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Praxus
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I was just reading something about the issue. It said that since 1/infinite = 2/infinite then if you cancle out the two infinities you get 1=2. I didn't quite understand what they meant.

Thanks for the clear and VERY concise answer:)

Infinity in mathematics is a mere symbol which means a quantity larger than any you can think of. If this quantity is say 100, then infinity is anything larger than 100. If it is 10000000000, then infinity is anything larger than that. Therefore, infinity has no definite identity and as such cannot be used in mathematical equations. You can use it in symbolic calculation, for defining limits. But you can't use it in an equation, bacause infinity has no definite identity, and can't have it. It doesn't exist; it is a mere construct of the mind.

So, the above example, wherever you took it, is a load of nonsense.

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  • 4 weeks later...
  • 1 year later...
I was just reading something about the issue. It said that since 1/infinite = 2/infinite then if you cancle out the two infinities you get 1=2. I didn't quite understand what they meant.

1) 1/∞ = 2/∞

2) (∞/∞)(1/∞)=(∞/∞)(2/∞)

3) 1∞=2∞

4) ∞=∞

The error involves simply canceling out the infinity denominator, when one needs to multiply each side by infinity over infinity (which reduces to 1/1 or 1).

Now you have 2*∞ = 1*∞ which is true mathematically.

Reduce to ∞=∞.

Q.E.D.

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You cant divide things by infinity, it isnt a number. 1/infinity makes no sense. You could write something like

Lim(x->inf) 1/x
which would be equal to 0.

Similarly,

Lim(x->inf) 1/x = Lim(x->inf) 2/x
is true, since it reduces to 0=0. Edited by Hal
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You cant divide things by infinity, it isnt a number.
Hmmm. But I've seen people talk about aleph numbers. Aleph-null ha a successor Aleph-1 which has a successor Aleph-2 and so on. And I thought that Aleph-1 = 2**Aleph-0. So maybe you can explain the concept "divide" and "number" such that this doesn't seem like raw stipulation.
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Hmmm. But I've seen people talk about aleph numbers.
Yeah, you can define transfinite cardinals in terms of sets (aleph-0 is the cardinality of the set of integers etc), and then specify arithmetical operations on them in terms of the underlying sets. However, no operation of division is defined in standard cardinal arithmetic.

For example in naive set theory, when you talk about multiplying cardinal numbers, what youre doing is talking about the cardinality of the cartesian product of the sets they represent. For example, lets say we want to multiply the cardinal numbers '2' and '3'. If '2' is a cardinal number, then there is some set A corresponding to it with 2 elements, say A = {a,b}. Similarly there is a set B = {x,y,z} corresponding to '3'. Then, 2*3 is defined to be the cardinality of the cartesian product AxB.

The cartesian product is {(a,x),(a,y),(a,z),(b,x),(b,y),(b,z)}, which corresponds to the cardinal number '6' (ie there are 6 elements in it). This definition of multiplication can cover transfinite numbers - if you take the cartesian product of a set of 2 elements and a set of aleph-0 elements, then the cardinality of it will be aleph-0. So we can say 2*aleph-0 = aleph-0.

The problem with defining division is that theres no obvious inverse operation to cartesian products. Hence it is generally not specificied. That isnt to say that it CANT be specified though - in some constructions of numbers such as John Conway's 'surreal number' idea, all numbers including transfinite ones obey the same rules. Theres a brief essay here about this.

And I thought that Aleph-1 = 2**Aleph-0.
It does, but again cardinal exponentian is defined in terms of the underlying sets correponding to the cardinal numbers. If set A has cardinality '2' and set B has cardinality aleph-0, then 2^Aleph-0 is defined to the the cardinality of the set of all functions from A to B. This can be shown to have cardinality aleph-1.

Despite all this, 'infinity' isnt a number. You can construct transfinite cardinals like aleph-0, but none have the name 'infinity'. For instance, 2*aleph-0 makes sense within cardinal arithmetic, but 2*infinity is meaningless.

edit: theres a decent introduction to transfinite arithmetic here

Edited by Hal
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1) 1/∞ = 2/∞

2) (∞/∞)(1/∞)=(∞/∞)(2/∞)

3) 1∞=2∞

4) ∞=∞

The error involves simply canceling out the infinity denominator, when one needs to multiply each side by infinity over infinity (which reduces to 1/1 or 1).

Now you have 2*∞ = 1*∞ which is true mathematically.

Reduce to ∞=∞.

Q.E.D.

