I know a guy who claims that his brand of Eastern religion is superior because it holds to the idea of the infinite, way before it was “proven” by science, which is taken to mean that his religion pre-dated math/science and is therefore extra valid. Whatever. But what about infinity? If I claim that infinity exists in theory, do I run into a snag if I attempt to prove it logically? I mean, no computer can keep adding one to the previous number over and over, it would run out of room to store the number. Likewise, if every possible subatomic particle in the universe was assigned a number, assuming the universe was the storage for such a concept, it would theoretically run out of room to continue. So, is infinity limited by reality in any way if it is taken to be a scientific concept? Regardless, it seems like infinity is still also a philosophical concept.
Partial Proof: The Hubble telescope was once pointed into a bare spot in the sky and took a snapshot of distant galaxies therein. It was claimed on camera (in a documentary) by the astronomers running the experiment that the photo saw to the end of the universe, and the astronomers crudely calculated the number of galaxies to be about 50 billion. Even assuming a billion plus stars in each galaxy, and so forth, it seems likely that all known subatomic particles to exist can theoretically be expressed within the highest number on the average scientific calculator. Does this make infinity absurd? What are we to think of infinity in practical, theoretical, or religious terms?
Practical terms, errrr… no answer there yet.
Religious terms. They believe in miracles. Infinity shouldn’t be a stretch.
Theoretical terms, now we’re talking. Infinity comes into the universe through Quantum Mechanics, if we assume that QM accurately describes the universe (which itself is debateable). Every particle is an infinity of particles, and each area of a vaccuum is an infinity of particles. The is what the theory tells us.
But, ah!–you say–the math involved already had infinity built into it. So it is no suprise that we’ll find it in our theories that use math!
True true. So here we exit the realm of the theoretical and enter the realm of the philosophic. Infinity is, as many say, impossible to visualize with a finite mind, thus in cannot exist in our mind. Whatever we think of when we think of infinity, it isn’t infinity. Eh, I disagree. Our finite mind connot contain infinity, to be sure, but the operations involving infinity would have certain properties. For example, assuming infinity exists in number theory, there are an infinite amount of natural numbers. This is axiomatic.
Calculus uses infinites in a definitional sense to accomplish inproper integrals and, in differential calc, to use infintesimals instead of the messy epsilon-delta definition of limits (which removed infinites there, but not anywhere else so it always seemed like a waste of time for such a counter-intuitive definition).
So I guess, philosophically, we really have no problem using infinities. Practically, they apply to maths which we then apply to the real world through whatever epistemological assumption we make there.
Realistically, I would agree that infinity is a useless, if overused, concept that cannot be applied to everyday life, or any reasonable facsimile of one.
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Likewise, if every possible subatomic particle in the universe was assigned a number, assuming the universe was the storage for such a concept, it would theoretically run out of room to continue.
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I think about this all the time, lol, in creating a “real math” which is basically constructionist and involves no infinities. As far as size goes, if QM is to be belived there is also a smallest unit of time, space, etc etc, in which anything can be said to “happen”. So we know there is a lower bound. I think there is some consensus that the universe is also bounded and does not contain an infinite amount of anything, which makes it entirely plausible to use this new math.
After all, it might be nice to know which magnitude of infinity we are supposed to be discussing.
I don’t see this as an inescapable consequence of quantum mechanics. Can you explain it?
Yes, if both space and time are quantized it is not possible to point to any example of infinite scope in space or time if the Universe is closed. If the Universe is destined to expand eternally then the future has infinite scope. If the expansion/contraction is part of a cycle, and that cycle is uninterruptible, then the progression of events is likewise infinite, though the standard language for expressing time breaks down.
It is quite likely to contain the concept “infinity” in a finite mind. The proof lies in this very thread. It is not possible to contain “an infinite number of _____” in a finite _______. Or an infinite ________, for that matter, since “an infinite number” is an abuse of vocabulary.
Limited? I don’t see any way to make sense of this. I know of no uses of infinity as a practical scientific concept outside of mathematics. Are you saying that the inability to map the number of subatomic particles to an infinite cardinality has some consequence for practical science?
Plausible is a strange word to apply to a math. Certainly, I think your system will have great difficulty in solving such arcane problems as mapping trajectories or dividing 1 by 3.
