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We can't positively prove infinity, ever, about anything, because that would be trying to prove a negative. What we can do (as we did in mathematics) is to prove that the possibility of any upper bound whatsoever, on whatever, is not possible.

 

I don't think that it's comparable to proving a negative. It's impossible to prove that something is infinite because an infinite object has no bounds which means it has no identity. Proof itself is premised on the law of identity and so 'proving the existence of infinity' means using the law of identity to show that the law of identity does not exist.

 

With respect to proving a negative, one is never required to disprove a claim to knowledge when there is no evidence for the claim because a lack of evidence means that the 'knowledge' has no known basis in reality.

 

The two things are both wrong to do but it doesn't seem like the reasons why are the same.

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How did we discover mathematical infinity? We didn't count to it; the concept of "infinity" MEANS that you can never do that. Rather, we inferred it from the knowledge that every single number is 1 smaller than the next number. Whether it's 5 or 50000000000000, you can always add 1.

I don't know what "prove" infinity even means. It is plenty proven as a valid concept, as an abstraction like time is. Infinity is more complex than just boundlessness. It's not just "you can always add 1". I don't know a lot about the mathematical concept, but it's not supposed to be something as obvious as addition. It is a representation to deal with abstractions, just as circles aren't actually directly based on seeing a circle. They require perception to be valid, yes, but it's meant to represent parts of reality so we can deal with problems that aren't problems of perception. I don't mean to say they're "synthetic" concepts, just that concepts like this are concepts of method or tools to help concepts of method.

 

Related to the problem in the OP. To say there is 'physical infinity' is to say that there is no stopping point of reduction, and also that there is no real stopping point with which to say we've stopped being able to reduce. But it isn't anything to do with positing a "size" to the universe. You can divide a square into pieces for eternity, yet the square isn't itself any larger or smaller. A planck length might make sense, that there is a smallest way to express a measurement, but I don't see how that translates to an end of divisibility even if it were true.

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In response to Harrison D's remarks about infinity being inferred from unboundedness, I would like to make a fine distinction. One might think of "unboundedness" as being the very definition of "infinity", but one must be careful then not to use "infinity" as a noun. One may also be careful not to collect things like natural numbers into a "set" in order to keep "infinity" from becomming a noun, or from possessing the properties of other sets that are closer to how perceivable concepts are organized. "Infinity" is not perceivable. In the extended real numbers infinity is added as a noun and certain arithmetic operations are defined for it. Perhaps this should be thought of as a convenient fiction that is not directly traceable to perception. People use "infinity" in very flexible ways that often have no particular meaning. Therefore, one must be careful about the use of this term.

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I see no problem with referring to abstractions which are unbounded with nouns.  The set of real numbers between 0 and 1 are a continuum which, since mappable on 0, infinity is just as "infinite" in "number" of "points" ... the only danger is trying to say something about reality is directly parallel to mathematics...  but some things in reality, such as relational definition of space, must be thought of in ways very parallel to a continuum (real numbers)

 

So, your point is well taken but I would restrict your restriction to discussion of existents which fall outside of "continuum" type things... e.g. a relation like space (not an existent) can have, as a potentiality (and over time actuality), an infinite number of "values".  In fact an object moving from, relationally 1m with respect to another object to 2m with respect to that other object, has traversed an infinite "number" of distances (values of distance).  Assuming space as a relational thing only is a continuum (not discretized) we don't even need to refer to it in the hypothetical/potential... it has in fact been over the last duration of time at an infinite "number" of distances.

 

Maybe this raises the issue of whether we can even talk about "number of distances" when talking about a continuum... but if one assumes at any instant (accepting such a thing) the things have a distance (identity) then over time, assuming no finite granularity of space as something moves it has traversed (relationally) an infinite number of distances, because "distance" at no time can have non-identity, it must be what it is at that time, and over time it must have been at every possible value going from 1m to 2m.. which is an infinite number of identities of distance.

Edited by StrictlyLogical
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It's impossible to prove that something is infinite because an infinite object has no bounds which means it has no identity. Proof itself is premised on the law of identity and so 'proving the existence of infinity' means using the law of identity to show that the law of identity does not exist.

I think you may be right. Thank you.