Infinity over infinity is not 1. It is indeterminate -- i.e. it could be anything, from 0 to infinite and anything in between (plus all negative numbers. Not sure how it works for comlex numbers, as there are probably two infinities, one for the real part and one for the imaginary, but I'm just guessing now). To illustrate:

If you have (x^2+1)/(x+1) as x goes to infinite, then the limit is infinite (not one).

(x^2+1)/(x^3+1) as x goes to infinite, will be zero.

And (3x^2+1)/(x^2+1) has a limit of three as x goes to infinite.

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  • 7 months later...

*** Mod's note: Merged with an earlier topic. - sN ***

 

Can somebody please explain to me why it is an invalid argumentation of physicists to say that some sub-molecular laws of physics violate against the law of identity? I am wondering if physicists who work with special equipment to experiment in such areas of study as atomic physics don't have the right or maybe even not the possibility to find a law that contradicts the law of identity. What if that new law were only to be valid in atomic systems where different forces than those in our directly (without special equipment) perceivable environment are important? If we're only granted objective certainty how can we say it's impossible? Objectivists always claim that philosophy is possible without extra equipment, isn't it possible that in areas of study where special equipment is needed different laws apply which have no reciprocation to the rest? confused2.gifnuke.gif

GP

Edited by softwareNerd
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Hi - I am a new user, just registered, so please forgive if I am not getting the point here. However, I think I understand what you are getting at.

Physicists have tried to mechanically separate subatomic particles, to blow apart the identity of quarks, so to speak. What they have managed to observe is that when quarks are forcibly separated, the tiny meson particle is revealed, which is either a smaller component of a quark, or a factor in what holds quarks together. However, the meson only lasts for a trillionth of a trillionth of a second and then disappears. If such a meson is incapable of existing in this physical world of ours after being forcibly brought into existence by man-made mechanical means (I think this was done at Fermilab), then perhaps we cannot consider this as part of our physical reality, it is not meant to be, and we must look to other non-physical disciplines to explain what this sub-atomic exercise reveals to us. Am I on the right track here? Need some help here, people.

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What does "brought into existence" mean?
Somethat that did not exist previously, and (by some causal mechanism) does exist later is "brought into existence". An example is this reply of mine: before the time when I finish up the message and press "add reply", it didn't exist. Of course there are a number of things that are relevant to the post which did exist before, such as my computer, this forum, my thoughts, the post of yours which I'm replying to. At the lower physical levels, the little bits of iron and the magnetic field that are the concrete substratum of my post did exist, but their identify has (or will have) changed. So if smashing things together creates mesons, that would probably mean that some other existent was changed in some fashion where a meson exists which did not exist previously.

As for Betty's "unnatural conditions" question, the idea of "meant to be" implies that there is some purposive being out there with a plan and that exploiting physical law so as to create mesons is somehow a violation of god's intent -- well, I just don't buy that. If you can create mesons, even if it is just for nanoseconds and it requires massive expenditures of energy to do so, then it was "meant to be". Invoking god in the search for "higher meaning" is really a futile waste of time. I can't think of any reason why a "life-span" of 10^-8 seconds would indicate anything unusual. (Oh, also, mesons are not created artifivially, they are created naturally. Man is one aspect of nature, so anything that man does is natural).

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So, you are saying that scientists could find something that isn't itself?

I am just wondering if it is logically possible that a law of physics which is gained by using special equipment is true for example in the area of sub-molecular study and has veto power on metaphysics which is gained by direct sense-perception wíthout tools...

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I am just wondering if it is logically possible that a law of physics which is gained by using special equipment is true for example in the area of sub-molecular study and has veto power on metaphysics which is gained by direct sense-perception wíthout tools...
I don't see any reason to prefer unaided perception over aided perception. There is no serious question that there are objects smaller that a tenth of a millimeter, that radio waves exist, or that there can be sound waves at frequencies above 30 Khz. An insistence on "unaided perception only" is the antithesis of the idea of hierarchical knowledge. We know that a telescope is a valid instrument (we have built on our knowledge of the laws of optics); we know about abstract electrical notions such as "charge" which allow us to construct mechanical detectors.

The only valid question is whether we grasp the laws of physics well enough to construct the instrument in question. Exactly what error in instrumentation are you suggesting is at issue?

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As for Betty's "unnatural conditions" question, the idea of "meant to be" implies that there is some purposive being out there with a plan and that exploiting physical law so as to create mesons is somehow a violation of god's intent -- well, I just don't buy that. If you can create mesons, even if it is just for nanoseconds and it requires massive expenditures of energy to do so, then it was "meant to be". Invoking god in the search for "higher meaning" is really a futile waste of time. I can't think of any reason why a "life-span" of 10^-8 seconds would indicate anything unusual. (Oh, also, mesons are not created artifivially, they are created naturally. Man is one aspect of nature, so anything that man does is natural).