So, can one say that infinity is somehow, in someway, limited, thereby contradicting itself? If it is only used in math, fine, but I’m still surious as to how and why and under what contraints and what to think about it. What do I say to a devotee of the infinite? It is limited to math only?
Also, the idea dividing one by three of any real thing still intrigues me as a microcosm. Let’s say we are dividing a lead rod into thirds in a futuristic nuclear facility, so we calculate its third exactly, which is to say that we reach a limit by running out of atoms to quantify as it approaches the umpteenth place value. Conversely, if we are dividing one by three in a speculative sense and call it infinite, then it is fantasy anyway. Still curious if this argument requires a universe or not.
between 2 points 1 meter apart there are an infinite number of points. a point has no size it is only a position. but in physical reality there is a limit to how precisely we can measure positions. in mathematics
1 = 1.0000000000000000000000000000000000
but in engineering those are different numbers. in engneering you have to deal with how precisely you can manufacture, measure or cut something. the more precise the higher the cost and there will still be some upper limit to the precision. in 1000 years we will probably have higher precision.
Dirac proposed, genius that he was, that the reason electrons don’t fall below the zero state of energy was because that state was already occupied. That is, the vacuum of space is actually full. Of particles. And to whip up one of these particles, we needed to give it enough energy to come out of its state.
So an electron with mass m is stopped from sinking into a lower state by a similar electron with mass m. Interestingly, it would actually seem to be -m. So the energy of this “blocker” we want to find is -mc[sup]2[/sup]. To bring this guy around, “out” of the vacuum, we hit it with 2mc[sup]2[/sup].
Enter the antielectron. Then enter every other damn anti-particle for which we have a name for, except the photon which is, interestingly enough, its own antiparticle. Anyway…
So the entire vacuum of space, and any space not occupied by a particle, is occupied by a particle.
With the HUP, every particle’s indeterminate energy state allows it to temporarily gain energy, disassociate into more elementary particles, exchange photons with itself…anything, provided it does so in a small enough amount of time to keep the uncertainty below some value (h cross divided by two pi? I can’t remember exactly). Not only do we not know what particles are doind when we don’t look at them, but we don’t even know that they are the particles we think they are.
As Feynman explained, there are an infinite number of mathematical possibilities afforded to the quantum nature of particles. The sum of these particles states is what determines the probability of any particular state in reference to all possible states.
This is all I can remember from my head. At home I could gather a few references and some quotes if you’d like.
As well, I don’t disagree that it is possible to contain the concept “infinity” in a finite mind. As I noted in the sentence directly following what you quoted,
And so we would conceptualize infinity without containing it.
How to limit infinity If the known and postulated knowable universe contains no infinities…that is, it is finite and bounded, Quantum Mechanics gives way into string theory which removes probabilities and renormalization, and size has a lower bound, then it would seem to me that theories involving infinity are just mathematical oddities. As it stands, these mathematical oddities do some wonderful and accurate predicting in the real world, and so it seems that infinity, for now—if not forever–is an acceptable and real concept rooted in both musings and what we consider to be reality.
Constructionist mathematics has indeed gone the way of the dodo, but I find that to be the only honest way to handle things if it turns out that the universe is not infinite as we understand the term. If it turns out to be impossible for us to create a constructionist mathematics, then so be it (as it certainly seems to have been). This is only when I fancy a finite universe with quantanized everything (not necessarily using QM though). I don’t always fancy this to be the case.
I have no idea what a “devotee of the infinite” is? Someone who comprehends higher math? Are there other mathematical concepts that you would like to eradicate because you cannot point to them in the material world? Irrational numbers? Complex planes? Geometries of more than 4 dimensions?
Apparently you are using the word exactly in a manner I am unfamiliar with. If the number of atoms does not happen to be divisible by 3, then you do not have 3 equal parts. To me, this implies that you have not divided it “exactly” into thirds.
Fantasy is a dismissive term. Do you use it to apply to all abstract concepts (for instance, philosophy)? Under your definition, pi is a fantasy. Zero is a fantasy. Negative numbers are a fantasy. Thought is a fantasy.
