Even Logic Has Limits

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  • #1
Messiah
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The laws of nature reflect the intrinsic properties of everything which exists. Logic is the interpretation of those laws. By observing, defining and comparing the nature of that which we seek to understand, we derive knowledge which can be applied to familiar circumstances to predict outcomes. These mental equations, which are simultaneously solved for all known variables, produce conclusions. Valid conclusions usually fit all of the parameters of our empirical observations.

There is; however, an attribute of nature which does not readily lend itself to rational analysis - Infinity. Infinity is not an existence per se, it is a concept which defies logical interpretation. It is not exempt from the laws of nature and it is not contrary to logic, but it lies beyond the domain of logic because it is not defined - and logic requires definition. It is hard to fathom that although there is a finite distance between every two points in the Universe, there is no furthest point; and the very fact no ‘point of infinity’ exists serves only to validate the concept.

On the other extreme, we have 'nothing'. In its absolute sense it does not exist - it has no attributes, so it, also, is not defined. There is a relative - or logical - definition of 'nothing'. It is Ø or the empty set.

Logic is a derivative of reality. When you integrate a derivative (basic calculus) you lose something in the translation (usually a constant).
 

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  • #2
Hurkyl
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Infinity is quite well defined mathematically, thank you. The only problems that arise are when you naively attempt treat an infinite quantity/collection/whatever as if it was finite.


There are several mathematical fields that deal with infinite / infinitessimal things routinely, and quite beautifully... though I have a hunch that http://www.ugcs.caltech.edu/~shulman/pub/writings/core1/nonstandard.html would appeal to you the most.
 
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  • #3
Messiah
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Originally posted by Hurkyl
Infinity is quite well defined mathematically, thank you. The only problems that arise are when you naively attempt treat an infinite quantity/collection/whatever as if it was finite.


There are several mathematical fields that deal with infinite / infinitessimal things routinely, and quite beautifully... though I have a hunch that http://www.ugcs.caltech.edu/~shulman/pub/writings/core1/nonstandard.html would appeal to you the most.
'Xcuse me, but infinity - by its very definition - is UNdefined (else tell me where it is and how much of it I can gather).
"If, for every X there is an X+1" is a derivative - NOT a definition. What is the highest number??

Similarly, there is a concept of 'nothing' which is UNdefined. To define 'nothing' in absolute terms is NOT to define. In its relative context 'nothing' is Ø or the 'empty set', but in reality nothing is...
 
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  • #4
Laser Eyes
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There is one example of infinity in nature which everyone knows and seemingly accepts without question. We generally accept that future time is infinite. Time will continue running on and on and on without end (ie infinitely). Nobody seems to have a problem with this.
 
  • #5
Messiah
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Originally posted by Laser Eyes
There is one example of infinity in nature which everyone knows and seemingly accepts without question. We generally accept that future time is infinite. Time will continue running on and on and on without end (ie infinitely). Nobody seems to have a problem with this.

Yes - this is a concept - not a definition. If the end of time is not limited it is UNdefined (undefined=without limit or boundary)
 
  • #6
Hurkyl
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(undefined=without limit or boundary)

That sounds like a definition to me! :wink:




But seriously, if that is your criteria for being undefined, then I retract my objection, since your usage of "undefined" has no relationship to the mathematical usage of "undefined".


I'm not even sure your definition of "undefined" means what you wish it to mean: For example, I can easily define the sequence of alternating 1's and -1's by saying the n-th term is (-1)n... but this sequence does not have a limit, so it is undefined by your meaning.

Also, the surface of a sphere is a topological space without boundary, but are you comfortable calling surfaces of spheres undefined?
 
  • #7
Messiah
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Originally posted by Hurkyl
That sounds like a definition to me! :wink:
...Also, the surface of a sphere is a topological space without boundary, but are you comfortable calling surfaces of spheres undefined?

No - such surfaces are defined.

Every point on (or, for that matter IN) the sphere can be defined by a specific x/y/z position relative to some other given point in space.

Infinity - being UNdefined, cannot be so described.
 
  • #8
Mentat
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First off, Messiah, I agree that Logic has limits.

Secondly, I'd like to point out that one cannot logically find the flaws in logic, because they would be using that which they denounce to denounce itself (Godel's Incompleteness).

