, that human beings are not subject to the same limitations. What the Logician Kurt Gödel did in the 1930s was to show that the full theory of arithmetic-the theory of arithmetic we are all familiar with, which includes addition, subtraction, multiplication, and division-contains a self-reference statement equivalent to: "This statement is unprovable." If it is true, then the statement itself is unprovable, and arithmetic is incomplete--a theory is said to be incomplete if it contains a true statement which cannot be proven from the axioms of the theory. On the other hand, if the statement is false, then, since it is equivalent to a statement of arithmetic, arithmetic would be logically inconsistent. A farther consequence of this argument is that arithmetic, if consistent, must be incomplete, and hence must be undecidable--a theory is said to be undecidable if there is no algorithm which, given any statement in the theory, can tell you whether the statement is true or false(pi24)
Few of us would have been surprised to learn that there is no algorithm-no effective criterion, no outright test- for truth in elementary number theory. Such a test would make short work of unsolved problems such as Goldbach's conjecture and Fermat's Last Theorem; too good to be true. On the other hand, Godel's theorem came as a shock, for we supposed that truth in mathematics consisted in demonstrability. Substantially the same argument that establishes Godel's theorem, however, establishes something stronger: that truth for elementary number theory cannot be defined in protosyntax at all, either by proof procedure or otherwise(slp236)
http://www.hipforums.com/newforums/showthread.php?t=312319 "...I love this Colin Leslie Dean guy, if he even exists. I saw this is sciforums and physicsforums soooo long ago. and Trust me people, its not true.
This guy did not prove Godel wrong. If he did, and I wish he did because Godel's work is one of the greatest downfalls of my beautiful mathematics, that If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent. It almost shatters the foundations of mathematics from a philosophical standpoint.
Basically what Godels Incompleteness Theorem is saying, is that with our current axioms or hypothesis, we will never be able to prove everything, there will always be statements and theorems which are true but cannot be proved with our set of axioms. The only solution would be to add another axiom, but that would lead to more conclusions which cannot be proved within itself.
Colin Dean did not prove Godel wrong, though how I wish he did. Godel's work is amazing, but how GH Hardy said, there is no permanent place in the world for ugly mathematics. I hope in the future, there is a way to show a counter-example to Godel's theorem, and make the discipline of mathematics consistent and complete. But that hasnt happened yet, and definitely not by Colin Dean who supposedly has like 5 Ph. D.'s, and none of them in mathematics lol. read the first page of the pdf file....."
"... Godels theory makes total sense, and has very little to even do with mathematics.
If something can be proven to be true, then it could be said that it is just another way of putting something. If I use A, B, and C to prove D, it could be said that D is nothing but a rewording of A, B, and C. If If everything true could be proven true, it would mean that everything is merely a rewording of somethign else that is a rewording of something else, etc. It seems more reasonable that there must be certain facts that are unprovable - they simply are true. they are the handle that a reality holds on to. Although we would never know when we come to them, and may not come to them anyway, they must exist, or else nothing can be said to primarily exist...."
Stick With Me, Stick With Me, Oh It Goes Deeper
http://measureofdoubt.com/2012/09/12/colbert-deconstructs-pop-music-finds-mathematical-nerdiness-within/ Stick With Me, Stick With Me, Oh It Goes Deeper Let’s analyze the dilemma a bit further: She can’t KNOW she’s beautiful because, as Stephen points out, that leads to a logical contradiction – she would no longer be beautiful. She can’t KNOW that she isn’t beautiful, because that also leads to a logical contradiction – she would be beautiful again. It’s impossible for the girl to know that she is or isn’t beautiful, so she has to be uncertain – not knowing either way. This uncertainty satisfies the requirements: she doesn’t know that she’s beautiful, therefore, she’s definitely beautiful and can’t know it. It turns out she’s not in a flickering state of hot and not, she’s perpetually hot – but she cannot possibly know it without logical contradiction! From an external perspective, we can recognize it as true. From within her own mind, she can’t – even following the same steps. How weird is that? Then it hit me: the song lyrics are a great example of a Gödel sentence!
Gödel sentences, from Kurt Gödel’s famous Incompleteness Theorems, are the statements which are true but unprovable within the system. Gödel demonstrated that every set of mathematical axioms complex enough to stand as a foundation for arithmetic will contain at least one of these statements: something that is obviously true from an outside perspective, but isn’t true by virtue of the axioms. (He found a way to coherently encode “The axioms do not prove this sentence to be true.”) This raises the question: what makes a mathematical statement true if not the fact that it can be derived from the axioms? Gödel’s findings rocked the world of mathematics and have had implications on the philosophy of mind, raising questions like:
What does it mean to hold a belief as true? What are our minds doing when we make the leap of insight (if insight it is) that identifies a Gödel sentences as true? How does this set us apart from the algorithmic computers, which are plagued by their own version of Incompleteness, the Halting Problem?