I never said that infinity was relegated to math only, BTW. I said that I had only seen it introduced into science through math. The concept of infinity has certainly been applied to philosophy, theology and art as well.
Again, I am confused yb your position. Are you really saying if I can’t point to it then it doesn’t deserve consideration?
You do appreciate the essential absurdity in using the theoretical consequences of a mathematics containing infinity to decide that only non-infinite mathematics can be “honest”, don’t you?
Infinity in nature: we can know the precise position of an electron at a given time, as long as we know nothing of its velocity. (Do I have that right? My physics is sinfully weak.) Then we can do the same, five seconds later.
In what direction is the second point from the first? And, more to the point, how many possible directions are there for the electron to travel in?
Not the essential absurdity. If it so happens that we cannot show that infinities exist in nature, then it seems that the next step would be to remove the infinities from the math that we use to describe nature. If we can’t do that, then there is something at work that needs a good philosopher to muddle through.
Complex numbers, for instance, are used in the study of fluid mechanics (or so it says here in my analysis book). So do complex numbers exist? Is the flaw with our observation, our understanding, or our math itself?
It is possible to define limits without using infinity. Not all limits may be evaluated without it, however. It is not clear that induction may be used with a finite number system as the whole “induction hypothesis” as I understand it relies on an aleph null sized set of natural numbers. It would be “trivial” if the largest Natural (universe-based finite set of numbers) number were, say, 5.67 x 10[sup]39[/sup] to simply test every theorem for every number (theoretically).
If we have a theory of mathematics which involved infinities and then we determined that nothing in the universe is infinite using that math, it would seem only correct to attempt to remove the infinites from the math…to streamline it, so to speak, to be a more correct model for the universe. Again, from what I understand about the constructionist view it is largely impossible to do and maintain most of the theorems we have in math.
In which case, there would always be the OPs question… what’s the deal with infinity?
As usual, I am presenting a point that I don’t necessarily agree with. I believe there are infinites in nature no matter what context we create to match the data we obtain through inspection. But I do not suppose that it is necessary that infinites exist in nature…and if someone came along and showed that they don’t then I wouldn’t have a problem with that either. I mean, come on…who likes L’Hopital’s Rule?
I use infinity to model a Universe without infinite scope.
So I decide that my math should not have infinite scope.
Now my demonstration that the Universe lacks infinite scope is invalid.
The Universe, if quantized, has no concrete examples circles. Shall we remove the circle from geometry. There are no finite expressions of irrational numbers, does your “honest” math require jetisoning them, too? You say you do not necessarily agree with this viewpoint, and I am glad to hear it, but what possible value do you see in debating whether we should reduce the scope of mathematics to accounting?
dirac
Pressed for time tonight, so:
Are you certain of this as an example of infinity? The Fermi-Dirac distribution calculates the probability that a given quantum state will be occupied for a fermion. At temperatures of absolute zero it reduces to a 1/0 dichotomy with the cutoff occurring at e[sub]f[/sub] (Fermi energy). But the number of potential quantum states within a volume of space is not infinite (see “particle in a box” equations). So, given a finite number of potential energy states within any volume of space and the fact that fermions must obey the Pauli exclusion principal, the number of possible energy states in a finite Universe must remain finite.
Are you absolutely certain of this? I think it is the first I have heard of a negatively massed particle. If you are certain of the concept, are you certain you wish to introduce it into a discussion where you are arguing for a constructionalist mathematics/model of the Universe? Are we more justified talking about negative mass than we are about infinity?
Fantasy is a dismissive term. Do you use it to apply to all abstract concepts (for instance, philosophy)? Under your definition, pi is a fantasy. Zero is a fantasy. Negative numbers are a fantasy. Thought is a fantasy.
For the record, my fantasies are real. But, my point was, does infinity exist in nature or redefine it in some way? How so if so? It is obviously a euphemism at some point to some people. I don’t care if it exists to facilitate someone’s religion or belief in God or higher mathematics, I just want to know if anyone proved it to exist or uses it to prove other things, disjunctively or otherwise. Seeing people get passionate over infinity would humor me to no end. So, is it valid as real? What do we tell the children?
Dismissive or not, I could manage to argue surrogately from exerience with the anti-math potato-head mentioned in the OP, that with an undefined infinity people may find it convenient to dismiss reality itself.