Lastly, I'd like to point out that infinity is, in fact, defined. It may not be conceptualizable (if that's a word) to you, or any other human (perhaps), but it is still defined, otherwise we couldn't speak of "it" (IOW, we would have no concept to refer to as "undefined", unless we had a definite concept...paradox).
 
  • #9
Messiah
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Originally posted by Mentat
First off, Messiah, I agree that Logic has limits.

Secondly, I'd like to point out that one cannot logically find the flaws in logic, because they would be using that which they denounce to denounce itself (Godel's Incompleteness).

Lastly, I'd like to point out that infinity is, in fact, defined. It may not be conceptualizable (if that's a word) to you, or any other human (perhaps), but it is still defined, otherwise we couldn't speak of "it" (IOW, we would have no concept to refer to as "undefined", unless we had a definite concept...paradox).

Infinity is not an existence, per se. It is a derivative - a concept - not an entity. Entities have quality, quantity and position in the Universe. Logic is simply the observation, definition and comparison of those attributes - from which we forumlate 'knowledge' in order to predict 'outcomes' (cause & effect).

For every entity in the Universe, you may calculate coordinates of location, you may determine a volume, you may describe its physical properties (even inertness is a property). You cannot point to a location, distance or volume in the cosmos and say "this is infinity" - it is UNdefined.
 
  • #10
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Originally posted by Messiah
Infinity is not an existence, per se. It is a derivative - a concept - not an entity. Entities have quality, quantity and position in the Universe.

Oh really? Then the Universe is not an entity either, is it?

Logic is simply the observation, definition and comparison of those attributes - from which we forumlate 'knowledge' in order to predict 'outcomes' (cause & effect).

Erm...nuh-uh. Logic is the use of a reasoning system (to put it basically). Science is the observation, definition, and comparison of those attributes.

For every entity in the Universe, you may calculate coordinates of location, you may determine a volume, you may describe its physical properties (even inertness is a property). You cannot point to a location, distance or volume in the cosmos and say "this is infinity" - it is UNdefined.

Don't you see that you are defining properties of that which you say cannot be defined. Even if all we could do was say what "infinity" wasn't (though we actually can say what it is, and I'll attempt that in a moment), we would still be defining it. Much like true "irrationality". It can only be defined by what it isn't, but it is still defined (see last few pages of "I think therefore I am", and the thread, "what is irrational?").

Now, the definition of infinity: That which has no end.

Pretty simple, isn't it? I don't see anything wrong with that definition, do you?
 
  • #11
Messiah
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Originally posted by Mentat
Oh really? Then the Universe is not an entity either, is it?
If 'an entity' denotes singular, then of course not. It is ALL entities.

Erm...nuh-uh. Logic is the use of a reasoning system (to put it basically). Science is the observation, definition, and comparison of those attributes.
Science and mathematics are both ways encoding logic to facilitate the transfer if ideas and information.

Now, the definition of infinity: That which has no end.
Pretty simple, isn't it? I don't see anything wrong with that definition, do you?
Yes. The term 'that' indicates a thing - a physical reality. A circle has no end, yet it is not infinite - it is definable. The best definition of infinite is 'not finite' - which also means not defined (from the same word root)

FINITE
Function: adjective
Etymology: Middle English 'finit', from Latin 'finitus', past participle of 'finire'(to finish)
Date: 15th century
1 a : having definite or definable limits <finite number of possibilities> b : having a limited nature or existence <finite beings>
2 : completely determinable in theory or in fact by counting, measurement, or thought <the finite velocity of light>
 
  • #12
Zantra
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Originally posted by Laser Eyes
There is one example of infinity in nature which everyone knows and seemingly accepts without question. We generally accept that future time is infinite. Time will continue running on and on and on without end (ie infinitely). Nobody seems to have a problem with this.


That is mind boggling isn't it? If we were to accept that there is no infinty. That everything has an ultimate end. Such as space, we would then have to accept that time also, has an ending. That there will come eventually a finality where time ends and stands still. When all things cease to exist.

I have to agree here, that though we give a definition to infinity, it is by definition the connotation of things we cannot put a definitive answer on. We say that something is infinite because we cannot find a finite solution. If we were to travel to the edge of the universe and discover the end, then the universe becomes finite. Until we do find the end of something, it is infinite. It's just the answer to all things we cannot answer.