One may also be careful not to collect things like natural numbers into a "set" in order to keep "infinity" from becomming a noun, or from possessing the properties of other sets that are closer to how perceivable concepts are organized.

I don't understand what you mean, by that.

Since a "set" is just a particular grouping of things, I don't see the significance of referring to the "set of natural numbers"; I've always parsed it as nothing more than a fancy term for "numbers". So whatever it is that you intend to say, there, I don't get it.

Perhaps this should be thought of as a convenient fiction that is not directly traceable to perception.

Yes; that's my primary point.

People do frequently use "infinity" in very flexible ways. If we want it to be cognitively useful then we have to pay close attention to its referent, because such abstract things (like "infinity", "freedom", "economy", etc) tend to be slippery.

To that end, a useful idea which doesn't actually apply to any concrete thing (a convenient fiction) is exactly the right way to think about it.

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I don't know what "prove" infinity even means. It is plenty proven as a valid concept, as an abstraction like time is.

Yes, it is a valid concept. I meant proving its application to any other particular concept (like "time", "space" or "numbers"); not proving the concept, itself.

That would just be silly.

Infinity is more complex than just boundlessness. It's not just "you can always add 1".

Alright; let's take the "set of natural numbers". What does that refer to?

"Numbers" refers to every single number, considered as a single group. The concept of each number refers to a particular way to group *things* together, according to a certain pattern. Yes, that pattern is a quantification, but that (and everything else that implies enumeration) would be a circular definition; that a number refers to groups of that number of things.

The only good way to define a "number" is to do so in terms of counting; that "five" refers to anything that we can count "one", "two", "three", "four", "five" of; one repetition for each member, no more and no less.

In that way, you're absolutely right that numerical infinity is a concept of method; that we can repeat this method as many times as we like and never run out of distinct ways to group things.

However...

It's not just "you can always add 1".

If numbers are just abstractions of counting then I don't see how it could be anything other than that.

To say there is 'physical infinity' is to say that there is no stopping point of reduction, and also that there is no real stopping point with which to say we've stopped being able to reduce. But it isn't anything to do with positing a "size" to the universe. You can divide a square into pieces for eternity, yet the square isn't itself any larger or smaller. A planck length might make sense, that there is a smallest way to express a measurement, but I don't see how that translates to an end of divisibility even if it were true.

Whether or not it is possible to continue moving in a straight line forever, without stopping or looping, definitely has something to do with the size of the universe. Again, a concept of method as viewed recursively.

The same goes for the divisibility of matter, except that in order to turn it into a conceptual loop you'd have to know the exact difference between the matter you started with and the matter you end with; hence you'd need a theory of everything (like String Theory).

Edited by Harrison Danneskjold
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The only good way to define a "number" is to do so in terms of counting; that "five" refers to anything that we can count "one", "two", "three", "four", "five" of; one repetition for each member, no more and no less.

Corvini chews on "number" nicely in her "Two, Three, Four, and All That" series.

In that way, you're absolutely right that numerical infinity is a concept of method; that we can repeat this method as many times as we like and never run out of distinct ways to group things.

However...

It is because we understand that repeating this method that gives us our grasp of infinity, in this context.

Whether or not it is possible to continue moving in a straight line forever, without stopping or looping, definitely has something to do with the size of the universe. Again, a concept of method as viewed recursively.

The same goes for the divisibility of matter, except that in order to turn it into a conceptual loop you'd have to know the exact difference between the matter you started with and the matter you end with; hence you'd need a theory of everything (like String Theory).

Once again, consider the Axiom of Archimedes. If I'm not mistaken, it is in conjunction with this that Aristotle posited the potential and actual Strictly Logical suggested interjecting.

Edited by dream_weaver
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the only danger is trying to say something about reality is directly parallel to mathematics...

Exactly. And I mention numerical infinity, not because it necessarily relates to anything at all in reality, but because it's a fairly well-established and valid application of "infinity" which basically any Elementary schooler is capable of grasping.

It doesn't do anyone any good to bicker over particulars before we've actually defined what our question is.

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Every volume is either specific, or unspecified. Every distance is either specific or unspecified...

In an All S is P statement, to state the universe is infinite sets the universe as the subject and infinite as a predicate. What is the identity of infinity?