So, I will rephrase it to avoid stepping on religion. What is the necessity of infinity to anyone? What do you say to someone who says it is proven or real. Do we indulge misunderstandings as flattering or religious?
I don’t think so, but I’m open to being educated. How, for instance, do you define the limit of your basic Calc 1 example, f(x) = (x[sup]2[/sup]-4)/(x-2) as x–>2 ?
x doesn’t really ‘approach’ 2; there are only a finite number of x-values between, say, 1.9 and 2. We can plug those into f(x) in sequence, and I suppose you could define something on the resulting values of f(x) and call it a limit, but it’s a fundamentally different thing.
Does any abstract concept exist in nature? Does 3? Does -1? Does i?
Infinity exists in exactly the same way that every other mathematical concept does: independent of any real world examples. Mathematics does not define reality, though it may model it. In some cases, a mathematical model is so successful that it causes inspires people to change their understanding of nature.
Can I get a duh for that, too?
Several mathematical infinities have been proven to exist. They have also proven very useful in solving a multitude of real world problems. It is very much like the concept of “circle”.
Again, I suggest reading Cantor, then.
Tell me, do you find amusement in everything people get passionate about?
And this demonstrates what? That arguing with potato-heads is silly? That undefined qualities should not be used as the basis of arguments? That some people find it convenient to dismiss reality?
When did religion enter this?
Infinity is necessary to solve a wide variety of problems in a wide variety of fields. Without infinity, science and technology would both be severely retarded.
It is mathematically proven. It is as real as 3 or circle or pi.
I try my best not to indulge musinderstandings on this board. I am rarely flattering. I am never religious.
RT
I am afraid that I do remember it. [sub]don’t tell Izzy[/sub]. We should warn anyone following the link, though. The thread predates the vB transition, so teh posts appear out of order. The OP was by Libertarian, proposing a proof that mathematical infinity was fallacious.
Pauli Exclusion Principle…indeed. The number of fermions in any one volume of space occupiable by a fermion is one…to not put it eloquently. We might be ld to wonder, however, if the photon is a fermion. If virtual photons may interact with the virtual particle then there may be an infinite number of virtual photons in any given volume of space. The PEP doesn’t escape infinity, only infinite fermions. The idea behind renormalization is that the infinites in quantum mechanics is ignorable, not that they aren’t there.
Actually, I’m not certain what the hell I’m arguing about. I feel that infinites are inherent in the universe. If that is not the case, however, I feel we should attempt to adjust mathematics and philosophy accordingly.
Hardly. It is possible to show that the first ten prime numbers exist without introducing the idea of infinite sets. The question is if we can introduce the ideas behind the laws of the universe without bringing infinities into it. If a mathematical theorem holds throughout infinity, it clearly would hold for any subset of infinity, hence the idea behind constructionist mathematics in the first place.
But you knew that.
Welllll, this is where things get mucky. Constructionist mathematics is like what mormons are to christianity. If I may make so bold an assumption.
At any rate, the typical epsilon delta definition of a limit is a response to resistence about infintesimals. The idea of an infinitely small number which is both equal to and not equal to some other number is completely outside the scope of constructionism. Limits are merely viewed as approximations since we can never “reach” infinity(response to the infinity doesn’t exist idea). This view is what is normally taught in calculus, even though operations involving infinities are obviously allowed via infinite sums, indefinite integrals, improper integrals, and so on. I find it a sorry state of affairs that we remain inconsistent in the way infinity is viewed in much of math.
I would second Spiritus on the value of reading Georg Cantor, if your goal is to encounter a great mind who is passionate about infinities.
The epsilon-delta definition of limit, as I understand it, was developed as a rigorous way of doing limits. We need Kimstu in here - she’s the math historian - but my understanding is that for a century or so after Newton and Leibniz, limits still had some problems, and even the best mathematicians had to occasionally do some hand-waving. The epsilon definition deals with that in a comprehensive way.
What’s taught in the Calculus series, btw, doesn’t reflect the full understanding of mathematicians; it’s watered down considerably. (There’s a gaping hole in the standard Calc 1 ‘proof’ of the Fundamental Theorem of Calculus, for instance, that takes a couple of pages of serious grunging in an Analysis text to fill in, but it’s beyond the level of Calc 1 students, and nobody ever notices it anyway.)