But alas for these researchers, trying to unify Quantum Mechanics and General Relativity, Godel clearly showed decades ago that a complete mathematical model for a ‘theory of everything’, such as Quantum Gravity, String Theory and M-theory attempt to be in trying to reconcile Quantum Mechanics with General Relativity , was impossible to construct in the first place, so their efforts are in vain. That is in vain save for further verifying what Godel proved ‘mathematically’ in the first place:
THE GOD OF THE MATHEMATICIANS – DAVID P. GOLDMAN – August 2010 Excerpt: we cannot construct an ontology that makes God dispensable. Secularists can dismiss this as a mere exercise within predefined rules of the game of mathematical logic, but that is sour grapes, for it was the secular side that hoped to substitute logic for God in the first place. Gödel’s critique of the continuum hypothesis has the same implication as his incompleteness theorems: Mathematics never will create the sort of closed system that sorts reality into neat boxes.
Falsification is the answer in certain contexts but not in all, specifically that we will always have to assume something we will never be able to prove.
Ultimately my premises aren't falsifiable which prevents my thinking from entering infinite regress. My evidence for Gods existence is faith and faith is defined as being unfalsifiable because faith is the evidence for things not seen. God called those things that be not as though they were. Every atom is held together because God assumes they will be held together. If God himself has to assume something, then surely we as his creation aren't exempt .
All arguments will eventually reach the point where they are going to have to assume something about the description that will never be proved(incomplete).
Paradoxes such as predestination, Crocodile Dilemma will be resolved by some process or something outside of your knowledge context, since our knowledge is forever locked inside of a circle and we will have to assume that there is something greater outside of our knowledge circle.
On Jul 2, 10:29 pm, "Mike Dworetsky" <platinum...@pants.btinternet.com> wrote: > Ron O wrote: > > On Jul 2, 2:09 pm, backspace <stephan...@gmail.com> wrote: > >>http://www.gutenberg.org/files/19192/19192-h/19192-h.htm > > >> ''.....Natural selection, unguided, submitted to the laws of a pure > >> mechanism, and exclusively determined by accidents, seems to me, > >> under another name, the chance proclaimed by Epicurus, equally > >> barren, equally incomprehensible; on the other hand, natural > >> selection guided beforehand by a provident will, directed towards a > >> precise end by intentional laws, might be the means which nature has > >> selected to pass from one stage of being to another, from one form > >> to another, to bring to perfection life throughout the universe, and > >> to rise by a continuous process from the monad to man. .......'' > > >> If natural selection is non-random, why did Prof. Janet say it is > >> determined by accidents? > > > Because he was wrong? Except for its historical value why would > > anyone read about a misconception someone had in 1874? > > The gutenberg text is from a book summarising arguments on both sides of > Darwin's theory. > > Janet's book was published in 1867, when, as you say, none of the genetic > concepts we have today existed. All Darwin (and Janet) had to go on was the > fact that complex or multicellular life generally looks like its parents, > for example a child looks like a mix of both its parents. > > Why on Earth should anyone be raising a quotemine from a French philosopher > [not a scientist] from nearly 150 years ago as if it proved anything about > evolution?
Because you are using their exact terminology. If your views differ then change your terminology or at least define what you mean with selection. The Wikipedia Selection article states: http://en.wikipedia.org/wiki/Selection
...This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2006)...
http://en.wikipedia.org/wiki/Selection_(user_interface) This article discusses Design.
What is a design?
http://en.wikipedia.org/wiki/Design ....Philosophies for the purpose of designs In philosophy, the abstract noun "design" refers to a pattern with a purpose. Design is thus contrasted with purposelessness, randomness, or lack of complexity.......
A design is the usage of patterns as the representation of something other than itself. It has attributes such energy conversion to some goal, during predictable or decidable times as defined under http://scratchpad.wikia.com/wiki/Irreducible_Functionality.