Then of course the question arises: Mathmatically aside, How can we disprove infinity? It becomes Paradoxical, because if we find the answer, it is no longer infinite, but as long as we cannot find a solution or and end, it still remains infinite. It is by definition of itself, the unproven variable. And if something is truly infinite, it can never be disproved. And thus, we except it.
 
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  • #13
wuliheron
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Originally posted by Laser Eyes
There is one example of infinity in nature which everyone knows and seemingly accepts without question. We generally accept that future time is infinite. Time will continue running on and on and on without end (ie infinitely). Nobody seems to have a problem with this.

Cosmologists and countless others have a problem with this. Some believe, for example, that time began with the big bang and will end with the big crunch. Infinity is commonly accepted by the public at large, but then, they accept a lot of demonstrably irrational ideas. Notably, one of the most common sources of belief in infinity is among the religious. Whether or not infinity is a rational aspect of nature is a question of debate, especially among physicists who consider infinities cropping up in their equations to be a sign they did something wrong.

There is, however, one pointed limit to logic as a description of nature and that is the power of reason. My computer, for example, can perform logical operations way beyond my human capacity but it has no capacity to reason. No capacity to give anything meaning and context.

Logic itself is derived from reasoning and is ultimately based on reductio ad absurdum, reduction to the absurd. Without the context and meaning of the absurd logic has no structure much less meaning. To assert that logic and infinity describe everything then, is to imply that infinity itself is absurd. That I can agree with.
 
  • #14
Laser Eyes
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Some believe, for example, that time began with the big bang and will end with the big crunch.
Will that be something like the king-size Crunchie - mouth watering honeycomb surrounded by luscious chocolate?
 
  • #15
Hurkyl
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Every point on (or, for that matter IN) the sphere can be defined by a specific x/y/z position relative to some other given point in space.

Infinity - being UNdefined, cannot be so described.

First off, your definition has nothing to do with having a boundary.


Secondly, it's very easy to give coordinates to spaces that include coordinates for point(s) at infinity. For a simple example, consider the extended real numbers; that is the ordinary real numbers plus two extra points, +&infin; and -&infin;.

I can re-coordinatize the real numbers as follows: for every real number r, I assign it the coordinate y = arctan(r).

This recoordinatization condenses all of the real numbers into the interval (-&pi;/2, &pi;/2)... and it maps +&infin; to the coordinate &pi;/2 and -&infin; to the coordinate -&pi;/2.

Now, if we're infinityphobes, this construction maps +&infin; and -&infin; down to ordinary finite numbers and we can manipulate them as such... though I imagine it's much easier to manipulate them according to the definition of the extended real numbers.
 
  • #16
Messiah
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Originally posted by Hurkyl

First off, your definition has nothing to do with having a boundary.


Secondly, it's very easy to give coordinates to spaces that include coordinates for point(s) at infinity. For a simple example, consider the extended real numbers; that is the ordinary real numbers plus two extra points, +&infin; and -&infin;.

I can re-coordinatize the real numbers as follows: for every real number r, I assign it the coordinate y = arctan(r).

This recoordinatization condenses all of the real numbers into the interval (-&pi;/2, &pi;/2)... and it maps +&infin; to the coordinate &pi;/2 and -&infin; to the coordinate -&pi;/2.

That would be GREAT; however &infin; is NOT a point. If it was a point, it would be defined. &infin; is a symbol, not a value.
 
  • #17
Hurkyl
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I've defined it as a point, thus it is a point. :smile:

But I think the point of your objection is that you think that your concept of infinity does not correspond to what I've defined.


So I ask you, what is wrong with it? As I've defined it, +&infin; is to the right of every point on the (ordinary) number line, and -&infin; is to the left of every point on the number line. Any sequence of real numbers that increases without bound converges to +&infin;, and any sequence of real numbers that decreases without bound converges to -&infin;.

In the context of spatial position (in which you were using the term "infinity"), what does my definition of &infin; lack?
 
  • #18
Zantra
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Originally posted by Hurkyl
I've defined it as a point, thus it is a point. :smile:

But I think the point of your objection is that you think that your concept of infinity does not correspond to what I've defined.


