I’m not qualified to discuss quantum mechanics, but with Cantor I’m comfortable. In regard to explicitly demonstrating the existence of infinity (or at least one of many), I hope this helps.
It is a standard proof in introductory analysis classes that the square root of two is irrational. By this, it is meant that said number is not rational, i.e., it’s not expressible as the ratio of two finite whole numbers. Generally, we say that irrational numbers are such that their decimal expansion is neither repeating (in a finite block of digits) nor terminating (just a special case of a repeating block, namely 0). The square root of two exists; in fact, it is constructible by Euclidean methods. This implies that something other than finitely long decimal expansions or ratios thereof exists. For this something else, we bring in infinity. (To invoke that famous Sherlock Holmes quote, once you rule out the impossible (square root of two being rational), whatever remains (infinity), however improbable, must be true.)
I agree that this has a standard mathematical connotation. Below is more what I’m looking for. In my search I found a list that mentioned a paper by a Korean named Woosuk Park who submitted in 1997 one by the title “The Logical Status of the Axiom of Infinity.” or something close to that. I will read it when I find it. By the way of “introduction” (although I’m not proving anything) here is some stuff on the subject, which is scant.
Thus logic has an efficiency criterion, which makes the principle of parsimony concrete.
The technique shows clearly the reason infinities are meaningless, for no infinity can be produced by any physical gate array.
Open-ended iteration, endless repetition, loops without exit, are quite obviously pathological states caused by poor design, so every infinity introduced into logic circuitry is clearly an error.
Introduction of race states, endless loops, and indeterminate states can be classified as preventable error if it is predictable.
Logic has no room for infinite operations.
Unfortunately, even Russell himself recognized that the axiom of infinity was an arbitrary rule not supported by the theory of logic which he, and those who went before him, created. It was assumed. Because it was assumed, there is no definition within the system to support it and, therefore the system was still incomplete. The undefined element in the system was the axiom of infinity. Logic, and mathematics, were still unsupported. The theory of types represents a level of development shared with nearly all of the systems of symbolic logic now employed.
I am not mad at you. I do get frustrated when you blithely ignore the consequences of what you say (apparently because it is not what you meant). For instance, you stated that If a mathematical theorem holds throughout infinity, it clearly would hold for any subset of infinity. I gave examples for two properties (which might be axiomatic or derived, depoending upon your starting point) which clearly hold for the infinite set but not for the finite subset. You responded, so?
So your statement was wrong.
No. I am telling you that your statement is not correct. Some properties hold or infinite sets that do not hold for finite subsets.
Actually, the set of integers is closed under multiplication. But that is not important. What I am saying is your statement was wrong. Any reaoning derived from that statement is invalid. Any argument you were trying to support by that statement remains unsupported.
more examples
Then why bring it up? Why wonder if a photon is a fermion if you know it is not one? Why introduce this tangent from an obviously inalid base.
Why, if what you really want to argue is that photonic interaction can represent a “natural” infinity did you start by introducing dirac’s work with fermions? If you want to talk about photons, develop an argument for photons.
Noweher in your reference to dirac, in fact nowhere before the quoted post, had you introduced any argument for virtual interactions. I think you are saying what you think wrong.
IF indeed. I know of no mathematicians who make such a claim. It makes a fine hyperbole, but if you seriously want to argue from that postulate you have a lot of work to do.
My response is that this statement is absurd. A model is not the thing modeled.
Conceivable? Is that your test now? It is conceivable that infinities do exist in nature, therefore mathematic should include them.
I remain amazed that you are unable to see the circularity involved in using an infinite mathematics to argue for a finite universe to argue that infinite mathematics is invalid.
It obviously serves no purpose for me to once again point and say "look at the circle, but I am perversely unable to stop.
Look at the circle.
Read what I said! Please note that I sat absolutely nothing about creating the Universe. Please note that you have responded by claiming I am talking about creating the Universe. Why do you do this?
Understanding, arl. The word I used was understanding.
In every way. What other way is there to be irrelevant? We can define a cyclic algebara over a finite set, too. That fact has no relevance to anything we have been discussing. We can also bake apple pies, scratch ourselves, and fill posts with non sequitors. None of those facts are relevant to this discussion.
Look at the circle!
What we have idscovered is that our most accurate mdels of reality employ infinite mathematics. If we abandon the math, then we abandon the model.
No. Theories do not develop by abandoning the foundation that provided the data. If the foundation is invalid, then the data is invalid. Theories do not develop to account for invalid data. [sub]well, good theories do not[/sub].
Euclidean geometry is an infinite math. What is the ratio of a circles radius and diameter? What is the length of the hypotenuse for an isoceles right traingle with legs length 2?
The links you posted do not seem to suport your idea of “limiting” infinity.
The first deals explicitely with digital/mahine logic. It has no bearing on broader mathematics, logics, or philosophies. It does not claim that the behavior of electronic gates is a model for reality.