If I follow, you're saying that "infinity" is an unspecification; an omitted measurement, which renders its application to reality meaningless. I think that's close to it (very close), but not quite.

Let's look at numbers, again. When I say that "numbers are infinite", I don't mean that there could be some unspecified "biggest number" which has no successor; I mean to specify that there simply is no such boundary. So I think it's more (but not by much) than an unspecification; it's the specific negation of any particular quantity, at all.

It is because we understand that repeating this method that gives us our grasp of infinity, in this context.

:thumbsup: Edited by Harrison Danneskjold
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Since a "set" is just a particular grouping of things, I don't see the significance of referring to the "set of natural numbers"; I've always parsed it as nothing more than a fancy term for "numbers". 

Consider sets of flowers. There are many different sets of flowers. For example, the set of flowers in my back yard, the set of flowers given to my mother over the years, the set of flowers whose only members are the largest flower in the White House Rose Garden and the flower I just saw on my co-worker's desk.

 

These are:

 

{x | x is a flower in my back yard}

 

{x | x is a flower given to my mother during her life}

 

{x | x is the largest flower in the White House Rose Garden}

 

{x | x is the largest flower in the White House Rose Garden or x is the flower on my co-worker's desk}

 

And there is the set of all flowers:

 

{x | x is a flower}

 

And with numbers:

 

{x | x is a natural number less than 5}

 

{x | x is a solution to the equation x^2 = 4} 

 

{x | x is an even number}

 

{x | x is a natural number}

 

And {x | x is a natural number} is the set of all natural numbers (or we just say 'the set of natural numbers' as it is tacitly understood that we mean 'all').

 

So "the set of natural numbers" is not the same as just saying "numbers". It's not even grammatical to take them as the same:

 

"The set of natural numbers has no greatest member" is not even expressed grammatically by "Numbers has no greatest member".

 

/

 

As to infinity, one will not get a coherent discussion by mixing up two separate contexts: Objectivism and standard mathematics. I don't opine about Objectivism, but I should say what 'infinite' refers to in standard mathematics:

 

First, there is no object called 'infinity' (putting aside the "point of infinity" on the extended real line, which is something else). Instead there is 'infinite', which is an adjective not a noun. So it makes sense to say "the set of natural numbers is infinite" since the adjective 'is infinite' is true of the set of natural numbers. But we don't say "infinity exists" or "infinity is a number" or any of that.

 

Now, for any set S (such as the set of natural numbers) "S is infinite" means "S is not finite", which in turn is equivalent to "there is no 1-1 function between S and a natural number" (another definition that is equivalent with the axiom of choice is "there is no 1-1 function between S and a proper subset of S)".

 

Meanwhile, while there is no object called 'infinity', we sometimes loosely say "the first infinity" to mean "the first infinite cardinal number" (which is itself a set) and things like that. But that is a loose way of speaking and we don't actually mean that there an object called 'infinity', just that there is an object that is the first infinite cardinal and other infinite cardinals and infinite sets in general. 

 

Put another way, a set may have the PROPERTY of being infinite. But there is no THING that we call 'infinity'. 

 

/

 

Granted, there are mathematicians (called 'strict finitists') who do not allow systems in which there are infinite sets. But in the usual instances, mathematicians regard infinitude along the lines I just described and they do accept that there are infinite sets, or at the very least that the axioms of set theory prove the formula that is rendered into English as "there exist infinite sets". 

Edited by GrandMinnow
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  • 4 weeks later...

"The set of natural numbers has no greatest member" is not even expressed grammatically by "Numbers has no greatest member".

Does the sentence "numbers have no greatest member" render properly into English?

http://grammar.about.com/od/basicsentencegrammar/a/prperfectense.html

I believe I said that "there is no biggest number". I further thought that the relation of the "biggest" being towards any other number was implicit in the actual meaning of the words that I chose to string together.

Let's not argue semantics, though, when we can argue syntactics.

http://grammar.about.com/od/basicsentencegrammar/a/prperfectense.htm

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  • 8 months later...

I remember an interesting idea about time being infanately expansive. And our lifespan being just a blip in the eternal spectrum of time.

If a mere 100 years of a humans life represented even a glimmer of time on the grand time scale, you would quickly realize that your chances of existing today, right now at this moment, is overwhelmingly unlikely.