A flagellum *decides* when to generate energy, towards a goal, during a time. The mousetrap *decides*(represents the intent of the designer) to strike at a certain spot, during a certain time. Thus a design is the symbolic epresentation of If/then/what conditionality's It was shown that an IC or IF system is the interaction of objects or Key Functional Parts to symbolically represent purpose enacted in terms of energy, place and time. These three dimensional attributes can't be reduced further , but the KFP can be increased (Rube Goldberg) or reduced(Calvin and Hobbs) or made just perfect IOF(irreducible optimized functionality) in the context of aesthetics. The communists build slaps of concrete for people to live in, expressing their soul crushing views. A house can be built to express the sense of aesthetics or to express a crushed spirit.
What is a purpose? http://en.wikipedia.org/wiki/Purpose - Purpose1 Wikipedia scrapped their previous article on the concept of 'purpose' after it was pointed out that it contains Meaningless sentences. Purpose1 has the contents of this scrapped article
- ..Purpose is a result, end, mean, aim, or goal of an action intentionally undertaken,....
What is a goal or Intention?
- The term intentionality was introduced by Jeremy Bentham as a principle of utility in his doctrine of consciousness for the purpose of distinguishing acts that are intentional and acts that are not. The term was later used by Edmund Husserl in his doctrine that consciousness is always intentional, a concept that he undertook in connection with theses set forth by Franz Brentano regarding the ontological and psychological status of objects of thought. It has been defined as "aboutness", and according to the Oxford English Dictionary it is "the distinguishing property of mental phenomena of being necessarily directed upon an object, whether real or imaginary".
- A goal or objective is a desired result a person or a system envisions, plans and commits to achieve—a personal or organizational desired end-point in some sort of assumed development. Many people endeavor to reach goals within a finite time by setting deadlines. It is roughly similar to purpose or aim, the anticipated result which guides reaction, or an end, which is an object, either a physical object or an abstract object, that has intrinsic value.
What is an abstract object?
- An abstract object is an object which does not exist at any particular time or place, but rather exists as a type of thing (as an idea, or abstraction). In philosophy, an important distinction is whether an object is considered abstract or concrete. Abstract objects are sometimes called abstracta (sing. abstractum) and concrete objects are sometimes called concreta (sing. concretum).
What is philosophy? http://en.wikipedia.org/wiki/Philosophy
- Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language.  It is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational argument. The word "philosophy" comes from the Greek φιλοσοφία (philosophia), which literally means "love of wisdom".
We represent symbolically with the integer numbers system a physical concept, that of two apples or three apples. Godel's theorem thus states that there is something about our experiences that we know be true , but cannot prove or experience as this would induce an infinite regression of experiences, logic enables the expression of the experience but cannot be experienced itself. It is logical to bootstrap that adding an apple to another results in two apples. This ties in with the resolution of Agrippa's trilemma under Logical fallacies , infinite regress is avoided by knowing everything or believing something from God who knows everything. Hence God who is outside of infinity itself is that which we know be true (law of non-contradiction) , but cannot prove by necessity or the proof in turn would need proof ad-infinitum.
Our concern with any philosophical or spiritual question is what prevents regression and rhetorical circularity. Agrippa should be the central theme in all philosophy courses, instead it is simply ignored in the undergraduate courses and Christianity's virtuous circularity equivocated with Adaptationism's rhetorical circularity.
- Knights and knaves explanation to be added . Clarifies what Godel was saying. Integers represents our physical experiences symbolically, there is something about a physical knight we know to be true but cannot prove.
Kurt Gödel – Incompleteness Theorem – video https://vimeo.com/92387853 from Bornagain77 , http://www.uncommondescent.com/atheism/jeff-shallit-surely-the-right-analogy-is-santa-claus-to-jesus-christ-both-are-mythical-figures-spectacular-fail-at-history-101/#comment-516158
BRUCE GORDON: Hawking’s irrational arguments – October 2010 Excerpt: This transcendent reality cannot merely be a Platonic realm of mathematical descriptions, for such things are causally inert abstract entities that do not affect the material world,,, Rather, the transcendent reality on which our universe depends must be something that can exhibit agency – a mind that can choose among the infinite variety of mathematical descriptions and bring into existence a reality that corresponds to a consistent subset of them. This is what “breathes fire into the equations and makes a universe for them to describe.” Anything else invokes random miracles as an explanatory principle and spells the end of scientific rationality.,,, Universes do not “spontaneously create” on the basis of abstract mathematical descriptions, nor does the fantasy of a limitless multiverse trump the explanatory power of transcendent intelligent design. What Mr. Hawking’s contrary assertions show is that mathematical savants can sometimes be metaphysical simpletons. Caveat emptor. http://www.washingtontimes.com/news/2010/oct/1/hawking-irrational-arguments/