The second in fact introduces Cantor’s work as a counter to your position. The final line (the one you chose to quote) is semantically all but emtpy. It follows imediately the statement of Cantor’s result that there exist an infinitely many infinities.
The third link is a critique of Russell’s claim to have derived a complete Peano axiomitazation of arithmetic. Basically, the author is simply saying that since the axiom of infinity was required for the generation of Russell’s arithmetic but not present in the ramified Theory of types, the claim that a solid logical (RT) basis for arithmetic had been developed had not been met. The author is making no claims about the validity of infinity as a mathematical construction/conception.
Interestingly enough, I pointed arl to that very page some time ago in reference to using RT as a context for discussing epoistemology.
I am still hoping that you will clarify what it is about infinity rather than circles which drives your curiosity about the limits of mathematical abstractions.
I find no reference to the necessity of dealing only with fermions. I referenced electrons which have half-integral spin, yes. I then referenced the HUP, which allows electrons to “borrow” energy. Energy, as far as electrons go, takes the form of electromagnetic radiation. Photons. Which do not obey the exclusion principle. I then referenced the statistical work of Feynman to point out that there are an infinite number of possibilities that we must “sum” to obtain the statistical macro world view.
You seem to feel that showing the unclosed nature of a finite set of integers under addition violates this. I would quibble with your interpretation of throughout.
Obviously infinite sets can have properties which do not hold for finite subsets…namely, the magnitude of the number of elements. Trivial, I know. And you’re right, though I don’t like your example. It took me writing a huge paragraph here to see your point. A much more concrete example, to me (and what I was thinking about when I realized my mistake), was the mean-value theorem. Clearly this sucker is wrong if we don’t allow for a continuous number system, but it does hold for any continuous subset. Anyway, noted.
Integers are closed under multiplication, yes, if we consider division as a seperate operation as they are not closed under division. Rational numbers are closed under division, and complex numbers are closed algebraically. Fantastic.
Because you seemed so dead set on discussing fermions, and I couldn’t understand why you were ignoring photons.
Are we talking at different angles here? I thought you weren’t a Platonist. If I ask you, what is the universe?, you’ll show me the model. Is the universe the model? If not, then I’ll ask again, what is the universe?
You clearly stated that definitions are tautologies(in some other thread). The model for existence IS existence. By definition.
Is general relativity not a theory, then? It abandoned the idea that gravity was a force, and instead suggested that it was just the curvature of space. Is quantum physics not a theory, then? It abandoned the idea that particles are not waves. Both developed because the data conflicted with the existing theories.
Man, you are killing me. Your resistence to finite math is almost as religious as the resistence to non-euclidean geometry, or to pythagoras’s irrational numbers, or to my roommate’s feeling that pi is not transcendental, or to my feeling that anything not capitalist must be socialist (figured I’d toss that in for good measure). Resist away. I agree with you that that form of math is very likely impossible, but that’s because I think that infinities are inherent in nature, not because finite math is impossible to know a finite universe with (though this may be true as well).
I love it. I swear to God, hail Eris, and anything else that I am NOT DELIBERATELY being dense, trolling, etc. You must know this. Please allow me to elaborate a bit more here.
We use mathematics which invloves infinities. We also use mathematics which involves 1000 dimensional spaces (not that we do, but it is possible). If we develop a theory of the universe that does not involve solutions which depend on 1000 dimensional space, then clearly we may cut that away from the math without affecting the integrity of our understanding.
If we develop a theory about reality which does not involve infinities as answers or steps, then the question becomes, do we need the infinities there? We should, for rigor, attempt to utilize only the concepts necessary to develop the theory. If it turns out that infinity is one of these concepts, then clearly infinity exists in nature. If we cannot remove it from our model, and the only thing we know about the universe IS our model, then what our model says about the universe is what the universe is unless you are prepared to argue from some mystical position.
No, but you introduced them with extraneous comments (i.e. not leading to the conclusion you were trying to make) about the dirac sea and about whether photons are femions. I say again, if your point depends solely upon the infinite possibilities for photons then why did you bother introducing fermions? It gives the impression that you are throwing darts at random in the hope that one might stick.
The “wonder if a photon is a fermion” line still cracks me up. If you know that it is not, why would you even spew such chum into the water?
Excuse me? You thought the way to turn the conversation to photons was to make a patently false musing about the nature of a photon?
Would it have been that difficult to introduce the concept of the photon with , say, a reasoned argument why current models necessitate an infinite scope for it?
Yes, you mentioned HUP. With the conclusion, 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. How you intended that to be a demonstration of infinite scope is beyond me. How you intended a reader to immediately discern that you wanted to talk about photons is also beyond me.
you also mentioned Feynman. You said, “there are an infinite number of mathematical possibilities afforded to the quantum nature of particles.” There are also mathematical infinities present in the equations which present the quantized model of spacetime. Can you see the inherent contradiction in saying that one is a valid expression of truth while the other is not?
