Even Logic Has Limits

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Hurkyl

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As I mentioned, [-∞, ∞] and [-π/2, π/2] are identical topological spaces, except for the names of the points. Actually, any two closed intervals in the extended real numbers are identical topological spaces.

So how can ∞, in the way I'm using it here, have any less meaning than the symbol we use for any particular real number?


A symbol to be useful and have meaning must stand for some thing that can be resolved into a number, place or value in the end.
What do you mean by resolve? It's not clear why other symbols would resolve as such and ∞ would not. It's also not clear it captures other useful and meaningful ideas like sets, collections, or even basic things like "+".


How can we talk of the unlimited with that which is limited by it's very nature?
If we can talk about limits, and we can talk about limited things, why couldn't we talk about "unlimited" things?

Anyways, I don't even think "unlimited" is the right term to describe infinity; in every colloquial use of the term I've heard, describing infinity as something like "bigger than any (real) number" or "further away than any point in space" means exactly the same thing... and it's quite easy to precisely define things that satisfy those meanings when you step into more sophisticated mathematical constructs.
 
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Originally posted by Hurkyl
As I mentioned, [-∞, ∞] and [-π/2, π/2] are identical topological spaces, except for the names of the points. Actually, any two closed intervals in the extended real numbers are identical topological spaces.

So how can ∞, in the way I'm using it here, have any less meaning than the symbol we use for any particular real number?
Again infinity is not a point in any system nor is it a closed interval. The symbols that we use for real numbers can be resolved to a value. A real number has value. Once we are through manipulating the symbols we can substitute a real value for the symbol and come up with a value for the equation. We cannot substitute a point value for the symbol &infin, it is not a point value and has no value that has any meaning. It is all points and all values. That is why it is called undefined, it has no definable value.



What do you mean by resolve? It's not clear why other symbols would resolve as such and 8 would not. It's also not clear it captures other useful and meaningful ideas like sets, collections, or even basic things like "+".
This I am not absolutely sure of, we should ask Tom; but, &infin is not and cannot be considered a set because it is undefineable.

Anyways, I don't even think "unlimited" is the right term to describe infinity; in every colloquial use of the term I've heard, describing infinity as something like "bigger than any (real) number" or "further away than any point in space" means exactly the same thing... and it's quite easy to precisely define things that satisfy those meanings when you step into more sophisticated mathematical constructs.
Yes it is easy and valid until you try to solve or resolve the terms to a value. How do you substitute a value for &infin? Until you do that math is nothing but the manipulation of sybols following strict rules but has no meaning or value. Until we assign values to the symbols and solve the final equation according to the rules math, any and all math are abstract thoughts with no real meaning or value. &infin is an abstract symbols to which no value can be assigned thus it is undefined......

In the simplest terms I can think of:

Take the equation A+B=X as an example. We can solve the equation and find a value for X for every possible value of A and B except it A or B = &infin . Because we can not assign a value to &infin. Other than a mental exercise the purpose of math is to determine values for unknowns. If it can do that then it is in reality meaningless and worthless.
 

Hurkyl

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How do you substitute a value for ??
Why would I? ∞ is a value in its own right.


Mathematics (and logic) goes far beyond mere arithmetic. It's also for expressing relationships, interactions, concepts, and ideas.


In any case, maybe a practical (numerical) application will interest you. Your video card does not use 3-d Euclidean geometry; instead it adds a plane at infinity to create 3-d projective geometry, and this is the geometry in which it works.

I'm sure you've seen in physical problems some simplifications like "Well, we can imagine the sun is infinitely far away so that the light rays that come from it are parallel", right?

One of the reasons your video card uses projective geometry is because the lines eminating from a point at infinity are parallel! While the above is a mere simplification in physical problems, it is an exact description within your video card. For example, unless your video card reverts to a 2-d mode to generate the display you are reading right now, it treats it as if your eye is located on the plane at infinity. (Incidentally, it also resolves a 0 / 0 indeterminate form in order to size the display properly)
 

Tom Mattson

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Originally posted by Royce
This I am not absolutely sure of, we should ask Tom; but, &infin is not and cannot be considered a set because it is undefineable.
LOL!

No, my dear Royce, Hurkyl is one of the people I go to when *I* get stuck.

To all:
You know, I really am surprised that this thread has focused on ∞ for so long. What does ∞ have to do with "the limits of logic" anyway? I would think that the discussion would focus on Goedel or Kant's Critique or some such thing.

*shrug*
 
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Originally posted by Tom
LOL!

No, my dear Royce, Hurkyl is one of the people I go to when *I* get stuck.

To all:
You know, I really am surprised that this thread has focused on ∞ for so long. What does ∞ have to do with "the limits of logic" anyway?
*shrug*
Yeah...I could go on about ∞ forever . . . OR I could change the discussion and we could talk about 'nothing' - which is also undefined. (I know....I know...everyone thinks they CAN define it, but they can't. To define 'nothing' in its absolute sense is NOT TO DEFINE. . . but they keep trying to plug in the relative 'nothing' which is defined as zero or the empty set.)

