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Post by Progenitor A on Jan 17, 2011 9:23:37 GMT 1
The concept of infinity is literally almost useless with one exception
When it ised in physics it is a red warning flag that whoever is using it : 1. Does not know what they are speaking about OR 2. If they do know what they are talking about are lost in their exaplanatory powers
Indeed to talk of 'infinte energy', 'infinite gravitational fields', 'infinite space' is unscientific.
Why?
Well science is concerned with hyptheses that are verifiable in principle. Infinity cannot be experienced, thus any 'scientific' topic that invokes infinity has stepped outside the parameters of science
Now there is an aparent contradiction in what I say, for how can I know that we cannot experience infinity?
I do not know, obviously, so the statement that I have made must be regarded as an hypothesis that cannot be verified and therefore is not a scientific statement
I accept that, but happily, anyone putting forward such an argument is simply confirming that infinity is unscientific and I will happily go along with that
Now here's an absurdity of infinity
We are told that 1/0=infinity (some sly mathematicians say it is indeterminate for very good reason as is demonstrated below, but we know really {don't we] that as x gets very small oi 1/x gets very big, and as x approaches zero, 1/x approaches infinity, so none of your sly word play mathematicians please)
1/0 = infinity
Multiplying both sides by zero
(1/0) x 0 = infinity x 0
(1/0) x 0 = infinity x 0
1 = 0
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Post by carnyx on Jan 17, 2011 11:56:55 GMT 1
NM,
I suppose the problem stems from the word 'infinity' being used as a label for many kinds of unquantifiable property, which can lead to paradox and aborted understanding.
You mentioned mathematics. Now this method of thinking is algorithmic ( i.e. procedural .. to do with processes ..if you like) , and dependent on axioms. And as we know from Goedel, there will be paradox in these kinds of thought process. For example, x/0 = uncountably large (aka 'infinite') whereas 0/x ought to = uncountably small ( aka 'infinite'). But this fact is elided via the use of the 0 symbol which really means 'null' or absence of anything.
But even when we dientangle this obvious source of paradox by inventing new symbols for null and infinity, we are still left with the fundamental idea that that any sequential mode of thinking will have paradox. And in the case of mathematics it is that the sequence can degenerate into an an endless process of defining the value of , for example, 0/x or x/0.
(I have to stop here, as work calls ... but I like one definition of infinity as being 'the infinite 'now'')
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Post by petergriffin on Jan 17, 2011 14:16:40 GMT 1
NM,
Infinity does have it use espcially in mathmatics, also infitiy can have different sizes. For example there are an infinite number of prime numbers, therefore there must a infinite number of integers, as all primes are integers but all integers are not primes thus the number of integers is a bigger infinite than the infinite number of primes. All the counting numbers include all primes and integers and the other numbers which are neither primes or integers then this number - you guessed it is infinite, but is bigger than the infinite number of primes and the infinite number of integer.
This is just one small area where infinity is both valid and has a real use as it undepins some of the basic axioms of math.
I do agree to some extent that it can be used in wrong context by some poeple, but to me it has a real meaning and a size.
I susspect the same objections were raised when zero was introduced to Western maths in the middle ages, but where would we be without zero now. So long live infinity, in fact infinite long life to infinity
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Post by speakertoanimals on Jan 17, 2011 17:28:51 GMT 1
you are mixing up two different things -- what can be TESTED, and what might actually be true. Just because something may not be capable of being directly tested doesn't mean that it may not be true. Indeed, we may require it to be true (even if not directly testable) based on other things which ARE testable.
Let's take this simple case -- suppose we have a sheet of stuff which, for various reasons, we think has a definite constant curvature. We can measure the curvature (or at least make a local estimate of the mean curvature).
Now what is, when we measure it, we get a value indistinguishable from zero -- that is okay, surely, it just means our sheet is FLAT.
Now rather than a sheet, make it space. EDGES to space make no sense, so we have two possibilities -- either space curves back on itself, so no edges, OR it doesn't curve back on itself, has no edges, hence is of infinite spatial extent.