Consider all the years past or years ahead of you. 

Of all these eons upon eons, what are the chances that you happen to exist today In a span of 100 or so years.

Do you catch my drift?

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4 hours ago, Marzshox said:

Do you catch my drift?

What is your drift? I don't think it's obvious from your post.

If you have a bag of a billion little lack balls and just one white ball, and it they're all mixed up properly, and you pull one out and it is white, what do you conclude from that? I can't see how you could conclude anything.

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23 minutes ago, softwareNerd said:

What is your drift? I don't think it's obvious from your post.

If you have a bag of a billion little lack balls and just one white ball, and it they're all mixed up properly, and you pull one out and it is white, what do you conclude from that? I can't see how you could conclude anything.

Well the odds you pick out a white ball is extremely unlikely. I think that's conclusive to my point.

The odds of living for a short time span during a period of infinite years, is not only not likely, but nearly impossible, since there are infinite times one could have been born instead.

I feel like I can explain this forever in multiple ways, but still find it hard to expand on this anymore cohesively than I have...

How is it that 100 years of life is occurring now, when there are infinite other opportunities to have been born (in the past and in the future). The odds that you are alive right now are 1 in infinity!

One way to explain this winning of the infinite "jack-pot" is to assume that in some form or another, you have always existed.

Alternatively, you would have to hit some unlikely odds :)

 

Edited by Marzshox
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1 hour ago, Marzshox said:

Well the odds you pick out a white ball is extremely unlikely. I think that's conclusive to my point.

Conclusive of what? If you picked the white ball, you picked the white ball, no matter what the odds. The probability that you picked it is 100%, now that you've picked it. 

To make it clearer, take the human out of the picture. Let's say you've mixed this up and put the bag of balls in a black box. This box has a little outlet which is shut right now. From the design, you know that one ball must be right there, but you do not know which one. In this case, the probability that it is white is tiny. 

However, from this example you can see that probability does not describe the reality of the balls in the container. Rather, it describes our own lack of knowledge. If the containers was transparent and we could see which ball was about to pop out, and we could see the white one right there at the door, we would say the probability is 100% that the ball is white. 

Suppose each of the million balls was numbered, and we tossed the container, spinning into the sky, and let it drop and burst. Then, we measure which ball was the furthest from some pre-determined point. The probability for any particular ball to be furthest is 1-in-a-million, but some fella has to be "it". And, if all the processes involved are non-volitional, then it follows that one particular ball had to be the one. 

Of course, there is no way we could predict it, with our lack of knowledge of all the variables involved and also not knowing how those variables interact. Still, if no hand of God is reaching down, the answer is determined 100% by the situation, and when we say the probability of 1-in-a-million, we are describing our ignorance. We are not describing a fact of reality purely outside ourselves. In this sense, probability is a concept from epistemology: describing our own state of knowledge/ignorance/uncertainty. 

In actual fact, one particular ball had 100% "probability" and all others had zero (except that it is an invalid use of the term "probability" to use it to describe events as such: we can only use it to describe as aspect of our knowledge.

Edited by softwareNerd
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Let me tell you how it was taught in most regular third grade classrooms, so the rest of us can understand what I mean.

If you have 999,999 white balls, and 1 black ball, all in a bucket, and as you instruct, begin to "grab the balls"...

Out of 1 million attempts, you should grab the black ball at least once, whether it's your first time grabbing balls, your 500,000th time. Or maybe your a millionth time grabbing balls.

Some asinine theory of probability, with intricate flatulous fallacy ways of thinking, does not change the laws of probability we're all familiar with.

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Actually I digress. For some reason my unfamiliarity with your logic makes it hard to understand. But if I grasp what you're saying

There is a 100% chance that One person is likely to be the one in a million. And there is a 100% chance that person has to be somebody

in the lager picture, you're saying that everyone alive is actually just that one black ball out of a million white balls?

if that is the case, you're supporting my idea that we are the unlikely winners of insurmountable odds for life

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  • 5 months later...

Infinity cannot exist per the Law of Identity.  Everything that exists is finite. Infinity in a metaphysical sense is an invalid concept; there is no actual infinity.  It would mean something without identity, something not limited by anything.  Everything that exists is limited. There is no such thing as an unlimited quantity or an unlimited number of attributes because "infinite" means a quantity without any specific identity.