What everyone seems to overlook is that there is an abstract and a relative connotation to the terms infinity and nothing. The relative (or defined) connotation is a 'first derivative' of the abstract and is the only concept which can be used in the tool of logic. When you integrate the relative connotation, you lose something in the translation - like the unknown constant in integral calculus.

Logic - itself - is a DERIVATIVE of reality. . .not reality itself, therefor it has limits when reality DOESN'T. Logic requires DEFINITION, reality is OFTEN not defined...i.e. ∞ and 'nothing'. (I think I'm getting close to my main theme here...)

GEEEEEZ, the communication code we call English used to only require conjugations and declentions ... now, it seems it requires integration, too
 
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Originally posted by Royce
Again infinity is not a point in any system nor is it a closed interval. The symbols that we use for real numbers can be resolved to a value. A real number has value. Once we are through manipulating the symbols we can substitute a real value for the symbol and come up with a value for the equation. We cannot substitute a point value for the symbol &infin, it is not a point value and has no value that has any meaning. It is all points and all values. That is why it is called undefined, it has no definable value.

This I am not absolutely sure of, we should ask Tom; but, &infin is not and cannot be considered a set because it is undefineable.

Yes it is easy and valid until you try to solve or resolve the terms to a value. How do you substitute a value for &infin? Until you do that math is nothing but the manipulation of sybols following strict rules but has no meaning or value. Until we assign values to the symbols and solve the final equation according to the rules math, any and all math are abstract thoughts with no real meaning or value. &infin is an abstract symbols to which no value can be assigned thus it is undefined......

In the simplest terms I can think of:

Take the equation A+B=X as an example. We can solve the equation and find a value for X for every possible value of A and B except it A or B = &infin . Because we can not assign a value to &infin. Other than a mental exercise the purpose of math is to determine values for unknowns. If it can do that then it is in reality meaningless and worthless.
APPLAUSE
YESSSSSSSSS!! By George, me thinks you have it.

Mathematics does; however, reveal some interesting comparisons when applied to the indefinite. Consider the fractions 1/2 and 1/99,999,999,999,999,999. As the denominator of a fraction increases, its value decreases. Though infinity is undefined and cannot be represented by a value, it is obvious that if the numerator of a fraction is finite, then regardless how large that numerator may be, the ratio approaches Zero as the denominator grows to ‘approach infinity’. In the relative context, the size of the entire zone of the cosmos detectable to our technology has no quantitative value compared to infinity.

Using any given point in space as an X,Y,Z axis, one may theoretically extend equidistant lines to infinity through the spectrum of polar coordinates. The procedure inscribes a sphere which theoretically encompasses the Universe. By definition, the selected point is the center of that sphere - and the center of the Universe. Since the same can be done for all points, it means that in the relative context every position in the cosmos is its center.

Logic defines reality using three basic criteria - quality, quantity and location. Compared to an infinite Universe, the size (quantity) and position (location) of any finite element would have no relative value. If the Universe is infinite and the qualitative value of each element in the Universe is also Zero, then the logical equivalent of ‘nothing’ may actually exist.

Is it possible the sum of QUALitative values within the Universe are countervalent?

(GEEEEZ I hope so, or the theory I've been exploring for the last 30 years is mush)
 
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Okay, Hurkyl, so I'm playing logical chess with a master. I've done it before and of course got beat badly at first; but it is the only way to learn. I soon got to the point that I could at least give him a good game. Do you want to keep playing or want me to resign?

I am perfectly willing to conceed that infinity is a logical and mathematical valid symbol that you can treat just as any other symbol. I still maintain that math and logic are abstracts that other than mental exercise have no meaning or value unless applied to reality by substituting real values for the symbols and that is not possible with infinity.

By the way can infinity be considered a set?

I seem to have a propensity for stepping in it then to make matters worse I end up with that foot in my mouth.
 
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Hurkyl

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I don't mind playing as long as people are listening; it's good exercise for all sides!


I am perfectly willing to conceed that infinity is a logical and mathematical valid symbol that you can treat just as any other symbol.
That's the limit (ha ha!) of the point I was arguing because...

I still maintain that math and logic are abstracts that other than mental exercise have no meaning or value unless applied to reality by substituting real values for the symbols and that is not possible with infinity.
I accept this, in the sense that while I don't buy it, all I can do is simply disagree.

I personally believe that no mathematical notion has a "reality" (whatever that means); I believe the most abstract mathematical notion and the number 1 are equally meaningful (or meaningless, if you prefer). I believe value (aka worth) is a matter of practicality, and I've found infinity to be an extremely practical concept and it has helped me understand and/or simplify a great many ideas, and I am actually quite fond of its use.

However, logically, I can't apply any of this to your beliefs because they're my beliefs. The way I determine meaning and value has no bearing on the way you determine meaning and value. IMHO, at this point in a debate, all one can do is find a contradiction in your opponent's beliefs, or to preach the merits of your own beliefs.