So I ask you, what is wrong with it? As I've defined it, +&infin; is to the right of every point on the (ordinary) number line, and -&infin; is to the left of every point on the number line. Any sequence of real numbers that increases without bound converges to +&infin;, and any sequence of real numbers that decreases without bound converges to -&infin;.

In the context of spatial position (in which you were using the term "infinity"), what does my definition of &infin; lack?

I think the point that is being made is that infinity, though definable in abstract mathmatical terms, has no real world definition.
 
  • #19
Hurkyl
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I think the point that is being made is that infinity, though definable in abstract mathmatical terms, has no real world definition.

I would accept mathophobia if Messiah wasn't directly claiming math couldn't define infinity. :smile: I've used infinity (and infinitessimals) countless times, and they behave exactly like I think I should behave (including disappointing me because they don't behave exactly like finite things).
 
  • #20
quartodeciman
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Amusing.

Both Messiah and Hurkyl are both right and wrong. It is an argument about language. Here are online Merriam-Webster dictionary entries for 'infinite' and 'infinity'.

---------------------------------------------------------------
Main Entry: in·fi·nite
Pronunciation: 'in-f&-n&t
Function: adjective
Etymology: Middle English infinit, from Middle French or Latin; Middle French, from Latin infinitus, from in- + finitus finite
Date: 14th century
1 : extending indefinitely : ENDLESS <infinite space>
2 : immeasurably or inconceivably great or extensive : INEXHAUSTIBLE <infinite patience>
3 : subject to no limitation or external determination
4 a : extending beyond, lying beyond, or being greater than any preassigned finite value however large <infinite number of positive numbers> b : extending to infinity <infinite plane surface> c : characterized by an infinite number of elements or terms <an infinite set> <an infinite series>
- in·fi·nite·ly adverb
- in·fi·nite·ness noun

Main Entry: in·fin·i·ty
Pronunciation: in-'fi-n&-tE
Function: noun
Inflected Form(s): plural -ties
Date: 14th century
1 a : the quality of being infinite b : unlimited extent of time, space, or quantity : BOUNDLESSNESS
2 : an indefinitely great number or amount <an infinity of stars>
3 a : the limit of the value of a function or variable when it tends to become numerically larger than any preassigned finite number b : a part of a geometric magnitude that lies beyond any part whose distance from a given reference position is finite <do parallel lines ever meet if they extend to infinity> c : a transfinite number (as aleph-null)
4 : a distance so great that the rays of light from a point source at that distance may be regarded as parallel
---------------------------------------------------------------

Some of these preclude assigning the words to a definite object and some allow it (infinite or transfinite number).

A dictionary tells us how words are actually used, not how they (for some reason) ought to be used.

George Cantor contemplated this and chose to go with the word 'transfinite'. He contemplated an absolute infinity that would transcend all actual sets, a perfect continuum. It doesn't really exist. A set theoretic approach handles this nicely. The ultimate infinity for ordinal numbers is ordinal but not a proper set. Therefore, one cannot add 1 to it{1}.

same source:
---------------------------------------------------------------
Main Entry: trans·fi·nite
Pronunciation: (")tran(t)s-'fI-"nIt
Function: adjective
Etymology: German transfinit, from trans- (from L) + finit finite, from Latin finitus
Date: 1902
1 : going beyond or surpassing any finite number, group, or magnitude
2 : being or relating to cardinal and ordinal numbers of sets with an infinite number of elements
---------------------------------------------------------------

{1} more accurately: 1 cannot be added to it and produce something ordinally beyond it, unlike actual ordinal numbers.
 
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  • #21
The only limits are the limits we imagine.
 
  • #22
Royce
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Originally posted by Hurkyl



So I ask you, what is wrong with it? As I've defined it, +&infin; is to the right of every point on the (ordinary) number line, and -&infin; is to the left of every point on the number line. Any sequence of real numbers that increases without bound converges to +&infin;, and any sequence of real numbers that decreases without bound converges to -&infin;.

In the context of spatial position (in which you were using the term "infinity"), what does my definition of &infin; lack?