Actually, I am not much of a religious person in daily life. I wasn't referring to looking to a higher purpose to explain the meson. I find it an interesting puzzle that when you blast apart a quark, a meson is observed for a nanosecond, then disappears. Like a meson cannot exist in a meaningful way (okay, okay, I know) without it being a part of something else. It has no identity of its own, really - except for the nanosecond. So that's what I mean by maybe it is not meant to be. Try as we might, and expend huge amounts of energy to do this, the meson is not going to stick around for us to analyze further.

However, I see where this thread may be going - we must invent the instruments that will analyze this elusive particle somehow, so that we can pick it apart even further. I have no doubt that mesons do exist in this physical world for a nanosecond, but my first thought was, what's the point?

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Hi - I am a new user, just registered, so please forgive if I am not getting the point here. However, I think I understand what you are getting at.

Physicists have tried to mechanically separate subatomic particles, to blow apart the identity of quarks, so to speak. What they have managed to observe is that when quarks are forcibly separated, the tiny meson particle is revealed, which is either a smaller component of a quark, or a factor in what holds quarks together. However, the meson only lasts for a trillionth of a trillionth of a second and then disappears. If such a meson is incapable of existing in this physical world of ours after being forcibly brought into existence by man-made mechanical means (I think this was done at Fermilab), then perhaps we cannot consider this as part of our physical reality, it is not meant to be, and we must look to other non-physical disciplines to explain what this sub-atomic exercise reveals to us. Am I on the right track here? Need some help here, people.

They did not create the meson, they created A meson, what they are doing is looking at what the state of things might have been at the big bang time. (other reasons to, but that's the usual one)

And when some physisists say that a quantum scale particle violates the law of identity, they're talking of when a (lets say proton) changes into either one or more different particle(s) then changes back VERY quickly, (also the particles that it changes into could also do the same type of switch-switchback). The cool thing is that sometimes this "virtual" particle(s) might have more mass/energy than the origional. The greater the discrepency, the faster the whole reaction. :D

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What does "brought into existence" mean?

I'd like to reffer you to the theory of elementary waves by Lewis E Little which succesfully resolves all contradictions of Ouantum Phenomena. Check on www.yankee.us.com/TEW/TEW96paper.html.

Albert Einstein said "You believe in a dice-playing God and I in perfect laws in the world of thing existing as real objects"

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  • 4 months later...

*** Mod's note: Merged with an earlier topic. - sN ***

 

I think that Leonard Peikoff unnecessarily and unjustly dismisses the uncertainty principle as irrational. For those who aren't familiar with the term, the uncertainty principle states that the more accurately one predicts a particle's velocity, the more innacurately on can predict that same particle's position and vice versa. As Stephen Hawking explains,

In order to predict the future position and velocity of a particle, one has to be able to measure its present position and velocity accurately. The obvious way to do this is to shine light on the particle. Some of the waves of light will be scattered by the particle and this will indicate its position. However, one will not be able to determine the position of the particle more accurately than the distance between the wave crests of light, so one needs to use light of a short wavelength in order to measure the position of the particle precisely. Now, by PLanck's quantum hypothesis, one cannot use an arbitrarily small amount of light; one has to use at least one quantum. This quantum will disturb the particle and change its velocity in a way that cannot be predicted. Moreover, the more accurately one measures the position, the shorter the wavelength of the light that one needs and hence the higher the energy of a single quantum.

Taken from Stephen Hawking's A Brief History of Time, pgs. 56-7


Leonard Peikoff is clearly in opposition to this position. I think that he arbitrarily dismisses the uncertainty principle. He says

Meanwhile, the younger phycisicsts - typified by Werner Heisenberg, whose Uncertainty Principle was announced in 1927 - were suggesting to the avant-garde that the traditional Newtonian-Einsteinian view of a universe fully accesible to man's mind is outdated, inasmuch as the subatomic realm is ruled not by cause and effect, but ultimately by chance (a viewpoint once confined to the age of pre-physics).

Taken from Leonard Peikoff's The Ominous Parallels, pg. 202


Peikoff's dismissal of the uncertainty principle and quantum mechanics, two theories that are observable and have empirical evidence, seems ignorant and irrational in my view. Can anyone explain this contradiction? Edited by softwareNerd
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