We don’t have any infinitesimally small numbers that are both equal to and not equal to some other quantity. That’s outside the concept of limit entirely, and even that much is usually correctly taught in Calc 1. The limit of f(x) as x–>n may differ from the value of f(n), but they’re two different things, not the same thing, so it’s prefectly alright for them to have different values.
And standard calculus/analysis doesn’t teach limits as approximations. They’re exact quantities.
I wasn’t trying to be impudent towards you, I value your advice as usual. My point was, as still is: Can it be said that the concept of the infinite is somehow limited? If so, then how? I’m sure you can see the implications to this seemingly contradictory statement. I’m not a mathmatician, but live among the potato-heads who have either embraced or accepted something without apodictic proof. Thanks for the tip on Cantor. It seems religion always factored into it, not only from my experience by Cantor’s as well:
*It was in that same year of 1874 that Cantor published his first paper on the theory of sets. While studying a problem in analysis, he had dug deeply into its “foundations,” especially sets and infinite sets. What he found flabbergasted him so much that he wrote to a friend: “I see it but I don’t believe it.”. In a series of papers from 1874 to 1897, he was able to prove among other things that the set of integers had an equal number of members as the set of even numbers, squares, cubes, and roots to equations; that the number of points in a line segment is equal to the number of points in an infinite line, a plane and all mathematical space; and that the number of transcendental numbers, values such as and e that can never be the solution to any algebraic equation, were much larger than the number of integers. Interestingly, the Jesuits also used his theory to “prove” the existence of God and the Holy Trinity. However, Cantor, who was also an excellent theologian, quickly distanced himself away from such “proofs.”
Before in mathematics, infinity had been a taboo subject. Previously, Gauss had stated that infinity should only be used as “a way of speaking” and not as a mathematical value. Most mathematicians followed his advice and stayed away. However, Cantor would not leave it alone. He considered infinite sets not as merely going on forever but as completed entities, that is having an actual though infinite number of members. He called these actual infinite numbers transfinite numbers. By considering the infinite sets with a transfinite number of members, Cantor was able to come up his amazing discoveries. For his work, he was promoted to full professorship in 1879.
However, his new ideas also gained him numerous enemies. Many mathematicians just would not accept his groundbreaking ideas that shattered their safe world of mathematics. One great mathematician, Henri Poincare expressed his disapproval, stating that Cantor’s set theory would be considered by future generations as “a disease from which one has recovered.” However, he was kinder than another critic, Leopold Kronecker. Kronecker was a firm believer that the only numbers were integers and that negatives, fractions, imaginary and especially irrational numbers had no business in mathematics. He simply could not handle “actual infinity.” Using his prestige as a professor at the University of Berlin, he did all he could to suppress Cantor’s ideas and ruin his life. Among other things, he delayed or suppressed completely Cantor’s and his followers’ publications, raged both written and verbal personal attacks against him, belittled his ideas in front of his students and blocked Cantor’s life ambition of gaining a position at the prestigious University of Berlin.*
http://www.shu.edu/projects/reals/history/cantor.html
*One of the greatest revolutions in mathematics occurred when Georg Cantor (1845-1918) promulgated his theory of transfinite sets. This revolution is the subject of Joseph Dauben’s important study the most thorough yet written of the philosopher and mathematician who was once called a “corrupter of youth” for an innovation that is now a vital component of elementary school curricula.
Set theory has been widely adopted in mathematics and philosophy, but the controversy surrounding it at the turn of the century remains of great interest. Cantor’s own faith in his theory was partly theological. His religious beliefs led him to expect paradoxes in any concept of the infinite, and he always retained his belief in the utter veracity of transfinite set theory. Later in his life, he was troubled by recurring attacks of severe depression. Dauben shows that these played an integral part in his understanding and defense of set theory.*
I do not believe that is correct. The Pauli exclusion principle demands only that fermions cannot have identical quantum states.
Are you suggesting that a photon is a particle with half-integer spin?
It is not.