If we measure the curvature, and find that it is indistinguishable from zero, we have no choice but to say -- possibly very, very big, perhaps infinite.
We CAN'T logically rule out the totally flat case, certainly not on the grounds that we can't test it! What is being limited here is NOT what the universe can be, but whether or not we can directly verify all properties of that universe.
There is ANOTHER role for infinity in physics, which is where theories give infinite values for physical variables. This then just indicates that our theory is probably not complete.
But I really don't see why you want to try and constrain the universe to not use infinity, just because we maybe can't directly verify it if it is. There is no actual physical reason WHY the universe can't be infinite, just that if it is, we can't verify it directly.
Its' realy no more mysterious than the fact that the integers can be continued without end. And the usefulness in maths terms comes from the fact that we can have different sizes of infinities.
Nope. You're wrong about Godel! JUst means that there are statements which you cannot prove as bieng true or false, and in fact you can add either the true or false statement as a new axiom, and generate a new system.
A PARADOX is what you get if you try and have both A is true and A is false provable within your system -- then you can prove anything is true, and you might as well give up, your system is fundamentally flawed! All that Godel says, in effect, is that some things aren't subject to proof, and you can take either A or not A and still get a sensible system. Not the same as a paradox.
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Post by carnyx on Jan 17, 2011 18:39:00 GMT 1
STA,
Who are you gainsay that Goedel's work informs me of the idea that in systems of axioms, there will exist undecidable propositions? That these undecidabilities can take the form of statements that are both true and not-true? And are therefore PARADOXES? It is perfectly plain what I meant; why are you having trouble with it?
As these 'systems of axioms' mean procedures ....the point I made was that the addition to Arithmetic of the axiom that 'zero' would denote a null AND ALSO an infinitely small number, created scope for paradox.
This could be rectified via an additional axiom, that whilst '0' would denote a null, the symbol 'B' (Bloody tiny) should be added in situations where the number was too small to be defined by the arithmetic procedure in the time we have available to us .. and in the same way that would use the 'infinity' sign to show a number that is too large to define by the arithmetic procedure in the time we have available ...
So, with regard to the OP, 'infinite' means we don't have time to quantify the number, and so these infinities represent a real subjective limit to our dickerings (as the Yanks say)
Added to Goedel's idea that axiomatic ( aka procedural) thinking will always bump into undecideabilities, and to Perose's idea that such linear thinking will never be able to produce Artificial Intelligence, (aka creativity) .... it seems that both Mathematical and Physical thought-processes have certain inherent constraints to their importance in human terms.
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Post by Progenitor A on Jan 17, 2011 18:40:10 GMT 1
Hi Peter NM, Infinity does have it use espcially in mathmatics,.... I will take your word for that , as I am not a mathematician ... also infitiy can have different sizes.... I have heard this before and think that it is logically flawed for the reasons that I will explain shortly For example there are an infinite number of prime numbers, therefore there must a infinite number of integers, as all primes are integers but all integers are not primes thus the number of integers is a bigger infinite than the infinite number of primes. Very succinctly expressed Peter, but I will now explain why the logic is flawed. Imagine Mr Integer and Mr Prime are standing before God In front of them are two (rather large) bins labelled INFINITY INTEGER and INFINITY PRIME. God puts a big pile of marbles besides Mr Integer, each of them labelled 0, 1, 2, 3......(to infinity) God puts a big pile of marbles beside Mr Prime labelled 1, 2, 3, 5, 7......(to infinity) He then instruct them to put one marble belonging to them in the bins marked INFINITY INTEGER and INFINITY PRIME. They must each keep pace with one another so that if one puts in a marble in his bin then the other puts a marble in his bin God will tell them when they have reached infinity. So they each together put one marble into each of their bins This goes on for a very long time until God shouts' STOP, YOU ARE THERE!' (God speaks very loudly because he is a bit bossy) 'EACH BIN NOW HOLDS AN INFINITE NUMBER OF MARBLES!' And looking down Mr Integer and Mr Prime magically see that the pile of marbles they each had has disappeared! Now, does the bin labelled INFINITY INTEGER contain more marbles than the bin labelled INFINITY PRIMES? If so where did the extra integers