The specific examples are meaningless. The point has been proved. Your statement about finite set maths was incorrect. If you understand this, then why babble on about these individual cases? If you understand your error, and since you claimed this was “the idea behind constructionist mathematics” then why don’t you connect the dots?
You are killing me, since you seem unable to comprehend statements that I think are trivially clear. EUCLIDEAN GEOMETRY IS AN INFINITE MATH How do you get from that statement to a religious resistance to finite maths on my part. READ WHAT I WRITE Yes, I am shouting. I am frustrated in the extreme that a simple declarative sentence can be so insidiously misunderstood. Good God, I even gave you two classic examples of where infinite scope is reqiured by Euclidean Geometry. Did you even think about those questions? Did you think I posted them for no reason (like you wondering about photons with half-integer spin, perhaps)? Believe me, I do not ask questions in a debate unless I think the answers are relevant.
Hell, Pies ‘R’ Squared even discussed the classic derivation of infinity from a Euclidian construction of 2[sup].5[/sup]. What, you think we both made it up?
Again!!! I have nothing against finite maths. I simply do not think they are useful for mapping the real world whether or not it is quantized fully. At one point, it seemed you might try to argue that a finite math could usefully model a quantized universe. That is an interesting thought, actually, but I have yet to see anything approaching a case for it.
Platonist? Do you even read this stuff before hitting submit? How in the world do you go from the distinction between model and object to Platonism? Do I need to spell out that not only is the model not the Universe, the shadow of the model is not the Universe either?
f(x)=x is not a line. It is an equation which, if graphed in Cartesian coordinates, will create a line. How much simpler can I make this? How can you possible misunderstand my position?
The toy car on my shelf is not a car. The equation is not the graph. The image I have of you at this moment is not you.
No, I wouldn’t. But if you asked me what teh Universe “looked like” I might show you the model. If you asked me what my dog looks like, I might show you a picture of my dog. As astonishing as this might be, the picture is not the dog. the picture is also not the platonic ideal of Dog.
No.
What we measure.
Your definition makes absolutely no sense to me. Can you justify it? It certainly is not a standard definition of model.
For the last time, truly, READ WHAT I WROTE!
Did physicists rely upon the existence of gravity as a force to develop the data that suggested gravity was not a force?
Did they rely upon the dichotomy between wave and particle to generate the data that quantum phusics was created to explain?
Please read that carefully. Was the theory being overturned necessary for the data collected to be valid?
That is what you are arguing when you attempt to use the predictions of an infinite mathematics to justify removing the infinite from mathematics. LOOK AT THE CIRCLE!
I do believe you. If I did not then I would have ceased responding to you civilly before now. I have to ask you, though, if/when you respond to this post please read it carefully. Please make a particular effort to understand what I am saying and reply only to what I am saying (or open a new topic of conversation, of course). Please.
This example is not apt. Must I say this one more time? The math being used to argue for a quantized universe is dependent upon infinities.
QM needs infinities.
We use QM to declare infinities invalid.
~ QM is not valid.
LOOK AT THE CIRCLE!
But we have not done so. I see absolutely on reason to suppose that we will ever do so. If we find a theory of reality that only requires multiples of 3 we might consider abandoning all other numbers.
arl, “the Universe is the model” is a more mystical position than almost anything I can imagine. How you go by knowlege==model to Universe==model is beyond me. It is an unjustified leap of incredible magnitude.
Spiritus, we’re gonna take a huge breath here. We get into what I feel are some serious debates, but they always seem to revolve around either me being wrong, which I’ll admit, or you misunderstanding me, which may or may not be my fault.
So we’re gonna back up a bit. My original post in this thread has downright agreed with, I believe, everything you have said in this post, right up until the last paragraph. This is where it all went awry.
Now, we both understand that QM is not finished. It does not contain all the forces of nature, the standard model has a lot of renormalization (which many find to be, well, distasteful). Thus it is conceivable that as QM develops more these infinites will disappear and the entire theory will be unified by a discrete numbering system. Without reference to infinites, without necessitating continuous number systems (ie-containing infinitely small numbers), this possible QM comes to us in a hypothetical sense.
Here you go completely ape-shit at me because I feel that the math we use to describe this system should attempt to remove anything not necessary to developing the ideas.
This is, you say, circular.
Now we will take another step back here and consider exactly how proofs work, which you know damn well, but I’d like to make it perfectly clear.
We’ve all seen proofs of the Pythagorean Theorem. My personal favorite, and the one I remember, deals with the areas of squares and the triangles that result from a square inscribed in a square. If I come to find that the square is an unecessary part of my mathematics, if I remove it I remove the Pythagorean Theorem.
I think we are in agreement there.
BUT it is possible to develop the pythagorean theorem without using squares! Thus, we still have the same principles but we could also remove something deemed unnecessary and superfluous.
I am suggesting that, given the possibility that we will come to a QM conclusion that the universe is really quantanized we should try and develop all the necessary concepts involved in QM using some “quantanized” math.