Any sequence of real numbers that increase without bound DIVERGE to infinity not converge. Infinity is not a point nor a limit. Any infinite system is ether on a flat endless plane of n dimentions or on a hyperbolic plane of n dimentions, saddle shaped is the usual example.
Between the real numbers 1 and 2 are an infinite number of real numbers and between each of those numbers are an infinite number of numbers none of which repeat. The series of real numbers is infinite as is the series of imaginary numbers and irrational numbers of infinite length, ad infinitum, ad nausium.
Infinitly cannot be expressed or defined in either math or logic other than expressing it as an undifined symbol. We can use the symbol in algebraic operations but they are meaningless. We can use the sybol in Booleen Logic and possibly others but it is meaningless.
Heisenberg's Quantum equations were full of infinities and thus rendered meaningless and useless until Feynman threw the infinities away and ignored them in a method that even he said was not valid and shoddy; but, the equations became very useful and extremely close to obeserved phenomina.
When we manipulate and play with the symbols for infinity we may be under the impression that we are actually doing something but in reality we are just playing with an undefined meaningless symbol
 
  • #23
Mentat
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Originally posted by Messiah
If 'an entity' denotes singular, then of course not. It is ALL entities.

You couldn't refer to it as "it" if it weren't a singular entity, could you?

Science and mathematics are both ways encoding logic to facilitate the transfer if ideas and information.

No, science is a logical framework, that only works after having accepted certain premises.

Yes. The term 'that' indicates a thing - a physical reality.

That's because it (notice my use of the word "it") is a physical reality.

A circle has no end, yet it is not infinite - it is definable.

Ah, but a trip along the surface of a circle would have no end, and thus it is - at least in some sense - infinite.

The best definition of infinite is 'not finite' - which also means not defined (from the same word root)

No it doesn't. To define something is to deny yourself the ability of calling it undefinable.

FINITE
Function: adjective
Etymology: Middle English 'finit', from Latin 'finitus', past participle of 'finire'(to finish)
Date: 15th century
1 a : having definite or definable limits <finite number of possibilities> b : having a limited nature or existence <finite beings>

Yes, being limited, having definable limits. It's existence can be definable, without being finite, but it's limit's cannot.
 
  • #24
Hurkyl
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Any sequence of real numbers that increase without bound DIVERGE to infinity not converge.

Converge is the correct term; recall I'm considering the topology of the extended real numbers instead of that of the ordinary real numbers, so the points -&infin; and &infin; are part of my space, so things can converge to them.


When we manipulate and play with the symbols for infinity we may be under the impression that we are actually doing something but in reality we are just playing with an undefined meaningless symbol

What makes &infin; an undefined meaningless symbol and not every other symbol in mathematics?
 
  • #25
Royce
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Originally posted by Hurkyl
Converge is the correct term; recall I'm considering the topology of the extended real numbers instead of that of the ordinary real numbers, so the points -&infin; and &infin; are part of my space, so things can converge to them.

Maybe in your space they converge; but, you are again and still concidering -&infin; and +&infin; as points. They are not points. That is my point. In math and/or logic to consider them and treat them as points is invalid. That is why I say that the symbols are meaningless and undefined unlike every other symbol. 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. Infinity can not be resolved at all. It is by definition, if one exists, unresolvable.
I find it extremely hard to write in a clear concise and understandable way about this as our language, or at least my command of it, is so limited. How can we talk of the unlimited with that which is limited by it's very nature?
 
  • #26
Hurkyl
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As I mentioned, [-&infin;, &infin;] and [-&pi;/2, &pi;/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 &infin;, 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 &infin; 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.
 
  • #27
Royce
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Originally posted by Hurkyl
As I mentioned, [-&infin;, &infin;] and [-&pi;/2, &pi;/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 &infin;, 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.
 
  • #28
Hurkyl
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How do you substitute a value for ??

Why would I? &infin; 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)
 
  • #29
quantumdude
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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 &infin; for so long. What does &infin; 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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  • #30
Messiah
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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 &infin; for so long. What does &infin; have to do with "the limits of logic" anyway?
*shrug*

Yeah...I could go on about &infin; 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. &infin; 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
 
  • #31
Messiah
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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)
 
  • #32
Royce
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0
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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  • #33
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
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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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  • #34
Zantra
781
3
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.
 
  • #35
Royce
1,514
0
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?
 

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