Well, since you intoruced fermions as the means by which infinity is “realized” in quantum mechanics, it seemed appropriate for me to rebut that argument. I might also note that feynman’s construction for positrons does does not require the dirac sea at all.
Please make up your mind. On the one hand, you argue that just because infinities in mathematics are useful does not mean hey are “real” and that an “honest” math should do away with them. Then, you point to infinities in a particular math that vanish through normalization and declare that they are “real”.
QM IS MATH. The infinities contained therein are exactly as “real” or “unreal” as the infinities whih underlie calculus and geometry.
Mathematics does not exist solely to model the material universe. Frankly, since our understanding of the universe is shaped by the mathematical models which have proven useful for predicting its behavior, the very process you propose is circular.
Hardly?!
Do you even understand what you are arguing? The model of teh universe which you build that might disqualify infinite scope in space or time is entirely dependent upon mathematical infinities. If you use this model to decide that mathematical infinities should be disallowed, you have declared your model invalid. Thus, you have no basis for declaring mathematical infinities invalid.
The ability to determine a finite number of primes is entirely irrelevant.
Several centuries of increasingly predictive mathematical models argue strongly that we cannot.
Your premise is incorrect. In the set of integers, every number has an additive inverse. In the set of “first ten primes” that result does not hold. Integers are closed to addition; the first ten primes are not. etc. etc. etc.
The idea behind constructionalist mathematics produces few intersting and fewer useful results.
And my point, still, is that you have introduced nothing to argue that the concept of infinity is "limited’ that does not apply equally to any abstract concept and particularly to any mathematical concept.
This is a limitation in exactly the sense that any definition is a limitation.
Beyond that, please see my response to arl concerning the circularity of using quantm mechanics to draw any conclusion about the “appropriateness” of infinity in our models for the Universe.
Cantor’s results are as apodictic as any mathematical proof.
Regarding religion: it has no bearing on the mathematics and is not intrinsic to the concept of infinity. Infinity is, however, intrinsic to many religions, and this has perhaps colored the motivations of some who explored the concept mathematically. None of that makes inserting religion into this debate inevitable (or desireable). You may certainly do so, but it there is little to discuss unless you actually draw a connection.
No. Which is why there may be an infinite amount of activity around a particle, because photons may exist in the same point in space.
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Well, since you intoruced fermions as the means by which infinity is “realized” in quantum mechanics…
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I think you’re seeing what I said wrong. Around any fermion are an infinite number of virtual interactions going on.
First of all, I am leaving the matter as an open discussion because it is not clear either way, so I cannot conclusively argue for either. I feel that infinities are inherent in the universe, but am willing to accept an alternate posssibility. Really, spiritus, you getting mad at me now for not making blanket assertations?
And if that math corresponds on a one-to-one level with the universe, then THE UNIVERSE IS MATH. What we are calling a model is an accurate translation of the universe into a set of properties. And if infinities exist in the math, then they exist in the nature math is based on. There is no difference between an accurate model for the universe and the universe itself. Though I’d be interested to hear other opinions on the matter.
No, and who is stopping people from continuing to delve into improper integral forms and all the pleasure they bring? I am merely stating that it is conceivable that infinities do not exist in nature, and so in any mathematical model we have to describe nature we should remove the infinities.
Well, this makes it sound like we create the universe through math instead of discover its properties. I don’t think you are that subjective, man.
I hope you didn’t choke on your coffee.
In what way? I really don’t get it here. It is convenient for our existing math to have the real number system as continuous, but if QM is correct, then time and energy are not continuous, they are quantanized. So you tell me which is more incorrect, applying a mathematics based on continuity of the number system on a discontinuous set of objects or accepting what we’ve discovered and adjusting the theory to more accurately represent the data.
This is how theories grow and develop. We do not invalidate all discoveries of an empircal nature just because we start fooling around with the math.
Well, I think this is still a bit harsh. Euclidean geometry was held as perfection for centuries. If you’re raised to think in real numbers, a discontinuous number system would obviously be an idea you would resist as nonsense.
So? What are you trying to tell me here, that we cannot work on finite sets without having infinite sets? I’m sorry, you must find me really dense here, but what the hell are you saying? Of course they aren’t closed. So what? The set of integers isn’t closed under division(multiplication), do we thus invalidate number theory?