Here you go nuts again. Infinities are inherent in QM. Yeah, I know, which is why I mentioned it in my original post. That’s what I said, that’s what I felt. BUT many people felt that Euclidean geometry was perfect and true as well, until non-euclidean geometry came along and messed everything up (who gives a fuck if they contain infinities?).
By analogy I am suggesting by hypothesis that if there is a non-QM QM to come out of this that doesn’t have infinities in it that we should try and make sure our model, as well, doesn’t have infinities in it. We should try and develop the same theory along a different path.
Re-proving the pythagorean theorem does not invalidate it, and doesn’t make it some circular argument. There really is more than one way to prove the pythagorean theorem. By hypothesis I felt that it may be possible to remove infinites from math which described a non-infinite system. OBVIOUSLY WE ARE NOT DISCUSSING THE SAME QM. Trying to reach the same conclusion using a different method is in no way dishonest, circular, or whatever else you would like to throw at it. It is the struggle for consistency.
IF it should so prove to be the case that it is impossible to remove the infinities from our math to re-prove our new QM, then it seems that, indeed, even still, there are infinites in nature.
Man, I really don’t think I’m doing anything wrong here. We’ll leave the model question for later, I think. Our hands are full here.
A complete sentence semantically meaningless? Regardless, I would prefer to digest all three quotes together and say that math hasn’t nailed down compound infinity, because it may be a (patho)logical error assumed as undefined. It doesn’t bother me that all three quotes are talking about logical infinity directly, if that is not allowed in math, then math has better things to do I suppose. To me, the circle remains an illusion of infinity (the infinite circle isn’t worth the paper it’s printed on if time consumes it). We are missing an element of reason to bridge the ideas. It reminds me of the identity of indiscernables, where “two identical” orbs are imagined to sit in space. Are they different or are they the same? Two schools of course. Regarding infinity, I realize that by defining it, it becomes deinfinity. I will wait to read Woosuk Park’s paper, anyone with a name like that deserves to be famous.
. . .
In my naivete, I now assume there is a phenomenological problem with an idea like infinity, because it can be easily “imagined” by anyone without practical application and without understanding the proof for it. In other words, the problem is that I cannot imagine infinity in too simple of terms, such as “indefinity” or “interminable” or “endless” (and other words associated with torture). As such, it becomes synonymous with nothingness, and everyone knows nothing is perfect through non-existence of nothing (not perfection), and therefore infinity is perfect. Fine, but if as perfect, infinity therefore can’t be grasped through imperfect means (I’m not just being frivolous here, unless you think infinity is frivolous). The other side of the coin is that it is ineffable, but obvious to all, except me.
ARL has a point, if infinity is functionally meaningless, then, hypothetically, anything relying on it could be in error at some point. It seems the contingency of space and time, and hence causality, render infinity problematic to any field that explains reality. If it’s okay with everyone, I’m going to start putting the infinity symbol in quotation marks.
This is what you said in your first post: 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.
These are some other things you have said in this thread: Infinity is, as many say, impossible to visualize with a finite mind, thus [it] cannot exist in our mind.
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.
We might be [led] to wonder, however, if the photon is a fermion.
If a mathematical theorem holds throughout infinity, it clearly would hold for any subset of infinity
I think you’re seeing what I said wrong. Around any fermion are an infinite number of virtual interactions going onsee first statement There is no difference between an accurate model for the universe and the universe itself.
Euclidean geometry was held as perfection for centuries.in a response strongly implying that Euclidean geometry did not contain infinity
Now you say:
Well, it might help if you would stop saying so many things that are wrong. I respond to what you say–the actual words typed. they may not accurately convey what you intended to say, but they are the only guide I have. If I have misunderstood what you wrote then I apologize. I would appreciate it if you would point out specific examples of my confusion. If you are uncomfortable with the actal statements that you have made, then perhaps you should retract them and rephrase.
Well, I also disagree with the statement I referenced at teh beginning of this reply. And with the statement that an infinite scope cannot be held within a finite container.
Still, you are correct that we are in agreement on teh major philosophical points.
No. Infinity is inherent in every single piece of the mathematics that QM relies upon, from volume equations to differentials to summations. You seem fixated upon those areas where an infinite scope is explicit in the “final equation”, ignoring (as with Euclidean Geometry) the infinities inherent within the structures of the math.
QM will never “evolve” into a discrete mathematics. You might conceive of a discrete mathematics developing independently which proves applicable to the same problems.
I called this circular because you were using the results of QM to argue for invalidating teh basis of QM. That is circular. If you had always intended to be speaking of a hypothetical situation in which a new mathematics had been developed which had QM’s predictive power without any reference, explicit or implied, to infinity then I did indeed misunderstand you.
I do not think you made such a plain statement, though. I think you argued that our insight into the quantized nature of the Universe, gained through QM, should be used to eliminate infinities from our mathemtics (if said insight found no “real” expressions of infinity, of course). I based this upon that final paragraph:
To me, the above indicates that your development of a non-infinite math would be spurred by insights into the nature of reality derived from an infinite math.