come from? The answer is that each bin has exactly the same number (if infinity is a number) of marbles Here is another logical dissection of the same problem If there are more numbers in th infinite set of Integers than there are in the infinite set of Primes, then how much larger is the set of Integers? If anyone can say how much bigger the set of integers is than the set of Primes, then we can easily 'top up' the set of Primes by adding some more, can't we. And if you say to me, but then we can 'top up the Integers so there are more of them than Primes again. Yes But then we can top up the Primes again..... there is an UNLIMITED supply of both sets, aren't there? And there's the puzzle of infinity - it simply does not stop Here's something else to consider Numbers are an abstraction with no physical reality OK, you might counter that six apples have a physical reality and so they do, but thephysicalreality is the the apples, and the numbering of the apples is an abstraction that we impose upon them You might think I am being daft here, but I happen to think that it is true For example the integer 2, how many are there? Is there an infinite number of 2's, so that people can use 2 at any time no matter how many people want to use it? Or is there just ONE 2 that we all somehow time share? But what is the mechanism of time sharing? No there must be (abstractedly) an infinite number of 2's But of course there are not - we are just talking about an abstraction that has no reality. Just as any set of numbers have no reality. Now as infinity only exists mathematically (there is no physical entity of which we are aware that is infinitely large) and if the numbers that are used as the basis for mathematics have no reality, then in reality there is no infinity. It is simply a (useful/useless?) mathematical convenience Just as -1 (0.5) has no reality but is useful in mathematical manipulation. This is just one small area where infinity is both valid and has a real use as it undepins some of the basic axioms of math. Sorry Peter, but unless you can disprove my first two logical constructions then the result you express so clearly is false. I do agree to some extent that it can be used in wrong context by some poeple, but to me it has a real meaning and a size. I certainly agree with you about the abuses of infinity, and I wish I could share your feeling of real meaning and size I susspect the same objections were raised when zero was introduced to Western maths in the middle ages, but where would we be without zero now. So long live infinity, in fact infinite long life to infinity ;DNow zero I am all for - I have great admiration for zero and would not want to see its good name besmirched (even if it is not real - I think!)
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Post by speakertoanimals on Jan 17, 2011 19:05:50 GMT 1
Because what you say is WRONG!
I am able to gainsay you because I am one of the few people on here (probably) who has read godels original papers and understood them.
Undecidable is more subtle than you think -- it means that you can either take it as true, or as false, and STILL get a consistent set of axioms. This ISN'T a paradox, just means that the axioms you started with are insufficient to determine the truth or falsity of the extra statement.
So, a prime example is geometry. you can assume parallel lines never meet, and get euclidean geometry, or assume they may, and get non-euclidean geometry. No paradox, just different types of geometry, distinguished by whether the parallel postulate is taken to be true or false.
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Post by speakertoanimals on Jan 17, 2011 19:25:06 GMT 1
Axioms AREN'T procedures, you have amgled Godel again. The procedures are the way you manipulaten symbols in order to generate new true statements from previous ones.
As regards zero, zero and an infinitely small number are actually DIFFERENT concepts.
Zero can be defined for integers, perfectly well defined, but there are no infintely small numbers, only 0 (zero), or 1 (smallest you can have other than zero).
What 1/epsilon is in the limit that epsilon tends to infinity is a DIFFERENT concept, depending on exactly HOW epsilon tends to infinity. So, I can have 1/n, where n is an infinitely-large integer. This number becomes infinitesimally small, yet although the limit of this sequence is zero (ie you can get arbitrarily close to 0 using this sequence), that isn't the same as saying zero. There is a subtle difference here.
Except mathematicians perfectly happy that it is logical, hence who should I trust? I'll go with the mathematicians................
the problem here is that you have failed to think carefully enough about what 'having more marbles' means in the case of infinite numbers. So, for finite numbers, 'having more' means I can take away as many as I have, and still have some left over.
The same does not apply to infinite numbers. Because what we mean by 'being the same size' has to be defined in a way that includes infinite numbers.