(Of course, once the discussion steps back into the realm of logic / mathematics, I get to use all of the tools of those fields to defend my position and attack opponents' positions)


By the way can infinity be considered a set?
Yes...

But everything in mathematics can be considered a set (except for proper classes)... but this would require delving deep into logic to discuss the idea of a logical model, and I don't quite think that is the point of your question.

And it really depends on how you want infinity to behave. I've been focusing on infinity as a point in this thread beacuse that was where the discussion seemed to be focused. However, infinity as a point is totally different from infinity as the size of a set, or infinity as a hyperreal number, or infinity as a number of repetitions, or...



Back to Messiah:

Logic defines reality using three basic criteria
Logic defines nothing but logic. Another field (like physics or philosophy) may use logic as a tool to assist in the definition of reality, but it is the other field doing the defining, not logic.


Anyways, I'll presume you define reality by those three basic criteria. I'm curious why you find it a problem with there being a zero relative value between the finite and the infinite... the finite things still have the "correct" relative value to each other, don't they? The zero relative value problem just means that different scales of observation may behave differently, and it may be more difficult to make useful observations relating finite to infinite.

Interestingly, one of the most important advances in mathematics was performed by considering the same problem in the opposite direction; that infinitessimals have zero value in relation to finite values! Newton and Leibniz had the insight to consider the relative value that infinitessimals have to each other, and thus developed the field of differential calculus.

Also, relatively recently, a new field of numbers, the hyperreal numbers has been discovered and is the subject of Nonstandard Analysis. These numbers are interesting because they, thus far, have been the most successful system allowing arithmetic of nonfinite values; for example, finite values do not have zero relative value to infinites, they have infinitessimal relative value.
 
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Man this is more entertaining than watching the olsen twins mud-wrestle.(ok maybe not, but it's damn funny!)

Royce and Hurkle you both have good arguments from my perspective. It's amazing how the broadest statement can be whittled time and time again down the finest points, until semantics are all that is left.
 
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Zandra, we have just begun. Thats the fun of infinities and infinitessimals, you never run out of things to play with.

Okay Hurkyl, one last post tonight. we agree to limit the field of play to abstract as reality as a subject is closed. Your choice math or logic?
 

Hurkyl

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We might want to start a new thread, I don't want to hijack Messiah's thread. Either math or logic is fine!
 
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Yes, I agree. Actually I think I've really sadi all I can say on the subject. Any more and I'd just repeating myself. I am obviously no expert but am willing to continue on another thread if you think it worthwile.
 
Take a break and listen to this lecture by Sir Penrose, which includes his views on philosophy of mathematics and natural sciences. Specifically, at slides 09 and 10 is something called Goodstein's theorem. It defies standard induction technique but yields nicely to general ordinal arithmetic.

link --->

ITP public lecture: Science and the Mind by Dr. Roger Penrose from Oxford University
(audio file and colored lecture slides)

There are many interesting lectures on this site.
 
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jammieg

That very interesting where Penrose suggests that thoughts and free will may stem from principles in physics, but it's seems very fringe science. I don't think I would have a clue what they were suggesting without the diagrams.

Does logic require a concrete defintion of a thing to justify that it can be deduced logically? Is there ever a perfect defenition of anything?
Logic is slow like science but accurate, in my view that is the main impeding limit of logic, there is likely much more that we have a notion of but haven't worked out logically or mathematically yet because many things are so abstract and some causes and effects seperated by months or years that we sense them in other ways long before they are awaringly reasoned out.
 
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Messiah

Logic is a derivative of reality.

Some short observations about logic.In fact logic is the branch of philosophy which deals with correct reasoning that's it with the correct deduction of conclusions from a given set of premises.Observe that there is no need for the premises to be true empirically.Basically logic is a feature of human reasoning,very useful in epistemology indeed,but there is no necessity that nature should obey the rules of logic.A good example is quantum physics,more specifically the violation of Bell's inequalities problem,where there is still valid an option often overlooked by a majority of scientists (for pragmatic reasons):that usual logic could be invalid when applied to explain phenomena at that level (in the sense that it cannot give us a good description of reality since the nature does not follow its rules).

I do not know whether you are accustomed with the philosophy of religion but one of the main objections at all deductive arguments pro/con God hypothesis is exactly that:even if we had a sound,irrefutable logically,argument pro/con God there is no necessity to believe/disbelieve without having empirical arguments beyond all reasonable doubt proving that God does exist/not exist.So that even if an argument pro God were sound that does not imply that God does exist in reality (valid also for the arguments against God's existence).Sounds strange but even if the premises of very succesfull scientific theories are all true (empirically) there is no necessity that all their conclusions are also true empirically.That's why for example it is still reasonable to doubt all predictions made by a very succesfull otherwise scientific theories not 'confirmed' practically with arguments beyond all reasonable doubt.For example General Relativity predicts that in the singularity of a black hole it ceases to give an accurate description of empirical facts,still we do not have yet sufficient arguments which to compel us to believe that,so that skepticism about this is still rational.
 
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