This is from your last post, which seems to agree with my interpretation. You now supply the additional detail, though, that you would require such a math be able to produce all results possible in QM.
That does make a difference to the usefulness of the new math. It also raises an interesting question: what if it can’t? What if QM shows a fully quantized Universe but it is impossible to describe that Universe accurately with a finite mathematics? Do you maintain your stance that the only “honest” thing to do is banish infinity from our mathematics?
Non-Euclidean Geometry does not “mess up” Euclidean Geometry. They are separate fields. Neither is a complete description of reality. As to caring whether they are infinite, I care only when you make an statement implying an incorrect answer to the question. Saying it was a mistake. Refusing to acknowledge the mistake was annoying. Trying to shrug off the error as unimportant is unworthy of you. Ignorance is what this board is supposed to fight against.
Well, I think if we have a non-infinite non-QM math that is as or more predictive than QM we will already have out model, don’t you? And, yes, I agree with you that new theories should take the new mathematical model into account.
This, again, is a radical statement.
Well, I think you are saying many things that are untrue. I think when called on these statements, you are evasive rather than upfront about them. I attribute that to an imprecision of expression on your part. I think you are so focused upon the larger argument you intend to make that you do not pay close attention to what you (or I) have actually written. I may be wrong, of course.
I also think that I have not understood the larger argument you are trying to make, which is extremely hypothetical in nature. This might be due to a failure of comprehension on my part, a failure of expression on your part, or both. Certainly, such a hypothetical exposition was not at al what I expected in response to the OP, so my own predispositions might have interfered with my receiving your message.
Further, I think that you attribute the friction which sometimes enters our exchanges to the second trait, while I attribute it to the first.
Finally, I think that your position on model==reality is so extreme that you really should expound upon it. I can see no possible justification for the many statements you have made along those lines, and I am very curious to see why you think such statements are sensible.
Yes, a complete sentence can be semantically all but empty. It can also be very misleading when removed from context. Of course, the link itself was practically devoid of content. If it gave you any insights into infinity I cannot imagine what they were. I suspect that you have simply latched onto the final phrase.
You may certainly do so, but you would be incorrect. Math has “nailed down” many infinities. And the three threads were not talking about the same thing. Taking the quotes together is like agregating The day is hot, That girl is hot, and The items in the pawn shop are hot to arrive at a conclusion about heat.
They are not. As I explained above, the first is talking about infinite expressions within a digital logic. The second is nothing more than a cursory report of some mathematical treatments of infinity (with a misleading final sentence). The third deals with the logical derivation of a peano arithmatic through Russell’s Ramified Theory of types. It’s mention of Russell’s axiom of infinity is tangential to the thesis, and the probems the author mentions are not inherent withing the axiom but aply to the inclusion of the axiom within a structure that was supposed to be rigorously derived from RT.
That may not bother you, but it should prevent you from pretending that they provide a unified statement about either mathematical or logical infinity.
Who said anything about infinite circles? I was talking about a plain ordinary Euclidean circle. Pick a size, any size. If you don’t like circles, pick a triangle. Or a number. My point was simply that these are all mathematical abstractions, just like infinity. They are exaclty as real or unreal.
Excuse me? Can you explain this?
How is this different from any abstract concept? There is a phenomonological “problem” with abstraction, if you chose to see the ability to formulate a concept absent material example as a problem.
I do not know it. Furthermore, I see no reason to assert either that infinity becomes synonymous with nothingness or that nothingness is more intuitively or precisely “imagined” than any other abstraction, including infinity.
Excuse me? Do you have any reason for making this statement? It depends, I suppose, on what you mean by “grasped”. Nevertheless, it seems more of a mystical assertion than a reasoned one.
I do not believe arl ever claimed that infinity was functionally meaningless. If he did, he was in error. Infinity is functionally quite useful.
Quite the opposite. To date, infinity has proven literally indespensible to any field that explains reality using mathematical models.
It is okay by me, but I might laugh at you when I see it.
You seem to be throwing away one of the most useful concepts in mathematics based upon, well, nothing so far as I can tell, nothing well-founded certainly. Shall I assume you will also put “circle” and “pi” and “derivative” in quotations?
Discrete (and, individually, finite) combinatorial problems on small sets produce much larger numbers of possibilities to the age-old question, “in how many ways can this be done?” In how many ways can you arrange 15 distinct objects? The answer is over a trillion.
In discrete mathematics, you always need more, bigger numbers out there to be the solutions to your problems involving smaller numbers of things. Consequently, a system with a last and highest number shuts down discrete mathematics in an extremely awkward way.
You don’t need the continuum to need infinity. (But once you’ve got infinity, you need the continuum.)
Hmm. I’m not sure how or why a discussion on the meaning and/or existance of infinity has been sidetracked into a discussion on quantum maths. But there you go.