In mathematics, we must embrace the possibility of an infinite set of numbers but neither numbers nor sets of numbers exist in the world as such; there is no actual infinity.  All multiplicity is finite. An infinite number would have to be one that could not be reached by counting (infinity to the power of infinity +1, 2, 3....).

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16 hours ago, NameYourAxioms said:

Infinity cannot exist per the Law of Identity.  Everything that exists is finite. Infinity in a metaphysical sense is an invalid concept; there is no actual infinity.  It would mean something without identity, something not limited by anything.  Everything that exists is limited. There is no such thing as an unlimited quantity or an unlimited number of attributes because "infinite" means a quantity without any specific identity.

In mathematics, we must embrace the possibility of an infinite set of numbers but neither numbers nor sets of numbers exist in the world as such; there is no actual infinity.  All multiplicity is finite. An infinite number would have to be one that could not be reached by counting (infinity to the power of infinity +1, 2, 3....).

Agreed that to speak of a "number" of things as "infinite" is a contradiction, if there were no upper bound to how many things there are we would be in error to ascribing a "number" to it, because no number would be ascribable.

Question:  How would you address the statement:

"In the 10 seconds it took my car to accelerate from 0 mph (relative to the pavement) to 55 mph, my car travelled at different speeds, in fact an infinite number of speeds between 0 mph to 55 mph."

 

Is the statement true or false or meaningless and why?  Could it be corrected by tweaking language?  Is there something fundamentally different about trying to speak of aspects of something continuous versus speaking of discrete things?

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37 minutes ago, StrictlyLogical said:

Question:  How would you address the statement:

"In the 10 seconds it took my car to accelerate from 0 mph (relative to the pavement) to 55 mph, my car travelled at different speeds, in fact an infinite number of speeds between 0 mph to 55 mph."

 

Your question is epistemological.  We must draw a distinction between the metaphysical and epistemological. Mathematics, in Newton's view, is only a tool devised by men to help answer questions about matter (mathematics is a man-made epistemological device).

In mathematics we must embrace the the possibility of an infinite set of numbers. MPH is a ratio and, mathematically, man may choose to divide it in an unlimited number of infinitesimally small units depending on his desired precision requirements. Mathematical infinity is valid epistemologically but infinity in the metaphysical sense, as something existing in reality like an unlimited universe, is an invalid concept.   

 

Edited by NameYourAxioms
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3 hours ago, NameYourAxioms said:

Your question is epistemological.  We must draw a distinction between the metaphysical and epistemological. Mathematics, in Newton's view, is only a tool devised by men to help answer questions about matter (mathematics is a man-made epistemological device).

In mathematics we must embrace the the possibility of an infinite set of numbers. MPH is a ratio and, mathematically, man may choose to divide it in an unlimited number of infinitesimally small units depending on his desired precision requirements. Mathematical infinity is valid epistemologically but infinity in the metaphysical sense, as something existing in reality like an unlimited universe, is an invalid concept.   

 

Are you saying "speed" is epistemological rather than metaphysical?

What about "distance" (as a relationship between two entities)?

Edited by StrictlyLogical
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Speed is a measurement (distance divided by time, such as miles per hour) and, therefore, epistemological.

When I look at a distance as being 5 miles, I am looking at it numerically (measurement requires human consciousness so it is epistemological).

If I say that this particular pencil is 3 times longer that particular pencil, I am looking at it geometrically. My focus is on the quantitative relationship between the 2 pencils (the relationship exists metaphysically independent if human consciousness).

The concept, "place", refers to a relationship among bodies.

Edited by NameYourAxioms
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On 8/10/2016 at 2:09 PM, NameYourAxioms said:

Infinity cannot exist per the Law of Identity.  Everything that exists is finite. Infinity in a metaphysical sense is an invalid concept; there is no actual infinity.

But the exact same thing can be said for the finite.

Both "finite" and "infinite" are standards of measurement, established by Man to serve a very specific purpose in science and mathematics.  They are not ontological.  They are epistemological.

Is a pencil long or short?  It depends.  Is a whale heavy or light?  It depends.  Is 50 MPH fast or slow?  It depends.

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