So, the definition Cantor used is that two sets same size if they can be put into one-to-one correspondence with none left over. Then if take integers (and even integers), we have as many of each. Then we have to get our heads tound the fact that we can have n = 2n (all integers versus even ones), yet still have n same size as 2n. Basic rule is -- x = 2y meaning that x is bigger than y no longer holds for infinite numbers.
We hence have A DIFFERENT definition of bigger than for infinite numbers, just as we had to have a different definition of same size as for infinite numbers.
All you have shown is that if you try to aply the rules for finite numbers to infinite ones, you get nonsense, but I could have told you that before you started! Because the rules you are using only apply to finite numbers! If you wantt o deal with infinite numbers as well, you need new rules which then apply to finite AND to infinite numbers.
your problem is that you have a very narrow view of what logic and maths is, ande need to learn something more than basic arithmetic. You are happy with finite numbers, and think that is enough to understand stuff. yet as soon as you go to the continuum (a concept most people are fairly happy with), your simple constructions aren't enough to cope. you can have any position on the real line between 0 and 1, you can be anywhere between here and there, yet your limited maths can't even COUNT the possibilities in any sensible way! Whch is why Zenos paradox caused so many problems for so many years. Stick to your lines of reasoning, and Achilles can never overtakle the yourtoise, or motion is impossible.................................
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Post by carnyx on Jan 17, 2011 20:06:19 GMT 1
STA,
You seem to read things that are not in fact there. I am not talking about successively developing sets of axioms, but the results that flow from applying the procedures prescribed from USING ONE GIVEN SET OF AXIOMS.
And now we come to your ignorance of the meaning of the word 'paradox' . It is defined as a true statement or group of statements that leads to a contradiction. Whether or not the situation can be subsequently resolved by adding an extra axiom, rider, rule, or what have you .. will not stop a paradox from having occurred IN THE ORIGINAL SYSTEM OF AXIOMS. Rather, all that has happened is the problem is fixed via the creation of a newer and more complicated system of axioms.
And finally, the idea from Goedel that "in systems of axioms, there will exist undecidable propositions" as I said earlier, is consistent with the idea that no system of axioms is ever going to be complete .. as you pointed out with your geometry example.
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Post by speakertoanimals on Jan 17, 2011 20:39:01 GMT 1
Wrong. The original system of axioms DOESN'T lead to a contradiction, so no paradox. It just has nothing to say, in effect, about the Godel statement, it cannot say whether it is true or false.
In fact, it goes further than this, and says that if you incluide it as either true or false as a new axiom, you still get a consistent set of axioms either way.
Hence we have the undecidability, NOT paradox.
No paradox anywhere, whether or not you think I know the meaning of paradox. You have misunderstood Godel.
completenes sin't the same as having paradoxes (which is totally pathological and once you have a real paradox you can prove anything, so you might as well give up!).
It realy does seem to me that you have totally missedthe point as regards undecidability, and mistakenly latched upon paradox. Whereas the defining criterion is that whatever you ADD (the statement being ture or being false), you DON'T get a paradox.
Rather than being paradoxical, it in fact says maths is much more open-ended and much more interesting that was thought beforehand.
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Post by carnyx on Jan 17, 2011 23:31:50 GMT 1
STA, despite all of your fireworks and huffing and puffing;
1. IRL, the word Paradox is defined as a true statement or group of statements that leads to a contradiction. An undecidable proposition is one such an example. Here is one; "This statement is false".
3. The idea from Goedel's work that "in systems of axioms, there will exist undecidable propositions" as I said earlier, is consistent with the idea that no system of axioms is ever going to be complete .. as you pointed out with your geometry example.
4. And as you say, this leaves Mathematics, and so Physics, as "0pen ended" ... in other words limited and partial methods of human thought that can never provide complete explanations or 'ultimate truths' of the universe, and reality.
A bit sad, really, don't you think, for those who think Science and Maths are the ways to 'the truth'?
43?