Personally, when I think infinities, I think statistics. Admittedly this may be because I am a statistician. Anyway. Statistics is real-world and it is riddled with infinities. It’s hard to make sense of a probability distribution without them. I guess it comes down to the fact that there are often an infinite number of choices one can make.
Ok, I really don’t want to continue arguing for things that we agree on.
Second, I clearly said that I was wrong for stating that thing about finite subsets of infinity.
Third, the comment about euclidian geometry was meant to point out that what is inconceivable to some is by no means impossible or nonsensical. A finite math, to you, seems nonsensical. In the 1600’s, non-euclidean geometry seemed nonsensical. That comment had nothing to do with whether or not infinites exist.
That is, such a math is nearly inconceivable to apply to the laws of nature, just like non-Euclidean geometry was nearly inconceivable to mathematicians in the 1600’s. That doesn’t mean it is impossible.
There is no doubt that a maths exist which do not hold an infinitely divisible number system: number theory itself, dealing only with integers. But can we develop number theory to the point that we can derive laws about nature (would nature turn out, by inspection, to be completely quantanized)? Could we remove the other end of infinity–infinitely large–from number theory without a problem? And so on.
I said. You then suggest that infinity as a concept can exist in our minds since we, obviously, are discussing it. That’s fine…we have the concept of infinity, but not infinity itself. I mentioned just after that quote that we can deal with its properties, obviously having a grasp on the concept.
Unless you’d care to side with me that the model is the thing, in which case we’d clearly be able to contain infinity in our mind. No?
Ok, this one is a little over the top, since I left out a second option, which would be that there was a negative energy scalar there, namely, (-1). The complete solution is actually E= +/- mc[sup]2[/sup]. The speed of light, as a constant, is not negative…and even if it were, it is squared in that equation, so unless the speed of light has an imaginary component that we’re not including in all other equations about it, this leaves us with two choices. Negative mass, or some negative scalar quantity. :shrug: You pick. I’m sorry I even brought it up as it is not a supported statement. It was merely historical context to explain why the vacuum was not empty.
This is a taunt, man. You brought up the Pauli Exclusion Principle as a counter to the infinities involved around any area of space. I figured anyone who knew what the PEP was would know that fermions exchange photons, and around fermions are virtual photons, and photons don’t obey the principle, in which case the PEP was not the correct counter.
Conclusion: In any area of space there are an infinite number of particles. You may still feel this is unsupported. This is why I brought up Feynman. All things that can happen do happen…an infinite amount of things occur at every point in space. These virtual occurances cancel each other out(through interference) to lead to a statistical macro occurance.
Because QM is riddled with infinities this is not the time to do that. If it turns out that some Theory of Everything shows that the entire universe, from space to time to energy to whatever, is quantinized(which is the hypothesis), then would be the time to try and develop, as I suggest, the same concepts with a math that does not have infinitely small numbers or infinitely large ones.
2+2=4 regardless of whether we work with the real number system (infinitely small numbers) or the set of integers(discrete quantanized numbers).
Though I mentioned it above, I still prefer to not completely hijack the thread to address this. We might, perhaps, revive the epistemology thread since it would fit there pretty well.
Well, it sure messed up all the people who held that Euclidean geometry was perfect and correct as a model for the universe. This is a historical context to the ideas that surrounded math, which is what I thought we were discussing.
[aside]
Spiritus, I am not going to deny that there are many things of which I am ignorant. There are things that I have been misinformed about as well. But because of my ignorance and my misinformation, I don’t know that what I know is wrong, so I’m gonna say things which are wrong. You don’t hesitate to point that out, and I don’t mind at all. You get frustrated because of this, and I get frustrated because through pointing out my errors you don’t see the point I’m getting at, which is one that I think you you agree with. But believe me, every debate with you is a wake-up experience.
Reolved: from now on, I will explicitely retract statements that I correct from previous posts so the contradictions aren’t seen as weaseling. I’m no danielinthewolvesden! ;)[/aside]
I wasn’t supposing that infinity was useless in math, merely less usefull in logic or as a philosophical concept. I just learned that my position in math would be regarded as a “finitist”. Below is an excerpt from the same page to explain how I prefer to think of infinity, not for its usefulness, but for the problems it presents:
As an instance of the second sort of processes we referred to above, those about which no consensus has been reached as to whether they are supertasks, we can take the process which is described in one of the forms of Zeno’s dichotomy paradox. Suppose that initially (at t = 12 A.M., say) Achilles is at point A (x = 0) and moving in a straight line, with a constant velocity v = 1 km/h, towards point B (x = 1), which is 1 km. away from A. Assume, in addition, that Achilles does not modify his velocity at any point. In that case, we can view Achilles’s run as the performance of a supertask, in the following way: when half the time until t* = 1 P.M. has gone by, Achilles will have carried out the action a1 of going from point x = 0 to point x = 1/2 (a1 is thus performed in the interval of time between t =12 A.M. and t = 1/2 P.M.), when half the time from the end of the performance of a1 until t* = 1 P.M. will have elapsed, Achilles will have carried out the action a2 of going from point x = 1/2 to point x = 1/2 + 1/4 (a2 is thus performed in the interval of time between t = 1/2 P.M. and t = 1/2 + 1/4 P.M.), when half the time from the end of the performance of a2 until t* = 1 P.M. will have elapsed, Achilles will have carried out the action a3 of going from point x = 1/2 + 1/4 to point x = 1/2 + 1/4 + 1/8 (a3 is thus performed in the interval of time between t = 1/2 + 1/4 P.M. and t = 1/2 + 1/4 + 1/8 P.M.), and so on. When we get to instant t* = 1 P.M., Achilles will have carried out an infinite sequence of actions, that is, a supertask T = (a1, a2, a3, . . . , an, . . . ), provided we allow the state of the world relevant for the description of T to be specified, at any arbitrary instant, by a single sentence: the one which specifies Achilles’s position at that instant. Several philosophers have objected to this conclusion, arguing that, in contrast to Thomson’s lamp, Achilles’s run does not involve an infinity of actions (acts) but of pseudo-acts. In their view, the analysis presented above for Achilles’s run is nothing but the breakdown of one process into a numerable infinity of subprocesses, which does not make it into a supertask. In Allis and Koetsier’s words, such philosophers believe that a set of position sentences is not always to be admitted as a description of the state of the world relevant to a certain action. In their opinion, a relevant description of a state of the world should normally include a different type of sentences (as is the case with Thomson’s lamp) or, in any case, more than simply position sentences.
…
As a corollary it may be said that supertasks do not seem to be intrinsically impossible. The contradictions that they supposedly give rise to may be avoided if one rejects certain unwarranted assumptions that are usually made. The main such assumption, responsible for the apparent conceptual impossibility of supertasks, is that properties which are preserved after a finite number of actions or operations will likewise be preserved after an infinite number of them. But that is not true in general. For example, we saw in section 1.2 above that the relevant state of the world after the performance of a task T = (a1, a2, . . . , an) on the relevant state S was logically determined by T and by S (and was an(an-1(an-2(. . . (a2(a1(S))). . . )))), but we have now learned that after the performance of a supertask T = (a1, a2, a3, . . . , an, . . . ) it is not (that is the core of Benacerraf’s critique). The same sort of uncritical assumptions seem to be in the origin of infinity paradoxes in general, in which certain properties are extrapolated from the finite to the infinite that are only valid for the finite, as when it is assumed that there must be more numbers greater than zero than numbers greater than 1000 because all numbers greater than 1000 are also greater than zero but not viceversa (Galileo’s paradox). In conclusion, if some supertasks are paradoxical, it is not because of any inherent inconsistency of the notion of supertask. This opinion is adhered to by authors such as Earman and Norton (1996).
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The postulate of permanence seems to characterise our world at least as evidently as the principle of continuity. Notice in particular that certain physical bodies (particles) may dematerialise, but that is not inconsistent with the postulate of permanence since such a dematerialisation leaves an energy trace (which is not true of Black’s ball). Consequently, we can see that the case Black’s infinity machine is one in which the principles of continuity and permanence turn out to be mutually inconsistent. As long as we do not give up any of them, we are forced to accept that such an infinity machine is physically impossible.
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In the first case, we will know at t = 1/3 P.M. that Goldbach’s conjecture is false; in the second case we will know at t = 1/3 P.M. that it is true. Weyl (1949) seems to have been the first to point to this intriguing method -the use of supertasks- for settling mathematical questions about natural numbers. He, however, rejected it on the basis of his finitist conception of mathematics; since the performance of a supertask involves the successive carrying out of an actual infinity of actions or operations, and the infinity is impossible to accomplish, in his view. For Weyl, taking the infinite as an actual entity makes no sense. Nevertheless, there are more problems here than Weyl imagines, at least for those who ground their finitist philosophy of mathematics on intuitionism à la Brouwer. That is because Brouwer’s rejection of actual infinity stems from the fact that we, as beings, are immersed in time. But this in itself does not mean that all infinities are impossible to accomplish, since an infinity machine is also ‘a being immerse in time’ and that in itself does not prevent the carrying out of the infinity of successive actions a supertask is comprised of. It goes without saying that one can adhere to a constructivist philosophy of mathematics (and the consequent rejection of actual infinity) for diferent reasons from Brouwer’s; supertasks will still not be the right kind of objet to study either.
As Benacerraf and Putnam (1964) have observed, the acknowledgement that supertasks are possible has a profound influence on the philosophy of mathematics: the notion of truth (in arithmetic, say) would no longer be doubtful, in the sense of dependent on the particular axiomatisation used. The example mentioned earlier in connection with Goldbach’s conjecture can indeed be reproduced and generalised to all other mathematical statements involving numbers (although, depending on the complexity of the statement, we might need to use several infinity machines instead of just one), and so, consequently, supertasks will enable us to decide on the truth or falsity of any arithmetical statement; our conclusion will no longer depend on provability in some formal system or constructibility in a more or less strict intuitionistic sense. This conclusion seems to lead to a Platonist philosophy of mathematics.