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Post by speakertoanimals on Jan 18, 2011 14:27:16 GMT 1
Except, you numpty, an undecidable statement DOESN'T lead to a contradiction.
'this statement is false' is NOT an undecidable proposition.
You have confused the liar paradox with Godel -- similar but not the same, which is kind of the point.
So, this statement is false, is self-referential, and the proof of the Incompleteness theorems uses such self-referential statements in the proof, but those are not the statements (and we have some), that have been shown to be undecidable within various axiomatic systems.
the point about 'this statement is false' is that the statement can be shown to be NEITHER true nor false, because IF you assign a truth value to it, that leads to a contradiction.
But that is not quite what Godel used, he used instead -- this statement is not PROVABLE (within the axiomatic system being considered), a subtle but powerful difference.
The point about the undecidable statements, rather than truth or contradiction or paradox, is that you can add either the statement OR its contradiction as a new axiom, and either system still works. So, rather than being true but unprovable (as you claimed for the liar), or contradictory if true, we instead have the much more interesting result that it works if true, and it works if false, BOTH choices lead to prefectly good extended systems of axioms.
As I suspected, you don't know enough about Godel, and have just lifted some statements about the liars paradox from some pop-sci account, and thought that was the whole thing..............
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Post by abacus9900 on Jan 18, 2011 17:25:27 GMT 1
But do you?
To really understand the problems Godel's ideas address you have to have experienced the insight that any axiomatic structures put in place will necessarily be co-dependent on the level of reality they are supposed to be referring to at the time. Now, I do not think it is such an outrageous idea to suggest that we have not yet exhausted the richness of reality so that our mathematical structures to date will have to be adapted in the future to reflect levels of reality yet to be dreamed of. In fact, only last night in the Horizon programme one idea put forward by Max Tegmark about the nature of reality was that it is entirely mathematical, which is why maths is so powerfully able to describe the processes of the universe!
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Post by speakertoanimals on Jan 18, 2011 17:57:21 GMT 1
Bollocks. Formal axiomatic systems (ie maths) has bugger all to do with reality, there is no co-dependance as you seem to want to claim. And there is certainly no reality that a formal axiomatic system refers to, that is kind of the point!
So, I can formulate an axiomatic system that would be non-euclidean geometry, whether or not there is a 'real-world' out there that USES such a geometry for its physics. Ditto other weird and wonderful mathematical systems, that may or may not be part of the physics that describes our real world.
Again, a few snazzy sounding phrases from you, that are actually totally meaningless.
Of course, there has to be 'a reality' for there to be a 'me' that is doing the maths, but that doesn't mean that I can only do the maths that is the maths to be directly used in a physical description of my reality. The reality constrains the person doing the maths, not the maths that person can do.
Actually all reduces to your initial claim about 'axiomatic structures' and the reality they are supposedly referring to. The latter doesn't necessarily exist, maths and axiomatic structures can be formulated and studied without supposedly referring to ANY reality. Again, that is the whole point of them!
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Post by abacus9900 on Jan 18, 2011 18:38:14 GMT 1
No, you've misrepresented what I have said.
I was trying to point out that axioms that have been established in the past will inevitably become outdated in time due to their dependence on the logical structure of reality pertaining at that time. For example, we now know that at the quantum level of reality things may be in many places at once, which directly contradicts the old idea of things existing as distinct objects and reflected as such by arithmetical structures accepted for eons over human history. Quantum computing can hardly use the concept of serially occurring 'bits' any longer, having to replace this idea with the 'Q-bit.' Nobody used to think that objects could exist in more one location at a time but now we have to reflect this fact in the mathematics.
What you keep overlooking is that no axioms can be used to predict what developments will occur in the future stemming from them because axioms themselves are a human construct and therefore an act of creation, so maths itself is fundamentally an expression of human creativity, albeit promoted by environmental experiences. Imagine, if you will, a set of simple arithmetic axioms existing without any human beings also existing. How would such a set of axioms be able to be expressed and developed? The simple answer is that they could not, so we can now see that axioms are not an independent set of instructions but have to interact with human intelligence, which provides the driving force of new ideas.
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