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What I meant was that if everyone asssumes that nothing is perfect, and that infinity is something perfect, then we have a strain on the word ‘perfect’ obviously, but I was even allowing that nothing was also infinity, but being conceived of imperfectly (as something perfect). If infinity is not to be conceived or fathomed, then we should consider the word to be indefinity or deinfinity to describe the useful infinity apart from the misconceived one. Your objection to infinity being bandied in specific circumstances is the same as my objection to math purveying it as perfect from logic.
You realize, of course, the humor in bringing up probability and statistics as seperate from QM?
brian
…that infinity is something perfect…
Eh? Is this in RE to the eastern religion guy mentioned in the OP? Or in relation to the successful application of infinite maths to scientific understanding?
Sorry, I got so caught up in my own arguments I seemed to have missed something.
I know you are not DITWD. It is because I believe we are both honestly exploring the same issues that I find these conversations so frustrating sometimes. I also find them rewarding, since they prompt me to examine in detail ideas that I am often tempted to gloss over.
I repeat: a finite math does not seem nonsensical to me. Hell, cyclic algebras were one of my favorite things back in Abstract Algebra. A finite math to model reality, however, has little hope of being useful.
Since you introduced the concept immediately after I mentioned that history had shown the lack of utility in finite maths, and since when I responded that Euclidean Geometry was an infinite math you accused me of having a “resistence to finite math” I believed you were proposing EG as a finite math. Since you say that wasn’t the case, the point is moot.
I will note, however, that EG was not replaced as a mathematical model. What was shown to be untrue was that EG was a complete description of all geometries. Since Euclid specifically restricted his geometry to the plane, I really don’t think anyone ever thought it was a perfect model for higher dimensional geometries. But if anyone did, then spherical and hyperbolc geometries certainly proved them wrong.
Please note the bolding. If you do not mean visualize in the most restricted sense (imagine a visual image) then if we model infinity we have visualized it. If you do mean it in the most restrictive sense, then if I imagine a circle I have visualized infinity.
Or negative energy.
Well, I know that the sentence you seem to have dropped reference to was every particle is an infinity of particles. I also know that virtual particles are a “whole other beast” as it were. It truly did not occur to me that you would use virtual particles as an example of a “real infinity” that could be pointed to in a constructionist sense, since they cannot, in fact, ever be pointed to.
Fermions do exchange photons. they do not exchange an infinite number of photons.
Well, this is one of the things I find most annoying about arguments that “QM says ____”. There are many methematical models in QM. there are many competing theories to explain why those models work. Making pronouncements about what QM tells us must be the nature of reality is unwarranted.
For instance, you have mentioned the Dirac and Feynman. Dirac is most closely associated with matrix mechanics, and his model of the positron is a “hole” in the dirac sea. Feynman is (of course) associated with Feynman diagams and his model of a positron is an electron travelling backward in time. There is also wave mechanics, most closely associated with Schrodinger. These three formulations are mathematically equivalent (i.e. achieve the same results) but the “models” are quite different. As a rule, each formulation tends to be emphasized in distinct areas of physics.
Well, I agree that it would be worthwhile to try. I do not agree that it is dishonest to use the mathematics that provides the most useful prdictive model of reality.
Well, it really seems more a question of ontology, but I agree that it should probably be shifted to its own thread. I’m going to be out of town next week, though, so if we start another thread I will be only sporadically present for the first few days.
But there’s a big difference between messing up the way people think about math, and messing up the math itself. Replacing the historical Parallel Postulate with either of the alternative versions simply creates alternative geometries.
The only difference now is that we don’t regard Euclid’s geometry as the only possible geometry. But Euclidean geometry is still as valid as ever.
RT
I said it messed “everything” up, which is a pretty vague statement. Given the context of the posts, I can see how “everything” might take on a less ambiguous meaning and refer to the math itself, which I hope I’ve cleared up.
spiritus
hmm, visualize. Yeah, DAMN. You know when Alice was at the Tea Party with the Mad Hatter? I am often reminded of a similar situation. I know what I mean, and when I read my own words I still know what I mean and I think they mean that because of it, but you’re right. They sure don’t.
But still…I don’t know that visualizing a circle visualizes infinity. I’m now entirely unclear on how to say it…
When we visualize a circle we do just that, but we don’t actually visualize an infinite amount of points between every point…we just “know” that they are there.
Similarly, when we visualize the real number system on a line, we don’t visualize all the numbers down to infinity. We simply “know” that between any two points is another point.
Er, no? If Energy is mass times the speed of light squared, the only way energy can be negative is if there is a scalar negative applied to certain situations not mentioned in the equation, if the mass is negative, or if the speed of light is a pure imaginary number no real component). No?
